MYP
Mathematics
A
concept-based
approach
4&5
Standard
Rose Harrison • Clara Huizink
Aidan Sproat-Clements • Marlene Torres-Skoumal
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v
Each
topic
ends
with:
Summary
key
of
the
points
Mixed
practice
–
summative
assessment
facts
and
of
the
skills
learned, including
problem-solving
questions,
questions
range
of
and
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contexts.
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context
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global
the
on
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topic.
the
Table
of
contents
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your
suppor t
website:
www.oxfordsecondar y.com/myp-mathematics
Problem-solving
2
9
Space
9.1
1
Form
1.1
2
speaking
•
The
language
9.2
mathematics
interiors
Volumes
of
Are
we
A
20
Relationships
2.1
Spacious
38
related?
•
Functional
relationships
40
parable
Logic
3.1
62
But
can
you
coordinate
prove
it?
•
Using
logic
functions
10.1
A
frog
10.2
A
thin
10.3
into
a
Representation
A
whole
range
of
things
•
Working
of
data
Getting
prince
with
divides
your
more
and
ducks
in
a
row
•
data
that
linear
What
are
•
Scatter
regression
A
model
of
the
chances?
More
than
Seems
•
one
6.2
City
I
and
I’m
irrational?
irrational
exchange
rational
12.1
Seeing
12.2
Growing
numbers
this
please?
12.3
So,
•
178
Histograms
to
I’m
How
do
units
and
7.3
Going
7.4
Which
positive
they
212
measure
reasoning
around
and
segments
triangle
around
and
is
up?
•
Converting
quantitatively
13
the
452
forest
right
14
geometric
what
do
trees
•
Making
a
•
given
pattern
454
Arithmetic
sequences
you
think?
•
462
Drawing
conclusions
475
492
Well-rounded
ideas
•
Using
theorems
It
strikes
a
chord
494
•
Intersecting
chords
508
you?
•
power
14.2
Like
14.3
Decisions,
240
for
The
516
of
gentle
cosine
exponentials
•
functions
ocean
518
waves
•
Sine
and
functions
534
decisions
•
Inequalities
557
and
ratios
256
15
Systems
More
than
572
likely,
less
than
certain
•
270
Probability
to
from
predictably
Exponential
15.1
comes
next?
•
Finding
to
work
the
beginning
backwards
systems
574
patterns
sequences
Back
the
Models
14.1
•
Patterns
8.2
and
Justication
224
sectors
just
triangles
trigonometric
in
441
•
value
Right-angled
What
•
210
absolutely
Absolute
8.1
me
methods
189
13.2
8
423
•
Measurement
Circle
•
160
circle
7.2
problem
•
conversion
skylines
Yes,
408
a
Generalization
13.1
7.1
solve
176
Currency
7
to
forms
reasonable
Can
way
158
Quantity
6.1
•
transformations
Simple
and
6
384
136
saying
Rational
proportion
equality
generalizations
you
•
122
Simplication
Are
time
graphs
12
5.1
less
102
happen?
probability
5
372
in
406
Equivalent
4.4
•
Working
11.3
and
done
indirect
Equivalent
did
us
346
82
grouped
How
•
functions
Equivalence
11.2
4.3
321
fractions
Equivalence
4.2
space
with
11.1
sets
2D
80
11
4.1
line
Getting
Direct
4
in
•
344
Algebraic
64
306
parabolas
Change
in
geometry
shapes
about
Quadratic
10
3D
Transforming
3
•
18
Mathematically
of
304
Answers
594
Index
662
272
•
Using
patterns
289
v i i
Problem-solving
Global
context:
Objectives
●
Applying
any
●
type
Selecting
Inquiry
Pólya’s
of
problem-solving
steps
to
●
solve
F
problem
and
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CIGOL
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to
applying
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context
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if
solve
appropriate
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the
C
makes
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questions
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are
Polya’s
steps
in
solving
a
mathematical
problems
solution
technical
problem?
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strategies
and
sense
in
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the
problem-solving
most
strategies
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the
problem
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can
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2
strategies
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useful
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is
problem
it
to
in
real-life?
estimate?
Y
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●
should
use
already
know
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1
how
Find
to:
the
length
of
27.2
side
x.
cm
x
21
●
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of
the
basic
perimeter
and
area
2
cm
Find the perimeter and area of each shape.
shapes
a
b
3
4
5
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the
prisms
volume
and
of
cuboids,
3
Find
cm
cm
cm
the
volume
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each
solid:
cylinders
a
b
8
10
cm
4
6
●
write
and
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equations
4
cm
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triangle.
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to
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angles
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this
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x
F
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problems
simple.
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the
may
The
challenging
problem-solving
are
Polya’s
method
involve
same
for
a
to
solve
lot
question
steps
of
a
in
solving
problem
complex
could
be
a
+
20
steps
problem?
isn’t
immediately
mathematics,
considered
x
easy
but
for
obvious.
others
some
may
people
Some
be
very
but
others.
Problem-solving
3
Reect
Are
the
and
discuss
following
questions
1
easy
or
challenging?
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everyone
agree?
2
●
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1.5
x
+
liter
m
●
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George
4x
per
and
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highest
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are
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diagram
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problem-solving
to
at
to
tank
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of
height
tank?
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the
professor
steps
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of
use
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mathematics.
for
all
problems
in
subjects.
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Pólya’s
1
problem-solving
Understand
a
Make
problem
2
sure
you
understand
b
State
what
you
need
c
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what
you
already
d
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the
e
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any
f
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the
g
the
words
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conditions
picture
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help
you
understand
a
plan
a
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b
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c
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d
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e
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f
and
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you
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each
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e
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sure
your
b
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worked? What
you
(perhaps
already
you
have
have
didn’t
found
better)
will
answer
the
answers
the
Check
that
a
range
been
4
(1945).
How
to
that
can
someone
follow
it
your
make
intermediate
sense
c
If
you
are
d
If
you
get
stuck, tr y
another
way
work?
answer
in
another
of
strategies
taught,
for
others
solving
you
may
problems
have
1
George
so
work
question
way?
have
plan
step
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answers
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strategies
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problem?
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3
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c
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Princeton
Problem-solving
University
Press.
in
mathematics.
discovered
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stuck, devise
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plan
Exploration
‘The
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Solve
these
problems
using
your
usual
method.
For
each
one,
write
down
main
the
teaching
steps
you
use
from
Pólya’s
175
123
2
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problem
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a
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In
to
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35
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+
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67)
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6
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area
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each
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triangle.
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76°
necessary,
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tenth.
a
b
7
5
9
8
7
7
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6
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answers
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decimal
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cm
7
7
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together
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car,
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car,
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should
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be
will
car
6
quicker
take
12
than
that.
minutes
to
wash
together.
Problem-solving
c
Make sure your nal answer answers the question
In
of
Example
incorrect
1,
most
of
solutions
the
work
is
in
step
2:
Devise
a
plan.
The
main
causes
If
are:
a
●
rushing
to
step
3
and
starting
to
do
calculations
without
a
clear
your
pencil
leaving
out
step
4,
and
not
checking
answers
in
the
context
of
the
problem.
kg
weighs
or
travels
a
at
1
should
plane
20
common
Practice
these
problems
using
Pólya’s
steps.
Show
your
plan
and
your
tell
Remember
to
check
that
your
answers
make
sense
in
the
you
must
working
be
clearly.
km/h,
sense
something
Solve
is:
plan
50
●
answer
context
wrong
in
your
of
solution.
the
1
2
problem.
Light
travels
could
one
travel
Under
for
A
the
km
Sidney
80%
create
5
sold
after
her
a
cover
row
40
acid
and
for
her
swimming
is
$721.
If
the
radius
times
no
in
area
Determine
of
around
the
the
Earth
Earth
is
light
every
how
day
many
and
days
cover
it
will
an
take
pond.
when
against
rowing
the
with
current.
the
current,
Determine
but
the
only
team’s
current.
solutions:
jewelry
dropped
Find
at
miles,
for
double
the
how
the
at
the
how
a
A
is
much
a
50%
of
bazaar.
price
many
7:22 am.
cycling
acid
each
solution,
solution
he
and
B
needs
is
to
They
$7.50
they
She
112
to
sold
crossed
miles
and
sold
a
some
piece.
at
In
for
total
they
$9.50.
the
nish
running
lineat8:07 pm,
26.2
miles.
Find
race.
Problem-solving
C
year.
many
solution.
triathlon
speed
of
hours
acid
sold
then
2.4
2
can
Determine
68%
and
one
how
days.
rowing
there
Justine
pieces
average
a
2
half
in
dierent
of
piece,
began
km
solution.
ml
algae
and
just
when
when
two
in
nd
year.
month
hours
has
200
90
Ilhan
2
1
to
speed
Corinne
$9.50
6
can
in
miles
miles,
conditions,
in
algae
rowing
an
in
ideal
pond
team
16
4
trillion
4000
entire
3
six
approximately
strategies
One
●
The
next
Which
examples
Theymay
help
problem-solving
show
you
get
some
started
strategies
strategies
when
you
you
are
are
can
the
use
most
in
useful?
denition
problem-solving.
‘knowing
do
stuck.
when
know
what
you
what
you
with
example,
it
1.
take
For
1
person
can
to
simplify
how
a
problem
much
complete
a
does
1
by
replacing
item
cost?
one
Or
of
how
the
to
numbers
many
hours
does
task?
Problem-solving
is
to
don’t
Strategy: Take 1
Sometimes
of
problem-solving
7
do’.
Example
Three
to
n
painters
paint
=
2
the
number
house
in
6
can
house
of
paint
in
6
a
house
in
8
hours.
How
many
painters
painters
to
paint
the
1
hours
Understand
b
c
3
painters
n
>
take
8
are
needed
hours?
State
State
the
what
what
problem
you
you
need
to
already
nd
know
hours
e
3
Identify
any
Assume
To
2
paint
all
in
conditions
painters
less
on
paint
time,
you
the
at
unknowns.
the
need
same
more
rate.
painters.
Devise a plan
Strategy: Take 1
1
painter
takes
n
painters
t
hours
Find out how long it takes one painter to paint the house.
take
6
hours
Work out how many painters you need to paint it in 6 hours.
3
painters
→
8
hours
1
painter
→
24
3
Carry out the plan
hours
One painter takes 3 times as long as 3 painters.
4
painters
n
=
4
painters
→
6
4 painters take
hours
the time of 1 painter.
4
4
are
needed
Look back
to
The nal answer satises n > 3,
paint
the
house
in
6
hours.
which ts with expectations.
Strategy 2: Use easy numbers
Replacing
method
to
numbers
use
Example
Potatoes
How
x
=
weight
x
>
1
in
the
the
problem
original
with
‘easy
numbers’
can
help
you
see
the
problem.
3
cost
many
in
$1.56
per
kilograms
of
kilogram.
of
potatoes
Shona
did
spent
she
$1.95
on
potatoes.
buy?
potatoes
1
Understand the problem
Shona paid more than $1.56, so she bought more
than 1 kg.
Assume
and
a
potatoes
customer
cost
$2
bought
per
$10
2
kg,
Devise a plan
Strategy: Use easy numbers
worth.
Rewrite with numbers that are easier to work with, to
That
means
the
customer
bought
5
kg
see what operations to use to solve the problem.
of
potatoes.
10
= 5
2
Continued
8
Problem-solving
on
next
page
=
weight
in
kg
3
=
4
Look back
a
Shona
bought
1.25
kg
of
Carry out the plan
Check your answer
b
potatoes.
Does it make sense?
1.25 kg satises x >
1.
$1.95 is less than $3.12 for 2 kg,
so 1.25 kg is a reasonable answer.
Reect
●
In
●
How
Example
are
Practice
Solve
these
3,
the
how
did
‘Take
3
substituting
one’
and
‘Use
to
check
many
Jordan
using
that
students
Pólya’s
your
has
an
would
average
a
Find
the
score
b
Find
the
average
average
c
to
be
Determine
Jordan
score
easier
easy
numbers
numbers’
help
solve
strategies
the
problem?
related?
2
steps.
answers
It takes 6 students 2 hours to stu
how
2
discuss
problems
Remember
1
and
she
the
high
needs
score
on
on
she
of
the
in
two
the
strategies
context
of
seen
the
so
far.
problem.
envelopes for Back to School Night. Determine
needed
82%
one
sense
to
her
the
nish
rst
next
needs
on
the
four
test
the
job
in
a
quarter
of
an
hour.
tests.
for
her
next
average
two
tests
to
for
be
her
84%.
overall
85%.
minimum
wanted
as
of
be
Use
make
to
as
have
number
an
possible
of
average
on
all
tests
of
that
90%,
remaining
could
and
be
left
assuming
to
take
that
she
if
could
tests.
2
3
Two
area
painters
in
5
work
minutes
for
a
while
painting
the
other
company.
one
takes
One
8
of
them
minutes
to
can
paint
paint
the
a
200 m
same
2
area.
Find
painted
4
Light
from
our
5
it
it
would
3 000 000
sun,
nd
take
them
to
paint
an
area
of
1300 m
if
they
km
how
in
ten
many
seconds.
minutes
it
If
Earth
takes
for
is
150
light
million
from
the
kilometers
sun
to
reach
planet.
After
installing
ll
Pump
it.
1
a
Determine
b
If
both
a
ll
how
pumps
nd
new
can
pool,
a
fast
start
how
the
at
long
Kedrick’s
pool
8
pool
the
it
in
can
same
will
family
hours
take
be
while
lled
time,
to
ll
but
the
is
going
pump
using
pump
pool
to
2
both
1
(in
use
can
two
ll
it
pumps
in
12
to
hours.
pumps.
breaks
down
after
an
total).
Thien lives on the 11th oor of a building. He has to climb 48 steps just to get to
the
7
long
together.
travels
the
hour,
6
how
third
There
80
oor.
were
days.
many
enough
After
Assuming
the
more
Determine
20
many
provisions
days
rate
days
how
of
the
into
food
food
at
the
a
steps
space
mission,
consumption
will
he
needs
station
10
to
to
feed
astronauts
doesn’t
take
20
to
get
people
returned
change,
to
to
his
oor.
for
Earth.
determine
how
last.
Problem-solving
9
8
Diagonals
polygon,
in
polygons
excluding
its
are
line
segments
that
connect
the
vertices
of
the
sides.
a
Find
how
many
diagonals
there
are
in
a
20-sided
b
Find
how
many
diagonals
there
are
in
an
polygon.
n-sided
polygon.
Strategy 3: Draw it!
It
often
helps
Example
A
5
water
m
of
draw
The
lawn
a
diagram,
especially
if
the
problem
involves
geometry.
4
sprinkler
wide.
the
to
is
is
placed
sprinkler
watered
in
the
sprays
by
the
middle
water
to
of
a
a
rectangular
distance
of
2
lawn
m.
6
What
m
long
and
proportion
sprinkler?
0.5
m
1
Understand the problem
f
1
5
2
m
m
1
Draw a diagram to help you
m
understand the problem.
m
A fairly accurate drawing can
help you see whether your
answer makes sense.
0.5
6
m
m
b
State what you need to nd
c
p
=
proportion
of
lawn
watered
by
Identify any conditions on the unknowns.
sprinkler.
Based on this fairly accurate diagram,
2
To
nd
the
proportion,
you
Devise a plan
rst
need
to
nd
2
A
=
r
where
r
=
2
m
the
c
area
of
the
circle
and
the
area
of
the
rectangle.
A
=
l
×
w
where
l
=
6
m
and
w
=
5
m
l
2
A
=
r
2
=
×
2
=
4
c
3
A
=
l
×
w
=
6
×
5
=
30
Carry out your plan
Always
use
exact
values
until
the
nal
answer.
l
Do
4
p
=
0.4188790205
p
=
0.42
Look
In
of
the
3.14
situation,
for
π
nal
answer
it
is
to
reasonable
the
nearest
round
percent.
Final
lawn
is
watered
by
the
satises
sprinkler.
c
10
to
42%
answer
42%
use
back
this
the
=
not
Problem-solving
Answer the question.
Practice
1
A
6
cm
length
2
The
long
5
cm.
square
side.
a
3
Its
prism
Find
base
slant
of
the
a
base
that
is
a
right-angled
triangle
with
two
sides
of
volume.
the
height
Determine
its
has
its
is
Great
pyramid
at
Giza
measures
230.4 m
on
each
186.4 m.
ratio
of
the
height
of
the
pyramid
to
the
perimeter
of
base.
b
Explain
c
Discuss
how
this
whether
ratio
you
is
related
think
this
to
is
the
a
number
.
coincidence.
Strategy 4: Work backwards
Some
problems
question.
asked
and
For
how
asked
The
you
example,
much
to
Example
give
one
nd
its
an
you
answer
are
orange
side
and
given
costs,
or
ask
the
you
total
you
are
to
nd
price
given
of
a
part
of
oranges
the
area
of
the
and
a
are
square
length.
5
diagonal
of
one
side
of
a
cube
is
11.3
cm
to
3
s.f.
Find
the
cube’s
volume.
1
Understand the problem
s
f
d
Draw a diagram
s
s
c
d
=
11.3
State
what
you
know
cm
b
State
what
you
need
to
nd
3
Volume
of
Need
nd
to
cube
=
s
s
2
2
s
2
+
s
d
=
d
=
11.3
=
127.69
=
63.845
=
7.990
2
2s
Devise
2
Pythagoras’
theorem
Use
s
volume
to
Carry
nd
out
the
the
with
of
the
d
and
s
to
nd
s
cube.
plan
2
The nal answer should be rounded to 3 s.f.
only at the end. All previous answers must be
2
2s
plan
Use
3
2
2s
a
2
=
rounded to at least one more place as needed.
2
s
b
s
cm
3
Volume
=
s
The
2
=
7.990
volume
of
the
your
side
s
answers
is
shorter
make
than
sense.
the
diagonal.
3
=
510.1
cm
3
The
Check
cube
is
510
4
Look
back
cm
c
Answer
Problem-solving
the
question.
11
Practice
1
The
4
height
of
a
cylinder
is
twice
as
long
as
its
diameter.
Its
volume
is
3
500
2
cm
The
Find
diagonal
shown
3
.
in
the
of
the
cube.
A
box
with
of
the
a
area
cube
diagram
no
top
of
(that
at
has
its
a
base.
passes
right)
height
through
the
measures
17.32
that
times
is
six
2
middle
cm.
its
of
Find
length
the
the
cube,
as
volume
and
a
width
that
2
is
of
its
height.
If
the
surface
area
is
576 cm
,
nd
the
dimensions
of
the
box.
3
4
When
by
5
It
milk
which
takes
with
30
freezes,
its
lawn
Find
how
electric
it
the
long
as
it
increases
decrease
long
mower.
Then
with
volume
will
twice
electric
minutes.
the
volume
Nathan
the
its
to
One
broke
when
mow
day,
down.
by
the
he
He
it
one-fteenth.
melts
lawn
used
took
the
20
back
with
into
the
electric
minutes
Find
a
the
liquid
fraction
state.
pushmower
mower
to
than
for
nish
mowing
pushmower.
takes
Nathan
to
mow
the
whole
lawn
using
only
the
mower.
2
6
The
surface
volume
7
of
Suppose
a
source
of
the
you
of
of
a
cube-shaped
box
without
a
top
is
.
252 cm
Find
the
box.
have
a
running
4L
container
water,
and
explain
a
how
9L
container.
you
could
If
you
measure
had
out
access
exactly
to
6L
water.
Reect
●
area
and
Compare
Which
●
How
●
Is
D
can
the
the
ones
discuss
methods
were
used
dierent
best
4
you
and
most
methods
strategy
always
lead
the
When
is
‘good
enough’?
●
How
can
an
you
others
to
the
to
solve
problems
so
far.
the
same
solution?
simplest?
estimate
nd
used
often?
a
solution
information
you
need
to
that
solve
a
is
problem
in
real-life?
●
In
to
the
How
practice
solve
the
there
is
a
there
is
not
12
problems
problem
lot
estimates.
useful
of
is
it
you
was
enough
estimate?
have
given
information
Then
to
in
solved
the
so
question.
and
you
have
information
and
you
your
answer
will
also
Problem-solving
far,
be
to
In
the
to
information
real-life
choose
have
an
all
what
nd
estimate.
you
situations,
you
extra
need.
need
sometimes
Sometimes
information,
or
make
L
=
liter
Exploration
A
square
room
2
has
four
walls,
each
3
m
long
and
2.2
m
tall.
Paint
comes
in
2
10-liter
one
Pedro
Mia
and
or
door
5-liter
calculates
calculates
out
Pedro’s
the
the
Reect
which
situations
of
four
paint
m
the
When
is
it
better
of
and
covers
its
one
an
area
of
window
walls
and
ignores
the
walls,
and
subtracts
is
1.5
door
the
m
1.5
.
m
and
areas
The
by
1
room’s
m.
window.
of
the
door
down
For
Find
2
Amy’s
For
the
she
area
ve
per
it
tins
is
of
better
used
in
paint
to
this
needed
calculate
using
Mia’s
accurately
or
and
to
situation.
better
be
as
to
estimate
exact
as
than
to
make
exact
calculations?
possible?
than
to
under-estimate?
have
problem,
of
this
friends
Amy
too
state
is
much
which
missing
in
information
information
order
for
and
is
you
some
do
irrelevant,
to
solve
not
and
have
write
it.
parallelogram.
come
buys
to
visit
four
and
a
her
pizzas
tub
of
by
at
bus.
$5.75
icecream
Each
each,
for
pays
500
$4.75.
g
$4
of
Find
for
a
ticket.
strawberries
how
much
at
change
gets.
42-inch
where
would
she
and
Jack
wants
24
gets
stop
like
television
to
inches
Determine
bus
of
over-estimate
kilogram,
Elizabeth
tall
to
it
information
supper,
$2.75
paint
to
problems
each
which
1
sizes
5
these
enough.
and
whether
is
important
up
at
400
a
a
television
diagonal
the
that
television
for
her
apartment.
measures
measures
42
30
She
inches.
inches
knows
The
wide,
that
cabinet
30
inches
deep.
7:00
m
buy
put
whether
is
to
has
or
not
am
from
the
and
his
television
needs
house.
to
could
catch
the
Determine
t.
bus
at
to
school
what
time
at
he
7:53
needs
am.
to
The
leave
house.
Josip
the
of
0.8
discuss
it
the
liter
by
of
of
is
Some
5
Justify
amount
and
Practice
4
area
When
a
m
area
number
areas.
estimate
3
the
the
One
1.8
window.
Work
In
pots.
measures
just
bottle
bought
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lightweight
of
its
height.
cylindrical
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full,
water
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bottle
weighs
450
where
the
grams
and
radius
holds
of
a
3
volume
6
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770
way
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cm
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angle
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Problem-solving
13
Summary
Pólya’s
problem-solving
1
Understand
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Make
sure
b
State
what
you
need
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d
Identify
e
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to
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help
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you
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make
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sense
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stuck, devise
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simplify
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your
question
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does
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other
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possibilities
formula
make
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c
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c
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sure
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the
steps
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person
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cost?
to
by
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often
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it!
draw
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geometry.
how
complete
Strategy
4:
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backwards
task?
Some
to
Strategy
2:
Use
easy
numbers’
in
the
numbers
can
original
14
help
a
part
of
give
the
you
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answer
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and
ask
example,
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numbers
are
Replacing
problems
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much
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length.
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a
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helps
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.
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of
Kim
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leaves
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km/h.
travels
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40
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house
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academic
co-authors,
mathematician
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away
tree.
highly
who
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giving
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her
plus
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yet
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she
then
mathematical
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collaborated
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hamsters.
together.
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posing
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around
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mathematician
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highly
fruit
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apple
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hamsters
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apart.
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month.
hour
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after
km
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lives
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one
speed
hours
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Tom
Tom.
school
average
speed.
from
to
leaves
an
of
average
papers
than
history.
Problem-solving
15
000.
much
for.
Many
just
people
enjoy
mathematical
mathematicians.
interesting
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discoveries
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puzzles,
puzzles
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mathematical
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mathematician
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solve
Rice
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cover
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as
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immediately,
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would
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1883.
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King
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would
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on.’
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replied:
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game
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sized
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c
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=
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e
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of
rice
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total
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grains
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the
rst
row
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the
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the
entire
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rows
chessboard.
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minimum
number
of
moves
for
a
tower
of
n
n
disks
take
i
16
of
is
to
2
1.
move
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the
tower
minimum
of:
time
it
would
3
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around
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English
William
Part
1:
added
Wrap
to
2:
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make
need
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above
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of
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at
make
inch
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from
points?
rope
the
rope
all
around
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hover
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inch
points?
information
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to
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diameter
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Game
1702.
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equator
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all
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mathematician
this
rope
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1988,
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at
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equator
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rules
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rope
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Choose
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amount
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down
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e
the
nd
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enthusiast
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that
needs
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another
numbers
and
with
brackets,
mathematical
to
make
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total
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problem-solving
strategies
to
make
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from
set:
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(r
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sphere.
whether
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to
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sphere
answer.
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you
Now
surprised?
Puzzle
and
recreational
showed
how
to
mathematician
arrange
the
Sam
Loyd
seven
gures
7
.
and
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expression.
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this
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Problem-solving
17
you
1
Form
Form is the shape and underlying structure of an entity or piece of work,
including its organization, essential nature and external appearance.
The
language
of
mathematics
Mathematics
is
written
using:
●
numbers
●
words
●
symbols
●
diagrams
●
letters
●
shapes
What
forms
help
mathematics
you
when
recognize
you
see
them?
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Are
there
different
thought
innite
different
sizes
of
room
What
if
a
number
1
moves
moves
●
free,
And
what
at
an
moves
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2
moves
the
the
one
and
in
no
to
free.
but
the
the
so
with
an
motto
on.
is
are
room
n
4,
in
room
room
Room
all
–
2,
of
1
2
is
the
and
rooms
guests
rooms
the
moves
odd
already
guest
in
innite
happy.
but
room
the
the
number
time
to
all
guest
problem
room
all
hotel
hotel
–
the
even-numbered
occupied,
are
but
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innite
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2,
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with
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and
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–
arrives,
occupied?
room
18
one
room
if
David
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room
now
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No
innity
imagined
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to
of
guest
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of
innity?
for
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Innity
forms
experiment
number
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●
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guest
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are
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now
numbered
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symbol
word
for
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lemniscate
is
is
called
of
Latin
a
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origin,
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means
on
‘pendant
cur ve
is
Lemniscate
the
known
of
as
the
Bernoulli;
Car tesian
plane
its
ribbon’.
equation
2
(x
is:
2
+
y
2
)
2
=
2a
2
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2
−
y
)
y
x
●
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think
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slider for the variable a from
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effectively
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was
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designed
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function
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purpose
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form
innity?
and
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of
and
the
appearance
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job
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attractive.
●
Which
do
you
think
is
the
best
chair
●
Which
do
you
think
is
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best
graphic
design?
design?
19
Mathematically
speaking
1.1
Global
context:
Objectives
●
Classifying
the
Scientic
Inquiry
dierent
kinds
of
real
numbers
and
technical
innovation
questions
●
What
●
What
kinds
●
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are
●
How
is
a
set?
F
●
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the
●
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interpreting
solve
MROF
●
and
problems
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kinds
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real-life
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of
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are
there?
operations?
do
you
represent
sets
and
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mathematics
●
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set
sets
C
contexts
sets
of
Using
the
language
of
sets
to
model
real-life
D
How
in
useful
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sets
real-life
and
Venn
diagrams
problems?
problems
●
ATL
Communication
Organize
20
and
depict
information
logically
How
does
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function?
N U M B E R
Y
ou
●
should
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know
types
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how
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to:
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set(s)
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7
Some
spaces
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is
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examples
subject
of
teachers
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set
your
is
a
set.
You
use
capital
You
Sets
7
f
c
0.1
sets
sets
are:
year,
=
C
=
{my
the
the
of
are
set
set
0
d
g
1.5
h
there?
of
of
objects.
to
denote
describe
described
three
{vowels
{seven
solar
D
of
irrational
students
your
in
your
favorite
class,
books,
in
a
set
Each
a
set,
using
member
and
put
words,
of
the
or
words:
=
favorite
in
the
largest
a
set
is
set
set
of
colors}
English
moons
A
alphabet}
in
the
=
writing
{blue,
e,
i,
1
and
21}
of
all
your
parking
=
{a,
C
=
{Ganymede,
=
{2,
3,
o,
set
of
by
element
in
its
curly
elements.
listing
elements:
green}
u}
Titan,
Moon,
4,
an
the
list
described
B
D
of
a
orange,
Earth’s
between
called
members
by
Sets
system}
{integers
the
the
supermarket.
letters
can
of
their
=
member
set?
collection
the
brackets.
a
this
local
of
B
a
numbers,
Sets
F
A
is
which
etc.).
b
e
it
state
…,
Callisto,
Europa,
Io,
Triton}
20}
…
The
symbol
means
‘is
an
element
The
of ’.
three
dots
The
symbol
∉means
‘is
not
an
element
or
if
the
set
A
=
{16,
25,
36,
49,
64},
then
49
A
but
81
∉A.
used
by
Greeks.
The
order
in
which
you
write
the
elements
of
a
set
is
not
=
{a,
e,
i,
o,
u}
is
the
same
as
B
=
{e,
o,
u,
i,
a}
a
class
there
are
ten
students
aged
16,
and
four
students
aged
of
set
their
written
The
of
students
ages
four
has
in
the
only
class
two
has
14
elements:
elements
{15,
16},
(the
not
students’
16
ancient
With
written
names).
ten
is
sets,
writing
the
used
out
elements
The
times
is
cardinality
of
of
elements
the
in
a
set
A
is
written
n(A).
It
is
called
of
impractical.
set
and
15
times.
number
the
rst
15.
set
The
was
the
ellipsis
when
all
In
It
important.
the
B
so
‘continue
pattern’.
example,
‘and
of ’.
on’
For
ellipsis
mean
the
set.
1.1 Mathematically speaking
21
a
For
example
if
number,
then
You
write
can
Natural
the
the
set
set
A
A
=
is
common
{knife,
a
nite
number
numbers
fork,
spoon}
then
n(A)
=
3.
If
n(A)
is
a
real
set.
sets

=
{0,

=
{…,
using
1,
2,
3,
set
notation.
…}
The
Integers
3,
2,
1,
0,
1,
2,
3,
symbols
numbers
+
Positive
integers

=
{1,
2,
3,
often
…}
are
innite
sets,
as
they
contain
an
innite
number
of
are
to
as
struck’
elements.
because
be
Practice
sets
referred
‘double
These
for
...}
they
produced
could
by
1
double-striking
1
List
a
the
A
is
elements
the
set
of
of
these
the
days
sets,
of
and
the
nd
the
number
of
elements
in
each
set.
a
character
a
typewriter,
on
‘blackboard
b
B
is
the
set
of
months
c
C
is
the
set
of
factors
d
D
of
the
year
not
containing
the
letter
they
r
were
with
of
the
set
of
positive
integers
less
than
30
that
are
multiples
of
4.
Describe
each
set
in
edge
chalk
on
J
=
{1,
3,
5,
7,
of
the
a
instead
point,
to
words.
dierentiate
a
as
written
blackboard
of
2
the
bold’
12.
the
is
or
week.
9}
from
other
them
bold
characters.
3
b
K
=
c
L
d
M
=
{isosceles,
{right
=
Given
{4,
the
8,
equilateral,
angle,
12,
sets
A
obtuse,
16,
=
…,
{4,
right-angled,
acute,
scalene}
reex}
40}
6,
8,10},
B
=
{1,
8,
27,
64},
C
=
{1,
3,
4,
7,
11}
and
ATL
D
=
If
the
a
c
e
Set
In
{0,
4
±1,
±4},
false,
state
whether
write
the
8 ∈D
addition
variable,
or
a
to
describing
set
vertical
innite
the
set
of
a
builder
line,
sets.
notation,
=
is
each
correct
b
7
∉
d
27
f
n
statement
is
true
or
false.
statement.
C
∈ A
(C ) = n ( D )
notation
using
builder
A
A
is
1 ∈ A
described
set
±3,
∈ A
builder
nite
±2,
statement
For
words
any
x
would
the
that
set
look
|
such
or
This
by
x
is
∈
a
curly
on
A
and
x
listing
uses
restrictions
example,
which
values
of
in
notation.
and
{x
all
set
=
the
{1,
read

natural
number
its
elements
brackets
variable.
2,
3,
like
…
It
9}
it
can
enclosing
can
can
be
be
be
a
used
written
for
in
this:
,
and
1
x
≤
is
1
x
≤
9}
between
and
9
inclusive.
Tip
For
is
0
2 2
1 Form
natural
numbers,
1
≤
equivalent
<
x
<
10.
x
to
≤
9
N U M B E R
Using
to
set
know:
builder
the
Rational
notation
rational
you
can
dene
another
number
set
that
you
need
numbers
numbers:
We
don’t
have
p
In
words,

is
the
set
of
rational
numbers
(numbers
of
the
form
)
such
that
a
symbol
for
q
both
p
and
q
are
integers,
and
q
does
not
equal
0.
irrational
They
All
of
the
the
set
integers,
rational
numbers
and
irrational
numbers
together
real
numbers,
.
The
real
numbers
can
be
represented
on
numbers.
numbers
form
that
of
are
the
cannot
expressed
number
be
real
as
line.
a
7
fraction,
for
√22
example:
√8
4
or
.
π
x
5
Example
Write
a
S
is
b
P
=
c
M
these
the
{2,
=
3
2
1
0
1
2
3
4
5
1
sets
set
3,
{2,
4
of
5,
4,
in
real
7,
6,
set
…,
8,
builder
notation.
numbers
between
0
and
1
37}
…}
Use
a
the
‘less
than’
b
symbol
There
c
is
no
because
special
0
and
symbol
1
are
for
not
prime
included.
numbers.
Even numbers are multiples of 2, so they can be written as 2n.
Example
Write
out
2
the
elements
of
each
set
in
list
form.
a
b
Describe
set
G
in
words.
c
a
E
=
{−2,
1,
0,
1}
3
and
2
are
not
included.
b
There is no rst real number greater
c
G
is
the
set
of
real
numbers
greater
than
0
and
less
than
1.
than 0, or last real number less than 1, so
you cannot list the elements of this set.
1.1 Mathematically speaking
23
Practice
2
ATL
1
Write
nite
2
out
or
a
{x
c
{a
a
e
{c
c
x
Write
the
∈ , − 2
∈ , a
∈ , c
each
=
{1,
<
x
a
+ 3
the
multiple
set
T
c
U
is
the
set
of
real
d
V
is
the
set
of
rational
in
0,
81,
10,
set
builder
odd
b
The
set
of
multiples
c
W
{1,
2,
4,
The
universal
The
empty
From
set
the
State
whether
cardinality.
b
{
d
{b
f
{
y
y
b
p
∈ ,
y
>
∈ , b is
p
each
set
is
0}
a
factor
∈ primary
28}
of
colors }
notation.
…}
between
numbers
1
and
between
0
2,
including
and
2.
1.
notation:
of
for
U
numbers.
8,
set,
set,
example,
form.
its
solving
set
universal
list
5}
numbers
The
For
in
state
…}
20,
a
=
of
builder
b
Write
49,
10,
set
nite,
8}
<
S
{…,
each
is
5}
<
using
25,
of
set
a
=
9,
If
is
set
Problem
3
elements
innite.
is
{
}
the
the
universal
16,
U,
of
32}
is
or
set
set
the
,
W
set
of
U
3.
=
is
set
that
the
of
set
with
winter
months
no
months
of
{yellow,
contains
the
red,
all
sets
being
elements,
and
set
S
so
of
considered.
n(
)
=
0.
summer
months,
the
year.
blue}
you
can
make
these
sets:
In
J
P
=
=
{yellow,
examinations,
red}
the
K
=
{yellow}
S
=
{yellow,
set
blue}
to
L
=
{red}
T
=
{red,
universal
will
you
be
in
the
blue}
question,
M
=
All
set
{blue}
these
and
U
sets,
U
which
can
itself,
are
A
subset
be
called
made
subsets
=
{yellow,
from
of
the
red,
elements
of
U,
set
symbol
if
for
The
all
is
for
x
a
subset
∈
empty
A
⇒
set
is
is
of
⊆
.
a
set
B
if
Written
x
∈
B,
then
a
subset
of
A
including
the
empty
U.
every
in
⊆
any
element
in
mathematical
A
is
also
in
B.
form:
B
set.
So,
for
any
set
A,
The
set
24
is
a
subset
of
itself.
1 Form
∈
A
⇒
then
A
read
as
⊆
x
∈
B’
‘If
B,
can
for
elements
in
A
elements
are
B,
then
subset
of
the
also
A is
B.’
be
all
.
in
Every
when
necessary.
blue}
‘x
The
given
a
N U M B E R
Two
Some
●
If
sets
are
further
A
=
{1,
equal
if
they
examples
5}
and
B
of
=
contain
subsets
{1,
2,
3,
exactly
the
same
elements.
are:
4,
5},
then
⊆
A
B.
n
●
If
C
=
{3
n
0
D
=
{3
If
E
=
∈
1
,
3
,
3
and
}
2
D
{1, 3, 9, 27} ,
=
then
⊆
D
C,
because
3
,
3
}
2
●
In
You
other
can
(when
in
A
=
this
The
use
is
then
case
as
}
at
it
A
⊆
E
be
1
Consider
down
a
all
⊆
then
to
element
⊆
E
⊆
similar
for
an
A
it.
subset
symbol
are
denote
of
underneath
improper
the
⊂
F
and
⊆
F
E
are
both
true.
subset.
that
A
of
⊂
B,
is
B
or
not
If
in
means
B
is
two
B )
A
an
sets
then
is
a
are
you
not
can
proper
improper
equal,
use
subset
subset
of
the
of
A,
B
and
.
in
the
way
improper
or
they
work
proper
to
≤
and
<.
If
in
doubt
subset.
1
set
of
one
used
Exploration
{ −1, 1},
symbol
line
an
use
=
F
the
a
and
can
=
least
is
you
F
and
use
without
symbol
⊆
= 1
always
⊂
B,
x
words,
there
symbol
If
{
x
containing
its
two
elements,
for
example,
A
=
{1,
2}.
Write
The
subsets.
the
2
Now
consider
a
set
containing
three
elements,
for
example
B
=
{1,
2,
set
3
Do
4
Based
down
the
5
Test
same
on
subsets
all
your
Reect
a
its
again
your
of
of
rule
for
with
on
and
a
of
are
any
subsets.
a
ndings,
set
and
set
3}.
subsets
Write
itself
empty
n
set
containing
suggest
a
rule
four
for
given
set.
Later
in
elements.
determining
the
number
of
elements.
set
containing
discuss
ve
elements.
1
your
mathematics
●
For
the
set
A
=
{1,
2},
is
every
element
of
∅
a
member
of
A?
course
Is
there
any
element
of
∅
that
is
not
in
learn
A?
key
Use
your
answers
to
explain
why
∅
is
a
subset
of
A,
and
of
any
of
generalization
specic
or
general
rule
is
‘a
general
statement
made
on
the
basis
What
specic
concept
Compare
Did
you
always
●
Why
the
examples
rule
you
all
get
the
true
for
any
should
you
and
related
did
you
use
in
Exploration
of
the
concept
have
found
same
be
in
result?
number
of
cautious
Exploration
Is
that
elements
when
1
with
enough
in
the
to
of
Until
justied
1?
your
●
the
Justication.
examples’.
you
●
will
set.
Logic
A
you
about
others
say
that
original
in
your
this
set?
class.
rule
is
general
statement,
cannot
will
be
you
certain
always
be
generalizing?
1.1 Mathematically speaking
it
true.
25
Practice
3
ATL
1
2
Determine
whether
a
{1,
2,
3}
b
{1,
2,
3,
c
{16,
17,
give
{2,
5}
{2}
c
{3} ∈{ 2, 3, 4 }
d
4
e
Given
⊆
⊆
{ 2,
{ 2,
A
false,
=
a
If
A
b
If
R
c
If
p
d
If
A
if
a
⊆
=
a
;
less
{rational
these
than
equal.
6}
numbers
statements
are
greater
true
or
than
false.
15}
If
the
statement
is
why.
set
and
that
D
the
2,
⊆
⊆
to
B
=
{2,
3,
4},
below
are
true
then
A
=
B
d
Q,
⊆
32
or
a
then
(nd
an
diagram
A
⊆
or
example
false.
that
For
makes
those
the
that
statement
helpful.
C
R.
then
p
∈
Q
A.
subsets.
not
set
a
set
can
have
exactly
24
subsets.
you
operations?
represent
sets
operations
intersection
3}
A
of
that
and
and
sets
B
B
A
=
and
contains
complement
of
{2,
3,
without
intersection
B,
the
and
to
their
operations?
manipulate
numbers,
sets
also
have
of
of
4,
5},
set
.
1 Form
the
repeating
and
sets
a
sets
is
set
any
that
of
contains
them
is
C
all
=
the
{1,
2,
elements
3,
4,
5}.
that
Set
C
is
written A ∪ B
elements
as
2 6
a
and
operations.
both
set
C,
arithmetic
Intersection:
The
⊆
B
has
do
●
union
P
are
Union:
the
5}
operations
●
{1,
<
drawing
S
whether
use
and
in
B
then
How
are
nd
then
●
Union
statements
S
What
you
x
counter-example
and
●
own
The
numbers
∈ , 1 <
B
B,
Set
is
...}
x
the
∈ P
Determine
C
{x
may
⊆
5
=
are
solving
You
Write
A
sets
3, 4 }
give
4
their
of
3, 4 }
Determine
as
pairs
1}
reason
b
these
∈{2, 3, 4 }
false).
If
19,
2
Just
not
{prime
a
are
3,
whether
a
Problem
3
;
18,
Determine
false,
;
or
A
and
A
is
that
B,
both
and
denoted
is
by
sets
have
in
common
is
D
=
{2,
3}.
written A ∩ B
A′,
and
described
in
set
notation
N U M B E R
For
of
the
the
So
universal
set
E ′
=
E
is
{1,
If
U
=
{1,
made
2,
Example
set
3,
U
=
up
{1,
of
2,
the
3,
4,
5,
6},
elements
in
and
the
set
E
=
{4,
universal
6},
set
the
that
complement
are
not
in
set
E.
5}
3
2,
3,
…,
10},
A
=
{2,
4,
6,
8},
B
=
a
b
c
d
{2,
3,
5,
7}
and
C
=
{1,
a
5,
9},
nd:
A and C have no elements in common.
b
B ′
=
{1,
4,
6,
8,
9,
10}
First
A
c
∪
B ′
A′
=
A′
∩
=
{1,
{1,
3,
2,
5,
4,
7,
6,
9,
8,
9,
=
{3,
5,
7}
(A′
d
A
∩
∪
(A
B) ′
B
∪
∪
B
=
C
∪
Practice
{1,
=
2,
{1,
C ) ′
=
4,
2,
6,
3,
8,
4,
9,
5,
nd
∩
are
(A′
.
10}
First
B
nd
10}
B)’
not
is
.
made
in
up
of
the
elements
in U
that
.
10}
6,
7,
8,
9}
{10}
4
ATL
+
1
If U
B
=
a
=
{x
{1,
x
3,
∈
5,
, 1 ≤
…,
19}
b
A ∩ B
x
≤
20
},
A
C
=
{x
and
=
A ∪ B
{2,
x
c
4,
8,
…,
20},
∈ primes, 1 ≤
d
A′ ∩ C
x
≤
20} ,
nd:
′
′
( A ∩ C )
e
( A′ ∪ B )
e
( R ∩ S )
+
2
If
U
=
and T
a
 ,
R
= {x
=
x
The
b
three
●
n(F
●
F
●
∩
∩
G ′
=
Reect
For
any
their
=
{ x
,
x
≤ 15},
< 10
},
S
=
{x
−5
<
x
<
5}
nd:
c
R ∪T
set
=
R′ ∩T
d
S ′ ∩T
′
∩T
is
U
F,
=
{11,
G
and
12,
H
13,
such
14,
15,
16,
17,
18,
19,
20}.
that:
2
∅
x
is
odd,
and
two
union
x
subsets
G)
H
∈
solving
universal
Write
x
∈ ;
R ∩ S
Problem
3
{x
∪
≤
x
discuss
sets,
A
11
B
A
and
B,
≤
19}
2
explain
Tip
why
the
intersection
A
∩
B
is
a
subset
of
Try
with
sets
A
you
any
and
B.
two
Can
generalize
your
result?
1.1 Mathematically speaking
27
Venn
In
a
Venn
Circles
It
to
is
diagrams
diagram,
represent
important
include
brackets
the
a
rectangle
subsets
that
you
listing
the
use
universal
when
of
the
set
the
represents
in
universal
forms
your
elements
of
a
universal
set.
set.
representation
Venn
of
the
diagrams,
correctly
for
–
example,
remembering
or
using
curly
set.
U
A
Venn
diagram
for
a
set
A
and
A
its
complement
A’
U
Venn
diagram
of
the
intersection
A
of
the
two
sets
A
and
∩
B
B:
U
This
Venn
diagram
intersection
A
of
the
shows
two
that
sets A
the
and
B
B
is
the
Sets
A
empty
and
B
set:
are
.
disjoint.
U
This
one
sets
A
shows
and
B.
the
The
union
of
unshaded
two
area
A
outside
the
A
the
circles
complement
and
B:
(A
∪
of
Draw
3
Venn
set
represents
the
set
of
;
(A
∪
in
in
B) ′.
the
the
rst
one
second
Describe
the
shade
the
diagram
region
shade
relationship
the
that
region
between
that
these
two
mathematically.
Draw
two
Venn
diagrams:
represents
the
set
represents
the
set
(A
∩
B) ′;
.
in
in
the
the
rst
one
second
Describe
the
shade
the
diagram
region
shade
relationship
that
the
region
between
these
that
two
mathematically.
Draw
a
Venn
intersection
was
union
diagrams:
the
sets
the
2
represents
sets
2
two
B
therefore
B) ′
Exploration
1
is
easier
drawing
2 8
a
–
diagram
is
proving
a
to
this
diagram?
1 Form
show
subset
that
of
using
for
their
any
union
reasoning
in
two
sets,
.
A
and
Which
Reect
and
B,
do
the
you
discuss
2,
think
or
by
N U M B E R
Reect
and
discuss
The
3
you
●
How
many
dierent
ways
have
you
represented
the
set
have
far
by
diagrams
called
What
can
you
say
and
about
De
How
new
has
representing
information
Example
sets
about
using
a
Venn
diagram
helped
you
the
discover
Venn
4
Augustus
De
They
used
a
B
a
B
A
to
represent
these
sets
or
relationship
between
b
c
d
of
Morgan.
in
the
circuitry
sets.
electronics,
well
e
of
⊆
are
extensively
diagrams
after
British
and
⊆
named
mathematician
sets?
elds
Draw
are
Morgan’s
?
Laws,
●
using
2?
Venn
●
discovered
in
so
Exploration
relationships
as
in
the
as
eld
logic.
U
A
A
B
Draw
set.
a
Set
rectangle
B
is
representing
completely
the
contained
universal
in
A.
U
b
A
B
Shade
the
area
that
that
not
in
together
is
A,
contains
everything
with
all
of
B
U
c
The
the
shaded
area
complement
represents
of
B
have
what
in
A
and
common.
d
A
The
shaded
intersection
area
of
shows
all
three
the
sets.
B
C
U
e
A
Shade
and
B
B
the
rst,
intersection
then
shade
of
all
C
1.1 Mathematically speaking
2 9
A
of
C
Tip
When
drawing
vertical
total
lines
area
Venn
and
shaded
the
diagrams
other
and
the
it
using
can
be
useful
horizontal
intersection
of
the
to
shade
lines.
sets
The
is
one
set
union
where
the
using
will
be
lines
the
cross.
U
A
Practice
B
A
∩
B
A
∪
B
5
ATL
1
Draw
Venn
∩
diagrams
to
represent
each
B)′
set.
a
(A
c
(A′
e
A
∪
(B
∩
C )
f
g
A
∩
(B
∪
C )′
h
∩
B)′
Problem
b
A
d
(A′
(A
∪
(A
∪
∩
(A
∩
∩
B)
B ′)′
B)′
∪
C
B)
∪
(A
∩
C )
∪
(B
∩
C )
solving
Venn
are
2
Write
the
set
that
the
shaded
part
of
each
Venn
diagram
U
b
U
English
mathematician,
John
A
A
B
after
represents.
an
a
diagrams
named
B
This
glass
in
Venn.
stained
window
Cambridge
University
commemorates
achievements.
c
U
A
d
U
B
C
3 0
1 Form
his
N U M B E R
U
e
U
f
C
C
g
h
U
U
C
C
You
are
already
operations
determine
of
familiar
addition
which
intersection
and
of
By
drawing
statements
these
the
properties
multiplication.
properties
are
of
In
also
real
the
numbers
following
valid
under
under
the
binary
exploration
the
set
you
operations
will
of
union.
Exploration
1
with
and
a
3
Venn
below,
diagram
show
for
that
each
the
side
of
following
the
equals
properties
sign
hold
in
for
the
set
operations:
2
a
A
∩
B
c
A
∩
(B
e
A ∩
The
the
=
B
∩
∩
C )
(B ∪ C )
additive
inverse
A
=
=
(A
∩
(A ∩ B)
identity
of
n
B)
is
n
of
∩
C
∪ (A ∩ C )
the
since
real
n
+
of
m
Determine
B
Find
E
3
sets
and
if
and
C
∪
B
sets
if
and
and
are
F,
since
whether
operations
Find
is
the
E
the
×
not
∪
B
n)
is
=
there
=
d
A
∪
(B
f
A ∪ (B ∩ C )
1
×
(
n)
since
m
is
an
B
=
=
=
B
∪
C )
is
0
since
+
n
=
m
1
×
(m
identity
∪
n
A
=
=
+
(A
∪
B)
(A ∪ B)
0
=
0
+
m
=
m
∪
∩
n
Commutative
laws
Associative
laws
Distributive
laws
C
(A ∪ C )
=
n
and
0.
1
≠
=
1
×
and
the
0).
and
inverse
for
the
∩
such
two
and
they
Summarize
C
or
m
A
numbers
(
m
inverse
b
F
that
A
∪
B
∪
A
=
E
∪
A
and
A
∩
C
=
C
∩
A
=
A.
identities.
exist
exist,
such
are
that
the
properties
of
A
∪
E
=
A
=
B
and
A
∩
F
=
F
∩
A
=
C.
inverses.
the
set
operations
union
and
intersection.
1.1 Mathematically speaking
31
D
Venn
Venn
How
useful
●
How
does
diagrams
research,
be
diagrams
●
are
biology,
are
form
used
and
sets
to
and
Venn
inuence
model
social
in
solving
real-life
problems?
function?
and
science,
diagrams
solve
problems
where
in
elds
‘overlapping’
such
as
market
information
needs
to
sorted.
Example
A
market
5
research
company
surveys
100
students
and
learns
that
75
of
Set
them
own
a
television
(T )
and
45
own
a
bicycle
(B).
35
students
own
by
a
television
and
a
bicycle.
Draw
a
Venn
diagram
to
show
this
theory
Find
how
many
students
own
either
a
television
or
a
bicycle,
or
online
Data
both.
for
b
Find
how
c
Explain
many
own
neither
a
television
nor
a
or
a
bicycle,
your
but
diagram
not
shows
that
50
on
students
own
either
a
television
so
search
get
∩
T )
35
U
n (B)
and
so
=
n
n (B
45
=
100
45
(B
∩
35
n (T )
∩
T )
T ’)
=
=
35,
and
=
so
10
=
n
75
(B
n (T
75
∩
∩
35
+
35
100
+
85
40
=
=
15
85.
students
remaining
a
The
is
b
c
n
number
(B
∪
T )
of
=
The
number
n
∪
(B
The
both
10
+
T )′
number
can
40
32
=
of
=
be
students
who
own
either
a
television
or
a
bicycle,
or
both
85.
students
who
own
neither
a
television
nor
a
bicycle
is
15.
of
students
seen
as
the
50.
1 Form
who
own
union
of
either
the
two
a
television
sets
less
the
or
a
bicycle
but
intersection,
or
not
T )
B ’)
=
15
10
split
a
that
into
40
=
=
35,
your
query
will
response
quickly.
(B
products
is
bicycle.
both.
n
shopping
engines.
sale
sets
how
used
information.
search
a
is
both
more
N U M B E R
Practice
6
Objective:
v.
organize
These
to
a
Communicating
information
questions
organize
Venn
1
C.
encourage
information
using
a
logical
students
logically.
to
use
The
str ucture
Venn
students
diagrams
should
be
(and
able
a
to
table
draw
in
Question
and
1)
interpret
diagrams.
i
For
the
Venn
diagram,
describe
the
Sea
characteristics
and
those
of
of
only
both
whales,
whales
and
of
only
creatures
sh,
sh.
WHALES
ii
Explain
how
the
diagram
has
helped
you
with
FISH
live
your
in
water
descriptions
more
and
whether
a
dierent
form
could
be
live
lay
useful.
have
ns
have
breathe
b
i
Below
is
whales,
that
a
table
sh,
illustrates
choose
an
describing
and
shrimp.
these
live
ns
✓
whales
universal
have
lay
eggs
✓
shrimp
whether
illustrate
2
In
2
a
in
year
both
many
3
In
a
of
4
a
47
5
two
school
Of
new
Art.
students,
chose
one
by
at
of
50
how
Physics
c
neither
30
students
cookery.
Venn
in
water
to
breathe
breathe
have
water
in
air
water
legs
✓
✓
✓
✓
✓
✓
or
the
Venn
enrolled
a
Venn
either
in
involved
many
in
diagram
Music,
diagram,
Music
involved
are
Of
or
form
✓
is
best
to
both.
are
26
in
Art,
determine
and
how
Art.
community
in
students
diagram,
a
the
archery
b
the
chess
2
2
up
nor
5
85
By
service
drawing
involved
in
a
either
one
in
signed
both
in
in
up
for
subjects.
up
three
are
and
soft
total
drink
drinks.
number
and
If
of
each
students
By
Mathematics
drawing
a
and
Venn
70
for
diagram,
for:
all
and
many
clubs
dierent
in
3
school
three
cookery
chess,
how
cookery
the
a
both
Physics.
chess
only
chose
chose
diagram.
signed
students
determine
and
for
students
students
determine
Venn
enrolled
are
are
a
93
25
drinks,
students
these,
club,
total,
water.
students,
Mathematics
cookery
live
are
lunchtime,
signed
many
Mathematics
cookery
in
are
62
how
these
university
only
the
75
drawing
only
are
sure
✓
drawing
and
bottled
b
and
By
18
enrolled
athletics,
a
There
students,
not
breathe
✓
form
swim
information.
determine
while
determine
table
can
activities.
chose
Physics,
6
350
in
cafeteria
150
and
cafeteria
students
the
50
are
diagram,
student
in
of
205
the
given
of
Music
school
these
In
group
students
projects,
Venn
the
of
diagram
Make
scales
air
set.
scales
✓
Explain
Venn
✓
sh
ii
characteristics
a
characteristics.
appropriate
births
some
Create
eggs
births
but
are
students
clubs,
not
only
are
clubs:
6
of
archery,
in
chess,
them
15
archery.
archery
are
only
belong
By
in
to
drawing
a
in:
only
club.
1.1 Mathematically speaking
3 3
7
In
a
by
class
bus,
to
school
the
total
by
bus
By
32
bus,
have
drawing
The
a
not
not
live
in
live
in
the
have
to
the
in
lunches.
school
school
13
Venn
diagrams
The
squash
below
students
bus,
students
determine
n(U
in
in
)
A
=
105,
or
B
Draw
b
Find
or
a
a
what
how
Justify
Apply
a
n(A)
Venn
of
represent
can
choose
to
many
participants
to
not
have
school
to
by
students
in
school
lunches.
travel
school
travel
live
in
school
bus.
total
have
select
How
have
●
Why
was
set
37,
and
it
diagram
in
to
the
n(B )
=
an
after
tennis
school
(T ),
sports
basketball
(B),
S
represents.
84.
showing
the
sets
All
A,
B
number
A,
following
to
B,
of
and
of
and
the
elements
of
U
are
either
U
elements
the
watch
situation.
support.
committee
Determine
discuss
in
A
∪
intersection
B
of
with
A
the
and
B.
how
A
Your
mobile
received
many
student
library
36
students
council
received
votes.
24
actually
47
held
votes,
students
a
and
voted
voted.
4
4:
you
students
both
=
charity
and
Exploration
What
in
table
rule.
rule
which
●
diagram
connecting
charities.
34
have
not
and
do
c
each
elements
your
your
on
Reect
the
do
school
4
neighborhood
both
students
students
travel
to
town
both.
rule
number
vote
9
S
words
a
c
9
travel
school
and
and
town.
and
the
(S ).
Exploration
●
by
b
Write
In
town
in
lunches.
town
school
S
2
live
solving
a
1
3
school
the
diagram,
school
lunches.
travel
school
Venn
live
lunches.
program.
or
do
5
school
do
students
Problem
8
but
students
16
and
school
have
town,
11
of
students,
they
by
school
lunches.
A
of
and
used
easier
to
sets
to
model
consider
sets
the
of
situation
students
in
step
rather
2?
than
considering
individually?
operation
models
charities?
1 Form
the
fact
that
some
students
voted
for
for
N U M B E R
Reect
Working
can
be
your
and
with
used
least
one
to
school
dierent
discuss
more
timetable,
subjects,
three
or
organize
circles
to
of
and
your
your
5
test
your
classmates,
categorize
some
household
schedules.
illustrate
the
chores,
Create
dierent
discuss
of
the
your
two
how
Venn
following
homework
Venn
situations
diagrams
situations:
for
diagrams
you
have
with
at
chosen.
Summary
A
a
set
set
The
is
is
a
collection
called
symbol
an
∈
of
objects.
element
means
‘is
of
an
Each
the
member
of
set.
element
Written
in
if
x
for
The
of ’.
set
The
symbol
∉
The
number
means
‘is
not
an
element
all
empty
It
is
called
the
elements
in
cardinality
a
of
set
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3 5
6
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table
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11
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sports
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many
out
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in
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university
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numbers},
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diagram
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39
this
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medical
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State
share
how
in
many
with
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105
each
genes
of
multiple
many
in
the
total
three
sclerosis,
genes
all
are
and
three
73
diseases:
stroke.
diseases
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a
c
the
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medical
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might
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concepts.
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can
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and
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is
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(some
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are
these
may
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and
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main
types
social
that
is
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environmental
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and
that
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by
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to
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all
and
and
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all
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and
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how
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2
diagram:
common.
these
is
sustainable.
1.1 Mathematically speaking
37
–
2
Relationships
Relationships are the connections and associations between proper ties,
objects, people and ideas – including the human community’s
connections with the world in which we live.
Subitizing
How
is
many
immediately
without
too
–
than
for
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penguins
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here?
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you
knowing
actually
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primitive
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mathematical
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are
subitize
is
less
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relationship
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creating
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like
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relationships
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numbers
subitize?
relationships
mathematical
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different
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and
or
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expresses
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variables.
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and
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concepts
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,
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s
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models,
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3 9
Are
we
related?
2.1
Global
context:
Objectives
●
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and
a
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the
dierence
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a
SPIHSNOITA LER
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function
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mapping
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interpret
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used
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all
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2.1 Are we related?
41
of
2
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3
other
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b
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B
Continued
4 2
2 Relationships
on
next
page
you
think
mapping
got
its
A lg e b r A
c
List
all
d
Using
the
the
ordered
ordered
coordinate
2
the
the
where
A,
same
a
Dene
set
b
Determine
ordered
c
A
if
in
possible,
ii
If
not
is
,
and
can
this
draw
possible,
the
any
equation
=
you
pairs
If
x
as
Consider
graph
real
A
=
{
the
2,
1,
a
List
b
Hence,
c
Draw
set
d
all
A
Using
of
1,
the
2}
=
x
is
draw
the
+
an
set
a
where
a
A
Use
=
,
your
equation
b
A
Dene
x
and
why
relation
relation
can
(as
in
Reect
●
What
to
●
What
a
are
on
a
x
of
is
an
set
element
of
B.
diagram
to
represent
all
the
diagram.
on
a
coordinate
plane,
where
y
y
is
,
where
an
(x,
of
y)
that
set
to
x
is
element
an
of
element
set
of
the
set
B.
represent
this
relation.
B
represent
the
relation
between
pairs
as
coordinates,
plot
these
points
on
a
plane.
same
and
equation
y
is
an
a
2
x
=
y
element
of
the
coordinate
,
where
of
graph
plane,
set
in
x
is
an
element
of
the
set
A,
B.
step
where
x
3
is
to
draw
any
the
real
graph
of
this
number.
B.
dened
and
are
points
2
=
diagram
either
Exploration
the
represent
these
B
knowledge
be
relation.
not.
equation
pairs
ordered
on
set
where
mapping
2
the
2,
element
mapping
elements
mapping
these
Consider
plot
this
equation.
this
ordered
coordinate
4
represent
number.
0,
list
a
y
explain
equation
the
and
that
coordinates,
y
2
3
y)
B
i
Draw
pairs
(x,
plane.
Consider
set
pairs
a
the
1)
or
discuss
advantages
by
by
listing
giving
the
a
set
of
specic
ordered
rule
(as
pairs
in
that
make
Exploration
a
2).
1
and
limitations
of
using
a
mapping
and
limitations
of
using
a
graph
diagram
relation?
advantages
to
represent
relation?
●
In
which
situations
is
a
mapping
●
In
which
situations
is
a
graph
diagram
better
than
better
a
than
mapping
a
graph?
diagram?
2.1 Are we related?
4 3
A
It
relation
has
●
a
three
a
set
of
ordered
pairs
{(x,
y) | x
∈ A,
y
∈ B}.
components:
relation
the
or
rule
that
maps
x
onto
y
for
each
ordered
pair
in
relation
●
a
set
A
●
a
set
B
A
is
that
that
relation
Practice
contains
all
contains
maps
set
A
the
all
onto
x
elements
the
y
set
B
of
each
ordered
pair
elements.
1
ATL
1
Determine
which
many-to-one,
a
A
=
A
=
each
c
A
=
in
2
Find
in
A
a
to
each
of
real-life
Dene
b
Write
set
the
of
your
in
to
{couples},
A
a
in
in
student
{students
student
type
in
your
their
B
=
sets
the
to
B
class},
diagram
best
=
their
mother
{number
number
B
=
in
B
and
B
each
as
relation
in
type
and
in
of
maps
For
the
and
the
in
the
relation
relations:
and
the
relation
B
relation
maps
maps
each
each
couple
diagram.
1
the
elements
in
set
A
to
the
elements
B
two
diagram
Hint:
one-to-many,
these
siblings},
siblings
mapping
2
3
of
of
B
question
that
of
each
{mothers},
grandchildren
for
(one-to-one,
matches
{grandchildren},
their
A
class},
A
situation
down
mapping
many-to-many)
{students
maps
b
or
equations
that
Look
best
back
y
=
x
describes
at
your
+
2
and
the
x
2
=
relation
answers
to
y
,
determine
between
set
Exploration
A
the
=
type

and
of
mapping
set
B
=
.
2.
x
A
function
one
A
function
value
a
is
element
there
function
can
is
a
in
relation
set
be
where
thought
only
each
element
in
set
A
maps
to
one
and
only
B
one
of
as
possible
a
number
output
machine,
value.
Such
where
a
for
number
every
input
machine
is
called
machine
y
The
A
function
can
be
written
as
f (x)
=
y,
letter
most
●
x
is
the
input
value,
x
∈
y
●
f
is
the
output
value,
x
often
∈
represent
function,
B
letters
is
the
name
of
the
function
that
maps
x
to
=
y
is
read
‘the
function
f
of
x
is
y’
or
just
‘f
of
x
is
y’.
h
such
used
are
way
of
writing
f (x)
=
y
is
f
:
2 Relationships
as
g
in
problems
more
.
than
4 4
other
often
involving
Another
a
but
y
and
f (x)
is
used
A
to
●
f
where:
one
function.
A lg e b r A
Linear
f
:
x
the
equations,

input
and
f (
If
a
the
Using
x
the
5)
ordered
to
=
−3,
pair
domain
maps
The domain
range
generates.
the
f
B
is
+
2,
3,
are
you
the
functions.
can
say
output
Function
of
called
function
function
The
x
is
elements
a
a
=
is
You
that
5.
notation
can
f (3)
You
uses
=
write
5,
could
both
f (x)
=
meaning
also
write
elements
x
+
that
f (4)
of
2
or
when
=
6
an
equation.
set
of
of
y
notation,
example.
one
and
as
this
function
for
in
function
The
such
+ 2.
x
range
is
is
is
set
A
the
the
the
also
onto
elements
of
set
B,
then
set
A
is
called
range.
set
set
of
of
called
input
all
the
values
output
set
of
that
the
values
images
function
that
of
the
the
can
take.
In
a
mapping
diagram,
all
elements
of
elements
in
domain
are
set,
all
and
elements
the
domain
is
specied
in
the
question.
If
not,
it
is
assumed
to
set
of
real
numbers,
.
If
a
function
is
dened
as
a
set
of
ordered
f (x)),
the
then
the
range
Example
Consider
is
domain
the
set
is
of
simply
output
the
set
values
of
(all
input
the
values
values
of
(all
the
values
of
the
the
set.
1
this
in
mapping
A
diagram.
B
words
what
the
the
18th
Leonhard
century,
Euler,
one
relation
of
could
in
f (x)).
Alice
Write
of
are
x)
In
a
one
pairs
other
and
in
the
be
range
(x,
the
domain.
Usually,
the
the
function
the
most
prolic
represent.
Charles
Baby
b
Justify
why
the
relation
is
a
Diego
c
State
the
domain
and
range
mathematicians
in
function.
of
history,
the
the
introduced
notation
f (x)
to
Gabriel
Toddler
function.
Pauline
d
Write
the
function
for
one
of
represent
of
a
function
x
the
Elsa
ordered
a
This
b
day
This
each
c
in
mapping
which
a
pairs
age
care
Domain:
diagram.
diagram
group
could
children
represent
belong
to
at
center.
relation
child
this
is
is
in
a
function
only
{Alice,
Range:
{Baby,
g (Elsa)
=
one
age
Charles,
In order to be a function, each element from the domain
because
must map to one and only one element in the range.
group.
Diego,
Gabriel,
Pauline,
Elsa}
Toddler}
g was chosen for the function, since it
represents the group a child belongs to.
d
Elsa
is
a
Toddler
toddler.
A
Practice
2
1
this
Consider
mapping
B
2
diagram.
3
a
Determine
b
Write
c
Find:
if
this
relation
is
a
function.
Justify
your
answer.
2
4
the
domain
and
range
using
set
notation.
5
i
f (1)
ii
f (2)
iii
f (5)
2.1 Are we related?
4 5
2
For
each
of
i
Write
ii
Justify
iii
State
iv
Write
the
in
following
words
why
the
what
the
the
relation
domain
down
mapping
one
and
relation
is
a
could
range.
specic
ordered
pair
in
b
brand
represent.
function.
Chocolate
a
diagrams:
the
function.
x
Name
Gender
Joanne
Girl
c
y
Countr y
1
1
1
Neuhaus
Belgium
Côte
James
d’Or
2
Leonidas
4
Jessica
2
Toblerone
Switzerland
Jennifer
Lindt
5
Hershey’s
25
USA
Boy
Joseph
5
Italy
Baci
Example
State
{(1,
the
2),
2
domain
(2,
4),
and
(3,
6),
range
(4,
8),
of
(5,
the
relation:
10)}
In
Domain
=
{1,
2,
3,
4,
a
list
of
ordered
x-values. The
In
Range
=
{2,
4,
6,
8,
a
list
of
domain
ordered
1
2
State
the
and
range
for
each
a
{(7,
4),
(2,
9),
(4,
6),
(8,
1),
(5,
b
{(6,
2),
(8,
8),
(3,
7),
(4,
4),
(4,
c
{(
d
{(a,
g),
(t,
e
{(a,
x),
(a,
3,
1),
Consider
the
all
(5,
c),
1),
(s,
y),
set
the
(
=
List
ii
Hence,
list
iii
Draw
mapping
iv
Use
v
Determine
your
w),
(b,
{
(3,
(
x),
2,
ordered
the
1),
(k,
z),
i
a
2,
g),
(a,
A
elements
of
diagram
mapping
to
or
not
(
7,
(p,
1),
(8,
1)}
p)}
z)}
1,
y)
2}.
that
the
diagram
whether
4)}
1),
0,
(x,
2)}
k),
(b,
1,
pairs
j,
relation.
For
range
(set
relation
this
a
to
c:
relation.
B).
the
determine
the
relation
represent
represent
to
each
relation
which
is
a
between
type
of
set
A
relation
and
it
y
4 6
=
x
b
2 Relationships
y
=
3x
+
3
function.
c
y
=
set
is.
2
a
the
domain
written
pairs,
range
3
domain
is
the
in
is
set
the
list
of
range
notation.
is
the
list
of
10}
y-values. The
Practice
pairs,
5}
x
B
is
written
in
set
notation.
A lg e b r A
Activity
With
some
range
as
by
input
this
functions,
simply
and
method,
the
it
thinking
kinds
copy
possible
about
to
the
values
kinds
that
complete
determine
are
this
of
2x
can
you
so
values
produced
be
can
the
any
double
f
convention,
function
range
square
number,
h
can
four
have
A
functions
on
natural
can
A
the
Practice
each
in
the
square
g
is
square
only
root.
{ y | y ≥
the
Sothe
0,
∈
}.
real
domain
of
.
the
domain
domain
so
any
of
the
gives
is
Activity
or
the
range
show
for
largest
the
dierent
dierent
possible
set
possible
restrictions
you
functions.
of
values
that
a
function
take.
restricted
State
is
used
Using
.
positive
can
be
value,
is
root
You
can
output.
since
any
of
that
as
and
Range
value,
domain
domain
table.
By
The
the
Domain
x
=
of
and
Function
f (x)
is
the
domain
is
a
subset
of
the
natural
domain
of
the
function.
4
largest
possible
domain
and
the
corresponding
range
9
3
for
function.
2
1
y
=
5x
3
2
y
=
−4x
+
x
y =
+
12
5
2
4
y
=
2x
2
5
y
=
2
+
5
6
y
=
3x
y =
8
y =
9
2x
x
3
1
15
32
7
x
y
=
x
+ 1
1
10
y
=
+ 1
x
2.1 Are we related?
4 7
Reect
Recall
the
and
discuss
denitions
●
Are
all
functions
●
Are
all
relations
●
If
you
●
What
a
relation
and
a
function.
relations?
functions?
answered
Using
C
of
2
‘no’
to
either
functions
are
the
question
to
similarities
above,
explain
express
and
dierences
why.
relationships
between
relations
and
functions?
●
How
are
function
So
●
far,
a
you
have
function
domain
value
●
a
seen
a
there
function
relation
is
is
3
a
a
special
For
each
Label
to
type
one
ways
one
exactly
investigates
of
and
one
one-to-one
of
determining
if
a
relation
is
a
another?
relation,
only
one
output
or
a
further
where
for
element
each
in
the
element
range.
of
the
(Each
input
value.)
many-to-one
how
you
relation.
can
determine
whether
or
not
function.
Exploration
1
dierent
that:
exists
generates
Exploration
a
is
the
related
3
mapping
each
graph
diagram,
with
its
draw
a
set
of
points
on
a
coordinate
plane.
Use
relationship.
a
new
coordinate
3
3
for
6
6
10
10
1
1
15
7
2
2
3
3
19
9
5
one–to–one
If
2
you
are
the
many–to–one
following
steps
to
guide
Determine
which
axis
represents
the
domain,
b
Determine
which
axis
represents
the
range,
Determine
a
b
Draw
of
your
from
the
Looking
a
graph
at
graphs
graph
your
of
y
show
a
whether
=
4x
graph,
function.
or
not
a
y
many–to–many
you:
a
determine
3
try
19
9
5
one–to–many
stuck,
x
=
f (x).
Suggest
relation
is
how
a
you
could
function.
9.
determine
the
type
of
relation
this
is.
Continued
4 8
mapping
diagram.
12
4
15
7
each
2
2
12
4
2 Relationships
plane
3
10
on
next
page
A lg e b r A
c
Draw
a
mapping
ordered
the
pairs
type
possible
d
Decide
of
Hence,
for
your
ordered
whether
pair
or
that
decide
it
for
not
your
to
represent
mapping
relationship
relationship
e
diagram
the
whether
y
=
You
mapping
mapping
graph
=
diagram,
represents.
this
y
4x
9.
until
do
Select
you
not
are
need
only
a
few
satised
to
list
with
every
diagram.
diagram
conrms
the
type
of
represents.
4x
9
is
a
function.
y
2
4
Here
is
the
graph
of
=
y
x
5
a
Determine
which
of
the
3
four
types
graph
of
relationship
this
1
represents.
2
b
Hence,
is
a
decide
whether
or
not
y
=
4
x
1
function.
3
5
You
a
●
have
function
with
a
used
two
mapping
many-to-one,
●
with
a
then
the
The
one
if
than
line
the
the
each
relation
vertical
called
diagram:
then
graph:
point,
more
dierent
(sometimes
is
a
if
for
function
a
deciding
from
mapping
relation
is
has
a
x
to
whether
or
not
a
relation
is
y’):
diagram
is
one-to-one
or
function
only
one
corresponding
y-value,
function.
test:
point,
‘a
x-value
if
relation
one
methods
is
no
a
then
vertical
line
function.
the
If
relation
intersects
a
is
vertical
not
a
a
graph
line
at
more
intersects
the
than
graph
at
function.
2
y = 0.5x
1.5 is a function, because no vertical
y
line intersects the graph more than once.
=
x
is
ver tical
not
line
y
a
function,
intersects
because
the
at
graph
least
one
twice.
y
4
4
y
=
0.5x
1.5
2
2
0
x
2
0
4
x
2
2
2
4
4
4
6
8
2.1 Are we related?
4 9
Practice
1
Use
the
5
vertical
line
test
to
determine
which
of
Example
Determine
c
y
5),
x
functions.
y
x
x
x
3
whether
(5,
6),
or
(2,
not
5),
these
(4,
relations
6)}
b
represent
{(1,
3),
2
=
show
y
x
{(2,
graphs
y
y
a
these
3
d
x
2
+
y
a
(2,
function
4),
(3,
from
3),
x
(4,
to
y
4)}
2
=
4
a
2
5
6
4
Draw
a
mapping
diagram
for
the
ordered
pairs.
5
5
6
2
This
relation
is
not
a
maps
to
both
5
and
5.
function.
This
is
a
one-to-many
relation.
b
1
3
2
3
4
This
4
relation
is
c
a
function.
This
is
a
many-to-one
function.
y
4
3
2
y
=
x
−
3
1
Draw
4
3
2
1
4
5
the
1
a
graph
vertical
and
line
use
test.
2
3
5
This
relation
is
a
function.
Therefore,
you
No
use
the
notation
y
=
x
3
or
f (x)
=
x
vertical
graph
Continued
5 0
2 Relationships
line
would
cross
the
3.
on
next
page
at
more
than
one
point.
A lg e b r A
y
d
6
2
x
2
+
y
2
=
4
2
Draw
a
graph
and
use
x
2
the
vertical
line
test.
2
You
This
relation
is
not
a
can
graph
Reect
and
Representation
which
●
How
did
●
How
does
a
●
Why
Decide
Give
is
you
relation
is
Practice
1
discuss
an
something
a
it
is
MYP
represent
a
of
this
vertical
circle
at
lines
more
that
cross
than
one
the
point.
3
related
concept.
It
is
dened
as
‘the
manner
in
presented’.
relations
representation
is
draw
function.
enable
and
functions
you
to
in
Example
determine
whether
3?
or
not
function?
important
to
distinguish
between
relations
and
functions?
6
which
reason
of
the
for
following
your
relations
are
functions.
answer.
b
a
A
1
2
3
4
A
B
4
2
0
2
B
c
{(3,
12),
d
{(9,
9),
(
16,
(8,
7),
10),
(7,
(3,
5),
12),
(5,
9),
(15,
(9,
3),
5),
(3,
(6,
12),
(
0
1
5
15,
1
1
5
7
5
3)}
12)}
y
y
10
9
8
2
7
6
0
x
5
2
4
3
4
2
1
0
1
2
3
4
5
6
7
8
9
10
x
2.1 Are we related?
51
y
y
6
4
4
2
2
x
4
4
2
Reect
and
Justication
evidence
●
is
used
Where
in
discuss
an
to
MYP
related
support
the
4
a
previous
concept.
It
is
dened
as
‘valid
reasons
or
statement’.
example
and
practice
problems
have
you
used
justication?
●
How
is
a
does
justication
Is
●
Can
it
not
important
think
you
of
a
1
Explain
2
If
think
to
justify
of
some
is
number,
why
an
Make
for
this
element
if
Writing
A
up
your
see
your
they
a
example,
●
f
is
the
●
x
is
an
●
f (x)
●
‘3x
+
get
the
is
is
whether
or
not
a
relation
relation
you
is
have
a
function?
worked
with
that
are
is
as
tells
‘I
by
a
the
of
you
then
square
the
result.’
of
a
of
a
write
puzzle.
notation.
is
called
(input
dened
Then
denition
mathematical
is
function,
number’
function
domain
f
this
this
function
you
dening
a
expression
values)
as:
as
Write
share
f (x)
=
and
3x
+
function
with
others
and
have.
specic
that
range
2,
the
it
function.
species
(output
the
values).
then:
domain
x’
what
denition
and
function
the
of
3
something).
think
same
corresponding
usually
=
mathematically
of
‘f
add
domain
f (x)
function
element
2’
5 2
a
it,
function.
the
the
dened
name
read
the
domain
a
equations
mathematical
between
if
a
of
in
write
is
is
own
function
function
This
whether
double
(i.e.
puzzle
relationship
For
support
4
mathematically
3
to
functions?
Exploration
x
you
function?
●
‘I
enable
of
the
function
element
this
specied.
specic
If
it
does
from
to
it
is
each
element
of
the
domain
to
range.
function
isn’t,
2 Relationships
the
f.
When
assumed
to
dening
be
the
a
set
function,
of
real
the
numbers.
A lg e b r A
Example
A
recipe
an
x:
states
extra
time
f (x):
40
using
the
4
chicken
function
weight
the
that
minutes.
of
the
cooking
needs
Express
the
notation,
chicken
time
(in
and
(in
to
be
cooked
relationship
dene
the
for
15
minutes
between
domain
weight
and
per
kg
and
plus
cooking
range.
kg)
Set
in
minutes)
Neither
the
weight
nor
the
the
the
input
and
context
cooking
of
output
the
time
variables
problem.
can
be
negative.
Domain:
The
chicken
will
take
at least 40
minutes
to
cook.
Range:
f (x)
=
15x
+
15
40
Check:
Check
f (1)
=
15
×
1
+
40
=
55
f (2)
=
15
×
2
+
40
=
70
Reect
Think
what
If
had
you
cooking
of
x
is
this
sense
in
a
real-life
sense
of
When
input
the
x
=
be
of
each
the
height
a
bad
sensible
meaning
Yet,
y
=
variable
the
that
=
15
cooking
the
x
×
function
=
0,
0
+
in
does
40
a
are
Good
of
a
of
2
chicken
kg
(x)
Do
needs
chicken
needs
plus
the
15
30
40
more
results
min
min
+
seem
40
+
minutes.
40
min
min
sensible?
=
=
55
min.
70
=
it
Example
make
40.
For
4.
sense
what
that
the
values
domain
(or
real-life)
or
an
denition
do
they
constraints
number
Why
but
do
not
make
chicken).
the
called
variable
age.
kg
worded
represents
Tina’s
possible,
0
elements
represent
problem.
and
f (0)
restrictions
variable
kg
values.
kg
intended?
with
restrict
to
for
mathematically
(as
These
variables
that
context
Tina’s
would
may
problem.
setting
are
all,
are
context
easy
per
5
denition
values
situation
important
at
minutes?
function
real-life
the
values
40
1
A
chicken
is
with
A
discuss
input
no
time
Sometimes
A
and
minutes
you
in
order
on
the
situation,
amount
make
it
is
within
examples
think
to
domain.
are
writing
x
=
Tina
denition?
2.1 Are we related?
53
min.
Practice
7
Objective:
i. use
C.
in
both
oral
In
these
questions
notation
to
for
explain
For
each
and
the
how
a
Express
b
Dene
c
written
you
the
the
(notation,
symbols
and
terminology)
function
a
and
notation
words
such
to
express
as
‘domain’,
the
relation,
‘range’,
set
builder
‘one-to-one’
function.
3:
domain
range
how
use
range,
is
relation
and
carpenter
to
to
and
relation
1
language
explanations.
need
domain
the
Explain
A
mathematical
question
domain
1
Communicating
appropriate
the
using
and
range.
based
on
relation
charges
his
function
is
clients
Decide
the
a
notation.
what
real-life
the
constraints
are
on
the
context.
function.
a
rate
of
€30
an
hour,
plus
a
single
€40
fee
The
for
each
job.
fee
charged
Express
the
relationship
between
time
spent
on
a
job
and
the
to
the
placement
€
A
bathtub
water
lled
empties
between
the
is
time
out
with
at
a
elapsed
120
rate
liters
of
since
25
the
of
water.
liters
plug
per
was
The
drain
minute.
pulled,
plug
is
Express
and
the
pulled
the
and
the
relationship
amount
of
water
On
in
bathtub.
a
tropical
account
size
of
parcel
a
Evaluating
a
corresponds
Example
If
f (x)
f (x)
=
=
3x
old
So
might
it
one
3x
f (4)
=
3
f (4)
=
12
+
×
f
(x )
=
for
of
land,
function
to
a
given
cost
of
a
parcel
additional
and
the
means
15%.
nal
in
the
land
is
Express
price
nding
element
of
that
the
$200
the
the
per
client
element
square
relationship
of
pays
the
4
+
for
range
meter.
between
2,
be
€37
country,
but
another.
Taxes
the
it.
that
nd
the
value
of
f (4).
Start
+
2
with
the
original
2
=
14
− 2
a
Describe
b
Draw
function
denition.
Substitute
8
a
currency.
domain.
2
4 x
in
words
table
of
corresponding
5 4
the
an
in
what
with
5
+
Practice
1
island,
fees
is
on
did
their
in
and
based
countries
37€
3
symbol
client.
often
2
of
the
what
values
output
this
that
function
shows
values.
2 Relationships
at
does
least
to
5
the
input
dierent
value.
input
values
and
the
4
for
x
A lg e b r A
2
2
g (x )
=
x
+ 2
a
Describe
b
Draw
a
in
words
table
of
corresponding
what
values
output
this
that
function
shows
at
does
least
to
5
the
input
dierent
value.
input
values
and
the
and
the
values.
2
h (x ) =
3
x
a
Describe
b
Draw
a
in
words
table
of
corresponding
4
5
If
f
(x )
a
f
(3)
If
a
In
a
p (3)
= 7x
− 3 ,
=
triangle
cases,
are
f
and
an
given
Solving
an
domain
input
two
5,
or
(
shows
at
does
least
value
will
the
1)
=
pentagon,
generates
see
that
corresponds
is
to
5
the
input
dierent
value.
input
values
when
5
Consider
functions
it
a
is
specic
is
to
called
means
element
output
the
possible
This
given
p (8)
generate
function
a
(0)
d
f
(20)
d
p (10)
nd:
one
values
value.
to
Exploration
f
c
input
output
equation
c
p (6)
more
you
only
that
that
function
values.
p (5)
b
function,
this
nd:
b
InExploration
you
values
output
p (4)
certain
what
same
nd
the
solving
nding
in
the
value.
And
output
input
the
the
in
value.
value
if
function.
element
of
the
range.
2
1
2
For
each
at
=
x
For
each
For
4
Use
a
f (3).
function,
nd
the
a
equation
each
your
Evaluating
one
nd
function,
for
between
Solving
f (x)
function,
the
each
value
two
=
7x
This
12
is
and
called
f (x)
=
x
evaluating
the
function
3.
solving
3
the
input
function
to
=
of
x
such
that
f (x)
=
16.
This
is
called
16.
whether
there
is
only
one
possible
input
value.
decide
value
function
f (x)
determine
output
ndings
an
when
value
and
(nding
(nding
if
a
an
the
the
function
output
value
value
of
always
value.
of
x,
is
a
one-to-one
Justify
your
f (x))
only
gives
given
f (x))
can
relationship
answer.
one
give
answer.
more
than
answer.
2
f(x)
=
7x
12
f(x)
=
x
23
64
8
5
8
evaluating
the
function
at
x
=
5
solving
the
function
when
f(x)
=
64
2.1 Are we related?
5 5
Practice
In
1
9
questions
h (x )
=
1
to
7,
evaluate
the
function
at
the
given
values.
2 − 4 x
In
a
h (0)
b
h (1)
c
h (
1e,
for
d
2
3
f
h (20)
(x )
e
= 3x
h (
f (0)
b
d
f (12)
e
(x )
f
x
in
the
a
function.
h (2x)
− 5
a
f
a)
substitute
1)
f (4)
c
f (3x)
f (
f
4)
f (x
+
1)
= 13
a
f (0)
b
d
f (
e
25)
f (19)
f (
c
x)
f (20,000)
f
f (12x)
2
4
f
(x )
=
x
a
f (0)
d
f (9)
+1
b
f (5)
e
f (x
c
+
1)
f (
f
5)
f (4x)
2
5
f
(x )
=
x
a
f (1)
d
f (7)
− 2x
+1
b
f (2)
e
f (
c
10)
f (0)
f
f (3x)
3
6
7
f
(x )
=
4 x
− 2x
+1
a
f (2)
b
d
f (3)
e
f
(x )
= 3x
f (
f (
1)
c
2)
f (0.5)
f
f (
x)
+ 5
a
f (5)
b
d
f (3x)
e
8
If
f
(x )
= 8x
9
If
f
(x )
=
10
If
h (x )
11
If
r (x )
− 4 ,
+ 12 ,
x
nd
f (x
the
nd
= 15 − 2 x ,
f (10)
the
nd
c
+
5)
value
of
value
the
of
value
f (2x)
f
a
b
of
such
such
c
f (3x
that
f (a)
that
such
2)
f (b)
that
=
=
h (c)
12.
12.
=
3.
2
=
,
x
nd
the
value
of
d
such
that
r (d)
=
36.
In
11
and
12
there
2
12
If
f
(x )
=
+ 5 ,
x
nd
the
value
of
e
such
that
f (e)
=
are
If
+
nd
the
value
of
p
such
that
k (p)
=
4
2
1
14
f
If
,
If
p (135)
that
p (s)
the
value
of
q
such
that
f (q)
=
5
solving
=
=
nd
5
Problem
15
1
(x ) =
x
south-east
and
p (180)
=
south,
nd
the
value
of
s
such
north-west.
In
16
If
child(Olimpia)
that
m
a
must
such
be
that
function.
5 6
possible
values.
,
k (x ) =
x
two
1
1
13
54.
=
Viola
satised
child(m)
Explain
in
=
and
order
you,
your
child(Arienne)
for
and
child(x)
reasoning.
2 Relationships
to
determine
Share
=
be
if
in
your
William,
a
state
function.
your
with
condition(s)
Suggest
family
ideas
the
a
child(m)
others.
value
=
you
of
is
16,
how
many
children can a parent
have
if
child(x)
function?
is
a
A lg e b r A
Recognizing
D
not
So
far,
●
a
●
if
an
is
You
What
●
Can
input
not
a
have
by
a
to
specic
a
be
that
are
generates
a
looked
function.
at
its
the
lists
at
how
However,
to
is
more
the
look
domain
than
decide
it
like?
one
and
range
output,
the
relation
if
a
possible
mapping
to
tell
if
a
diagram
relation
or
is
a
a
graph
function
equation?
6
of
equations
below.
NOT
y
x
4
For
A
2
3x
functions
between
FUNCTION
=
not
justied?
relation
relation
Exploration
at
relations
function.
looking
Look
are
that:
already
represents
just
is
do
inequality
know
function
that
functions
●
you
relations
each
relation,
x
∈
and
y
∈
FUNCTION
2
+
y
=
16
2
2x
+
5y
=
9
x
=
3(y
+
1)
7
2
y
=
2
2x
+
3y
2
=
5
2
+
x
4y
3x
4y
5
=
0
x
y
=
1
3
y
=
1
2(x
+
1)
Based
that
on
are
2
If
3
Based
8
the
possible,
on
●
What
input
●
Why
●
Are
not
●
above,
and
rewrite
your
arelation,
Reect
lists
functions
but
and
sort
of
each
results,
is
not
make
those
to
of
about
the
types
ofequations
not.
look
which
equation
discuss
relations
conjecture
are
equation
suggest
the
a
that
like
type
a
of
y
=
f (x).
equation
represents
function.
6
generate
more
than
one
output
value
for
a
single
value?
do
those
there
relations
other
generate
relations
that
more
have
than
similar
one
output
properties
value?
(and
are
therefore
functions)?
How
does
this
relate
to
the
‘vertical
line
test’?
2.1 Are we related?
57
Summary
In
an
an
ordered
object
pair
from
a
(x,
rst
y),
set
the
rst
and
the
term
Another
represents
second
The
represents
an
object
from
a
second
relation
is
a
set
of
ordered
y) | x
∈ A,
y
domain
that
The
has
three
a
relation
or
rule
that
maps
x
onto
y
for
range
set
A
pair
that
in
the
set
B
natural
contains
all
the
x
elements
of
values
that
relation
contains
maps
mapping
relation
oval,
in
are
and
the
set
A
diagram
paired.
lines
rst
all
the
y
of
is
set
pair
function
set
A
maps
function
or
to
set
B
onto
shows
Each
arrows
how
set
is
are
elements
in
the
is
to
a
can
be
x
is
the
input
●
y
is
the
output
restricted
in
the
elements
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drawn
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of
from
second
by
in
a
more
an
elements
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for
at
function.
the
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of
=
x
y
is
is
name
set
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input
function
is
also
the
the
set
function
of
all
the
called
the
set
generates.
in
the
of
images
domain.
a
is
the
function
largest
can
possible
set
take.
domain
is
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subset
of
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only
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value,
x
value,
of
read
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function.
line
more
If
than
a
test:
if
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vertical
than
one
point,
vertical
line
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point,
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the
line
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relation
the
relation
is
graph
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a
at
a
function
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dened
each
one
element
element
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domain
in
set
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by
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=
y,
the
species
a
mathematical
values)
the
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range
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is
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y
function
f
of
x
is
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or
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corresponds
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a
of
given
the
domain
element
in
range.
just
y’.
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1
the
take.
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the
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output
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writing
pairs:
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of
set.
values
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way
term
practice
Determine
prince
whether
each
diagram
princess
represents
a
relation
or
a
3
9
10
400
15
100
function.
Justify
your
answer.
1
2
lord
9
lady
20
5 8
10
2 Relationships
225
3
8
a
A lg e b r A
y
d
y
i
10
3
2
8
1
6
0
x
1
2
3
4
1
2
3
4
4
2
y
j
20
0
2
4
6
8
10
x
15
y
e
10
5
6
0
x
4
2
6
2
State
the
domain
and
a
{(1,
1),
(2,
4),
(
2,
b
{(3,
5),
(4,
5),
(5,
c
{(
d
{(1,
range:
4)}
2
y
2,
3),
(
3,
6)}
2),
(
2,
5)}
f
6
1),
(2,
3),
(5,
8)}
4
3
Find
the
a
f
(x )
c
f
(x ) =
largest
= 3x
x
6
4
2
2
4
possible
− 8
domain
and
b
f
(x )
= 3
d
f
(x ) =
x
6
x
4
2
4
f
x
4
4
e
range:
(x ) =
x
f
f
(x )
at
the
=
x
− 1
4
6
4
Evaluate
each
function
given
values:
y
g
2
a
10
a (x )
i
8
6
b
=
2 −
x
a (2)
b (x )
=
8x
ii
a (0)
iii
a (
2)
ii
b (3)
iii
b (1)
ii
c (5)
iii
c (0)
ii
d (4x)
iii
d (2
− 4
4
i
b (4)
2
c
c (x )
=
x
− 1
x
i
c (1)
2
d
d (x )
=
2x
+ 3
y
h
i
d (4)
4
3
2
1
0
4
3
2
x
1
1
2
3
4
1
2
3
4
2.1 Are we related?
5 9
x)
5
Solve
each
function
at
the
given
values:
8
The
table
shows
postage
rates
for
letters
and
parcels.
f
a
(x )
i
=
f (x)
4 x
=
− 2
18
ii
f (x)
=
0
iii
f (x)
=
Letters
2
Weight
less
than
Price
1
b
f
(x ) =
x
50
1
1
f
i
(x ) =
f
ii
f
(x )
=
(x )
=
2
£0.65
100
g
£0.92
200
g
£1.20
4
5
c
f
iii
(x ) = −
g
x
Small
i
6
A
f (x)
5
computer
returns
a
=
the
Justify
ii
f (x)
program
number
why
this
asks
of
=
1
you
letters
iii
for
in
computer
f (x)
any
that
=
9
word
Weight
and
less
parcels
than
Price
100
g
£1.90
200
g
£2.35
500
g
£3.40
750
g
£4.50
word.
program
is
a
function.
b
Write
down
what
this
computer
program
1
does
is
in
function
called
‘p’
for
notation,
where
the
program.
a
Draw
a
weights
c
p (horse)
ii
p (Mississippi)
iii
p (hi)
b
Decide
price
Find
x
such
mapping
of
i
p (x)
What
=
is
or
whether
the
3
ii
p (x)
=
arrows
show
7
A
For
8
c
p (three)?
what
hospital
ground
the
stops
at
has
the
Justify
b
Find
c
For
the
each
Justify
x
10
a
the
the
does
of
p (x)
ground
below
the
T (n)
elevator
function.
domain
domain,
elevator
hours.
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of
T (n).
suggest
stops
your
at
the
that
answer
pairs.
suggest
a
x?
(including
ground.
times
=
day.
possible
in
24
each
is
≤
above
number
times
your
≤
oors
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ordered
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3
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of
0
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largest
during
of
6 0
nth
why
number
list
9
and
average
a
oor
numbers
oor)
gives
d
diagram
and
the
to
represent
delivery
the
charge.
the
price
weight
determines
determines
the
the
weight.
possible
range
choice.
2 Relationships
of
T (n).
as
a
to
your
mapping
diagram
this.
Determine
function.
f
items
that
Add
e
£5.00
Evaluate:
i
d
kg
function
whether
this
relation
is
a
to
A lg e b r A
Review
in
Fairness
and
1
Many
context
development
universities
decisions.
paired
The
with
takers,
Family
results
family
shown
use
in
test
of
in
one
income
the
income
a
table
their
such
as
a
admissions
test
by
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variable
were
reported
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and
the
($)
Total
10 000
test
score
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Is
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1402
50 000
1461
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1497
90 000
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110 000
1569
130 000
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Does
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relation
a
150 000
Car
this
diagram
Explain
of
this
relation.
Explain
an
equitable
action,
if
any,
your
or
fair
do
reasoning.
you
think
should
be
age,
factors.
of
the
For
22
in
rate
rates
kind
vary
of
car
example,
one
by
city
greatly
you
male
can
depending
drive
drivers
calculate
multiplying
their
and
under
their
age
by
on
other
the
average
$1000
1604
180 000
subtracting
that
from
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1625
a
180 000
mapping
income
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function?
demonstrate
insurance
your
and
b
of
taken?
annual
a
a
mapping
this
What
age
family
range
1326
3
Draw
and
output
2400)
e
a
domain
the
relation.
test-
relationship?
than
and
here:
(max.
More
variable
1714
diagram
and
the
whether
or
to
total
not
this
down
annual
represent
test
Write
cost
based
on
b
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what
c
For
the
of
his
equation
insurance
age
for
(C )
the
for
average
a
young
male
(a).
the
score.
relation
is
a
males
average
function.
C (18)
under
cost
of
represents.
the
car
age
of
Then
22,
insurance
a
is
nd
the
C (18).
annual
function
ofage?
c
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reasons
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this
relationship
d
mayexist.
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charging
age
d
Should
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2
The
universities
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natural
in
use
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the
information
If
percentage
dierent
so,
of
countries
world
their
the
dierent
driver
a
rates
fair
depending
practice?
on
use
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State
f
Suggest
valid
what
for
C (21)
reasons
drivers
represents.
why
this
under
the
Then
function
age
of
nd
is
Percentage
of
world
population
Percentage
of
world’s
resources
Brazil
3
2
China
19
20
India
17
5
Indonesia
4
1
Japan
3
4
Pakistan
3
0.5
3
6
5
18
used
Russian
Federation
United
States
2.1 Are we related?
C (21).
only
24.
resources.
Country
the
Explain
in
how?
and
of
61
3
Logic
A method of reasoning and a system of principles used to build
arguments and reach conclusions
Transpor t
A
farmer
and
a
from
He
is
travelling
wolf.
While
eating
needs
puzzle
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to
he
to
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grain,
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market
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and
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the
in
with
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wolf
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he
bag
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from
of
grain,
prevent
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chicken,
the
the
chicken
chicken.
small
Of
only
one
other
passenger
at
a
course,
time.
overlook
How
the
it
the
out
to
river?
using
get
chicken
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Can
the
number
trips?
6 2
farmer
the
safely
fewest
of
the
grain,
wolf
of
can
other
you
himself,
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work
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sometim
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rst
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logicians
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Three
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waiter
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philosophy
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so
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give
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person
a
lunch
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each
himself
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have
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doesn’t
change
gives
thinking
looking
at
explanation
ring
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him
of
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once
for
on
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reinforce
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married
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facts
them.
skin
been
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reasoning,
known
you
nger,
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woman,
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63
But
can
you
prove
it?
3.1
Global
context:
Objectives
●
Masuring
and
●
th
twn
Finding
th
a
Orientation
Inquiry
distan
lin
and
distan
twn
a
two
●
points,
F
point
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two
How
CIGOL
Finding
th
shortst
●
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th
midpoint
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midpoint
formula
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distan
from
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point
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a
time
do
you
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th
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shortst
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How
do
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you
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distan
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lin
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two
and
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from
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space
questions
two
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points
in
How
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two
givn
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points?
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form
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formula
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Communication
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infrns
6 4
and
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proof
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G E O M E T R Y
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how
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plane
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points
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apply
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2
theorem
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join
shape
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4),
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them
up
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have
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5,
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drawn.
this
nearest
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millimeter.
h
5
9
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construct
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with
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cm
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a
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ruler
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traditionally
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do
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mathmatis
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ar
is
th
pross
of
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rigorous
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thr
parts
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prmiss
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fats
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th
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you
alulat
in
draw
stp
an,
th
th
shap
lngth
of
ACD
AC
and
ompar
it
to
th
3
Continued
on
next
page
3.1 But can you prove it?
as
sin,
psyhology.
Th
a
in
conclusion.
now
●
n
points?
6 5
7
Add
a
point
similar
to
your
diagram
8
Calulat
th
lngth
of
BC
9
Calulat
th
lngth
of
AB
stp3.
or
Explain
your
and
Masurmnt
dtrmining
How
●
To
●
What
●
Hav
In
alulat
th
lngth
of
BC
using
a
whih
you
and
ompar
it
and
ompar
it
think
ar
mor
your
to
masurmnt
your
in
stp
masurmnt
aurat:
your
is
dos
what
is
discuss
an
MYP
quantity,
in
masurmnts
mant
you
of
y
Rlatd
you
1
‘a
to
did
aurat
dnd
dimnsion
rlat
dnd
usd
Conpt,
or
auray
otaind
1
1
apaity
Exploration
dgr
Exploration
th
you
unit’
in
using
onpt
masur
th
as
a
of
mthod
of
unit’.
Masurmnt?
th
ontxt
‘a
dnd
lngths?
of
Exploration
1?
masurmnts?
Pythagoras’
y
B
thorm
points
to
on
nd
a
th
distan
oordinat
onstrutd
3
alulations.
Reect
●
to
pross.
a
grid.
right-angld
twn
First
3
you
triangl.
(4, 5)
two
A
skth
A
(
1, 2)
5
an
hlp
sids
of
you
th
nd
th
lngths
of
th
shortr
x
triangl.
B
O
A
and
B
ar
equidistant
from
O
if
th
distans
OA
and
OB
ar
qual.
AO
=
OB
A
Practice
1
1
Find
th
distan
from
(4,
2
Find
th
distan
from
(
3
For
2)
to
(7,
1)
to
6).
Mak
th
points
a
Find
th
b
Roin
PQRS
c
P (10,
distans
says:
‘th
must
Roin’s
13),

a
logial
3,
Q (17,
PQ,
QR,
lngths
pross
Premises:
Conclusion:
Explain
6 6
fault
in
3 Logic
and
QR,
a
four
13),
and
S (17,
11):
SP
RS
and
SP
ar
whthr
lengths
are
quadrilateral
Since
PQRS
th
R (24,
Dtrmin
The
process:
11).
all
qual,
Roin
is
so
th
shap
orrt.
was:
If
Reasoning
37),
RS
PQ,
squar.’
(2,
Roin’s
PQRS
is
a
has
has
four
square.
logi.
equal.
four
equal
equal
sides,
sides,
it
is
then
a
it
is
square.
a
square.
a
skth.
G E O M E T R Y
4
a
On
10
V (4,
mm
squar
graph
papr,
plot
th
points
O (0,
0),
U (22,
8),
&
and
To
23).
to
b
Zah
T R I G O N O M E T R Y
laims
that
th
triangl
OUV
is
‘show’
quilatral.
using
Without
masuring,
dtrmin
whthr
or
not
his
laim
is
or
tru.
xampls
diagrams;
‘prov’
5
a
On
suital
axs,
plot
th
points
A (41,
99),
B (99,
41),
C
(99,
99),
E (
41,
99),
F (
99,
41),
G (
99,
41),
and
H (
is
a
to
muh
41),
mor
D (41,
mans
dmonstrat
41,
rigorous
99).
produr.
b
Show
that
c
Mark
d
Dsri
e
Vrify
f
To
any
all
lins
th
that
what
points
ight
of
points
symmtry
rotational
would
ABCDEFGH
omputr
Problem
to
quidistant
of
th
otagon
symmtry
ABCDEFGH
xtnt
ar
it
is
not

a
of
th
rgular
appropriat
rprsnt
a
from
th
origin,
ABCDEFGH
otagon
on
(0,
0).
your
diagram.
ABCDEFGH
otagon.
to
rgular
us
th
otagon
oordinats
in
print
or
of
on
th
a
srn?
solving
Th
6
In
th
onstllation
hors’s
alld
ody
‘Th
ornrs
on
(
a
4,
of
is
Cartsian
1),
and
Pgasus,
rprsntd
Grat
Th
of
Squar’.
Grat
plan
(0,
y
Squar
at
(
1,
th
four
Th
of
an
5),

(3,
namd
stars
stars
at
onstllation
Pgasus
wingd
th
Grk
drawn
myth,
2).
killd
Show
that
Th
Grat
Squar
of
aftr
ths
proprtis
of
a
All
sids
ar
b
All
angls
th
th
th
Prsus
Gorgon
(whos
squar:
sam
was
snaks),
lngth
was
ar
to
whn
Pgasus
hair
a
in
mythology.
Mdusa
has
th
hors
Aording
2),
is
mad
of
Pgasus
orn
from
hr
90°
lood.
c
Opposit
sids
Exploration
ar
paralll
2
y
Th
diagram
shows
th
point
A(4,
7).
A
7
1
Writ
down
2
Liy
says,
th
distan
from
A
to
th
x-axis.
6
5
‘Th
point
(2,
0)
lis
on
th
x-axis.
4
Pythagoras’
thorm
tlls
m
that
th
distan
3
from
A
to
(2,
0)
is
units.
2
1
Thrfor
th
distan
from
A
to
th
x-axis
0
is
7.3
units.’
0
Explain
3
Writ
and
4
whthr
down
th
not
down
of
point
th
on
this
distan
oordinats
Writ
th
th
or
of
lin
x
a
from
th
distan
th
is
A
point
from
=
logial
1
A
that
to
th
on
to
is
1
2
3
4
to
A
5
x
argumnt.
th
th
lin
lin
lin
losst
y
y
x
to
=
=
=
5,
5
1,
that
and
is
losst
th
oordinats
A
Continued
on
next
page
3.1 But can you prove it?
67
5
6
Find
th
oordinats
Find
th
shortst
Find
th
oordinats
that
7
is
losst
to
of
distan
th
shortst
Find
th
oordinats
is
Find
losst
th
Reect
to
from
th
distan
on
A
point
to
th
lin
th
on
y
lin
th
=
y
lin
=
x
x
+
1
+
1.
with
that
is
losst
to
A
quation
of
from
th
A
point
to
th
on
lin
th
.
lin
with
quation
y
=
7
2x
A
shortst
and
of
point
A
Find
that
th
distan
discuss
from
A
to
th
lin
y
=
7
2x.
2
y
A
For
Exploration
7
2:
6
●
What
diultis
did
you
nountr
in
stp
7
5
ompard
to
stps
3
to
6?
B
4
●
Suppos
you
ar
trying
to
nd
th
shortst
distan
3
L
from
A
to
a
gnral
lin
L
whos
quation
you
don’t
2
know.
1
0
If
B
is
th
point
aout
th
lin
on
L
losst
to
A,
what
an
you
say
x
0
Th
shortst
masurd
You
it
is
an
Suppos
that
this
a
logi
to
you
you
an
you
that
lin
this
a
that
in
givn
disovr
rst
this
th
assumption
you
your
fat
1
2
3
4
AB ?
prpndiular
stalish
an
If
th
twn
lin
us
mak
assumption.
tlls
and
distan
along
us
tru,
L
lin
to
is
and
th
a
point
not
on
that
lin
is
lin.
tru.
On
you
hav
dtrmind
that
futur.
and
that
thn
som
assumption
som
of
was
valid
your
not
ddutions
ddutions
tru.
This
is
ar
asd
on
untru,
alld
proof
y
ontradition.
‘Proof ’
mans
Mathmatis,
rsult
will
a

dirnt
proof
always
susquntly
supports
vry
a

is
It
disprovd.
ut
vidn
is
things
roust
tru.
hypothsis,
ontraditory
a
is
In
in
Sin
logial
prmannt,
Sin,
whih
ould
and
Mathmatis.
argumnt,
whih
and
if
a
proof
‘proof ’
is
a
ody
susquntly

In
guarants
is
of
valid,
it
that
a
annot
vidn
ovrturnd
that
if
found.
ATL
Exploration
1
On
plain
2
Using
3
papr,
draw
ompasss
prpndiular
to
and
L
a
a
and
lin
L
rulr,
passs
and
a
point
onstrut
through
a
A
that
sond
is
not
lin
on
L′
–
th
that
lin.
is
A
Continued
6 8
3 Logic
on
next
page
G E O M E T R Y
3
Lal
th
To
show
not
th
4
If
that
P
losst
P
is
ls.
5
point
not
and
Triangl
th
Sin
losst
anothr
6
L
ABP
L
is
th
A
If
y
in
is
this
A
on
L.
P
L,
start
following
losst
Lal
y
supposing
that
P
is
point
it
must
li
somwhr
B
sntns:
triangl,
,
it
is
<
to
th
AB,
and
angl
∠APB
is
whih
AB
is
th
of
triangl
ABP,
sid.
ontradits
th
assumption
that
B
is
th
point
P
your
a
sid
longst
own
to
words
A
ontradition
ours,
Point
A
th
lin
a
losst
produing
Practice
1
AP
losst
proof
and
thn
on
is
mans
whih
to
mt
.
∠APB
Explain
lins
point
point,
point
omplt
Thrfor
on
two
losst
aus
whih
th
T R I G O N O M E T R Y
point.
th
Draw
Copy
is
whr
&
rsult
thn
th
lis
how
on
a
involvs
that
is
this
lin
applying
oviously
original
shows
that
th
prpndiular
a
fals,
assumption
logial
or
point
to
L
pross
ontradits
must
hav
n
on
through
to
a
your
A
hypothsis
assumption.
inorrt.
2
has
oordinats
(5,
8).
Construt
th
lin
through
A
that
is
You
prpndiular
to
th
lin
y
=
4x
+
an
dynami
Find
th
Lin
L
shortst
distan
from
A
to
th
lin
y
=
4x
+
has
quation
y
=
3x
for
L
has
quation
y
=
has
oordinats
8
or
a
GDC
3.
1
Lin
gomtr y
2.
softwar
2
us
2.
2x
qustions
and
2
3
2
Point
B
Dtrmin
Justify
whih
your
lin
(2,
9).
passs
losr
to
point
B
answr.
1
3
Draw
th
lin
L
with
quation
y
=
4
+
x
2
Add
th
points
Dtrmin
Justify
whih
your
Problem
4
Air
tra
airport
A(2,
9)
and
point
is
B (7,
losr
3)
to
to
your
diagram.
L
answr.
solving
ontrollrs
and
any
ar
othrs
in
that
harg
y
of
plans
ovrhad
on
taking
th
o
way
to
and
landing
anothr
at
thir
dstination.
Mak
Whn
plans
om
into
thir
airspa,
thy
trak
thm
using
a
sur
you
oordinat
us
logi
to
justify
systm.
your
Plan
A
is
at
(12,
5),
ying
towards
(21,
12).
Plan
B
is
at
(9,
8).
Y
our
a
Find
th
shortst
distan
from
Plan
B
to
Plan
A’s
answrs.
ight
path.
justiations
should
dout
b
Suggst
what
othr
information
(not
inludd
in
th
oordinat
air
tra
ontrollr
nds
to
prvnt
th
plans
gtting
too
to
no
whthr
systm)
your
th
lav
as
los
answr
is
to
orrt.
ah
othr.
3.1 But can you prove it?
6 9
Exploration
ATL
1
Using
2
Find
3
Compar
dynami
th
Crat
th
othr
th
5
Us
of
your
points
6
Vrify
7
Points
P
Vrify
of
(7,
Reect
11)
Q
your
and
Gnralization
th
midpoint
M
to
th
of
points
A (10,
6)
and
B (2,
8).
AB
oordinats
of
points
A
and
B.
noti.
th
stp
and
and
nd
twn
th
th
midpoint.
4
midpoint
of
oordinats
(Hint:
th
prdit
ah
of
onstrut
a
pair.
th
ndpoints
tal
of
th
formula
th
of
N,
th
midpoint
softwar.
P (a,
oordinats
givs
discuss
oordinats
15).
oordinats
for
mans
to
D (19,
using
hav
formula
that
of
points
of
in
answr
and
a
th
plot
midpoints.)
ndings
C
M,
rlationship
and
your
Suggst
8
pairs
th
you
oordinats
ndpoints
of
softwar,
oordinats
anything
Invstigat
and
gomtry
oordinats
Dsri
4
4
th
b)
and
of
th
orrt
Q (c,
d).
midpoint
oordinats
of
PQ
for
M
and
N
3
making
a
gnral
statmnt
on
th
asis
of
spi
xampls.
●
Whr
●
How
dos
●
Why
is
Th
in
Find
th
M

M
=
hav
of
(a,
b)
to
nal
vrify
and
you
(c,
d)
a
gnralizd?
you
to
prdit?
gnralization?
is
.
1
th
of
A(3,
midpoint
3 + 9
6 + 18
2
2
of
6)
and
B (9,
18).
AB

Use the midpoint formula M =



=
important
midpoint

4
gnralization
midpoint
Example
Lt
it
Exploration
(6,

12)
Practice
3
1
Find
th
midpoint
of
A(14,
2
Find
th
midpoint
of
C (5,
3
a
Plot
b
Find
c
Commnt
th
points
th
midpoints
on
quadrilatral
7 0
A(2,
your
7),
of
6)
and
3 Logic
20).
13)
and
D (10,
B (3,
10),
C (6,
AC
and
answrs
ABCD
B (18,
to
8).
11),
and
D (5,
8).
BD
part
b.
Explain
what
this
shows
aout
G E O M E T R Y
4
Triangl
a
PQR
Show
L,
M,
that
and
N
ar
Writ
down
c
Show
that
Show
with
that
vrtis
triangl
b
Problem
5
has
th
th
P (
PQR
2,
is
of
oordinats
LMN
Q (2,
3),
and
and
RP
R (6,
T R I G O N O M E T R Y
In
qustion
3).
isosls.
midpoints
triangl
3),
&
is
PQ,
of
L,
QR
M,
and
rsptivly
.
N
isosls.
solving
th
midpoint
quation
y
=
2x
+
of
points
U (7,
11)
and
V (
2,
1)
lis
on
th
lin
1.
that
6
As
you
saw
in
Prati
at
(
1
qustion
6,
th
ornrs
of
Th
Grat
Squar
th
Show
ar
that
Th
1,
5),
Grat
a
Diagonals
ar
b
Diagonals
ist
Using
C
(3,
2),
(
Squar
4,
of
1),
and
Pgasus
(0,
has
show
of
oordinats
Pgasus
5,
midpoint’s
satisfy
2).
ths
proprtis
of
a
squar:
th
quation
th
lin.
of
prpndiular
ah
logic
general
●
How
do
●
How
an
anothr
to
prove
claims
and
rules
you
form
you
logial
prov
th
argumnts?
distan
formula
and/or
th
midpoint
formula?
You
will
nountr
mathmatis,
‘If
…
hlp
you
you
to
Eah
why
then
…
to
stp
of
a
in
valid.
argumnts
vryday
is
snsil
othrs
logial
is
a
form
your
argumnt
If
a
of
stp
many
Any
logial
disions.
that
in
lif.
Th
idas
with
Thinking
xplain
your
not
th
just
in
strutur
logially
logi
will
will
hlp
valual.
ontain

to
situations;
pross
argumnt.
aility
ar
should
annot
dirnt
thought
a
justid,
simpl
you
justiation
won’t
know
if
for
it
is
tru.
pross
known
A
mak
stp
nssarily
logial
also
therefore…’
prsuad
th
Th
ut
as
of
justifying
a
laim
y
providing
stp-y-stp
rasons
is
deduction
deductive
ddution
Reect
argument
to
draw
and
Justiation
is
a
is
any
proof
or
olltion
of
rasoning
that
uss
onlusion.
discuss
dnd
as
4
‘Valid
rasons
or
vidn
usd
to
support
a
statmnt’.
●
What
●
Why
dirnt
is
typs
justiation
of
justiation
important
in
hav
you
sn
so
far
in
3.1?
mathmatis?
3.1 But can you prove it?
71
Exploration
ATL
You
hav
distan
for
th
5
alrady
twn
distan
shown
two
that
points.
twn
Pythagoras’
In
two
this
points
xploration,
P
(x
1
1
In
th
lvl
diagram,
with
P
point
and
is
Q
is
thorm
,
y
1
)
and
you
P
1

will
(x
2
horizontally
vrtially
an
,
y
2
usd
nd
to
a
nd
th
formula
).
2
y
8
alignd
1
with
P
7
2
P
(x
2
, y
2
)
2
6
a
What
typ
of
triangl
is
P
QP
1
Justify
your
?
2
5
answr.
d
4
b
Writ
down
c
Hn
th
oordinats
of
Q
3
writ
down
th
lngth
of
Q
P
Q
1
2
P
(x
1
and
P
Q
in
trms
of
x
2
,
x
1
,
y
2
,
and
,
y
1
)
1
y
1
2
1
d
Us
ths
lngths
to
nd
th
lngth
d
0
of
P
2
In
th
x
P
1
1
2
3
4
5
6
7
8
2
rst
diagram,
you
assumd
that
P
was
aov
and
to
th
right
of
You
2
P
.
Suppos
instad
that
P
1
was
low
and
to
th
lft
of
P
2
distan
formula
would
.
Show
that
skth
1
not
ould
th
th
two
hang.
othr
possil
arrangmnts
3
Suggst
othr
possil
positions
that
P
and
P
1
ould
tak
rlativ
to
ah
2
P
othr.
Explain
why,
for
any
positions
of
P
and
P
1
rmains
Th
(x
,
1
th
distan
y
)
and
formula
(x
1
Whn
,
y
2
Reect
P
and
●
Dos
●
What
)
is
stats
givn
that
th
distan,
d,
twn
y
P
discuss
ar
th
lvl
distan
happns
to
4
1
distan
th
a
(3,
11)
b
(
c
(103,
3,
and
8)
9)
Problem
Show
point
that
3
Show
72
two
points
5
horizontally:
formula
point
Q,
still
and
work?
th
triangl
P
P
and
and
twn
pair
of
points.
15)
(2,
(
ah
Q?
2
16)
17,
208)
solving
14),
Q (11,
12),
and
R (13,
8)
ar
all
th
sam
distan
from
7).
nd
that
(6,
P (7,
C (6,
Hn
formula
2
Practice
2
distan
.
1
Find
th
2
and
1
,
2
sam.
th
th
ara
points
of
F (
3 Logic
th
6,
irl
5),
G (
that
4,
passs
5),
and
through
H (
5,
9)
P,
Q,
form
and
an
R
isosls
triangl.
1
and
P
.
2
of
G E O M E T R Y
Activity:
Objective:
iv.
Proving
C.
the
midpoint
&
T R I G O N O M E T R Y
formula
Communiating
ommuniat
omplt,
ohrnt
and
onis
mathmatial
lins
of
rasoning
The
proof
midpoint
sections
Th
to
Q
(c,
d)
Copy
gaps
is
you
you
give
give
clear
a
complete
and
concise
and
coherent
explanations
proof
and
of
the
details
in
the
in
this
proof
that
laims
th
midpoint
of
points
P (a,
b)
M
has
that
and
if
help
complete.
th
proof
prov
will
formula,
you
Complt
and
outline
two
M
is
parts:
rst,
quidistant
omplt
th
proof
to
prov
from
P
outlin
that
and
M
lis
on
th
lin
PQ ;
sond,
Q.
low.
Theorem:
Draw
Th
midpoint
of
points
P (a,
b)
Th
Q (c,
d)
is
a
diagram
to
show
P
and
Q
on
a
grid.
and
position
of
P
and
Q
on
th
graph
is
not
M
important
–
just
don’t
put
thm
in
th
sam
pla!
Proof:
PMQ
and
is
th
a
straight
gradint
lin
of
if
th
MQ
ar
gradint
th
ofPM
sam.
Show
larly
that
PM
and
MQ
hav
th
sam
gradint.
Sin
a
th
gradints
straight
ar
qual,
PMQ
is
lin.
Us
Thrfor
P
and
Sin
and
M
is
M
is
quidistant
th
distan
formula
to
show
MQ
_____________________________________
______________________________________,
th
midpoint
of
PQ.
A
squar
marks
th
nd
of
a
proof.
5
a
Find
th
midpoint
of
A(3,
b
Find
th
midpoint
of
A(14,
9)
and
2)
B(7,
and
3).
B(6,
th
proof
11).
also
Show
that
th
points
A (2,
5),
B (4,
19),
C (14,
29),
and
D (12,
15)
nd
you
s
short
a
=
from
At
2
MP
Q.
Practice
1
that
of
a
might
‘Q.E.D.’
for
quod
erat
form
demonstrandum,
a
rhomus.
maning
b
Find
th
andDA
c
Show
d
Find
oordinats
E,
F,
G,
and
H,
th
rsptivly.
that
th
Hn
of
EFGH
lngths
dsri
th
EG
of
AB,
BC,
CD,
whih
‘that
was
to

dmonstratd’.
forms
of
midpoints
a
paralllogram.
and
shap
FH
EFGH
3.1 But can you prove it?
73
3
a
Plot
i
th
points
Show
non
ii
Find
that
of
b
Show
Considr
8),
of
sids
B (
th
ar
DA
4,
sids
of
E,
Find
RS,
that
EFGH
four
points
ii
Prov
Reect
●
How
or
●
4
forms
P (x
Whn
of
th
(7,
that
and
is
is
th
thy
sit
,
a
y
th
33),
Dtrmin
F,
ah
G,
and
th
D (28,
sam
lngth,
Givn
th
H,
th
midpoints
to
At
),
Q (x
,
y
2
T,
U,
a
6
Copy
and
of
AB,
BC,
),
R (x
2
V,
,
y
3
and
W,
),
and
S (x
3
th
,
y
4
midpoints
).
4
of
PQ,
QR,
paralllogram.
rlatd
and
twn
idntify
th
3600
hight
to
nding
th
avrag
(
Wst
bce,
36,
th
sit,
th
and
thy
whr
It
ar
has
lngth
statmnt
is
3b?
mark
artfats
Knnt
thr
41).
twi
a
3a
Long
thr
n
of
tru.
o
with
found
Barrow
in
hamrs
said
its
it
ar
that
oordinat
kp
England,
at
thy
a
and
th
a
a
grid
rord
grav
oordinats
form
an
isosls
as.
Justify
your
answr.
solving
points
th
that
paralllogram.
A(a,
b)
and
B (c,
d),
prov
that
th
point
lin
sgmnt
omplt
th
AB
in
th
following
ratio
1
:
sklton
2a
+
c
2b
+
,


divids
and
othr.

5
28).
6
xavat
asily
whthr
Problem
of
midpoint
rlationship
an
with
ar
and
man?
ak
(17,
triangl
ABCD
to
1
forms
discuss
forvr.
dating
9),
TUVW
nding
28),
rsptivly.
arhaologists
that
sit
oordinats
SP
arithmti
What
so
th
and
of
C (16,
rsptivly.
1
i
20),
paralll
oordinats
CD,and
iii
non
th
th
A (0,
3
d


3

2.
proof.
Tip
C
B
Two
said
if
points
to
thy

ar
ar
coincident
in
th
D
A
sam
thy
Givn:
lin
sgmnts
AC
and
BD;
th
midpoint
of
AC
is
oinidnt
with
pla,
hav
this
th
th
oordinats.
midpoint
Prov:
74
of
BD.
ABCD
is
a
paralllogram.
3 Logic
is,
sam
G E O M E T R Y
&
T R I G O N O M E T R Y
Theorem:
If
th
forms
midpoint
a
of
AC
is
oinidnt
with
th
midpoint
of
BD,
thn
ABCD
paralllogram.
Proof:
If
lins
AB
and
CD
ar
paralll
and
th
sam
lngth,
thn
ABCD
is
a
paralllogram.
Lt
th
midpoint
Thrfor
Lt
th
th
of
AC
hav
oordinats
oordinats
of
A
of

oordinats
th
(x
midpoint
,
y
A
By
th
midpoint
(X,
),
B

Y
of
BD
(x
A
).
,
y
B
ar
.
Think
)
Provid
th
information
snsil
you
notation
Us
th
midpoint
x-oordinat
.
for
Now
Th
gradint
of
AB
Th
gradint
of
CD
is
givn
y
nd
is
givn
y
thr
valus
y
.
will
mor
of
C
Thrfor
proaly
xprssion
arfully
of
C
formula
in
trms
to
xprss
of
x
givn
y
rsults
whih
dsri
and
AB
and
CD
Sin
for
to
th
show
nd
to
writ
gradint
that
it
of
a
CD
that
maths
th
simplify
gradint
you
may
nd
to
simplify
a
,
th
points
O (0,
0),
A(a,
a,
b,
c,
d
≠
0,
2d)
and
D (c,
d)
form
a
Presentation
●
Dos
●
How
ar
latr.
not
know
that
.
b),
B (a
+
c,
b
+
d),
and
C (c,
d)
form
c
≠
and
a
ar
w
solving
b
Assuming
mt
yt
two
xptd
nountrd
it
mattr
muh
dirnt
Thy
prsntd.
dirns
it
of
AB
ompliatd
paralllogram.
Hr
x
ompliatd
and
.
and
Problem
D
th
D
xprssion.
C (2c,
th
X
.
Again,
8
and
.
is
a
D
.
You
Prov
and
.
and
7
C
givn.
A
Similarly,
Thrfor
ar
B
formula,
and
aout
to
proofs
undrstand
worry
trm
if
th
points
A (a,
b),
B (2a,
2b),
the
as
is
prsntd?
nough?
Isosceles
you
th
som
congruent
that
proof
proof
is
of
prsntd
Don’t
th
ar
ar
‘proof’
prov
trapzoid.
of
how
,
d
triangle
might
proof
–
words
for,
for
nd
for
ar
theorem,
thm
now,
in
just
whih
–
What
you
txtook.
look
unfamiliar
xampl.
a
will
You
ar
at
how
th
proofs
you
may
not
hav
similaritis
and
thr?
3.1 But can you prove it?
75
Theorem:
those
If
sides
Proof
two
are
sides
of
a
triangle
are
equal
then
the
angles
equal.
1
C
α
β
D
A
Givn
Lt
In
∆ABC
th
with
istor
∆CAD
B
and
AC
=
BC,
ˆ
ACB
of
∆CDB,
mt
th
triangls
=
BC,
CD
=
CD,
ar
ˆ
A =
that
at
ˆ
B
D
(Givn)
(Common)
β ;
=
ongrunt
∴
Proof
AB
AC
α
∴
prov
ˆ
A
ˆ
B
=
Q.E.D.
2
C
D
A
Th
triangl
vrtx
Draw
CDA
CD
in
so
is
That
angl
as
th
th
ar
of
CA
to
two
AB
ommon
is,
ΔCDB
sids
any
=
isosls
CB.
ist
Prov
∠C ,
triangl,
that
forming
∠A
two
=
with
∠B
triangls,
CDB
istion;
sid
rprsnts
and
and
Thn
of
C
B
AC
to
two
qual
ΔCDA,
triangls,
=
oth
sids
to
is
and
=
and
∠BCD ,
CD
=
y
CD,
dnition
as
triangls.
and
two
∠ACD
givn;
th
sids
hn
inludd
and
th
th
two
angl
of
inludd
triangls
ar
ongrunt.
Thrfor,
triangls,
7 6
as
∠A
orrsponding
=
∠B ,
as
was
3 Logic
parts
to

of
ongrunt
provd.
this
opposite
to
G E O M E T R Y
Reect
Disuss
What
●
Whih
●
How
good
inlud
out
so
discuss
osrvations
strutural
proof
do
proofs
dirnt.
A
your
●
Both
and
you
know
inlud
Proof
proof
2
is
it
a
asy
think
th
to
oth
would
th
logi

th
Proof
usd
xampl,
proofs
proof
ut
wordy;
(for
othrs.
do
diagram,
dsri
is
with
whn
vry
justiation
that
you
T R I G O N O M E T R Y
7
lmnts
do
&
hav?
asir
is
to
undrstand?
nishd?
prsntation
1
to
has
far
mov
prviously
of
fwr
from
th
on
provn
two
proofs
is
vry
words.
stp
to
rsults)
th
and
nxt,
will
will

laid
follow.
Summary
●
Deductive
prviously
thm
●
A
togthr
and
OA
reasoning
known
B
and
ar
OB
to
or
is
th
mak
nw
equidistant
ar
pross
provn
fats
of
and
taking
●
A
putting
ons.
from
O
if
proof
logial
rsult
th
y
ontradition
pross
that
is
assumption
qual.
a
If
this
must
involvs
hypothsis
oviously
assumption.
distans
to
fals,
ours,
hav
or
th
Th
midpoint
of
(a,
b)
and
(c,
d)
is
B
original
a
Th
pross
of
stp-y-stp
AO
=
justifying
rasons
is
a
laim
known
as
+
c
b
+
d
,


O
●
y


2
2

providing
deduction
OB
●
A
A
deductive
of
rasoning
argument
that
uss
is
any
proof
ddution
to
or
olltion
draw
a
onlusion.
●
Th
lin
th
shortst
is
distan
masurd
twn
along
a
lin
a
point
and
a
prpndiular
to
●
Th
d,
lin.
distan
twn
formula
two
stats
points
(x
,
y
1
that
)
th
and
givn
1
Find
y
1
distan
twn
ah
pair
of
points.
3
Find
th
giving
a
(0,
0)
and
(5,
12)
b
(1,
4)
and
(4,
8)
a
2
(5,
d
(
Find
7)
3,
and
12)
th
(11,
3
x
−
x
2
1
)
y
)
is
2
+
(
y
−
)
y
2
1
and
(4,
distan
giving
signiant
a
(4,
8)
twn
b
(14,
c
(1.5,
8)
and
4.6)
your
(10,
ah
answr
pair
orrt
13)
(17,
and
whol
a
(2,
b
(
3,
c
(
3,
twn
answr
in
th
ah
form
pair
n ,
of
points,
whr
numr.
7)
and
(6,
15)
9)
and
(5,
2)
12)
gurs.
and
distan
your
7)
3)
(2.3,
1.8)
5)
and
(
9,
14)
of
4
points,
(
=
,
2
2
practice
th
c
d
distan,
(x
2
Mixed
a
your
inorrt.

●
a
produing
ontradits
thn
n
applying
and
Find
th
midpoint
of
ah
pair
of
points.
to
a
(4,
b
(
3,
c
(
3,
d
(2.3,

e
1
1
,


2)
4
9)
12)
(5,
and
5.9)
(
and


(6,
and
5)

3
and

and
1
8,
(
1
,


2)
3
14)
3.6,
11.2)


2

3.1 But can you prove it?
77
n
is
5
Prove
form
6
that
an
Thr
B (1,
points
4),
Show
7
Point
lin
th
and
that
A
points
isosls
hav
C (
th
has
(0,
0),
and
(3,
Th
4)
{1,
A(
4,
1),
a
6).
points
A
(6,
oordinats
5,
that
Sudoku
2,
3,
By
form
is
an
(3,
isosls
5).
to
onsidring
that
it
=
−2x
Find
y
8
=
A
th
−2x
has
(4,
+
qustion
us
th
st
is
th
not
positions
possil
of
to
th
1s
and
omplt
2s,
this
lin
4
2
2
1
3.
shortst
+
this
th
th
1
y
in
puzzl:
triangl.
Construct
prpndiular
puzzls
4}.
show
oordinats
through
0),
triangl.
3.
distan
Giv
your
oordinats
(
from
answr
1,
3)
and
A
to
B
to
1
th
lin
dp.
has
oordinats
1).
3
Determine
y
=
whih
point
is
losr
to
th
lin
x
2
9
Givn
th
points

th
point
3a
10
in
Point
th
A
c
3b
+
d
,


AB
+
A (a,
has
:
B
(c,
d),
prove
1
that
divids
th
lin
sgmnt

4
1
and


4
ratio
b)
b
Copy
this
grid:
3.
oordinats
(a,
b)
and
point
B
has
1
oordinats
M
is
th
(c,
4
3
d).
midpoint
of
AB
4
a
Find
th
oordinats
of
b
Find
th
distan
AB
c
Find
th
distan
AM
d
Hn,
M
1
prove
that
AM
=
AB
2
2
11
ABC
and
is
a
triangl
C (9,
with
vrtis
A (1,
0),
B (5,
0).
i
a
1
9)
Show
that
triangl
ABC
is
Explain
how
right-hand
b
Find
th
oordinats
of
th
midpoints
of
sids
of
Show
that
Hn
determine
th
midpoints
of
th
thr
sids
othr
ar
th
vrtis
of
an
isosls
Show
that
th
isosls
triangl
formd
8
midpoints
squar
of
th
sids
of
ABC
In
a
Sudoku
has
Most
Deduce
8,
9}.
puzzl,
puzzls
playrs
us
th
work
st
with
{1,
2,
a
3,
st
4,
Evry
78
th
th
possil
positions
of
grid.
answr.
th
digits
that
should
appar
in
th
lls.
row,
vry
squar
olumn,
must
st.
3 Logic
ontain
and
all
solving
of
5,
Point
X
has
oordinats
(4,
7)
and
of
AB.
Givn
that
A
has
vry
of
th
is
th
6,
(
in
th
in
units.
mdium-sizd
digits
2s
rmaining
midpoint
7,
top
ara
13
digits.
two
your
Problem
12
th
2.
from
grids
th
that
a
triangl.
iii
d
rtain
ontain
of
Justify
ABC

must
ABC
th
c
an
ornr
th
ii
thr
you
isosls.
3,
11),
nd
th
oordinats
of
B
oordinats
G E O M E T R Y
Review
in
ar
Short
in
a
luky
Jams
in
space
nough
Platinum
dirnt
T R I G O N O M E T R Y
context
Orientation
You
&
loation.
to
-
and
hav
found
sourg
Can
time
you
of
th
th
nd
only
svn
thm
all?
surviving
sas.
(On
H
th
opy
has
of
a
urid
map,
on
trasur
many
unit
map
pis
quals
of
on
lft
y
th
trasur
notorious
on
th
pirat
island
-
ah
lagu.)
14
13
Dolphin
12
Beach
Mermaid
11
Smuggler’s
Cove
Pool
10
Grey
9
Skull
Cliffs
8
Pirate’s
Rock
Forbidden
7
Port
Falls
Kaspar’s
6
Trader’s
Cave
5
Bay
Handyman
4
Hills
3
Falling
2
Palms
1
0
1
1
Th
Tsarina’s
Rok.
it
is
It
is
from
as
Tiara
far
is
from
Falling
2
3
urid
Skull
Palms
to
4
5
du
6
wst
Rok
to
Kaspar’s
7
of
th
8
9
Skull
Tiara
10 11 12 13 14 15 16 17 18 19 20
5
To
nd
your
as
st
Kaspar’s
Cav.
travl
Find
th
oordinats
of
th
pla
whr
Tiara
is
Th
wrk
of
th
halfway
is
to
Purpl
Porpois,
a
ship
that
200
gold
twn
Falling
douloons,
Palms
and
is
xatly
th
Handyman
6
Hills.
Th
oordinats
of
th
Purpl
If
Skull
Rok
is
th
losr
Falls,
two
point
Pirat’s
halfway
Clis.
Port,
From
and
from
thr,
thr
th
found.
oordinats
to
Tradr’s
to
thn
lagus
my
ast
of
prizd
Justify
pik
Bay
of
th
Cursd
Mdal
of
your
answr.
from
th
Bliz
right
is
urid
dp
-
so

pla.
th
rout
Silvr
Mrmaid
straight
from
Falling
Palms
to
th
than
Cutlass
Clis,
Pool.
If
stop
at
th
point
that
is
losst
to
is
Kaspar’s
urid

Bounty
Gry
Foriddn
th
Gry
snd
Porpois.
On
3
to
to
th
Caraas,
halfway
sur
Find
to
of
sank
Caraas.
arrying
navigator
Cav
Mdal
urid.
Find
2
Cursd
th
mdal
Tsarina’s
th
Cav.
Had
on
lagu
wst
and
dig
not,
dp!
it
is
urid
two
lagus
wst
of
Mrmaid
Pool.
a
Determine
th
position
of
th
Silvr
Find
th
distan
Kaspar’s
4
Th
Emrald
Crown,
whih
I
lootd
from
tom,
has
n
hiddn
in
a
If
you
at
an
on
nd
of
th
th
markd
orrt
sits
sit,
Find
on
you
th
will
th
distan
tru
loation
of
th
Kaspar’s
Cav
from
th
Clis.
th
Gry
Emrald
Explain
orrt
how
as
rown
th
is
as
Falling
Dtrmin
th
c
far
th
d
away
Palms
Hn
loation
you
know
Explain
lin
of
show
Falling
losst
ar
from
this
is
th
only
possil
e
Hn
from
that
Gry
Palms
how
to
midpoint
Crown.
loation.
Gry
Clis
to
Cav.
map.
nd
and
from
th
to
ruind
trasur.
Th
from
Palms
Cav.
Kaspar’s
uilding
Falling
an
b
anint
from
Cutlass.
this
tlls
Kaspar’s
Falling
of
th
Clis,
form
an
you
Cav
Palms
that
that
to
Kaspar’s
isosls
th
lis
Gry
Cav
triangl.
point
on
th
Clis
is
th
lin.
determine
th
loation
of
th
Bounty
Bliz.
3.1 But can you prove it?
79
4
Representation
The manner in which something is presented
The
Which
representation
is
b est
best?
may
Car tesian
coordinate
plane
tation
represen
Mapping
want
diagram
depend
to
what
on
show
or
to
you
nd
out.
y
0
4
1
2
2
0
3
2
x
y
2
0
2
2
4
x
6
2
4
Linear
y
=
equation
2x
−
4
6
Table
of
values
8
Which
representation
●
that
the
●
that
f(1)
●
that
the
function
=
best
is
x
0
1
2
3
4
5
y
−4
−2
0
2
4
6
shows:
linear
−2
y-coordinates
increase
by
2
for
every
increase
in
x?
C
Is
a
picture
thousand
ABCD
to
CD
Angle
●
is
and
How
a
70°.
does
you
Could
AB
perpendicular
=
Find
drawing
solve
you
a
words?
quadrilateral.
ADC
help
●
a
wor th
this
solve
it
to
is
BC
angle
a
parallel
B
DAB
diagram
D
70°
problem?
without
diagram?
A
8 0
Representation
in
symbols
Bayu-Udan
Jabiru
Challis
Pufn
Gove
T
oms
Weipa
Gully
Ranger
Browns
I N D
I A
N
Blacktip
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P A C I F I C
N
McArthur
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River
Island
Merlin
Argyle
Mt
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Garnet
N
Ellendale
Lajamanu
Centur y
Balcooma
Northern
Mammoth
Coyote
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Ernest
T
erritor y
Granites
Mount
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Isa
Y
arrie
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T
win
Y
uendumu
Hills
T
elfer
Nifty
rop
T
c
of
Capr
corn
Bowen
Deepdale
Brockman
Y
andi
2
Mt T
om
Paulsens
Mereenie
Kings
Canyon
Palm Valley
Mt Whaleback
Paraburdoo
West Angelas
Orebody
Basin
Queensland
Hermannsburg
Price
29
Plutonic
Western
Moomba-Gidgealpa
Magellan
Jackson
Australia
Moonie
Lake
Danot
Lawlers
T
allering
South
Peak
Sunrise
Golden
Ipswich
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Australia
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Creek
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CSA
Pound
Broken
Downs
St
Gunnedah
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ritton
Pit
Wilpena
Emu
Hill
Ives
New
South
Hunter
Middleback
Range
Northparkes
Singleton
Newcastle
Mount
Darling
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Cowal
Snowtown
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Range
Wales
Southern
Cathedral
Rocks
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Wattle
Point
Victoria
K
E
ACT
Y
Snowy
Bendigo
Mineral
resources
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Stawell
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Coal
Lake
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Challicum
Ballarat
(aluminium)
La T
robe Valley
Hills
Oil
Copper
W
Mountains
resources
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N
Queen Victoria
E
Market
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Diamonds
Natural
Gold
Uranium
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Solar
Strait
gas
S
0
200
400
600
km
Woolnorth
Beaconseld
Savage
ore
power
station
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Roseber y
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and
Wind
zinc
power
Lyell
Henty
Poatina
John
Butters
station
Gordon
Hydro
Silver
Maps
contain
terrain.
ones
symbols
Which
that
are
of
the
used
in
that
represent
symbols
places
below
other
power
features
do
than
you
T
asmania
station
of
the
landscape
recognize?
Are
or
they
maps?
81
A
whole
range
of
things
4.1
Global
context:
Objectives
●
Categorizing
Globalization
Inquiry
data
and
sustainability
questions
●
What
●
How
are
the
dierent
types
of
data?
F
●
●
Constructing
stem-and-leaf
diagrams
are
the
tendency
Calculating
quartiles,
interquartile
the
range
and
SPIHSNOITA LER
●
Constructing
ve-point
Identifying
●
Comparing
of
central
the
●
Giving
●
measures
range
●
a
dierent
calculated?
summary
of
box-and-whisker
a
set
of
data
C
diagrams
●
How
distributions
measures
data?
How
dierent
you
outliers
do
describe
do
compare
●
Should
●
How
we
dispersion
help
representations
data
ignore
of
you
help
sets?
results
that
aren’t
typical?
D
ATL
do
individuals
stand
out
in
a
crowd?
Communication
Organize
and
depict
information
logically
4.1
4.2
Statement of Inquiry:
How
quantities
are
represented
can
help
to
establish
underlying
relationships
6.2
and
trends
8 2
in
a
population.
S TAT I S T I C S
Y
ou
●
should
nd
the
and
range
already
mode,
of
a
median,
set
of
know
how
mean
1
1,
b
Representing
●
What
●
How
are
the
the
range
a
F
sets
dierent
of
types
of
mode,
of
3,
these
5,
3,
6,
median,
data
9,
8,
mean
and
sets.
3,
5,
6,
4
22,
21,
19,
20,
18,
19,
21,
19,
20,
19,
19,
22,
19,
21,
17,
16,
14,
22
data
data?
Data
are
the
dierent
measures
of
central
tendency
involves
collecting
raw
data,
organizing
the
data
into
is
the
and
analyzing
the
plural
Latin
word
visual
datum
representations,
the
calculated?
of
Statistics
P R O B A B I L I T Y
to:
Find
data
A N D
meaning
data.
‘a
piece
of
information’.
Qualitative
type)
two
●
using
data
describes
words.
a
certain
Quantitative
characteristic
data
has
a
(for
numerical
example:
value.
color
There
or
are
types:
Tip
Discrete
data
can
take
can
be
counted
(for
example:
number
of
goals
scored)
or
In
only
certain
values
(such
as
shoe
order
to
statistical
analysis,
continuous
●
Continuous
data
can
be
measured
(for
example:
weight
and
can
take
any
numerical
of
Categorize
state
each
whether
it
of
2
Time
3
Color
4
Numbers
5
Maximum
6
Heights
7
Dress
sizes
8
Daily
intake
9
Number
for
each
of
of
data
cars
a
car
degree
accuracy.
to
through
cross
eyes
in
girls
in
in
an
in
each
your
your
protein,
milkshakes
milkshakes
or
quantitative.
For
quantitative
data,
continuous.
in
intersection
intersection
your
class
in
family
Qingdao,
in
your
class
China
class
class
grams,
sold
sold
an
student’s
temperatures
students
of
qualitative
passing
children
of
as
or
student’s
daily
of
of
of
discrete
red
taken
of
set
is
Number
Flavors
rounded
certain
1
1
10
be
value.
toa
Practice
data
height)
must
and
perform
size).
in
in
a
a
for
members
of
a
sporting
team
cafeteria
cafeteria
4.1 A whole range of things
83
A
stem-and-leaf
can
then
be
diagram
is
a
visual
representation
of
ordered
raw
data,
which
Stem-and-leaf
analyzed.
diagrams
in
Example
are
Japanese
used
train
1
timetables.
Here
0,
1,
are
14,
the
11,
Construct
0
0
0
numbers
0,
an
1
6,
10,
1,
ordered
1
1
4
of
climate
0,
39,
1,
change
13,
stem-and-leaf
6
7
10
10
11
10,
surveys
26,
7,
diagram
4,
to
carried
17,
12,
out
58,
represent
by
22,
this
23
students.
15,
20,
17
data.
12
Sort
13
14
15
17
17
20
22
26
39
58
use
0
0
0
0
1
1
1
4
6
7
1
0
0
1
2
3
4
5
7
7
2
0
2
6
3
9
the
data
Thedata
the
in
ascending
ranges
tens
Construct
diagram
from
digit
a
to
form
order.
58,
the
so
stem.
stem-and-leaf
from
Remember
to
0
to
your
ordered
include
a
data.
key.
4
5
8
Key :
3
Reect
●
Is
●
What
In
a
a
|
9
represents
and
the
the
stem
●
the
leaves
●
the
key
Practice
diagram
similar
of
to
keeping
a
bar
raw
chart?
Explain.
data?
diagram:
represents
the
represent
tells
1
advantages
stem-and-leaf
●
sur veys
discuss
stem-and-leaf
are
39
you
category
the
how
nal
to
read
gure
digit(s)
the
of
each
data
point
values.
2
ATL
1
Here
are
the
three-week
35,
47,
34,
numbers
of
croissants
a
baker
sold
each
day
during
a
period:
46,
62,
41,
35,
47,
51,
59,
56,
73,
38,
41,
44,
51,
45,
60,
25,
35,
46
Use
Construct
an
ordered
stem-and-leaf
diagram
to
represent
the
the
number
2
The
masses
(in
grams)
of
11
Chinese
striped
hamsters
are:
the
22.4,
27.2,
30.5,
25.2,
23.1,
25.3,
21.3,
20.9,
24.5,
25.2
theleaf.
Construct
8 4
a
stem-and-leaf
diagram
4 Representation
to
represent
the
data.
part
stem
decimal
21.6,
whole
data.
and
part
as
the
as
S TAT I S T I C S
Problem
3
The
in
class
a
4
6
6
7
1
2
3
3
3
2
0
2
6
9
3
1
1
2
7
4
0
Write
2
|
0
least
ii
greatest
Find
the
c
Write
d
Students
are
180
184
173
166
distances
the
class.
in
kilometers
students
20
km
travelled
distance
number
of
how
who
travelled.
students
many
travel
percentage
the
the
the:
down
the
shows
school.
8
distance
b
Here
diagram
to
represents
down
i
Find
4
travel
0
Key :
heights
195
of
177
students
more
of
in
than
travel
18
students
one-year-old
175
173
km
who
13
get
get
apple
169
km.
cheaper
cheaper
trees
167
in
bus
bus
fares.
fares.
centimeters.
197
Use
183
161
195
177
192
161
the
Reect
●
What
for
●
●
any
set
single
and
●
data
about
of
Measures
is.
data
Range
is
sets
in
a
stem-and-leaf
the
two
stem.
diagram.
2
arise
with
there
central
are
of
data
as
when
using
stem-and-leaf
diagrams
sets?
value
mode
might
data
of
the
discuss
problems
Measures
a
represent
and
large
What
For
and
rst
166
digits
Organize
P R O B A B I L I T Y
solving
stem-and-leaf
one
A N D
that
all
is
are
small
two
measure
of
of
the
central
(spread)
of
of
(location)
typical
measures
range?
categories
tendency
most
dispersion
a
a
summary
summarize
data
set.
The
statistics:
a
set
of
mean,
data
with
median
tendency.
measure
how
spread
out
a
set
of
data
dispersion.
4.1 A whole range of things
8 5
Example
The
2
stem-and-leaf
every
day
for
17
diagram
days.
shows
Find
the
the
number
mode,
median
of
emails
and
received
range
of
this
by
Kirsty
data.
The
The
The
3
1
2
median
is
mode
the
is
middle
the
value
value
smallest
that
when
occurs
the
Key :
1
|
8
represents
18
13
a
set
is
the
9th
value,
which
is
20
=
of
40
4
=
36
Practice
The
n
data
stem-and-leaf
Wizards
during
Find
mode,
the
7
3
0
3
5
7
9
4
1
2
2
3
3
5
2
5
4
|
1
a
5
range
Find
day
the
over
a
mode,
7
8
2
0
2
2
5
7
7
3
0
0
0
3
6
8
4
0
1
1
5
1
|
8
9
5
shows
range
9
41
diagram
median
5
Key :
7
period
1
8 6
is
the
value
the
number
of
goals
scored
by
the
Welsh
tournament.
and
represents
stem-and-leaf
every
diagram
netball
median
2
The
values,
Here
3
Key :
2
in
order.
largest
value
is
the
median
is
40.
the
n
=
17.
dierence
between
emails
largest
1
are
often.
emails
The
Range
most
(04).
emails
value.
Median
4
emails
In
=
data
is
6
The
Mode
value
of
data.
goals
shows
range
15
the
number
of
days.
8
represents
this
9
20
and
of
cars
4 Representation
of
this
data.
cars
parked
in
a
car
park
and
the
smallest
value.
the
S TAT I S T I C S
Problem
3
The
P R O B A B I L I T Y
solving
back-to-back
students
A N D
stem-and-leaf
following
their
diagram
English
and
English
2
5
4
shows
the
Mathematics
marks
for
a
group
of
exams.
Mathematics
2
4
0
3
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1
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7
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5
8
3
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7
9
Key :
represents
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marks
2 | 4 represents 42 marks
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Mathematics
exam
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4
a
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down
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c
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down
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d
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the
median
e
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the
range
f
Justify
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the
number
whether
number
of
of
students
of
of
the
the
the
English
English
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minutes
who
a
took
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Mathematics
exams.
marks.
marks.
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sample
and
marks.
and
or
the
of
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marks.
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people
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median
statistics
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right
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of
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).
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4.1 A whole range of things
87
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the
data
set:
2,
5,
4,
The
3
median,
19,
6,
5,
8,
9,
median,
lower
10,
10,
Q
=
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and
11,
11,
13,
upper
2,
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15,
quartiles
20,
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9,
and
interquartile
range
of
this
8
20
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the
data
in
order.
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are
12
data
values,
so
n
=
12.
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2
The
Q
median
is
=
6.5th
value,
2
which
is
the
mean
of
9
and
10.
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2
4
5
|
6
8
9
|
10
11
13
15
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=
19
20
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lower
quartile,
lower
quartile
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1
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values
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than
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2
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1
2
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2
4
5
|
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8
9
|
10
11
|
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15
19
20
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upper
quartile,
=
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upper
quartile
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14
3
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2
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1
2
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2
4
5
|
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3
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8
9
|
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10
11
|
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range,
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IQR
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=
20
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Q
3
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In
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none
of
the
3,
the
number
quartiles
Exploration
1
Find
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,
Q
1
2
were
1
14
of
5.5
data
values
=
8.5
values
in
the
n
was
data
a
and
Q
4.
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this
case
for
these
data
sets:
3
a
2,
4,
5,
6,
8,
9,
10,
11,
13,
15,
19,
20,
22
b
2,
4,
5,
6,
8,
9,
10,
11,
13,
15,
19,
20,
22,
23
c
2,
4,
5,
6,
8,
9,
10,
11,
13,
15,
19,
20,
22,
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n
of
1
2
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multiple
set.
your
ndings
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4)
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2
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of
4)
+
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table:
(multiple of 4)
(multiple of 4)
(multiple of 4)
+1
+2
+3
multiple of 4
median Q
not
a
value
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in
original
data
Q
and Q
1
not
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value
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data
3
set
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set
when
Q
,
1
8 8
Q
2
and
Q
3
4 Representation
are
values
in
the
original
data
set.
S TAT I S T I C S
Reect
●
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data
Will
Q
and
set
,
has
Q
1
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any
the
●
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the
baker
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670,
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the
original
of
4)
data
1.
set?
of
the
lower
data
is
less
than
quartile
or
the
equal
upper
to:
quartile?
quartiles?
data
of
tell
SAT
lies
between
the
lower
and
upper
quartiles?
central
tendency
or
a
measure
of
dispersion?
you?
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28.4,
the
and
the
62,
the
(in
4
get
and
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to
median,
0
2
8
1
0
1
2
2
1
1
3
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3
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of
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interquartile
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the
grams)
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to
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number
41,
of
with
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and
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her
friends.
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lower
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doughnuts
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the
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upper
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students
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masses
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she
sells
each
day
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period:
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b
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percentage?
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called
the
a
range
a
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fraction
measure
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n
in
were:
b
35,
where
values
4
Calculate
A
what
are
as
a
does
a
a
4
this
compared
scores
3
they
is
the
4
the
fraction
Practice
2
set,
What
Cara
be
P R O B A B I L I T Y
3
data
are
What
1
values,
Q
2
What
Is
n
and
median
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●
discuss
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and
diagram
newly
73,
and
41,
lower
44,
51,
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43
quartiles.
range.
hatched
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38,
chicks
35.3,
41.3,
are:
49.9,
44.5
IQR.
shows
the
time
taken
(in
minutes)
by
a
class
of
school.
range
and
interquartile
range
of
the
times.
9
4
7
3
4
5
0
Key :
1
|
0
represents
10
minutes
4.1 A whole range of things
8 9
Problem
5
The
solving
annual
rm
are
salaries
shown
in
of
the
a
sample
of
9
back-to-back
senior
Senior
9
8
Key :
of
9
£59
|
5
represents
000
for
a
a
and
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employees
at
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law
diagram.
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3
5
8
4
0
2
8
0
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0
1
3
5
6
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2
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2
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stem-and-leaf
salar y
5
senior
|
9
0
£50
employee
represents
000
for
a
a
salar y
of
junior
employee
a
Find
the
median
salary
of
a
senior
employee.
b
Find
the
median
salary
of
a
junior
employee.
c
Find
the
range
Compare
of
both
the
senior
and
junior
salaries.
medians
measures
6
d
Find
e
Compare
Here
22,
are
24,
interquartile
the
29,
median
The
mean
The
upper
the
Write
The
the
is
is
people
36,
is
minimum
●
the
lower
45,
y,
senior
the
x,
y
and
data
47,
sets
of
a
with
the
upper
data
set
(Q
)
(Q
)
quartile
(Q
)
3
●
the
maximum
9 0
junior
an
and
junior
value.
4 Representation
is:
5
evening
and
salaries.
employees.
class,
z
median
2
●
53,
senior
z
value
quartile
median
and
attending
1
●
the
46.5.
summary
the
the
both
37.5.
of
●
of
of
35.
dierent
ve-point
12
x,
quartile
values
two
of
30,
age
age
range
salaries
ages
30,
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Find
7
the
IQR
7.
in
order:
the
and
of
the
spreads.
S TAT I S T I C S
Exploration
A N D
P R O B A B I L I T Y
American
2
mathematician
A
box-and-whisker
diagram
is
a
visual
representation
of
the
ve-point
John
summary.
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vertical
line
represents
one
of
the
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summary
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much
to
values.
the
eld
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came
both
10
20
30
statistics.
up
with
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whisker
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of
and
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40
and-leaf
diagrams.
Marks
ATL
1
Write
down
2
From
the
a
the
median
b
the
upper
5
values
information
of
how
you
can
4
Explain
how
you
calculate
5
Determine
6
Describe
The
ve-point
down:
what
the
situation
calculate
box
this
the
the
box-and-whisker
from
0
IQR
diagram
30
diagram
strawberry
2
4
Find
the
median
b
Find
the
maximum
c
Find
the
IQR
masses
could
10
of
a
Write
b
Calculate
c
Find
a
1,
1,
b
10,
c
10
do
diagram.
diagram.
the
whiskers
represent?
represent.
number
100
of
down
the
of
50
18,
000,
3,
range
15,
64
3,
24,
14
16
18
20
strawberries
strawberries.
dolphins
55
ve-point
number
3,
strawberries
range).
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45
ve-point
2,
of
were
represented
in
a
diagram.
the
the
numbers
strawberries.
number
male
the
12
of
60
Mass
a
What
represents
8
(interquartile
40
Write
this
this
plants.
6
a
box-and-whisker
3
from
from
represents.
Number
The
range
5
harvested
2
summary.
quartile.
Explain
a
the
write
3
Practice
1
the
of
and
4,
15,
000,
12
16,
75
80
(kg)
summary
dolphins
4,
70
of
the
data.
IQR.
summary
4,
65
for
who
each
weigh
data
52
kg
or
less.
set:
4
21,
000,
17,
11
15,
500,
13,
11
12
750,
12
250,
10
300
4.1 A whole range of things
91
4
Construct
property
a
box-and-whisker
items
found
Minimum
on
Q
a
diagram
for
Median
Q
1
15
5
The
of
data
on
the
number
of
lost
Maximum
3
22
number
this
train.
33
minutes
that
37
10
people
41
wait
to
be
connected
to
a
helpline
are
recorded.
5,
6
8,
13,
19,
22,
Find
the
median
b
Find
the
lower
c
Construct
The
heights,
30.1,
Construct
The
a
a
in
8
8
9
4
0
3
4
4
5
3
7
8
8
6
1
|
7
8
32.4,
of
diagram
down
b
Find
the
median
c
Find
the
interquartile
d
Construct
median
28.5,
data.
are:
32.0,
diagram
shows
the
the
34.6,
for
39.1,
the
28.9,
34.4,
24.7
data.
number
of
diners
at
a
restaurant
the
38
diners
minimum
number
number
of
of
diners.
diners.
range.
box-and-whisker
diagram
for
the
data.
solving
a
32,
trees
for
8
Write
Construct
diagram
period.
a
Problem
12
20.0,
represents
a
34
quartiles.
box-and-whisker
3
3
upper
meters,
two-week
Key :
30,
box-and-whisker
36.5,
a
27,
time.
and
stem-and-leaf
during
8
22,
a
29.2,
7
20,
box-and-whisker
minimum
18,
Q
diagram
23,
IQR
for
12,
a
data
range
set
C
A,
B,
C,
D
and
E
in
the
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stem-and-leaf
21,
and
●
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do
measures
●
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do
dierent
9 2
of
diagram
and
nd
the
dispersion
help
representations
4 Representation
you
help
9
2
B
6
8
3
C
3
4
values
here.
comparing
A
with:
1
of
1
sets
of
describe
you
data
data?
compare
data
sets?
D
8
E
S TAT I S T I C S
Exploration
A
doctor
simple
soon
in
is
they
seconds
In
one
saw
(to
2
a
how
task,
red
d.p.).
lack
Here
come
are
0.58,
0.53,
0.58,
0.43,
0.58
Set
2:
0.66,
0.10,
0.58,
0.58,
0.78
2
The
had
the
people
been
which
is
Reect
In
mean,
were
for
You
and
could
Why
●
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the
two
could
the
Distributions
clear
peaks
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The
for
sets
use
for
24
spent
Describe
b
Compare
0
a
typical
Set
this
the
of
data
set
of
from
the
Explain
set
the
press
is
do
button
as
measured
for
each
the
other
‘fully
reasons
to
task:
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for
a
were
range
readings.
which
to
times
interquartile
Decide
one
clear
reaction
value?
same
in
compare
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the
peak
for
set.
set,
rested’
your
they
and
decision.
time
What
for
Set
does
1
this
was
0.58
single
seconds.
value
tell
you
mean.
1
and
two
are
How
in
data
called
Set
well
2?
does
What
the
mean
other
measure
sets?
unimodal.
Distributions
with
two
bimodal.
stem-and-leaf
the
study
the
study
diagram
Mathematics
times
times
of
of
the
girls
the
girls
0
0
0
5
0
0
1
0
0
0
5
5
0
0
2
0
0
0
0
0
0
0
3
0
0
0
0
0
0
0
4
0
0
5
0
0
5
0
0
0
6
0
5
0
7
5
a
|
2
|
boy
0
in
shows
one
and
how
week.
much
All
time
times
are
a
group
in
of
minutes.
boys.
and
boys.
Boys
8
and
one
instructed
response
ability
1?
have
studying
0
Key :
and
person’s
5
Girls
5
The
a
4
a
5
were
sets
hours’.
situation
to
called
back-to-back
students
for
hours.
choose
real
with
are
that
we
times
you
rested
24
discuss
say
data
describe
median,
aects
3:
would
about
fully
for
‘awake
Exploration
●
mode,
awake
sleep
on.
two
1:
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of
participants
light
Set
1
P R O B A B I L I T Y
3
investigating
tasks.
as
A N D
0
0
0
0
0
5
5
5
represents
studying
0
for
a
girl
20
studying
for
25
minutes
minutes
Continued
on
next
page
4.1 A whole range of things
9 3
25 +
a
The
median
is
the
=
13th
value.
n
=
25
for
the
girls
and
for
the
boys.
2
The
median
time
for
the
girls
The
median
time
for
the
boys
For
the
=
Q
is
35
is
minutes.
20
minutes.
girls:
25
minutes,
Q
1
=
50
minutes
3
Use
The
interquartile
range
=
50
25
=
25
the
to
describe
each
For
the
=
Q
set
of
the
and
20
minutes,
interquartile
Both
the
About
Q
=
35
girls
The
distributions
30
median
for
of
the
time
for
35
only
boys
more
or
=
than
girls
15
one
spent
more
the
On
20
minutes
mode
25
(unimodal).
minutes
two-thirds
or
of
less
Use
the
the
stem-and-leaf
describe
the
shape
greater
than
the
is
average,
boys’
IQR,
so
longer
the
girls
than
the
spent
of
the
Objective:
move
C.
stem-and-leaf
you
has
dierent
will
diagrams,
between
Practice
times
a
the
larger
be
changing
and
dierent
forms
to
forms
the
of
spread.
mathematical
the
representation
representation
of
the
diagrams.
data
These
from
are
tables
to
examples
of
representation.
6
ATL
The
by
back-to-back
some
boys
stem-and-leaf
and
Girls
8
girls
to
diagram
complete
a
shows
simple
the
9
1
7
5
5
4
2
2
3
5
7
7
7
9
8
8
5
1
0
3
1
2
3
4
4
7
9
6
4
3
4
5
5
6
9
7
2
5
1
2
2
1
6
9
9
8
Key: 3 | 4 | 5 represents a girl who took 43 seconds
and a boy who took 45seconds
9 4
4 Representation
times
jigsaw.
Boys
9
measure
and
of
central
the
measure
the
girls
of
middle
box-and-whisker
of
distribution.
median
Communicating
between
practice
moving
girls’
each
to
more
dispersion
50%
of
diagram
studying.
tendency
1
of
data.
Compare
this
range
spread
minutes
have
minutes
boys.
=
the
Mathematics;
spent
time
range
two-thirds
studying
In
and
3
The
iii.
interquartile
center
boys:
1
b
median
minutes
taken
(in
seconds)
for
and
boys.
S TAT I S T I C S
a
Calculate
the
b
Describe
c
Compare
the
The
shape
the
Problem
2
median
and
of
times
interquartile
the
taken
range
of
times
for
the
girls
and
A N D
boys.
distributions.
by
the
girls
and
the
boys.
solving
average
monthly
temperatures
recorded
over
a
year
in
the
US
cities
of
On
Sante
Fe,
New
Mexico,
and
Saint
Paul,
Minnesota,
are
given
below.
the
temperatures
are
in
the
zero
data
in
a
back-to-back
stem-and-leaf
temperature
diagram.
produced
b
c
Describe
the
Compare
temperatures
the
represents
Fahrenheit.
the
Represent
Fahrenheit
All
scale,
a
P R O B A B I L I T Y
in
the
two
by
mixing
equal
weights
snow
and
of
cities.
common
temperatures.
salt.
Sante
Saint
3
The
Paul
heights
Apple
Pear
4
Fe
(in
F
M
A
M
J
J
A
S
O
N
D
44
48
56
65
74
83
86
83
78
67
53
43
26
31
43
58
71
80
85
82
73
59
42
29
centimeters)
trees:
trees:
a
Represent
b
Describe
c
Compare
The
J
the
the
one-year-old
apple
and
pear
trees
180,
184,
195,
177,
175,
173,
169,
167,
197,
173,
166,
183,
161,
195,
177,
192,
161,
166
171,
160,
182,
168,
194,
177,
192,
160,
165,
178,
183,
190,
172,
174,
170,
165,
166,
193
data
in
heights
the
of
a
of
back-to-back
the
apple
stem-and-leaf
and
pear
are
shown
below.
diagram.
trees.
heights.
box-and-whisker
diagrams
show
the
delayed
departure
of
two
trains,
A
and
B.
Train A
Train
B
0
10
20
Length
Compare
5
The
ages
the
of
a
lengths
of
the
sample
of
subscribers
Süddeutsche
Zeitung:
of
30
delay
40
50
(minutes)
delays.
to
two
German
newspapers
are
65,
36,
44,
25,
37,
29,
27,
19,
60,
46,
35,
20,
55,
64,
30,
31,
22,
48,
53,
67
shown
below.
24,
Use
Der
Tagesspiegel:
46,
18,
35,
20,
27,
25,
40,
24,
31,
18,
30,
19,
28,
21,
34,
54,
22,
27
29,
20,
63,
and
the
same
draw
Construct
two
box-and-whisker
diagrams
for
the
two
data
one
box-and-whisker
diagram
a
scale,
above
the
sets.
other.
b
Compare
the
ages.
4.1 A whole range of things
9 5
6
Patricia
shows
1
1
5
9
2
0
4
8
3
4
7
4
7
5
0
Ray
work
in
a
monthly
mobile
sales
for
phone
the
3
|
0
4
represents
34
mobile
diagram
10
phones
shows
20
Ray’s
Compare
Patricia’s
The
Should
●
How
outlier
of
is
a
the
IQR
and
Ray’s
eect
●
pattern
1.5×
stem-and-leaf
we
do
of
40
below
of
Q
of
the
,
or
of
phones
Exploration
foot
given
20,
1
lengths
the
same
50
year.
60
70
sold
results
a
that
stand
data
data.
A
greater
aren’t
out
set
a
which
data
than
in
point
1.5
×
typical?
crowd?
does
not
is
outlier
an
IQR
t
above
with
if
it
the
is
general
less
than
Q
3
4
(measured
to
the
nearest
centimeter)
of
19
students
are
below.
23,
22,
17,
24,
a
Find
the
lower
b
Find
1.5
×
c
Use
can
2
for
outliers
ignore
member
sales
sales.
individuals
rest
monthly
1
The
diagram
sold
30
Number
An
The
year.
8
box-and-whisker
D
store.
previous
7
Key :
The
and
Patricia’s
Find
the
be
the
length
mean,
excluding
your
9 6
interquartile
of
an
23,
46,
upper
range
18,
26,
25,
quartile
21,
and
24,
26,
30,
interquartile
24,
20
range.
(IQR).
outlier
to
determine
if
any
of
the
data
median,
any
an
mode,
range
and
interquartile
range
of
the
the
outliers
outlier.
results
in
statistics
a
points
outliers.
data:
b
of
24,
quartile,
denition
including
Which
27,
considered
a
Record
27,
table.
are
aected
4 Representation
by
the
outlier?
Justify
this
result.
foot
S TAT I S T I C S
Reect
In
and
Exploration
discuss
What
●
When
the
which
measure
might
What
When
a
are
set
of
data
result
occurred
data
has
then
decide
speaking,
it
can
any
be
a
the
identify
(2)
use
(3)
Practice
group
them
8,
to
2
and
of
17
23,
b
Determine
the
of
the
one
the
or
not
spread.
a
outlier.
If
to
you
use
measuring
If
it
is
a
them
think
a
device,
genuine
it.
indication
just
whether
and
misreading
keep
and
distribution?
of
the
extreme
dispersion
value.
You
of
a
may
set
of
need
to
range.
tendency
(center)
and
a
measure
of
and
shape.
young
set.
22,
a
range
children
set
of
25,
are
not
of
text
16,
it
19,
48,
mean
b
Find
the
interquartile
c
Determine
d
Find
the
excluded
and
any
range.
in
is
and
of
30,
tested
62,
31,
mode
better
for
to
your
sent
27,
median
aected
by
outliers.
to
see
how
bricks.
long
Their
it
took
times
(in
each
of
seconds)
31,
of
use
32,
the
the
32,
40,
48,
51
times.
mean,
median
or
mode
for
this
answer.
by
22,
17
34,
number
adults
31,
of
36,
texts
in
one
28,
week
30,
12,
are
shown
below.
20
sent.
range.
the
Give
your
measure
28,
messages
the
if
27,
reasons
Find
Which
were
interconnecting
median
whether
Give
8,
25,
a
e
decide
ignore
to
good
by
central
interquartile
mean,
number
40,
a
calculating
(spread)
the
24,
Find
36,
is
can
not
the
here.
a
The
or
tendency
outliers?
tendency
error,
central
describe
outliers
measure
assemble
16,
data
you
to
of
to
7
shown
12,
need
(typing
use
ignoring
central
aected
before
any
describe
median
range
of
measure
you
distribution:
(1)
a
you
of
whether
greatly
outliers
describe
are
outliers,
device)
then
which
would
results
measures
dispersion
A
included,
potential
the
outliers?
dispersion
measuring
exclude
1
any
error
but
The
for
human
Generally
To
6
by
point,
data,
is
of
some
calculating
broken
account
outlier
when
P R O B A B I L I T Y
4:
●
●
A N D
a
data
values
reason
why
can
any
be
considered
outliers
have
outliers.
been
included
or
calculation.
of
central
tendency
is
more
appropriate
in
this
case?
Explain.
4.1 A whole range of things
97
3
Look
back
at
a
Determine
b
Find
any
the
The
if
data
any
range
outliers
Problem
4
the
of
30
Male
8
8
Mathematics
the
data
study
been
values
times
for
included
study
can
the
or
times
be
girls
in
Example
considered
and
excluded
boys.
in
outliers.
Give
your
4.
a
reason
why
calculations.
solving
back-to-back
nearestkg)
of
of
have
on
stem-and-leaf
male
and
30
diagram
female
wombat
Female
0
8
shows
northern
the
masses
hairy-nosed
(to
the
wombats.
wombat
9
8
5
2
0
1
0
5
7
8
9
8
8
7
7
1
0
0
0
2
0
0
0
0
5
7
8
8
7
5
2
2
2
1
1
0
3
0
0
1
1
1
2
5
5
4
3
2
2
0
4
2
2
3
4
0
5
0
6
8
8
6
7
8
9
0
Key :
and
5
0
a
|
4
Decide
b
Describe
c
Compare
26
table
2
represents
female
a
The
|
wombat
whether
the
the
of
there
masses
masses
shows
cantons
10
the
a
male
with
are
of
of
mass
any
the
the
wombat
mass
40
kg
kg
outliers.
male
and
male
percentage
42
with
of
and
female
wombats.
female
households
wombats.
that
speak
English
in
the
Switzerland
Switzerland.
ocial
a
Construct
a
stem-and-leaf
diagram
and
determine
whether
there
outliers.
are
any
From
2010
Swiss
Federal
Statistical
b
Suggest
how
any
outliers
could
have
Describe
the
the
Oce
allowed
citizens
distribution.
to
indicate
than
ZH
BE
LU
UR
SZ
OW
NW
GL
ZG
6.4
3.0
2.9
1.6
3.8
2.8
2.6
1.2
8.4
as
SO
BS
BL
SH
AR
AI
SG
GR
AG
2.5
8.6
4.2
3.6
2.1
2.4
2.5
2.5
3.5
TI
VD
VS
NE
GE
JU
FR
TG
3.1
7.4
2.6
3.3
10.7
1.5
2.5
2.4
one
their
for
all
4 Representation
more
language
main
language
so
the
total
languages
exceeds
9 8
four
occurred.
has
c
has
languages.
100%.
S TAT I S T I C S
A N D
P R O B A B I L I T Y
Summary
●
In
a
stem-and-leaf
the
stem
the
leaves
●
diagram:
represents
the
represent
category
the
nal
A
ve-point
gure
digit(s)
of
each
summary
the
minimum
the
lower
data
quartile
of
a
data
point
(Q
set
is:
(min)
)
1
data
point
the
median
(Q
)
2
the
key
tells
you
how
to
read
the
values.
the
upper
quartile
(Q
)
3
stem
leaf
0
0
the
0
0
1
1
1
4
6
0
0
1
2
0
2
6
3
9
2
3
4
5
data
point
(max).
7
●
1
maximum
7
A
box-and-whisker
ve-point
7
diagram
represents
the
summary:
Whisker
Box
Whisker
Median, Q
2
Lower
Upper
Minimum
Key :
1 | 0
represents
Maximum
Quartile, Q
10
Quartile, Q
1
●
●
For
any
set
summary
of
data
there
are
two
categories
Qualitative
●
of
central
tendency
where
question
Quantitative
data
most
data
lies.
They
‘What
is
an
average
data
two
a
certain
(color,
animal).
of
dispersion
spread
out
the
between
the
‘How
(spread)
data
is.
much
data
has
data
They
scored)
is
)
is
value.
a
numerical
value.
There
–
can
or
be
can
counted
only
take
(number
certain
of
values
size)
there
(weight,
(Q
numerical
answer
variation
Continuous
quartile
a
describe
values?’
lower/rst
has
value?’
(shoe
question
–
types:
goals
The
words
answer
Discrete
Measures
●
describes
using
(location)
are
the
–
characteristic
Quantitative
describe
how
data
of
statistics:
Measures
the
3
the
median
data
height)
–
can
and
be
can
measured
take
any
value.
of
1
the
observations
arranged
●
The
set
of
to
the
left
of
the
median
in
an
●
observations.
upper/third
an
●
observations
quartile
arranged
The
set
to
of
interquartile
between
quartile
the
(Q
(Q
3
right
)
of
is
the
the
median
median
of
●
in
range
(IQR)
quartile
and
is
the
the
An
of
outlier
does
observations.
lower
– Q
the
distribution
behavior
3
the
The
not
of
the
or
an
all
is
t
a
It
a
data
data
the
could
inaccurate
set
of
the
general
be
describes
points
member
with
data.
of
the
an
in
the
data
set
pattern
anomaly
the
set.
of
in
which
the
the
rest
data
reading.
dierence
upper
●
).
A
data
1.5
1
×
point
IQR
is
an
below
outlier
Q
,
or
if
it
is
greater
less
than
than
1.5
×
1
above
●
The
median
quartile,
is
another
name
for
the
second
Q
3
Q
2
Mixed
1
Write
practice
down
quantitative
whether
these
are
qualitative
data.
a
The
color
of
b
The
prices
c
The
areas
d
The
numbers
of
graphic
public
of
2
Write
down
continuous
owers
of
or
calculators
parks
people
at
a
store
whether
these
are
discrete
or
data.
a
The
time
taken
b
The
number
c
The
heights
d
The
temperature
of
of
to
run
a
marathon
carriages
on
Humboldt
at
a
train
penguins
midday
4.1 A whole range of things
in
Iceland
9 9
IQR
3
The
masses
are
given
46,
52,
(in
kilograms)
of
a
group
of
people
Problem
solving
below.
8
The
box-and-whisker
diagrams
show
the
ages
of
ATL
64,
60,
82,
48,
72,
61,
70,
75,
59
people
a
Construct
a
stem-and-leaf
diagram
watching
two
dierent
lms
at
a
cinema.
to
F ilm A
represent
the
data.
F ilm
b
Write
down
the
number
of
people
B
who
0
weigh
less
than
60
10
20
30
40
Age
4
c
Find
the
median
d
Find
the
range.
During
a
one-hour
students
study
break,
the
amount
spent
Their
a
Compare
b
Deduce
on
social
media
sites
times,
in
minutes,
45,
37,
29,
48,
52,
41,
32,
26,
Calculate
the
median
b
Find
the
interquartile
c
Find
the
range.
Problem
which
Give
Sebastian
The
50,
32,
shoes
prots
11
made
years
time.
000,
$170
are
$38
range.
$560
000,
$62
5,
6,
7,
owned
7,
are
9,
9,
by
in
r,
calculated
was
9.5
by
as
as
a
family
answer.
a
small
business
during
the
follows:
000,
$1000,
000,
$250
a
Find
b
Determine
data
on
the
number
of
$85
000,
000,
$160
$3100,
000,
$120
000,
the
interquartile
if
any
of
range.
the
data
values
can
be
every
student
in
his
and
Find
the
range.
Give
a
reason
why
any
class.
have
been
included
or
excluded
in
order.
10,
that
the
outliers.
pairs
s,
13,
13,
the
upper
your
calculation.
Find
the
t
d
He
your
solving
collected
values
categorized
000
outliers
The
is
for
44
c
of
lm
reasons
considered
5
ages.
were:
$45
a
the
was
last
20,
70
of
9
recorded.
60
(years)
mass.
lm.
time
50
kg.
median
of
the
quartile
Q
data
was
set
is
13.
most
measure
of
appropriate
central
in
this
tendency
case.
which
Justify
your
choice.
3
a
Write
b
The
down
the
value
of
r
and
s
Problem
mean
of
the
data
set
is
10
Find
the
value
of
The
table
shows
All
the
dogs
the
masses
(in
kilograms)
of
t
ATL
Blackface
6
solving
10.
attending
a
veterinary
surgery
sheep
on
two
farms.
were
ATL
weighed.
Their
26,
18,
54,
90,
16,
5,
32,
27,
a
Find a
b
Construct
masses,
30,
18,
25,
3,
ve-point
in
6,
23,
kilograms,
32,
were:
Farm
A
63
48
60
55
45
49
19
55
Farm
B
54
71
68
57
62
70
54
49
Farm
A
65
49
57
56
64
43
64
48
Farm
B
66
68
72
50
56
49
70
64
43,
27
summary
for
the
data.
a
a
box-and-whisker
diagram
Construct
a
back-to-back
stem-and-leaf
for
diagram
to
represent
the
data.
thedata.
b
7
This
box-and-whisker
diagram
represents
Compare
on
results
you
from
when
a
survey
you
got
that
your
asked:
rst
‘How
the
old
the
two
c
smartphone?’
The
table
grazing
a
18
30
Age
The
interquartile
is
Write
b
Find
c
160
down
the
the
values
20
median
of
a
Poor
and
the
range
is
40.
shows
of
were
the
average
Blackface
people
10 0
who
and
hill
Deduce
b
surveyed.
were
adult
sheep
for
dierent
hill
30
45–50
kg
50–65
kg
70
kg
value.
Find
the
or
older.
4 Representation
the
sheep
grazing
on
number
your
of
sheep
conditions.
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the
people
Blackface
farms.
Upland
a
the
b
(years)
range
of
were
bodyweight
4
masses
the
answer.
the
conditions
two
farms.
available
Give
to
reasons
for
S TAT I S T I C S
Review
in
A N D
P R O B A B I L I T Y
context
Globalization and sustainability
1
The
by
table
forest
shows
in
19
the
percentage
European
of
Union
Austria
46.7%
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22.0%
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34.3%
land
states
2
covered
in
The
box-and-whisker
coverage
2010.
0
Republic
Denmark
11.8%
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53.9%
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73.9%
France
28.3%
Germany
31.7%
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29.1%
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21.5%
for
10
20
33.9%
Lithuania
33.5%
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10.8%
Poland
30.0%
Por tugal
41.3%
identify
the
any
3
Use
the
data
Slovakia
40.1%
Sweden
66.9%
more
Kingdom
11.8%
Construct
a
box-and-whisker
diagram
for
40
50
for
2010
in
the
forest
2012.
60
70
80
(%)
and
between
forestry
data.
representation
to
present
can
to
the
2012
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and
two
years.
if
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website
an
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draw
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Use
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your
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representation.
represent
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Europe)
of
you
b
states
coverage
changes
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Commission
a
30
EU
shows
9.7%
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same
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Ireland
the
diagram
there
are
any
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your
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this
data.
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and
How
you
have
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explored
the
statement
of
inquiry?
Give
specic
examples.
Statement of Inquiry:
How
and
quantities
trends
in
a
are
represented
can
help
to
establish
underlying
relationships
population.
4.1 A whole range of things
101
can
Getting
your
ducks
in
a
row
4.2
Global
context:
Objectives
●
Finding
from
●
a
the
frequency
SPIHSNOITA LER
●
mean,
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median,
frequency
grouped
mode
and
data
in
●
range
F
table
a
●
cumulative
the
frequency
summary
from
a
●
C
curve
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a
a
frequency
cumulative
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can
you
How
box-and-whisker
diagram
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it
can
can
you
does
be
nd
from
cumulative
ve-point
and
sustainability
questions
tendency
curve
cumulative
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grouped
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Globalization
a
the
represent
frequency
the
measures
grouped
type
of
represented
of
central
frequency
grouped
table?
data
in
data
and
aect
the
way
analyzed?
from
curve
●
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can
●
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do
real
data
ever
be
misleading?
D
individuals
stand
out
in
a
crowd?
ATL
Communication
Organize
and
depict
information
logically
4.1
Statement of Inquiry:
How
and
quantities
trends
in
a
are
represented
can
help
to
establish
underlying
relationships
population.
4.2
6.2
10 2
a
curve?
S TAT I S T I C S
Y
ou
●
should
understand
already
inequality
know
how
notation
1
Which
a
recognize
discrete
continuous
●
calculate
the
frequency
and
2
of
4.2
Is
data
these
write
a
each
construct
a
data
mean
from
a
3
Numbers
b
Heights
c
Shoe
d
Lap
The
data
summar y
and
4
a
of
quar tiles
divide
a
5
c
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can
When
into
you
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have
groups
can
a
before
with
you
frequency
●
c
6.4
d
discrete
6.0
or
nd
measures
laid
sunower
in
a
table
children
the
in
6
hens
plants
mean
1
shows
a
car
race
the
playgroup.
age.
1
2
3
4
5
Freq
0
2
6
4
3
a
a
for
Scout
box-and-whisker
this
data
13,
13,
13,
14,
17,
18,
18,
20,
21
Find
any
data
is
the
less
the
ages
15,
interquar tile
set,
what
than
lower
the
the
the
on
group.
12,
of
by
Formula
15,
grouped
the
eggs
Age
Write
Working
of
times
of
data
b
●
the
6?
11,
In
a
F
satisfy
sizes
diagram
set
of
Construct
box-and-whisker
how
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set
frequency
ages
b
identify
x
3.0
a
diagram
●
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values
continuous?
table
ve -point
3
b
Calculate
●
P R O B A B I L I T Y
to:
inequality
●
A N D
or
range.
fraction
equal
of
the
to:
quar tile
median
upper
each
quar tile?
fraction
as
a
percentage.
data
central
tendency
from
a
grouped
table?
you
large
represent
set
doing
of
any
data
grouped
it
is
statistical
data
often
in
a
cumulative
convenient
to
frequency
organize
the
curve?
data
analysis.
4.2 Getting your ducks in a row
10 3
ATL
Example
The
1
lengths
of
50
Eelgrass
leaves
were
measured.
The
results
are
given
Measuring
below,
to
46.3
the
nearest
35.6
tenth
75.2
of
a
43.7
52.7
57.3
49.0
57.3
leaf
42.6
47.1
50.3
50.7
gives
continuous
data;
this
34.2
58.0
37.0
59.3
has
to
55.6
be
56.0
a
centimeter.
68.1
rounded
to
a
59.9
certain
53.6
34.0
74.4
70.9
44.2
41.3
64.9
39.3
44.2
68.3
54.2
40.7
76.8
38.2
70.3
33.5
64.3
64.6
68.7
69.9
degree
accuracy
used
in
to
of
be
statistical
analysis.
48.9
51.6
51.7
a
Construct
b
Find
c
Write
a
minimum
the
range
a
grouped
class
down
=
46.0
=
that
modal
33.5
76.8
frequency
interval
the
cm,
33.5
52.0
=
45.6
table
contains
to
the
64.1
63.4
represent
the
45.1
62.0
data.
median.
class.
maximum
43.3
=
76.8
cm
cm
Choose
=
4.33
so
use
a
class
width
of
5
15
Length,
x
(cm)
Freq
a
class
width
to
give
between
5
and
cm
Cumulative
Write
equal
the
class
width
classes.
intervals
(In
using
this
case,
inequality
10.)
notation.
frequency
30
<
x
≤
35
3
3
35
<
x
≤
40
4
7
40
<
x
≤
45
6
13
45
<
x
≤
50
7
20
50
<
x
≤
55
<
x
≤
60
7
35
60
<
x
≤
65
6
41
65
<
x
≤
70
4
45
Add a Cumulative frequency column to help
nd the class interval that contains the median.
The
of
cumulative
the
and
including
cumulative
70
<
x
≤
75
3
48
75
<
x
≤
80
2
50
The
b
n
=
50,
so
the
The
class
The
modal
median
interval
that
is
the
=
contains
the
25.5th
median
is
data
50
<
x
25th
and
class
is
50
<
x
≤
≤
current
frequency
26th
is
every
is
values
3
the
one.
+
are
4
in
up
Here,
+
6
=
this
modal
class
is
the
class
to
the
13
class.
interval
55.
4 Representation
sum
class,
55.
with
10 4
the
for
value.
The
c
frequency
frequencies
the
highest
frequency.
S TAT I S T I C S
Reect
●
What
●
Why
and
does
can’t
frequency
●
How
The
cumulative
you
class
you
the
1
frequency
the
exact
estimate
range
width
minimum
nd
P R O B A B I L I T Y
represent?
value
of
the
range
from
median
from
a
grouped
table?
would
Estimate
discuss
A N D
is
of
the
possible
the
lengths
of
dierence
values
in
a
a
Eelgrass
between
class
grouped
frequency
table?
leaves.
the
maximum
and
the
The
interval.
has
class
class
because
An
of
estimate
highest
When
the
the
most
For
for
the
range
class
interval)
class
widths
data.
grouped
It
has
data,
a
(lower
are
the
you
from
–
equal,
highest
can
nd
grouped
bound
the
frequency
of
lowest
modal
class
table
class
is
the
is
(upper
interval
Practice
1
The
table
th
shows
the
heights
x (cm)
of
<
x ≤
1.30
4
1.30
<
x ≤
1.40
6
1.40
<
x ≤
1.50
8
1.50
<
x ≤
1.60
6
1.60
<
x ≤
1.70
6
Write
b
Find
Here
down
the
are
18
=
the
class
the
modal
interval
class
containing
class
interval
that
contains
the
median.
value.
data
modal
class
of
mode
one
has
a
group
of
students
on
their
14th
times
taken
for
contains
some
the
to
the
nearest
whole
value.
birthdays.
median.
students
to
complete
a
1500
m
race
second).
325
580
534
500
532
328
600
625
450
435
450
340
357
370
401
456
388
626
532
399
a
Construct
a
equal
and
size
b
Find
the
c
Determine
grouped
rst
modal
frequency
class
table
interval
300
for
<
x
this
≤
data.
Use
class
widths
of
350.
class.
which
class
interval
contains
the
median
value.
4.2 Getting your ducks in a row
a
instead
ATL
(measured
2.
bound
class.
that
20
interval).
Frequency
1.20
a
≤
2,
frequency.
the
the
x
1
Height,
2
contains
<
20
Grouped
class
18
width
10 5
3
The
heights
of
one-year-old
apple
trees
are
shown
below,
measured
to
the
ATL
nearest
cm.
180
184
195
177
175
173
169
167
197
173
166
183
161
195
177
192
161
166
a
Construct
a
b
Write
c
Determine
down
Problem
4
The
grouped
table
the
which
shows
w
contains
the
median
weights
(g)
of
apples
picked
Frequency
from
Cumulative
40
<
w
≤
50
7
12
50
<
w
≤
60
5
b
60
<
w
≤
70
c
19
70
<
w
≤
80
8
d
80
<
w
≤
90
3
30
the
that
values
the
back
at
you
Assume
the
mean
Now
the
all
class
the
What
to
the
the
in
d
the
class
50
<
w
≤
60.
data
in
For
all
the
30.
of
data
aect
be
data
estimate
midpoint
class
interval
in
the
assumption,
or
and
of
in
For
each
leaves
has
class
the
30 < x
calculate
class
example,
this
estimates.
all
upper
Eelgrass
in
Example
an
value
≤ 35
of
the
assume
estimate
in
interval
the
assumption,
has
class
the
for
30 < x
calculate
an
value
all
the
≤ 35
of
assume
estimate
for
the
better
of
in
a
the
How
each
would
estimate
than
the
class
mean
class
class
of
choosing
the
upper
a
dierent
4 Representation
value
mean?
or
lower
class
boundaries
interval?
from
mid-interval
lower
The
of
a
value
grouped
of
each
boundaries
interval
to
frequency
class
and
interval
dividing
calculate
table:
the
by
by
2.
mean.
on
next
page
lower
the
(not
Continued
10 6
1.
leaves.
would
midpoint
for
data.
example,
Using
the
two
the
table
raw
each
this
boundary.
are
the
leaves.
interval
an
the
and
frequency
Using
the
that
your
value
adding
Use
of
length
represent
Find
c
received
the
35.
class
class
calculate
not
boundary.
are
values
mean
b,
lies
grouped
all
assume
Compare
of
the
that
lower
a,
1
length
the
the
median
had
values
of
tree.
frequency
5
upper
value.
one
a
Imagine
●
interval
40
Look
●
class
the
Exploration
To
class.
≤
Verify
d
data.
w
b
c
this
<
Calculate
b
for
30
a
a
modal
table
solving
Weight,
1
frequency
boundary
interval
31
or
is
30.1,
30
etc.).
S TAT I S T I C S
2
To
calculate
for
mid-interval
Length,
x
an
estimate
value
(cm)
of
and
the
a
mean
column
for
for
Frequency
the
Eelgrass
mid-interval
leaves,
value
add
×
Mid-inter val
a
A N D
P R O B A B I L I T Y
column
frequency.
Mid-inter val
value
value
×
frequency
30
<
x
≤
35
3
32.5
32.5
×
3
=
97.5
35
<
x
≤
40
4
37.5
37.5
×
4
=
150
40
<
x
≤
45
6
42.5
45
<
x
≤
50
7
47.5
50
<
x
≤
55
8
52.5
55
<
x ≤
60
7
57.5
60
<
x
≤
65
6
62.5
65
<
x
≤
70
4
67.5
70
<
x
≤
75
3
72.5
75
<
x
≤
80
2
77.5
Total
frequency
(number
Sum
of
of
inter val
obser vations)
mid-
value
×
frequency
For
For
grouped
data
the
mean
is
dened
individual
points
as:
data
in
the
dened
Copy
You
can
and
also
complete
calculate
the
an
table,
and
estimate
of
calculate
the
mean
an
estimate
using
a
for
the
a
set
of
mean
is
as:
mean.
GDC.
RAD
1.1
A
B
C
D
1
Enter
the
mid-interval
values
and
2
32.5
3
frequencies
into
your
GDC.
3
37.5
4
42.5
6
47.5
7
4
5
B1
B11
RAD
1.1
A
B
C
D
2
37.5
4
42.5
6
47.5
7
Σx
52.5
8
sx
3
x
53.5
Σx
is
2675.
the
mean.
2
4
150063.
5
: =
Sn
…
11.9095
6
σx
D2
=
: =
σn
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11.7898
53.5
4.2 Getting your ducks in a row
10 7
Reect
●
In
and
discuss
Exploration
the
Eelgrass
1,
what
leaves?
2
does
What
the
does
mid-interval
the
value
mid-interval
×
represent
for
frequency
value
represent?
●
Why
can
you
frequency
●
original
data
Use
raw
the
Eelgrass
the
1
The
data
from
shows
videos
Time,
the
only
an
estimate
estimate
for
the
for
the
mean
mean
likely
from
to
be
a
grouped
one
of
the
in
Example
Compare
1
the
Exploration
to
calculate
calculated
the
mean
mean
with
length
your
of
the
estimate
for
1.
2
table
online
Is
values?
leaves.
mean
Practice
calculate
table?
t
in
the
number
one
(hours)
of
hours
a
group
of
50
students
spent
watching
week.
Frequency
Mid-inter val
value
Mid-inter val
value
×
To
nd
the
mid-
frequency
interval
the
0
<
t
≤
10
8
5
upper
<
t
≤
20
12
20
<
t
≤
30
16
30
<
t
≤
40
11
40
<
t
≤
50
3
and
Total
a
Copy
and
b
Calculate
online
2
The
complete
an
the
estimate
table.
for
the
mean
each
student
spent
watching
videos.
weights
of
Weight,
w
the
Irish
Wolfhounds
(kg)
<
w
≤
48
2
48
<
w
≤
51
5
51
<
w
≤
54
12
54
<
w
≤
57
13
57
<
w
≤
60
8
a
Calculate
an
b
Write
c
Find
d
Determine
down
the
estimate
the
for
modal
number
the
at
a
dog
show
are
shown
Frequency
45
10 8
time
of
class
the
mean
weight
of
the
dogs.
class.
dogs
who
interval
weighed
that
4 Representation
54
contains
and
add
lower
40
class
10
value,
kg
the
or
less.
median.
in
the
table.
boundaries
divide
by
2.
S TAT I S T I C S
Problem
3
The
table
P R O B A B I L I T Y
solving
below
Weight,
A N D
w
shows
(kg)
the
weights
Frequency
(w)
of
sh
caught
one
morning.
(f )
Many
scale
0.6
<
w
≤
0.8
types
can
be
of
used
16
to
0.8
<
w
≤
1.0
35
1.0
<
w
≤
1.2
44
1.2
<
w
≤
1.4
23
weigh
only
sh
some
but
are
recognized
by
International
Fish
<
w
≤
1.6
scale
between
Estimate,
correct
to
the
nearest
0.1
kg,
the
mean
weight
of
the
Find
the
modal
Any
sh
class.
caught
that
round
weighs
no
more
than
1.0
kg
must
be
two
angler
marks
returned
to
to
the
known
the
weight.
sea.
Find
d
Find
the
e
Estimate
f
A
sh
Cumulative
You
For
the
●
on
can
a
set
nd
n
estimate
the
range
doing
of
1.2
any
stay
x-axis,
that
the
kg
returned.
contains
the
median.
weights.
was
incorrectly
recorded
explain
in
whether
the
or
class
not
the
1.0
≤
w
mean
<
1.2.
and
same.
curves
curve
and
the
ve-point
data
sh
calculations,
the
frequency
the
of
interval
frequency
the
of
the
will
cumulative
plotted
number
weighing
median
A
the
class
Without
values
is
a
graph
cumulative
summary
on
a
with
the
upper
frequencies
from
cumulative
a
on
cumulative
frequency
class
the
boundaries
y-axis.
frequency
curve,
to
curve.
nd
for:
lower
quartile
Q
,
nd
the
value
on
the
x-axis
which
corresponds
to
1
25%
●
the
of
n
median
Q
,
nd
the
value
on
the
x-axis
which
corresponds
to
50%
of
n
2
●
the
upper
quartile
Q
,
nd
the
value
on
the
x-axis
which
corresponds
to
3
75%
of
If
is
must
down
heaviest
c
reading
sh.
the
b
for
records.
10
the
a
Game
Association
potential
1.4
the
n
4.2 Getting your ducks in a row
10 9
The
the
cumulative
Eelgrass
frequency
data
in
curve
Example
here
represents
the
grouped
frequency
table
of
1.
y
Upper
quartile, Q
3
50
45
75%
of
37.5
on
to
the
50
=
the
37.5th
value. Draw
cumulative
cur ve, then
a
frequency
down
to
the
line
axis
x-axis
from
across
and
40
ycneuqerf
read
37.5
off
the
value. Q
=
62
cm
3
35
Median, Q
2
30
evitalumuC
50%
of
50
=
25th
value. Q
=
53
cm
2
25
Lower
quartile, Q
1
20
25%
of
50
=
12.5th
value. Q
=
44.5
cm
1
15
12.5
10
Q
Med
Q
1
3
5
0
30
40
35
45
50
Leaf
0
30
35
40
45
55
60
length
50
Leaf
65
70
75
(cm)
55
60
length
65
70
75
and
can
use
take
it
to
●
Why
Q
,
ve-point
the
and
the
and
Find
●
Do
How
the
box-and-whisker
percentiles
the
cumulative
diagram
at
the
frequency
same
curve
scale.
3
25%,
50%
and
75%
used
to
nd
estimates
for
respectively?
ve-point
values
the
raw
could
number
than
from
3
the
from
summary
discuss
Q
2
●
●
the
are
Q
1
Maximum
draw
Reect
80
(cm)
Minimum
You
x
80
62
of
from
data
you
Or
the
in
use
leaves
cm?
summary
from
cumulative
Example
the
with
1?
the
between
50
less
and
Eelgrass
frequency
Which
cumulative
lengths
raw
frequency
than
62
curve
values
or
are
data
agree
the
curve
equal
to
Example
with
most
to
62
in
the
values
accurate?
estimate
cm?
1.
Or
the
greater
cm?
y
Practice
1
The
the
in
Find
the
university
estimate
of
an
of
the
curve
league.
for
the
median
players.
estimate
for
the
upper
The
90
lower
Find
an
70
Association
for
measured
while
2.31
the
range.
1.6
1.7
1.8
1.9
2.0
Height
4 Representation
the
meters
tallest
40
0
110
1.6
50
10
interquartile
history
60
quartiles.
estimate
National
Basketball
20
c
in
80
30
and
shortest
player
basketball
evitalumuC
Find
a
an
height
frequency
heights
ycneuqerf
players
b
100
cumulative
shows
a
3
2.1
(m)
2.2
2.3
x
meters.
was
S TAT I S T I C S
2
The
table
shows
the
distances
walked
(in
kilometers)
by
members
of
A N D
P R O B A B I L I T Y
a
ATL
walking
group
Distance,
d
in
one
week.
(km)
Frequency
Cumulative
frequency
0
<
d
≤
3
4
4
3
<
d
≤
6
14
18
6
<
d
≤
9
6
12
8
9
<
12
d
<
≤
d
≤
a
Copy
and
b
Construct
15
4
complete
the
table.
Plot
a
cumulative
frequency
curve
to
represent
this
data.
(3,
and
c
Find
estimates
for
the
median
and
the
4),
join
3
Find
The
estimates
table
shows
for
the
the
range
masses
of
and
the
18)
with
etc.
a
quartiles.
smooth
d
points
(6,
interquartile
Great
Danes
curve.
range.
at
a
rehoming
shelter.
ATL
Mass, m (kg)
65
<
m
Frequency
70
70
<
7
a
Construct
b
Write
c
Construct
a
≤
a
Problem
≤
75
75
<
m
13
cumulative
ve-point
a
m
for
box-and-whisker
80
80
<
9
frequency
summary
≤
curve
the
to
m
≤
85
85
<
m
12
represent
this
≤
90
9
data.
data.
diagram.
solving
ATL
4
For
the
sh
in
Practice
2,
Q3:
a
Construct
a
cumulative
b
Construct
a
box-and-whisker
c
Use
are
d
your
true
cumulative
or
frequency
sh
weigh
1.1
ii
60
sh
weigh
between
of
The
table
curve
to
decide
whether
these
statements
false.
90
25%
curve.
diagram.
frequency
i
the
quartile,
5
data
sh
nd
are
the
shows
kg
or
less.
1.1
classed
as
minimum
the
masses
kg
and
kg.
overweight.
weight
(in
1.5
at
grams)
By
which
of
using
a
sh
apples
in
50
≤
the
is
a
calculated
classed
as
upper
overweight.
box.
ATL
Mass, m (kg)
30
<
m
Frequency
a
Construct
b
Apples
Find
c
10%
of
40
40
<
m
5
a
estimate
the
less
for
apples
≤
50
<
m
7
cumulative
weighing
an
≤
the
weigh
45
g
more
curve
cannot
number
60
<
m
5
frequency
than
60
of
than
for
be
70
2
this
70
<
m
≤
80
80
<
8
m
≤
90
3
data.
sold.
apples
x
≤
that
grams.
cannot
Find
be
sold.
x
4.2 Getting your ducks in a row
111
6
This
box-and-whisker
diagram
shows
the
times
taken
by
100
people
to
ATL
complete
the
a
simple
word
puzzle.
Construct
a
cumulative
frequency
curve
for
data.
0
5
10
15
20
T ime
Problem
25
30
35
40
(seconds)
solving
ATL
7
The
cumulative
hour)
of
some
frequency
cars
on
curve
here
motorway
shows
the
speeds
(in
kilometers
per
A.
y
Only
140
ycneuqerf
on
three
the
countries
world
the
motorway
evitalumuC
limit
in
hour
rather
miles
60
kilometers
40
0
90
100
110
Speed
b
The
estimate
speed
Find
c
an
an
a
for
on
estimate
Construct
The
limit
the
the
for
median
the
70
speeds
Grouping
C
●
of
below
80
90
100
Speed
Compare
130
diagram
diagram
130
140
x
km/h.
cars
for
exceeding
the
shows
speed
the
the
data
speeds
speed
for
of
limit.
motorway
some
cars
A.
on
B.
60
d
is
percentage
box-and-whisker
120
(km/h)
speed.
motorway
the
box-and-whisker
motorway
How
does
of
the
cars
on
discrete
the
type
of
data
110
120
130
140
(km/h)
motorway
A
and
motorway
B.
data
aect
the
way
it
can
be
represented
and
analyzed?
When
data
is
you
represent
discrete
1 12
or
per
80
the
Great
the
United
and
Find
speed
100
than
per
data
in
a
continuous,
frequency
as
this
4 Representation
table,
aects
you
how
rst
you
need
write
to
the
decide
class
if
the
intervals.
hour:
Britain,
States
20
a
give
120
Burma.
S TAT I S T I C S
Continuous
boundar y
continuous
and
one
When
the
≤,
you
mean
For
data
has
values
inter vals,
for
written
with
one
Discrete
with
<
data
boundar y
sign
between
has
values
the
inequalities
no
class
are
≤,
there
for
≤
40
35
≤
x
≤
40
40
<
x
≤
45
41
≤
x
≤
45
continuous
on
way
data,
for
frequencies
the
as
class
for
intervals,
grouped
example
against
35
the
<
you
can
continuous
x
≤
upper
40,
40
class
gaps
Both
example:
x
decided
are
inter vals.
<
same
calculate
an
estimate
of
data.
<
x
≤
45,
plot
boundaries
40
the
and
All
45.
values
round
and
For
discrete
data,
for
example
35
≤
x
≤
40,
41
≤
x
P R O B A B I L I T Y
overlapping
and
35
the
cumulative
is
example:
have
in
overlapping
and
A N D
≤
45,
plot
the
all
interval
against
(40)
Example
and
the
next
point
lower
halfway
bound
between
(41),
at
the
upper
bound
40.5
to
values
40,
≥
40.5
cumulative
round
frequencies
<
down
of
up
to
41.
one
40.5.
2
ATL
Dieter
recorded
intervals
3
over
14
a
the
number
period
6
of
3
of
trucks
hours.
driving
His
results
past
are
his
house
shown
16
21
6
20
14
in
5-minute
below.
4
11
12
19
1
12
19
14
11
16
24
20
6
18
8
27
7
23
2
7
11
12
11
7
27
24
12
22
15
29
9
25
14
10
10
16
15
19
1
17
8
12
Construct
a
Number
cumulative
of
Freq
frequency
curve
New
for
upper
this
data.
Cumulative
Find
trucks
boundar y
for
1
≤
x
≤
5
5
=
5.5
the
new
each
halfway
class
x
≤
10
11
11
≤
x
≤
15
15
=
10.5
=
15.5
of
class
x
≤
20
10
21
≤
x
≤
25
6
26
≤
x
≤
30
3
=
20.5
=
25.5
=
30.5
one
the
interval
boundary
value
class
of
and
the
the
next.
31
a
Cumulative
column
≤
–
upper
16
Add
16
the
5
lower
≤
boundary
interval
between
boundary
6
upper
frequency
to
the
frequency
table.
41
47
50
Continued
on
next
page
4.2 Getting your ducks in a row
1 13
y
60
ycneuqerf
50
Plot
40
upper
boundary
evitalumuC
cumulative
the
y-axis.
The
20
10
5
10
15
20
Number
You
can
discrete
obtain
data
Practice
a
State
and
b
The
a
30
x
35
trucks
ve-point
exactly
whether
give
the
the
10
<
x
≤
20
20
<
x
≤
30
table
these
reasons
down
here
Number
a
in
of
25
summary
same
of
class
for
class
width
ii
shows
words
the
x
≤
18
19
≤
x
≤
23
number
17
5
≤
x
≤
9
25
10
≤
x
≤
14
30
15
≤
x
≤
19
12
20
≤
x
≤
24
10
25
≤
x
≤
29
3
30
≤
x
≤
34
2
35
≤
x
≤
39
0
40
≤
x
≤
44
1
114
an
your
Write
box-and-whisker
diagram
for
data.
the
estimate
answer
class
down
for
make
interval
the
continuous
or
discrete
data,
of
interval.
iii
words
in
6
<
x
≤
9
9
<
x
≤
12
each
of
100
iv
sentences
20
–
39
40
–
64
in
abook.
Frequency
4
c
a
continuous
represent
each
≤
≤
Calculate
of
14
x
Find
draw
for
answers.
≤
b
and
as
intervals
your
0
Does
way
4
Write
i
2
new
and
on
frequency
the
on
30
0
1
the
x-axis
the
sense
that
modal
mean
in
number
the
contains
class.
4 Representation
of
context
the
words
of
this
median.
in
each
sentence.
question?
rst
point
is
(5.5,
5).
S TAT I S T I C S
3
Beatrice
of
the
records
25
Number
goals
the
matches
of
0
number
in
≤
x
goals
scored
by
her
unihockey
team
for
each
season.
3
4
≤
x
3
a
Calculate
b
Explain
c
Find
d
Estimate
The
≤
of
P R O B A B I L I T Y
≤
7
8
≤
x
≤
11
12
≤
x
≤
15
16
≤
x
≤
19
scored
Frequency
4
the
A N D
estimate
why
the
table
an
this
class
the
5
for
data
interval
is
the
6
mean
number
6
of
goals
5
scored
per
match.
bimodal.
that
contains
the
median.
range.
shows
the
Extended
Essay
scores
of
50
DP
students.
ATL
Score
1
6
a
≤
≤
x
x
≤
≤
Frequency
5
1
10
9
11
≤
x
≤
15
15
16
≤
x
≤
20
15
21
≤
x
≤
25
5
26
≤
x
≤
30
5
By
drawing
and
the
b
Using
c
Find
d
Draw
a
new
table
cumulative
your
an
a
cumulative
estimate
with
for
correct
construct
frequency
the
upper
box-and-whisker
Problem
the
frequency,
curve,
and
diagram
upper
a
write
lower
to
boundary
cumulative
down
for
discrete
frequency
the
median
data
curve.
score.
quartiles.
represent
the
information.
solving
ATL
5
Brooke
month
recorded
in
Number
this
the
of
text
messages
she
sent
every
day
for
a
of
0
messages
x
≤
4
5
≤
10
Calculate
thetext
≤
x
≤
9
10
≤
x
≤
14
15
≤
x
≤
19
20
≤
x
≤
24
25
≤
x
≤
29
the
values
message
7
of
a
and
b
3
in
this
4
cumulative
frequency
3
b
Draw
table
c
Use
≤
boundar y
Cumulative
frequency
4.5
10
≤
9.5
17
≤
14.5
a
≤
19.5
24
≤
24.5
27
≤
29.5
b
≤
34.5
30
cumulative
your
Hence
x
≤
34
1
for
data.
≤
a
≤
2
9.5
means
values
Upper
30
sent
Frequency
a
number
table:
curve
to
represent
frequency
calculate
the
data
in
curve
each
a
for
value
this
in
up
including
to
‘all
the
and
9.5’.
data.
the
ve-point
box-and-whisker
summary.
plot.
4.2 Getting your ducks in a row
115
6
The
ages
(in
years)
of
the
rst
30
people
visiting
the
Tuileries
Gardens
in
ATL
Paris
on
a
day
are
shown.
25
65
34
48
4
55
32
45
23
43
23
37
45
36
26
39
43
45
29
15
15
20
64
37
61
25
17
23
45
36
a
Construct
a
grouped
b
Construct
a
cumulative
c
Use
your
visitors
d
7
certain
The
give
cumulative
who
Estimate
frequency
are
the
25
frequency
frequency
or
frequency
information
about
for
this
data.
curve.
curve
to
estimate
the
percentage
of
under.
percentage
cumulative
table
of
visitors
who
are
older
than
40.
curves
the
ages
Pedestrians
450
of
pedestrians
killed
Great
Draw
a
a
in
400
2014.
for
the
pedestrians.
box-and-whisker
diagram
for
the
pedal
350
300
evitalumuC
Draw
Britain
cyclists
box-and-whisker
diagram
b
pedal
ycneuqerf
a
in
and
cyclists.
250
200
Use
Compare
the
ages
of
the
same
scale
150
Pedal
c
cyclists
the
and
draw
one
100
pedestrians
and
pedal
cyclists.
box-and-whisker
50
diagram
above
the
other.
0
10
20
30
40
50
60
70
80
x
90
Age
Exploration
2
Use
the
number
1
In
your
class,
record
500
random
numbers
between
1
and
30
the
following
button
on
generated
your
in
random
calculator,
a
way:
spreadsheet
Use
technology
two
digits
to
produce
random
3-digit
numbers
and
record
the
last
formula,
of
each,
ignoring
any
over
number
For
example:
You
15
Random
number
record
951
a
class,
>
30
so
ignore
23
430
record
30
measures
2
51
record
a
Compare
intervals.
grouped
dierent
of
central
your
frequency
sized
are
class
table
with
intervals.
carefully
Use
chosen
technology
intervals.
to
calculate
with
the
As
the
other
students
who
chose
dierent
class
dierences?
Continued
116
nd
random
4 Representation
numbers
each
combine
them
one
tendency.
results
What
tables.
could
and
7
823
try
random
Action
307
Construct
or
30.
on
next
page
class
set.
into
S TAT I S T I C S
3
Discuss
class
4
how
and
Why
choosing
modal
would
it
dierent
class
intervals
aects
the
mean,
A N D
P R O B A B I L I T Y
median
class.
not
be
sensible
to
represent
this
data
in
a
stem-and-leaf
diagram?
5
Represent
whisker
6
Compare
Reect
If
a
●
set
of
Does
Does
and
●
and
data
D
is
the
grouped
the
of
choice
the
class
of
curve
and
box-and-
for
too
●
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can
●
How
do
gives
Switzerland
in
equal
class
interval
diagram?
and
width
aect
and
interval
many
real
others.
class
class
advantages
likewise
Age
frequency
4
in
median
box-and-whisker
table
cumulative
with
Misleading
The
the
modal
aect
intervals:
estimate
class?
the
of
the
mean?
Explain.
cumulative
frequency
curve
Explain.
disadvantages
of
having
too
few
classes.
classes.
statistics
data
ever
individuals
be
stand
misleading?
out
in
a
crowd?
3
the
ages
of
fathers
at
the
birth
of
their
rst
child,
in
2013.
Freq
Age
Freq
Age
Freq
Age
Freq
Age
Freq
≤ 12
0
22
388
32
5522
42
2203
52
189
13
0
23
598
33
5766
43
1773
53
146
14
0
24
958
34
5612
44
1487
54
128
15
0
25
1414
35
5516
45
1187
55
95
16
2
26
1815
36
5225
46
856
56
86
17
13
27
2365
37
4703
47
717
57
61
18
26
28
3073
38
4126
48
579
58
45
19
69
29
3792
39
3705
49
442
59
40
20
132
30
4382
40
3210
50
365
21
251
31
5108
41
2703
51
290
Draw
a
width
intervals
were
2
a
discuss
Exploration
1
in
results
choice
about
Discuss
Do
data
your
the
What
●
the
diagram.
grouped
under
Calculate
age
of
70.
an
1
frequency
≤
x
≤
What
estimate
fathers
from
10,
will
for
your
table
11
the
the
≤
x
to
≤
represent
20,
highest
mean
etc.
class
and
this
data.
Assume
the
interval
Use
that
all
≥
60
174
equal
the
class
fathers
be?
ve-point
summary
of
the
table.
Continued
on
next
page
4.2 Getting your ducks in a row
1 17
3
Peter
uses
the
class
intervals
classes
are
roughly
equal.
0
25,
26
29,
≤
51
x
≤
≤
x
≤
a
Calculate
new
Reect
●
In
30
≤
x
≤
frequency
estimate
34,
that
35
≤
x
the
≤
frequencies
38,
39
≤
x
≤
in
43,
all
44
the
≤
x
≤
50,
for
the
table
mean
using
and
his
the
class
intervals.
ve-point
summary
from
table.
and
discuss
Exploration
How
≤
grouped
an
summary
●
x
so
100
Construct
this
≤
below
that
should
3,
which
best
you
5
grouping
represents
group
data
the
so
of
the
data
gives
a
ve-point
data?
that
it
accurately
represents
a
data
set?
Consider:
●
–
the
–
class
–
the
Do
number
classes
width
upper
you
of
need
and
to
lower
boundaries
consider
the
size
of
of
the
the
class
data
intervals
set?
Summary
The
class
width
maximum
class
the
has
the
that
the
dierence
minimum
grouped
between
possible
the
values
For
in
grouped
the
equal,
containing
you
to
interval
calculate
are
the
the
modal
most
class
data.
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an
the
nd
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the
a
of
class
value
estimate
frequency
table
to
the
the
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cumulative
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contains
estimate
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interval)
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median.
column
class
grouped
nd
value.
mean
from
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the
each
class
or
mid-interval
value
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interval
by
adding
the
upper
lower
class
boundaries
and
dividing
by
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the
midpoint
of
each
class
interval
the
x
≤
and
a
C.
organize
the
mixed
logical
dened
(upper
frequency
40,
40
from
as:
curve
is
<
x
≤
against
45,
the
practice
to
you
will
represent
be
a
logical
For
discrete
≤
x
≤
40,
data,
41
≤
for
x
≤
data
4 Representation
on
with
the
horizontal
the
vertical
axis,
axis.
example:
plot
the
upper
cumulative
class
boundaries
example:
45,
against
plot
thepoint
the
cumulative
halfway
between
to
looking
given.
graph
the
class
interval).
2.
upper
bound
bound
ofone
(41),
at
str ucture.
specically
the
highest
calls
45.
Communicating
using
of
a
on
frequencies
for
grouped
lowest
plotted
data,
a
bound
of
inter val
mean.
information
structure
1 18
range
bound
continuous
lower
Objective:
is
cumulative
<
the
calculate
the
boundaries
frequencies
In
is
and
35
v.
mean
of
●
●
the
(lower
frequencies
table:
midpoint
table
–
class
the
35
a
for
cumulative
upper
40
●
data
a
frequency.
data,
contains
which
widths
interval
highest
frequency
To
class
class
the
For
is
the
interval.
When
is
and
at
organizing
information
using
40.5.
(40)
and
next
S TAT I S T I C S
Mixed
1
The
All
290
of
35
masses
455
desert
are
342
to
hedgehogs
the
465
nearest
480
are
b
listed
Estimate
of
gram.
400
c
500
fox
460
328
284
436
280
the
tails
Given
and
325
in
that
the
interquartile
the
the
shortest
longest
368
295
310
390
435
450
315
505
510
495
310
400
375
4
The
of
474
298
380
37
Construct
a
grouped
represent
the
463
State
modal
frequency
the
plot
frogs
at
a
zoo.
Determine
table
curve
2
The
table
hazard
the
an
shows
class
interval
learner
10
20
that
contains
for
the
mean
drivers’
marks
mass.
in
5
The
speeds
highway
a
for
30
c
3
m
≤
27
9
28
≤
m
≤
39
21
40
≤
m
≤
51
18
52
≤
m
≤
63
23
64
≤
m
≤
75
19
the
Construct
The
tails
of
a
a
a
range
of
the
diagram
were
cumulative
cumulative
random
measured
a
the
the
masses
cumulative
data.
40
50
60
70
80
90
100
(grams)
of
cars
passing
recorded
in
a
this
speed
camera
on
v
(km/h)
Number
of
cars
frequency
frequency
sample
in
cm.
in
the
≤
60
0
<
v
≤
70
10
70
<
v
≤
80
22
80
<
v
≤
90
61
90
<
v
≤
100
74
100
<
v
≤
110
71
110
<
v
≤
120
39
120
<
v
≤
130
17
130
<
v
≤
140
6
of
200
table.
curve.
adult
The
results
Calculate
cumulative
frequency
an
estimate
for
the
mean
speed
cars.
curve.
b
Here
is
same
a
cumulative
frequency
table
for
data.
200
Speed,
Number
Cumulative
175
v
of
cars
frequency
ycneuqerf
evitalumuC
v
≤
60
0
0
v
≤
70
10
10
v
≤
80
22
32
v
≤
90
61
93
125
100
75
50
25
10
15
20
25
Length
a
(km/h)
150
5
Estimate
the
median
30
35
40
45
50
v
≤
100
74
a
v
≤
110
71
238
v
≤
120
39
b
v
≤
130
17
294
v
≤
140
6
300
x
(cm)
length
of
fox
of
are
y
0
a
table.
60
the
represented
shows
marks.
a
foxes
a
data.
solving
are
v
≤
Construct
the
Frequency
16
b
for
the
Speed,
Estimate
cm
test.
m
a
11
label
class.
estimate
perception
Mark,
was
and
to
Problem
Calculate
length
data.
median.
d
length
draw
Construct
Mass
c
the
360
0
b
cm,
box-and-whisker
80
frequency
a
for
370
450
450
range
sample.
box-and-whisker
347
P R O B A B I L I T Y
practice
masses
here.
A N D
tail
in
the
sample.
Write
down
the
values
of
a
4.2 Getting your ducks in a row
and
b
1 19
the
c
On
graph
paper,
frequency
curve
construct
to
a
represent
cumulative
this
b
information.
The
cumulative
lengths
of
the
frequency
diagram
shows
the
halibut.
y
d
25%
By
speed
the
the
speed
upper
on
this
grams,
graph
i
the
median
ii
the
upper
of
to
a
quartile,
stretch
frequency
limit.
of
curve
selection
estimate
highway.
shows
of
the
200
160
evitalumuC
Use
in
the
limit
cumulative
weights,
a
exceed
ycneuqerf
The
cars
calculating
the
6
of
oranges.
estimate:
120
80
40
0
5
quartile.
10
15
20
25
Length
Give
your
answers
to
the
nearest
10%
of
the
oranges
weigh
more
than
x
40
45
x
50
the
interquartile
range.
grams.
c
Find
35
gram.
Estimate
b
30
(cm)
The
sherman
returns
any
sh
smaller
than
x
20
cm
to
the
river.
Calculate
an
estimate
for
y
the
number
of
sh
he
returns.
80
d
Fish
greater
than
28
than
or
equal
to
20
cm
but
less
70
ycneuqerf
cm
are
classied
as
small
sh.
Fish
60
that
are
28
cm
or
longer
are
classied
as
50
large
sh.
evitalumuC
40
Estimate
the
number
of
sh
in
each
category.
30
e
The
sherman
sells
small
sh
for
$6
and
large
20
sh
for
$10.
Estimate
how
much
money
he
10
will
0
50
55
60
65
70
Weight
75
80
85
90
95
earn
if
sherman
catches
all
the
sh.
8
The
cumulative
frequency
curves
show
the
ages
(grams)
foreign
male
Switzerland
A
sells
x
of
7
he
200
halibut.
The
in
and
foreign
female
residents
in
2013.
table
y
shows
the
nearest
lengths
of
these
halibut
to
the
cm.
1100
Male
Length,
x
(cm)
Frequency
1000
10
<
x ≤
15
24
15
<
x ≤
20
38
20
<
x
≤
25
53
25
<
x
≤
30
39
30
<
x
≤
35
29
35
<
x
≤
40
10
40
<
x
≤
45
7
900
Female
ycneuqerf
800
700
evitalumuC
600
500
400
300
200
a
Calculate
the
an
estimate
for
the
mean
length
of
100
halibut.
0
10
20
30
40
50
60
70
80
90
100
Age
By
constructing
plot,
and
12 0
4 Representation
compare
foreign
a
double
the
female
ages
box-and-whisker
of
the
residents.
foreign
male
x
S TAT I S T I C S
ATL
Review
in
A
marine
in
a
local
records
trout
and
biologist
river.
the
from
To
lengths
the
is
sustainability
studying
the
of
rainbow
nearest
a
trout
centimeter,
sample
of
100
d
Copy
and
this
she
75%
of
over
____
trout
The
the
rainbow
other
rainbow
cm
in
x
(cm)
trout
species,
lengths
of
<
x
≤
27
1
of
27
<
x
≤
29
5
29
<
x
≤
31
9
other
These
so
100
feed
the
31
<
x
≤
33
21
33
<
x
≤
35
29
35
<
x ≤
37
19
37
<
x
39
16
Male
0
i
≤
female
species,
are
sh
records
and
100
of
the
male
sh
shown
in
cumulative
frequency
the
the
same
river.
box-and-whisker
sh
4
8
12
16
20
24
28
table
32
Estimate the number of
Estimate
the
36
40
44
48
52
female sh that are
the rainbow trout in the river
.
number
of
male
sh
that
are
for
than
75%
iii
a
cumulative
frequency
Hence
of
the
rainbow
trout.
the
cumulative
frequency
estimate
curve
to
write
other
summary
Reect
and
How
you
have
for
the
species
in
percentage
the
river
that
of
the
75%
sh
of
the
a
rainbow
ve-point
the
curve.
of
trout
can
feed
on.
data.
discuss
explored
the
statement
of
inquiry?
Give
specic
examples.
Statement of Inquiry:
How
and
quantities
trends
in
a
are
represented
can
help
to
establish
underlying
56
(cm)
data.
Draw
Use
smaller
sh
smaller
c
sh
from
smaller than 75% of
ii
a
on
biologist
Length
b
are
diagrams.
Female
this
trout
length.
Frequency
25
Construct
statement:
rainbow
e
Length,
complete
river.
Rainbow
a
P R O B A B I L I T Y
context
Globalization
1
A N D
relationships
population.
4.2 Getting your ducks in a row
1 21
How
did
that
happen?
4.3
Global
context:
Objectives
●
Drawing
a
Identities
Inquiry
scatter
diagram
for
bivariate
●
data
F
●
Drawing
●
Understanding
a
line
of
best
t
(regression
line)
by
interpreting
the
questions
How
SPIHSNOITA LER
●
Using
sets
of
technology
lineofbest
can
you
describe
between
two
sets
of
How
you
draw
a
relationship
data?
do
and
use
a
line
of
correlation
best
betweentwo
relationships
eye
●
and
and
t?
data
to
obtain
the
equation
of
●
a
C
t
How
does
aect
●
Does
●
Do
our
the
way
ability
correlation
to
data
is
make
indicate
represented
predictions?
causation?
D
ATL
Information
Understand
and
use
help
to
and
clarify
want
to
be
like
everybody
else?
literacy
technology
systems
4.3
Statement of Inquiry:
Generalizing
I
representing
trends
among
relationships
can
individuals.
12.3
E4.1
12 2
S TAT I S T I C S
Y
ou
●
should
represent
on
the
already
data
by
Car tesian
know
how
plottingpoints
1
table
dierent
Plot
to
shows
calculate
the
mean
of
a
set
of
data
2
on
represent
identify
an
outlier
in
a
set
of
data
this
1,
Introduction
F
●
How
can
●
How
do
Investigating
Ifyou
can
valueof
the
you
a
Exploration
table
The
maximum
shows
to
a
and
relationship
use
between
relationship
variable
The
draw
predict
6,
(£)
5
4
2.1
1.70
4.5
3.60
3.8
2.90
2
1.60
2,
mean
4,
6,
8,
2,
of
7,
whether
can
be
Explain
6,
Price
8,
9,
1,
9,
any
set
4,
of
of
6,
2
these
classied
your
9,
this
as
answer.
11,
15,
20
correlation
describe
relationship
identify
one
you
to
6,
(kg)
the
plane
data.
2.70
values
outliers.
potatoes.
Car tesian
3.4
Determine
data
of
of
0.40
Calculate
3
a
cost
0.5
numbers:
●
the
weights
points
Weight
●
a
line
variables
between
the
of
value
two
of
between
best
is
a
sets
of
data?
t?
key
topic
variables,
the
two
you
in
Statistics.
can
often
use
the
other.
1
12
students’
possible
marks
score
on
for
each
Mathematics
test
was
and
Physics
tests.
8.
Student
A
B
C
D
E
F
G
H
I
J
K
L
Mathematics
8
6
7
5
4
8
5
6
7
7
3
2
Physics
8
7
6
4
4
7
5
5
7
6
3
3
1
Draw
scores
a
graph
on
the
For
(6,
2
example,
7),
and
Explain
between
so
with
Use
score,
for
on.
whether
the
Mathematics
y-axis.
(Mathematics
P R O B A B I L I T Y
to:
The
plane
A N D
a
Physics
score)
student
A,
Do
join
not
your
students’
scores
sensible
graph
plot
the
on
scale,
to
the
the
and
x-axis
label
represent
point
(8,
each
8).
and
the
For
Physics
axes.
Plot
student’s
student
points
scores.
B,
plot
points.
suggests
Mathematics
that
and
there
Physics
might
be
a
relationship
marks?
Continued
on
next
page
4.3 How did that happen?
12 3
3
Describe
patterns
4
5
The
teacher
also
got
Draw
a
a
line
the
on
When
as
Discuss
the
Correlation
the
one
a
on
for
in
the
in
Is
graph
close
the
is
Data
measure
the
relationship
of
the
two
there
are
the
association
as
to
is
the
two
the
these
one
subject
one
relationship
way
does
quantitative
and
the
bivariate
two
variables.
dependent
data
variables.
variable
decreases
increases,
there
x
of
the
The
the
is
correlation.
y
Positive
When
the
line.
called
other
negative
in
Which
between
the
y
mark
shows
x-axis
When
correlation.
whether
justied?
Physics.
variables
is
good
between
on
increases
increases,
a
claim
best
and
points
represented
from
got
this
that
Mathematics
Determine
class.
who
other.
your
data.
the
students
the
how
shows
y-axis.
is
in
variable
variable
other
positive
line
see
student
that
mark
marks
diagram
you
every
claimed
independent
variable
to
good
slope?
scatter
The
patterns
straight
between
A
any
apply
the
plotted
points,
closer
you
the
x
correlation
points
say
points
are
there
are
Negative
to
close
is
a
a
to
a
linear
straight
straight
line
correlation
line,
the
correlation
drawn
through
between
stronger
the
the
the
correlation
variables.
Strong
Moderate
correlation
correlation
12 4
4 Representation
No
middle
variables.
linear
correlation
between
Scatter
are
diagrams
also
called
scatter
plots
scatter
graphs.
or
S TAT I S T I C S
The
scatter
between
the
graph
higher
the
Practice
For
you
Mathematics
each
Physics
drew
and
in
Exploration
Physics
scores.
1
shows
The
strong
higher
the
linear
A N D
P R O B A B I L I T Y
correlation
Mathematics
score,
score.
1
scatter
plot,
describe
the
form,
direction
and
strength
of
the
‘Form’
association
between
the
variables.
Interpret
your
description
in
the
or
ofthe
is
linear
context
non-linear.
data.
‘Direction’
1
a
b
c
y
erocS
x
Height
Amount
e
gnitaeH
fo
thgieW
line
of
of
so
can
a
best
t
that
best
x-values
draws
ytilibaborP
llib
rac
line
points
a
is
a
the
line
line,
of
line
equally
through
mean
best
called
x
Temperature
straight
are
passes
and
draw
is
they
t
similar
x
mileage
of
t
the
x
revision
y
y
Gas
of
f
y
You
ni
eohS
t rA
d
is
negative.
tset
ezis
erocs
x
score
or
y
y
Mathematics
A
positive
drawn
through
distributed
the
the
by
on
point
the
either
middle
side
where
x
Height
of
is
of
the
the
a
set
line.
mean
of
data
The
of
the
y-values.
eye.
A
can
line
also
trend
GDC
regression
The
line,
or
by
graph
plotting
calculating
the
of
be
best
line.
software
points
mathematically.
You
can
x-value,
use
and
of
the
vice
line
of
best
t
to
predict
a
y-value
when
you
know
an
versa.
4.3 How did that happen?
t
called
12 5
a
Example
The
table
1
shows
dailysales
of
ice
Temperature,
Sales,
a
y
x
the
average
cream
(°C)
($)
Draw
a
in
daily
temperature
in
°C
for
a
week
and
the
dollars.
Mon
Tue
Wed
Thu
Fri
Sat
Sun
17
22
18
23
25
19
22
408
445
421
544
614
412
522
scatter
diagram
line
best
of
this
data
and
draw
a
line
of
best
t
through
the
points.
b
Use
the
temperature
of
t
to
predict
the
sales
of
ice
cream
on
a
day
when
the
is20°C.
a
Calculate
the
mean
Calculate
Ice
cream
temperature
the
mean
.
sales
.
sales
y
Plot
the
points
from
the
table
and
650
the
point
Draw
a
line
.
through
M
and
through
600
the
middle
of
the
other
points.
550
)$(
Tip
selaS
Use
a
transparent
500
ruler
so
you
can
see
M
all
the
and
points
below
it.
above
Pivot
450
the ruler through the
point
the
to
best
nd
position.
x
17
18
19
20
21
Temperature
22
23
24
25
°C
Draw
b
When
the
temperature
is
20°C,
predicted
sales
are
and
12 6
4 Representation
a
line
from
x
=
20
$460.
read
o
the
y
value.
S TAT I S T I C S
To
draw
●
Calculate
●
Plot
●
a
line
the
Draw
the
best
t
mean
by
of
line
The
x
and
on
the
the
through
points
P R O B A B I L I T Y
eye:
point
a
points.
of
A N D
scatter
and
should
be
mean
of
diagram.
through
evenly
y
the
middle
distributed
above
of
the
and
other
below
theline.
Practice
1
The
2
table
shows
●
Reaction
●
Height,
●
Weight,
●
Time,
Reaction
Height
in
seconds,
time
Draw
separate
Plot
height
ii
Plot
weight
iii
Plot
iv
Plot
height
Comment
1.68
Use
took
line
visual
responses
of
on
the
shoelaces.
E
F
G
H
I
J
5.3
4.6
7.7
3.7
4.6
3.6
4.2
4.8
5.3
4.6
1.64
1.50
1.67
1.75
1.45
1.35
1.87
1.67
1.56
1.70
60
60
64
76
48
45
75
75
52
60
9.1
10.5
13.4
9.6
10.5
9.2
8.8
8.5
10.4
6.3
scatter
the
diagrams:
x-axis
the
best
their
D
and
x-axis
distance
on
fasten
C
the
on
to
on
x-axis
t
on
form,
and
the
on
reaction
x-axis
and
each
weight
direction
and
y-axis.
distance
and
fastening
scatter
the
on
fastening
time
on
the
time
the
y-axis.
on
the
y-axis.
y-axis.
diagram.
strength
of
the
correlation
in
each
diagram.
the
m
on
reaction
c
Use
measure
B
(cm)
i
scatter
to
A
(s)
four
a
test
taken
(kg)
Draw
f
a
students:
kilograms
b
e
in
10
meters
distance
Fastening
d
on
(m)
Weight
a
distance,
in
in
data
line
of
best
t
to
estimate
the
weight
of
of
best
t
to
estimate
the
reaction
a
person
who
is
tall.
the
12
Explain
line
seconds
which
to
of
fasten
your
their
graphs
distance
of
a
student
who
shoelaces.
would
give
the
most
accurate
predictions.
4.3 How did that happen?
12 7
2
The
table
shows
information
on
road
deaths
and
vehicle
ownership
in
ten
countries.
Vehicles
per
100
Road
deaths
per
100 000
Countr y
population
population
UK
31
14
Belgium
32
29
Denmark
30
22
France
47
32
Germany
30
25
Ireland
19
20
Italy
36
21
Netherlands
40
22
Canada
47
30
USA
58
35
a
Draw
and
a
scatter
road
show
any
b
Draw
c
The
a
line
road
per
of
best
death
of
per
t
data
100
road
to
●
vehicles
per
population
for
How
does
deaths
the
represent
the
Switzerland
population.
Interpreting
C
with
100 000
100
on
population
the
y-axis.
on
Does
the
the
x-axis
graph
correlation?
25vehicles
number
diagram
deaths
per
Use
100 000
patter n
is
in
missing.
your
line
data.
Switzerland
of
population
the
best
in
t
to
has
estimate
Switzerland.
correlation
way
data
is
represented
aect
our
ability
to
make
predictions?
You
can
use
a
GDC,
graphing
software
or
a
spreadsheet
to
plot
scatter
ATL
diagrams,
models
add
the
a
the
process
value
of
line
of
best
relationship
Technology
nds
a
uses
vertical
called
the
linear
t
squares’
to
of
to
produce
the
regression
displacements
‘least
and
between
independent
an
equation
nd
the
the
points
a
line
line
of
from
that
best
the
will
y
P
6
1
4
P
5
3
P
1
x
1
2
1
12 8
the
4 Representation
3
4
This
and
predict
2
P
2
t.
line
x
P
for
line
that
variables.
nd
variable
P
the
y
method
then
for
a
uses
given
S TAT I S T I C S
Each
the
square
line.
for
the
try
to
the
in
The
sum
nd
of
the
squares
table
diagram
squares
these
line
and
Example
The
the
least
squares.
of
the
square
t
nds
With
some
yourself
by
the
of
sum
of
the
the
vertical
line
that
GDCs
or
plotting
a
their
distance
gives
the
graphing
line
from
a
point
minimum
software
through
(x ,
to
value
you
y ),
P R O B A B I L I T Y
can
showing
areas.
2
a
Use
b
Comment
c
Use
the
technology
on
weight
to
the
graph
Name
best
minimizing
gives
the
is
method
A N D
to
Height
and
draw
form,
a
scatter
direction
estimate
(cm)
height
the
of
students.
diagram
and
weight
Weight
10
and
strength
of
(kg)
a
162
of
cm
Name
Alex
168
77
Frida
nd
the
the
line
of
best
t.
correlation.
tall
student.
Height
(cm)
Weight
159
(kg)
87
Bea
149
45
George
195
94
Chip
173
65
Harr y
144
53
Dave
166
59
Inge
176
65
Erin
185
79
Jepe
180
87
a
1.1
1.2
Example
1.3
RAD
2
95
80
thgieW
User your GDC to create a line
65
of best t from the data.
50
y
140
150
160
=
0.7445 •
170
x
+
180
−55.09
190
Height
b
There
the
c
is
moderate
heights
Using
the
and
Practice
linear
correlation
between
weights.
graph,
approximately
positive
a
65
height
of
162
cm
would
correspond
to
a
weight
of
kg.
3
ATL
1
The
table
gives
the
height
and
weight
of
14
adult
mountain
zebras.
Height
x
1.37
1.17
1.2
1.34
1.42
1.42
1.37
1.48
1.51
1.23
1.57
1.29
1.30
1.44
257
171
185
214
315
271
242
329
314
185
356
228
230
285
(m)
Weight
y
(kg)
a
Use
b
Find
c
Comment
d
Use
with
technology
the
the
to
equation
on
line
height
the
of
draw
of
scatter
line
correlation
best
1.38
a
the
t
to
of
diagram.
best
t.
between
predict
the
the
height
weight
of
and
an
weight
adult
of
the
mountain
zebras.
zebra
m.
4.3 How did that happen?
12 9
2
There
leg
Leg
x
are
to
run
girls
1000
in
an
athletics
club.
The
table
shows
their
m.
82
91
85
93
86
95
70
87
90
75
91
80
245
210
243
220
240
225
250
243
230
248
220
240
run
a
Using
b
Find
c
Comment
the
The
table
cost
of
y
of
Plot
b
Find
machine
the
c
Use
your
could
to
when
●
Do
I
length
and
the
to
1000
time
taken
to
●
Measurement
●
Clerical
error
the
time
r un
m
for
student
a
cm.
of
the
cylinder
purchased
new
at
in
the
a
washing
same
machine
and
the
shop.
52
55
62
66
230
350
400
550
550
600
620
700
on
of
a
scatter
of
the
best
t
of
diagram.
line
to
50
of
best
estimate
t.
the
cost
of
a
washing
machine
with
cm.
predictions
want
to
predict
84
correlation
arise
due
leg
48
diameter
Does
be
t.
between
41
line
●
can
t
diameter
equation
cylinder
outlier
diagram.
best
36
Making
D
of
33
points
the
scatter
line
correlation
best
the
($)
a
a
the
measurement
shows
the
the
of
m.
x (cm)
Cost,
a
on
line
leg
draw
equation
1000
Use
Size,
technology
,
the
with
It
to
length
r un
An
15-year-old
time
(s)
d
3
and
(cm)
Time
y
twelve
length
to
in
a
be
like
variety
human
error
indicate
error,
(wrong
(writing
the
causation?
everybody
of
else?
ways.
categorized
by:
units)
results
incorrectly
or
misunderstanding
the
question)
●
Instrumental
However,
In
4.1
you
whether
If
you
think
An
it
not
think
it
is
could
used
or
a
at
the
13 0
the
to
the
be
is
(faulty
a
IQR
outlier
an
to
is
due
point,
of
point
determine
the
anomaly
distance
instruments)
legitimate
include
legitimate
outlier
look
error
the
outliers
to
it
in
which
the
in
human
must
the
data
be
4 Representation
outliers;
the
data
error,
kept
data.
point
caused
To
these
now
it
is
results.
your
decision
set.
remove
the
data
point;
if
you
in.
identify
from
the
outliers
line
of
best
in
bivariate
t.
data,
S TAT I S T I C S
Exploration
The
Olympic
medal
Year
P R O B A B I L I T Y
2
Games
distance
A N D
for
are
the
held
men’s
every
long
four
jump
years.
event
The
over
table
a
shows
period
of
the
gold
28years.
1956
1960
1964
1968
1972
1976
1980
1984
7.83
8.12
8.07
8.90
8.25
8.35
8.54
8.54
Use
‘number
of
Long
jump
years
after
(m)
onthe
(0,
1
Use
a
line
GDC
of
best
2
Describe
3
One
or
t
the
jump
graphing
for
the
software
draw
a
scatter
diagram
and
nd
x-axis.
8,
…)
the
data.
relationship
stands
to
4,
out
that
from
the
the
graph
others.
shows.
Is
it
much
further
from
the
line
of
Discover
best
t
than
the
other
points?
Is
it
an
How
are
does
based
this
outlier
aect
the
line
of
best
t
and
any
predictions
that
onit?
by
Find
the
line
of
best
t
without
the
anomalous
jump.
Use
this
line
the
gold
medal
long
jump
distance
for
that
Olympic
gold
for
scatter
but
it
graph
does
other
not
it
is
temperatures.
that
a
a
relationship
change
example,
over
the
a
fall
in
the
in
the
between
one
of
of
years
number
cause
and
two
variable
number
period
Demonstrating
linear
Reect
that
For
measured
unlikely
astrong
show
show
(causation).
temperature
But
may
have
of
eect
a
a
and
has
much
(correlation),
change
global
negative
pirates
is
variables
causes
pirates
this
in
the
average
correlation.
caused
harder
a
Bob
year,
In
video
called
Long
Olympic
Shadow
New
by
York
The
Times
higher
than
showing
correlation.
and
discuss
Exploration
You
1
cannot
2,
yo u
used
the
l ine
of
be st
t
to
p r e dic t
a
that
the
v al ue
relationship
within
in
search
Beaman’s
know
●
long
medal
year.
particular
A
jump
to
for
estimate
this
researching
the
jump
5
more
outlier?
about
4
1956’
the
range
of
d a ta
you
were
give n.
W as
it
appr opr iat e
you
to
can
see
between
dothis?
●
The
t
modern
to
predict
jumped
in
Olympic
Predicting
In
the
the
values
began
winning
original
Games
interpolation.
iscalled
Olympics
100
jump
Greek
years
inside
the
Predicting
in
in
1896.
in
this
Could
year?
Olympic
the
range
values
you
What
Games
use
your
about
1500
line
the
years
of
best
distance
earlier,
or
of
the
given
the
data
still
hold
or
lower
values
sets
for
will
higher
data
than
those
given.
future?
outside
two
data
range
is
of
called
the
given
data
extrapolation.
general,
interpolation
is
more
likely
to
give
an
accurate
prediction
than
extrapolation.
4.3 How did that happen?
13 1
Exploration
The
table
period
same
shows
and
time
Visitors
to
Revenue
the
number
revenue
of
from
visitors
to
registered
the
nail
Bahamas
salons
in
for
the
a
seven-year
US
during
the
period.
Bahamas
salons
the
3
2008
2009
2010
2011
2012
2013
2014
4.6
4.6
5.3
5.6
5.9
6.2
6.3
6.3
6.0
6.2
7.3
7.5
8.2
8.5
with
visitors
the
(millions)
from
nail
(billions
of
USdollars)
1
Draw
a
scatter
Bahamas
2
Describe
3
Draw
when
4
a
on
the
line
there
Does
it
3
of
the
factor,
You
an
5
to
t
that
and
the
in
use
it
make
from
the
to
visitors
the
there
is
of
other.
can
an
nail
on
the
to
the
y-axis.
variables.
the
from
revenue
the
occur
from
are
but
both
line
of
between
association
salons,
They
salons
nail
salons
Bahamas.
predictions
revenue
the
two
results,
nail
predict
to
relationships
Obviously
increase
these
revenue
between
and
to
illustrate
and
million
sense
shows
Bahamas
notcause
best
are
obscurevariables.
to
x-axis
correlation
make
Exploration
diagram
the
best
the
between
an
t?
most
the
increase
dependent
on
visitors
in
one
does
another
time.
cannot
Practice
use
correlation
to
show
that
two
variables
are
related
to
each
other.
4
ATL
1
The
average
consecutive
a
Use
prices
years
of
are
technology
to
a
liter
given
draw
of
milk
in
January
in
the
UK
tabulated
for
six
here.
a
scatter
diagram
and
line
of
best
t.
Month
Put
at
the
zero.
dependent
Then
24months,
b
c
Use
the
i
July
2009
ii
July
2012
iii
January
In
January
13 2
January
on
2010
is
the
12
x-axis,
with
months,
Pence
per
January2009
January
2011
is
Jan
2009
23.73
Jan
2010
24.67
Jan
2011
27.36
Jan
2012
28.08
Jan
2013
31.64
Jan
2014
31.52
etc.
line
Compare
variable
of
best
t
to
estimate
the
average
price
of
milk
in:
2015
2015,
this
to
the
average
your
price
prediction.
4 Representation
of
milk
Explain
was
any
24.46
pence
dierences.
per
liter.
liter
S TAT I S T I C S
2
Discuss
logical
a
what
a
c
period
human
strong
b
The
and
correlation
screen
medium
was
Between
you
the
Arm
Height
Use
and
Use
c
the
of
Discuss
in
any
Group
Group
1
between
the
Labradoodles
annual
in
number
Denmark
was
amount
and
the
of
time
quality
a
of
person
sleep
spent
that
looking
they
at
reported
a
that
there
number
oenders
was
TV
sent
have
and
of
to
span
strong,
positive
bought
in
linear
the
UK
and
prison.
longer
arm
a
licenses
arms
for
10
than
short
people?
students.
170
177
177
188
196
200
182
162
180
166
167
173
176
182
184
186
to
draw
a
y-axis.
is
t
diagram
the
predict
line
of
with
best
(interpolate)
arm
span
on
the
x-axis
t.
the
height
of
a
student
cm.
outliers?
Use
scatter
Find
to
171
If
this
so,
new
remove
line
to
them
predict
from
the
the
list
height
and
of
a
nd
the
student
new
whose
cm.
dierences
sets
2
2
grams.
between
b
and
c
drawing
Here
line
data
a
number
of
pins
and
24
28
29
31
33
5
7
11
10
14
15
16
9
11
20
two
groups
total
Use
scatter
that
and
which
a
weight
pins.
best
to
calculate
of
the
this
pins
on
the
Use
the
mean
drawing
method
diagram.
of
their
data:
24
a
of
the
20
to
weight
pins
technology
x-axis
and
calculate
and
to
of
drawing
divides
the
by
mean
draw
weight
a
on
a
pin.
the
weight.
scatter
the
y-axis.
t.
two
nd
drawing
weight
record
17
number
decide
their
and
13
the
of
is
pins
7
in
draws
with
of
of
drawing
Determine
mean
of
165
171
nds
unreliable
weight
the
people
best
work
of
the
Group
correlation
160
pins
students
Find
d
draw
156
(g)
diagram
c
you
solving
of
number
b
of
t.
is
the
weight
Weight
a
Can
155
the
any
weigh
Number
The
show.
negative.
the
height
span
best
span
Students
on
line
line
total
lists
arm
there
the
2005–2015
tall
technology
height
Problem
4
that
(cm)
Are
arm
d
below
population
day
and
young
(cm)
whose
each
years
of
here
span
the
between
think
table
years,
between
night
The
b
10
and
computer
Do
analyses
positive.
thenumber
a
of
births
correlation
3
correlation
P R O B A B I L I T Y
conclusions?
Over
of
the
A N D
of
a
the
new
weights
line
of
are
best
unreliable.
t.
Use
it
to
Remove
nd
the
the
mean
pin.
group’s
drawing
method
gives
the
most
reliable
estimate
of
the
pin.
4.3 How did that happen?
13 3
Summary
A
scatter
two
diagram
quantitative
variable
is
dependent
two
shows
variables.
represented
variable
variables
is
the
on
on
called
relationship
The
the
the
between
x-axis
bivariate
and
Data
line
the
independent
y-axis.
A
equally
the
line
from
of
between
is
two
increases
as
correlation.
a
measure
variables.
the
other
When
of
the
When
one
variable
of
t
a
is
set
distributed
t
is
of
straight
data
on
passes
the
the
a
of
line
either
of
the
so
side
through
mean
drawn
points
the
that
of
the
through
they
line.
are
The
point
x-values
and
is
the
y-values.
association
one
increases,
best
best
where
data
mean
Correlation
of
middle
variable
there
is
To
draw
a
line
●
Calculate
of
best
t
by
eye:
positive
decreases
as
the
mean
of
x
and
the
mean
of
the
y
other
increases,
there
is
negative
correlation.
●
Plot
●
Draw
the
point
on
the
scatter
diagram.
y
y
a
middle
be
line
of
evenly
through
the
other
M
and
points.
distributed
through
The
above
the
points
and
should
below
the
line.
An
outlier
outliers
Positive
correlation
Negative
closer
the
points
are
to
a
straight
in
line,
data
point
the
anomaly
correlation
between
the
data,
in
the
look
data.
at
the
To
identify
distance
from
the
line
of
best
of
t.
the
Predicting
stronger
an
bivariate
correlation
the
The
is
x
x
values
inside
the
range
of
the
given
variables.
data
is
called
outside
the
interpolation.
range
extrapolation.
more
likely
to
In
of
the
given
general,
give
an
Predicting
data
is
values
called
interpolation
accurate
is
prediction
than
extrapolation.
Strong
Moderate
correlation
correlation
Mixed
1
The
linear
practice
table
leaves
No
correlation
gives
that
fell
the
to
length
the
and
width
of
10oak
a
ground.
length
b
Length
(mm)
Width
Represent
Comment
38
c
By
width
146
44
119
38
149
53
a
Two
of
contain
on
a
and
scatter
width
correlation
diagram,
on
the
with
y-axis.
between
length
mean
best
length
and
mean
width,
t.
solving
these
a
of
leaves.
statements
mistake.
about
Explain
correlation
what
the
mistakes
are
36
and
38
a
how
the
‘There
is
income
151
51
147
43
123
33
128
42
b
‘There
of
c
a
high
statement
strong
objects
a
two
and
the
and
1999
time
the
between
correlation
the
correct.
gender.’
number
between
stations.’
be
between
their
correlation
and
positive
speed
between
could
correlation
workers
America
is
average
4 Representation
a
hot
‘There
third
of
is
Miss
using
13 4
the
the
the
line
Problem
2
135
data
x-axis
on
of
nding
draw
89
the
the
(mm)
and
103
on
of
the
age
murders
and2009.’
between
taken
by
a
the
train
S TAT I S T I C S
Objective:
ii.
select
D.
Applying
appropriate
mathematics
mathematical
in
real-life
strategies
A N D
P R O B A B I L I T Y
contexts
when
solving
authentic
real-life
situations
In
this
review
diagrams,
outliers,
you
using
in
in
Identities
Florence’s
seemed
height
Age
to
use
the
extract
mathematical
information
strategies
and
of
drawing
determining
the
scatter
reasons
for
context
relationships
parents
short
for
a
to
situations.
and
over
have
technology
real-life
Review
1
will
were
her
36-month
(months)
concerned
age
and
they
that
she
recorded
2
her
period:
36
48
51
58
66
72
86
90
91
93
94
95
There
10km
are
12
run.
of
students
The
of
hours
to
complete
training
the
Training
Height
(cm)
table
training
shows
per
for
the
week
Draw
b
Use
c
a
scatter
this
diagram
scatter
height
at
Could
you
diagram
age
42
use
to
of
diagram
this
to
charity
and
the
number
time
taken
run.
time
Time
to
(hours)
a
a
average
complete
run
(minutes)
data.
estimate
10
56
9
54
13
53
her
months.
this
predict
data
her
and
the
height
at
scatter
age
4
62
26
42
7
66
18?
a
11
54
6
66
7
68
22
39
5
70
10
67
Comment
training
on
time
the
and
correlation
between
the
complete
time
to
therun.
b
Estimate
student
c
Determine
to
predict
student
Reect
and
How
have
Give
specic
you
how
who
long
trains
the
18
whether
how
who
long
trained
run
would
hours
you
the
50
per
could
run
take
use
would
hours
a
week.
per
this
data
take
week.
discuss
expl o red
the
statem ent
of
i nq uiry?
exam pl es.
Statement of Inquiry:
Generalizing
to
clarify
and
trends
representing
among
relationships
can
help
individuals.
4.3 How did that happen?
13 5
a
What
are
the
chances?
4.4
Global
context:
Objectives
●
Representing
Identities
Inquiry
sample
spaces
in
tables,
lists
●
and
and
relationships
questions
What
are
the
dierent
ways
of
F
diagrams
●
Drawing
two-way
CIGOL
●
representing
tree
Calculating
and
diagrams,
Venn
diagrams
●
and
tables
probabilities
two-way
from
Venn
diagrams
●
How
do
of
event?
an
What
a
you
are
the
calculate
Using
with
tree
and
diagrams
without
Understanding
ATL
the
advantages
probability
to
calculate
probability
probabilities
of
the
and
dierent
representations?
replacement
●
●
space?
C
tables
disadvantages
●
sample
informal
ideas
of
randomness
Does
randomness
aect
the
decisions
D
we
make?
Communication
Understand
and
use
mathematical
notation
4.4
Statement of Inquiry:
Understanding
using
logical
health
and
making
representations
and
healthier
choices
result
from
systems.
15.1
E15.1
13 6
S TAT I S T I C S
Y
ou
●
should
calculate
already
know
simple
how
1
This
probabilities
blue
d
set
the
notation,
specically
complement
of
a
2
is
or
blue
not
a
Using
set
Let
c
diagram
3
is
the
on
yellow
set
the
5,
notation,
numbers
6,
7,
8,
Write
In
a
like
7
down
group
of
children
n(P)
a Venn
핌
=
Write
and
set
P
of
10.
{1,
2,
down
3,
the
n(P′).
children,
16
like
set
10}
the
than
P′.
24
bananas,
Draw
9,
of
write
less
Universal
elements
a Venn
once. What
lands
yellow?
prime
4,
draw
it
white
or
e
b
●
spun
that
yellow
c
use
spinner
blue
b
●
P R O B A B I L I T Y
to:
probability
a
A N D
of
them
10
of
like
them
apples
and
both.
diagram
to
represent
this
information.
Representing
F
Th
What
●
How
mathmatis
looking
and
●
for
ar
do
you
as
1
A
in
nw
rstaurant
prson
insid
pays
that
ways
alulat
vnts
vnts
th
of
proaility
involving
ar
rprsnting
rpatd
of
han
you
is
will
an
a
sampl
spa?
vnt?
alld
gin
Proaility.
to
s
how
By
to
prdit
proailitis.
Exploration
a
dirnt
hind
pattrns
alulat
th
outcomes
thir
ithr
town
ill,
says
runs
thy
‘Your
a
ar
promotion
givn
mal
a
today
to
gnrat
sald
is
usinss.
nvlop
fr’
or
with
‘Sorry,
a
Bfor
mssag
ttr
luk
nxttim’.
Th
rstaurant
During
and
a
thr
ownrs
on-hour
of
thm
laim
priod
gt
a
that
on
th
th
winning
han
rst
day,
of
a
ight
winning
mssag
ustomrs
pay
is
1
thir
in
6.
ill,
mssag.
Th
To
tst
that
th
rstaurant’s
laim
is
orrt,
you
an
st
up
a
1
Assum
that
thy
ar
orrt
and
a
winning
mssag
indd
appars
in
of
vry
6
sald
nvlops.
Th
proaility
of
winning
in
is
thrfor
6
is
tru
numr
as
rolling
1
Roll
a
di
2
Rord
8
a
six
tims
on
to
a
fair
numr
simulat
of
6s
di;
a
you
di.
8
popl
paying
thir
ills.
ould
rord
4s,
long
as
wr
th
for
on
th
standard
sam
of
1
any
out
proaility
simulation.
you
as
3s
or
you
onsistnt.
roll.
Continued
on
next
page
4.4 What are the chances?
13 7
3
Rpat
this
pross
30
tims.
Comin
th
rsults
for
othrs
ina
Your
GDC
othr
softwar
or
spradsht.
run
Basd
on
th
omind
rsults
of
th
simulation:
that
4
Calulat
th
prntag
of
outoms
whr
8
ills
paid
rsultd
prform
in
th
3
xprimnt
winnrs.
thousands
5
Calulat
th
prntag
of
outoms
with
no
vn
winnrs.
Ddu
th
most
ommon
numr
of
winnrs
in
8
ills
Try
Disuss
whthr
Do
think
you
Reect
In
●
and
Exploration
Asid
from
simulat
●
What
Simulation
prdit
is
onstruting
Th
sampl
outoms
Whn
you
of
di,
tool
happn
spa
a
who
had
printd
3
too
winnrs
many
wr
just
winning
with
luky.
in
mssags?
1
of
is
a
1
for
winning
8
futur.
th
ar
spa
an
two
using
is
th
You
of

possil
st
or
hav
usd
to
6?
likly?
also
nd
of
vnts
and
proailitis
y
situation.
omplt
list,
a
tal
outoms:
notation
you
proailitis
an
th
a
in
most
xprimnt
It
ould
mssag
popl
rprsntation
thr
sampl
way(s)
invstigat
th
xprimnt.
oin,
othr
from
to
in
spa
S,
an
what
winnrs
powrful
from
th
a
proaility
sampl
toss
Dsriing
rolling
might
a
popl
1:
th
a
8
rstaurant
discuss
numr
what
th
th
givs:
st
or
a
of
=
all
possil
diagram.
Hads
S
{H,
(H)
and
Tails
(T).
T}.
ATL
H
and
For
T
any
ar
sampl
outoms
In
A
in
Exampl
ours
So
P(A)
th
or
+
only
spa
th
2,
it
S,
sampl
A
and
dos
P(A′)
possil
=
A′
P(S)
outoms
=
spa
1,
aus
must
togthr
so
thy
for
ovr
ah
th
xprimnt
th
th
whol
sampl
P(S)
=
proailitis
in
Exampl
1
P(A′)
=
1
P(A)
=
rprsnts
proaility
of
A
2:
th
P(A)
+
P(A′)
5
+
=
not
=
1
6
6
proaility
of
spa.
th
aus
5
of
vnt
A
ourring.
ourring.
P(A)
13 8
spa,
=
1
6
P(A)
on
1
6
So
sampl
not.
1
Using
whol
our.
ovr
4 Representation
+
P(A′)
=
1
of
this
and
paid.
ompar
7
or
millions
tims.
6
an
simulations
P(A′)
is
th
ithr
th
your
rsults
ndings
Exploration
1.
S TAT I S T I C S
Example
Two
fair
P R O B A B I L I T Y
1
6-sidd
Rprsnt
A N D
all
di
th
a
a
list
b
a
tal
c
a
diagram.
a
S
=
ar
rolld.
possil
On
outoms
di
is
grn
and
th
othr
is
orang.
as:
{(1, 1), (2, 1), (3, 1), (4, 1), (5, 1), (6, 1), (1, 2), (2, 2), (3, 2),
(4, 2), (5, 2), (6, 2),
(1, 3), (2, 3), (3, 3), (4, 3), (5, 3), (6, 3),
As
(1, 4), (2, 4), (3, 4), (4, 4), (5, 4), (6, 4),
a
set
(1, 5), (2, 5), (3, 5),
(4, 5), (5, 5), (6, 5), (1, 6), (2, 6), (3, 6), (4, 6), (5, 6), (6, 6)}
b
1
2
3
4
5
6
1
(1, 1)
(1, 2)
(1, 3)
(1, 4)
(1, 5)
(1, 6)
2
(2, 1)
(2, 2)
(2, 3)
(2, 4)
(2, 5)
(2, 6)
3
(3, 1)
(3, 2)
(3, 3)
(3, 4)
(3, 5)
(3, 6)
4
(4, 1)
(4, 2)
(4, 3)
(4, 4)
(4, 5)
(4, 6)
5
(5, 1)
(5, 2)
(5, 3)
(5, 4)
(5, 5)
(5, 6)
As
Th
tal
most
is
a
th
ommon
rprsntation
sampl
6
(6, 1)
(6, 2)
(6, 3)
(6, 4)
(6, 5)
table
of
a
spa.
(6, 6)
c
As
Practice
Objective
iii
Mov
Make
in
1
all
sure
your
Th
roll
C:
Communiating
twn
you
list
dirnt
outcomes
representations
on
Rprsnt
suh
th
of
sample
gam
rsults
sampl
of
mathmatial
systematically
oard
di
forms
in
spa
a
for
and
include
rprsntation
all
the
possible
outcomes
spaces.
Daldøs
sor
of
rolling
uss
I,
4-sidd
II,
two
of
III
or
ths
(ttrahdral)
di.
Th
IV.
di
as
a
list
and
as
a
tal.
For
2
A
fair
oin
is
tossd
thr
th
sampl
spa
Q2,
mak
a
tims.
list
Dn
diagram
1
Sandinavian
of
a
for
this
xprimnt.
of
th
possil
outoms.
4.4 What are the chances?
13 9
3
Copy
and
rolling
omplt
two
4-sidd
this
sampl
di,
ah
spa
with
2nd
tal
fas
for
th
sum
numrd
1,
of
2,
th
3
sors
and
from
4.
die
In
2009
thrw
1
2
3
eid
a
ts1
2
strt
ar
–
slls
and
typs
typs
for
th
ttrahdral
di
(4-sidd)
ah
7
two
thr
spa
Eah
of
two
ox.
Thr
othr
two
multiply
th
a
a
of
th
v
sandwih
fruit
hois
a
v
to
2
togthr.
of
a
6.
a
di
th
a
draw
1
to
th
a
sampl
a
5.
pi
without
a
7,
world
stting
rord.
hans
1
in
1.56
of
this
trillion.
A
ard
to
or
a
th
fruit.
is
outoms.
drawn
from
outoms.
from
spa
of
possil
numrs
ard
List
rolld
th
possil
Gatsy
anana.
ar
rprsnt
ontains
Draw
Afrian
or
and
(6-sidd)
to
numrd
You
South
orang
sandwih
rprsnt
On
to
–
appl,
spa
ards
ards.
–
normal
sampl
spa
numrs
numrs
of
of
and
ontains
sampl
sts
ontains
possil
Construt
oxs
Draw
ar
di
7
simultanously.
6
of
5
vndor
Gyro
sampl
A
nw
Th
Grk
5
tims
throwing
4
A
woman
2
3
4
a
pair
4
154
1
a
0
to
ah
4
and
st
rprsnt
th
and
this
information.
Problem
8
Th
S
=
solving
sampl
{(1,
spa
H),
Dsri
(2,
th
Rprsnting
for
H),
an
(3,
xprimnt
H),
(4,
H),
is:
(1,
T),
(2,
T),
(3,
T),
(4,
T)}
xprimnt.
sampl
spas
allows
you
to
ount
spi
outoms.
Gams
hav
An
event
is
a
sust
of
th
possil
outoms
listd
in
th
sampl
of
n
han
playd
spa.
sin
anint
tims;
arhaologists
From
how
th
sampl
many
ways
spa
th
of
vnt
sors
on
‘gtting
th
th
two
sam
di
in
Exampl
numr
on
oth
1
you
di’
an
an
s
our.
disovrd
in
Turky
from
Us
apital
lttrs
for
vnts,
lik
A,
B,
C
and
ATL
A
=
gtting
th
sam
numr
on
oth
dn
th
vnt.
For
3000
14 0
of
vnt
4 Representation
dating
bce
and
xampl,
Mahjong
tils
from
bce
dating
di.
500
China.
Proaility
hav
di
in
S TAT I S T I C S
Example
You
roll
a
th
b
two
a
A
two
6-sidd
numr
dirnt
gtting
di.
on
Calulat
oth
th
proaility
of
gtting:
di
numrs.
th
sam
numr
on
oth
di.
Dene
1
2
3
4
5
6
1
(1, 1)
(1, 2)
(1, 3)
(1, 4)
(1, 5)
(1, 6)
2
(2, 1)
(2, 2)
(2, 3)
(2, 4)
(2, 5)
(2, 6)
3
(3, 1)
(3, 2)
(3, 3)
(3, 4)
(3, 5)
(3, 6)
Identify
same
4
(4, 1)
(4, 2)
(4, 3)
(4, 4)
(4, 5)
(4, 6)
5
(5, 1)
(5, 2)
(5, 3)
(5, 4)
(5, 5)
(5, 6)
6
(6, 1)
(6, 2)
(6, 3)
(6, 4)
(6, 5)
(6, 6)
A′
=
gtting
dirnt
numrs
on
th
two
the
set
Count
on
A′,
all
the
the
with
numbers
other
your
two
are
or
a
spas
from
Prati
1
to
answr
ths
Calulat
th
4-sidd
proaility
of
gtting
two
4s
whn
you
P(A′)
Two
th
4-sidd
proaility
of
not
gtting
two
4s
whn
you
roll
in
gams
ard
di.
ttrahdral
Calulat
rol-playing
and
di
ar
rolld
and
thir
sors
ar
th
proaility
of
gtting
a
sum
of
Calulat
th
proaility
of
gtting
a
sum
lin
hav
low
qual
to
A
fair
oin
a
thr
b
two
c
no
is
tossd
thr
tims.
Calulat
th
Th
and
proaility
a
9
dot
thm
what
or
to
valu
5.
thy
3
6
addd.
5.
not
trading
gams.
indiat
b
di
di.
oftn
a
1
ommonly
numrals
2
=
roll
usd
Calulat
two
. . .
qustions.
ar
b
A
2
sampl
two
the
of
use
+
Non-uial
1
in
outcomes
P(A)
Us
the
dice.
complement
. . .
Practice
event.
with
both
outcomes
dierent
di.
the
outcomes
number
The
b
P R O B A B I L I T Y
2
sam
=
A N D
of
rprsnt.
gtting:
Hads
Hads
in
any
ordr
Hads.
4.4 What are the chances?
141
4
Calulat
whn
5
Calulat
whn
6
Hr
52
th
you
th
you
is
proaility
roll
a
th
that
ttrahdral
sampl
playing
that
ttrahdral
proaility
roll
th
two
spa
th
di
for
sum
of
th
sors
will

a
prim
numr
of
th
sors
will

a
prim
numr
di.
sum
and
a
normal
hoosing
a
ard
(6-sidd)
from
a
di.
standard
dk
of
ards.
A,
K,
alld
ards.
Lt
A

th
vnt
Lt
B

th
vnt
‘gtting
a
Lt
C

th
vnt
‘gtting
an
a
pitur
ard’.
hart’.
vn
numr’.
a
Calulat
th
proaility
of
‘gtting
a
pitur
b
Calulat
th
proaility
of
‘gtting
a
hart’,
c
Calulat
P(C ).
d
Calulat
P(A′).
e
Calulat
P(B ′).
f
Calulat
P(C ′).
Problem
7
‘gtting
ard’,
or
or
P(A).
P(B).
solving
A
ag
ontains
A
ll
is
20
hosn
lip-on
at
iyl
random
from
lls.
th
Som
ag.
R
is
ar
rd
th
vnt
and
som
‘piking
ar
a
lu.
rd
ll’.
1
If
P(R′)
=
,
nd
th
numr
of
rd
iyl
lls
in
th
ag.
4
Probability
C
●
What
ar
th
proaility
Exploration
1
a
Writ
fair
b
down
diagrams
advantags
and
disadvantags
of
th
dirnt
rprsntations?
2
th
sampl
spa
for
rolling
a
4-sidd
di
and
tossing
Basd
and
on
your
tossing
sampl
spa,
alulat
th
proaility
of
rolling
a
1
Tails.
Continued
14 2
a
oin.
4 Representation
on
next
page
Q
and
th
J
ar
pitur
S TAT I S T I C S
c
How
and
2
a
Writ
b
Basd
vn
c
down
Stat
proaility
tr
diagram
possil
B)
a
on
vnt:
a
two
×
spa,
di
to
for
th
rolling
alulat
and
2b
to
proaility
of
rolling
a
1
Tails?
a
3
rlats
proaility
proaility
of
of
th
on
to
two
th
ithr
di.
proaility
th
rolling
4-sidd
sond
of
proaility
a
rolling
an
on.
of
rolling
an
3.
vnt
is
atd
y
th
vnt.
indpndnt
vnts
A
and
B
ourring
is:
P(B).
rprsntation
and
tr
oth
rolling
vnt:
th
rlat
P R O B A B I L I T Y
alulat
of
th
a
sampl
spa,
proailitis
of
so
you
mor
an
than
s
on
all
th
vnt.
3
drawing
numr
2nd
is
th
1b
spa
rst
answr
othr
P(A)
outoms
Example
1st
=
th
and
of
to
tossing
sampl
on
your
th
of
sampl
whthr
of
and
By
your
how
P(A
answr
th
numr
outom
A
on
Dtrmin
Th
your
proaility
numr
vn
3
dos
th
A N D
diagram,
of
two
th
rolling
1st
th
alulat
6-sidd
th
proaility
of
rolling
an
odd
di.
di.
2nd
di.
2nd
die
Draw
1
odd
a
pair
outcomes
of
branches
(odd
or
even)
to
represent
for
rolling
the
the
two
1st
die.
2
1st
die
odd
1
For
2
each
outcome
on
the
1st
die,
draw
the
two
1
2
even
possible
There
1
outcomes
are
three
for
odd
the
2nd
numbers
die.
(1,
3
and
5)
and
odd
six
possible
outcomes.
2
1
2
even
P(odd)
=
P(even)
1
=
1
even
2
Identify
P(odd,
odd)
event
tree
the
branches
P(odd,
odd),
diagram.
along
those
that
shown
Multiply
the
represent
in
red
in
the
the
probabilities
branches.
4.4 What are the chances?
14 3
Reect
and
discuss
Half
2
of
all
outoms
●
Would
it
hang
th
answr
in
Exampl
3
if
you
put
th
ranhs
2nd
di
rst
in
th
tr
What
do
you
ranhs?
noti
Explain
aout
why
th
this
sum
of
th
proailitis
on
any
pair
of
Why
do
Practice
1
A
ox
w
multiply
ontains
thr
proaility
that
a
rand-B
Copy
proailitis
and
that
rand-A
a
attris
rand-A
attry
is
omplt
faulty
th
tr
attry
is
on
and
ah
ranh?
is
svn
faulty
is
rand-B
0.3,
and
attris.
th
proaility
0.4.
diagram
to
show
all
th
proailitis.
0.3
faulty
0.3
A
not
faulty
faulty
B
not
b
2
3
A
attry
is
that
th
i
a
faulty
ii
not
Two
fair
a
faulty
a
Draw
b
Calulat
Jams
H
tr
starts
to
random
from
th
ox.
diagram
to
to
rprsnt
proaility
nd
draw
th
this
that
oth
proaility
tr
th
possil
oins
of
rolling
14 4
omplt
th
th
die
tr
proaility
two
Hads.
sixs
diagram.
second
not
Calulat
proaility
outoms.
show
6
and
th
tossd.
first
Copy
Calulat
is:
rand-A.
ar
th
wants
at
faulty
rand-A
oins
a
sltd
attry
of
6
diagram.
rolling
4 Representation
two
and
hav
numr
happns.
3
Th
a
th
di,
ths
di
●
numr
an
on
th
half
of
diagram?
1st
●
hav
for
odd
th
possil
sixs.
die
on
two
fair
di.
as
an
on
wll.
odd
th
2nd
S TAT I S T I C S
4
Nink
a
goal
goal
By
5
and
on
on
hr
Nink
b
nithr
drawr
grn
rst
rst
drawing
a
A
Mrl
hr
a
Calulat
b
A
andl
th
Mrl
and
3
Y
P(G)
is
A
th
Th
Th
alulat
oth
grn
and
ntall.
0.75.
proaility
proaility
that
that
Nink
Mrl
will
will
sor
sor
a
sor
will
and
2
vnt
th
a
proaility
goal
sor
a
yllow
piking
on
goal
thir
on
rst
thir
andls.
a
that:
G
yllow
is
attmpt
rst
th
attmpt.
vnt
piking
a
andl.
P(Y).
from
sond
th
drawr.
andl
is
Its
thn
olor
is
notd
and
it
is
rplad
in
pikd.
Baus
is
a
tr
diagram
to
rprsnt
th
sampl
still
pik
sond
c
Vrify
two
that
yllow
1
Calulat
2
a
you
piking
If
you
a
If
of
b
If
of
4
you
a
pik
you
do
not
a
a
a
pik
piking
Dtrmin
a
rd
lu
ard
rd
rd
a
a
ard
ard
rd
piking
you
tim.
grn
andls
is
mor
likly
than
piking
3
rd
pik
a
two
proaility
pik
piking
3
th
andls.
th
piking
b
andls
from
solving
Exploration
If
5
andl
thr
spa.
to
Problem
th
rplad,
ar
Draw
P R O B A B I L I T Y
0.82.
Mrl
is
pikd
drawr.
is
will
nor
ontains
a
is
diagram,
Nink
andl,
play
attmpt
tr
and
oth
attmpt
A N D
what
th
ard
th
and
nxt
ut
ut
from
to
a
rd
rpla
ard
it,
from
this
alulat
st.
th
proaility
of
tim.
thn
rpla
it,
alulat
th
proaility
of
tim.
do
from
happns
rpla
thn
nxt
ard
ard
piking
and
ard
ard
lu
rd
th
of
not
th
do
th
th
rpla
it,
rmaining
not
rpla
rmaining
alulat
th
proaility
ards.
it,
alulat
th
proaility
ards.
proailitis
for
th
sond
vnt
whn
ard.
4.4 What are the chances?
14 5
Tr
diagrams
Hr
●
ar
With
not
two
show
typs
ah
of
rplamnt
hang
for
outom
tr
–
th
and
its
assoiatd
proaility
larly.
diagram:
ths
sond
Tip
rprsnt
situations
whr
th
proaility
dos
Situations
vnt.
‘with
rplamnt’
produ
●
Without
rplamnt
–
ths
rprsnt
situations
whr
th
proaility
indpndnt
hangs
for
th
sond
vnt.
vnts;
‘without
rplamnt’
Example
ar
A
ag
from
ontains
th
vnts
4
ag
3
lmons
and
not
and
put
2
ak.
lims.
Thn
A
a
pi
sond
of
fruit
pi
is
of
pikd
fruit
is
at
dpndnt.
random
pikd
at
This
is
a
‘without
random.
rplamnt’
a
Draw
b
Find
a
tr
diagram
P(Lmon,
1st
a
to
rprsnt
Lmon)
and
outoms
P(Lim,
2nd
pick
th
and
thir
proailitis.
situation.
Lim).
pick
2
Lemon
4
The
3
upper
arm
probability
Lemon
of
of
the
rst
selecting
a
branch
lemon;
shows
the
the
lower
5
arm
2
Lime
represents
selecting
a
lime.
Since
there
4
is
no
replacement,
probabilities
for
the
the
rst
pick
second
aects
pick
,
as
the
you
3
Lemon
can
4
see
in
the
second
set
of
arms.
2
Lime
5
1
Lime
4
b
P(Lmon,
P(Lim,
Lmon)
Lim)
Example
=
=
5
ATL
In
a
ox
ar
3
rd
pns,
4
lak
pns
and
2
grn
pns.
You
pik
a
pn
This
at
random
and
thn
put
it
ak.
Your
frind
thn
piks
on
at
is
a
‘with
random.
replacement’
Calulat
th
proaility
that
you
oth
slt
th
sam
olor
pn.
situation.
R
B
G
=
=
=
hoosing
hoosing
hoosing
a
a
a
rd
pn
lak
pn
grn
Dene the events.
pn
Continued
14 6
4 Representation
on
next
page
S TAT I S T I C S
Friend’s
A N D
P R O B A B I L I T Y
pick
3
Red
9
4
9
Blue
2
Your
pick
Red
Green
9
3
3
9
Red
9
4
4
9
9
Blue
Blue
2
Green
2
9
9
3
Red
Green
9
4
9
Blue
2
Green
9
P (R,
R)
=
Find
P (B,
B)
the
color
P (G,
probability
G)
pen,
for
each
R)
+
P (B,
B)
+
P (G,
G)
proaility
Practice
1
A
ag
from
a
getting
the
three
the
same
colors.
that
th
oth
slt
th
sam
olor
is
P(R,
R)
+
(R,
P(B,
B)
R)
+
or
(B,
P(G,
B)
or
(G,
G)
.
4
ontains
Copy
you
of
=
is
Th
of
=
Probability
P (R,
for
=
ag
and
thr
and
at
lal
rd
it.
th
ppprs
You
tr
thn
and
four
tak
diagram
grn
anothr
low
to
ppprs.
tak
a
pppr
pppr.
show
2nd
You
all
th
proailitis.
pick
Red
1st
pick
3
Red
7
Green
3
Red
6
Green
Green
b
Calulat
th
proaility
you
at
c
Calulat
th
proaility
you
do
d
Calulat
th
proaility
you
at
two
not
at
rd
at
last
ppprs.
a
rd
on
pppr.
rd
pppr.
4.4 What are the chances?
14 7
G)
2
A
ag
ontains
1
whit,
4
rd
and
2
lu
ountrs.
Ros
piks
a
ountr
and
In
dos
not
rpla
it.
Sh
thn
piks
anothr
Q2,
two
a
Draw
a
tr
diagram
to
show
all
th
3
4
Calulat
Thr
oins
a
only
b
at
c
no
d
Explain
piks
a
9
two
ar
is
proailitis.
outoms:
a
sh
tr
piks
1
diagram
whit
and
and
1
rd
alulat
and
(whit,
(rd,
whit).
ountr.
th
proaility
that:
rolld
is
rolld
rolld.
rlationship
Th
ards
that
Draw
numrd
and
th
rolld.
Had
th
sid.
Copy
Find
on
ar
on
proaility
Had
Hads
Thr
on
ar
on
last
th
ar
possil
rd)
b
thr
ountr.
ards
othr
from
sid
th
omplt
proaility
twn
on
is
a
your
tal,
lank.
answrs
ah
Th
with
ards
to
a
ar
b
and
c
numr
all
1
lank
to
sid
9
printd
up.
Srna
st.
this
tr
2nd
diagram.
card
odd
that:
odd
1st
b
oth
c
at
ards
last
ar
on
card
vn
ard
is
even
odd.
odd
5
Of
a
will
group

of
v
sltd
pikingnams
studnts,
for
out
a
shool
of
a
hat.
two
trip
‘Mary
y
th
Th
even
v
studnts
Harryand
a
With
Find
th
th
aid
Alix
th
has
ox,
Th
a
of
Mary,
tr
Rafa,
diagram
or
ominations
proaility
that
Mary
a
of
tal
of
studnts
and
outoms,
that
Pitronlla
nd
ould
will

go
th
numr
of
sltd.
on
th
trip.
solving
ox
of
without
tal
a
possil
Problem
6
Jak,
Pitronlla.
dierent
b
ar
shows
rd,
yllow
and
grn
marls.
Sh
piks
a
marl
from
looking.
th
proailitis
Color
Probability
red
0.4
yellow
0.25
of
piking
th
dirnt
olors.
green
a
What
b
Thr
c
Alix
d
is
ar
piks
Draw
a
What
is
14 8
th
5
a
tr
th
proaility
yllow
that
marls
marl,
diagram
thn
to
proaility
Alix
in
piks
th
ox.
rplas
show
that
th
at
it
in
a
grn
How
th
marl?
many
ox
and
rd
marls
piks
on
of
th
marls
is
ar
anothr.
proailitis.
last
4 Representation
and
Jak’
is
even
rd?
thr?
sam
and
as
Mary’.
‘Jak
S TAT I S T I C S
Reect
Compar
and
your
discuss
tr
diagrams
●
Did
you
all
gt
●
Did
you
all
draw
●
For
Venn
qustion
a
th
do
3
for
sam
Prati
4,
qustions
2,
3,
5
and
6
with
othrs.
answrs?
sam
you
P R O B A B I L I T Y
tr
nd
to
diagrams?
draw
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diagrams
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In
6,
th
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group
6
of
30
hildrn,
9
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13
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ou
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Vnn
do
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random:
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9
5
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who
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like
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neither
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3
b
i
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1
In
strawrry
a
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not
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vanilla)
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=
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5
group
7
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40
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diagram
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a
this
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4.4 What are the chances?
14 9
2
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90
and
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in
(T ).
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60
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information.
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solving
univrsity
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information.
qustionnair
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diagram
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popular
26
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display
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ativitis.
4
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at
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oth
random
or
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of
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languags.
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study
Calulat
Arai,
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80
study
proaility
Chins,
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a
S TAT I S T I C S
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A
two-way
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150
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tal
shows
univrsity
information
th
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in
and
rows
and
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olumns.
sujts
of
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Science
Ar ts
Linguistics
Male
40
18
33
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15
20
24
th
P R O B A B I L I T Y
tables
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A N D
rows:
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columns:
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ar
91
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●
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55
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(40
+
of
(40
+
18
+
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ar
59
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+
20
+
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total
of
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a
b
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57
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+
of
+
total
20)
(33
+
+
38
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of
150
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and
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40
18
33
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15
20
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ii
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totals
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or
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at
random.
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57)
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59)
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38
studnts
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two
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33)
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students.
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59
female
20
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given
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picked
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students
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at
a
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female
random,
59,
not
in
total,
that
student
the
of
since
which
it
is
sample
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being
space
150.
4.4 What are the chances?
1 51
Practice
1
Th
6
tal
shows
ustomrs’
mnu
Miso-glazed
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Chicken
in
a
rstaurant.
stir-fr y
Lamb
kibbeh
salmon
Male
9
12
8
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1
14
6
From
this
th
numr
of
mal
b
th
numr
of
dinrs
c
th
proaility
a
Complt
aility
b
that
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who
a
in
th
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hosn
a
Th
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16
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28
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●
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wr
12
advand
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wr
10
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wr
of
tal
or
is
studnts
wr
sltd
an
was
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stir-fry
for
a
this
information
aording
to
on
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snowoarding
(M/F)
and
sta
in
total.
mal.
at
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shows
Under
40
is
60
thm
studnt
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dish
(advand/intrmdiat/ginnr).
●
A
stir-fry
dinr.
groups.
3
tal
a
fmal
2
two-way
ginnrs.
total.
studnts
random.
mal
in
Calulat
total.
th
proaility
prfrns
for
mid-morning
drink,
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Water
4
13
10
12
12
6
over
this
groupd
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40
that
snowoardr.
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ag.
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th
prntag
b
th
fration
c
th
proaility
undr
40
and
Problem
4
A
●
hoir
has
Thr
of
of
sta
sta
undr
who
that
drank
a
40
drink
watr
randomly
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of
51
ar
th
4
mmrs
mal
in
thr
ag
groups:
undr-15s.
undr-15s
ar
fmal.
3
●
26
●
7
fmals
mals
15 2
mmr
was
solving
2
●
sta
ta.
ar
and
ovr
11
15.
fmals
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ovr
4 Representation
20.
undr
15,
15
to
20,
and
ovr
20.
S TAT I S T I C S
a
Calulat
i
mal
iii
agd
ovr
Th
D
15
ondutor
What
do
‘Choos
fruit
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you
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has
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a
of
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mmr
pikd
at
random
is:
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a
mal
soloist
is
soloist
undr
at
random.
Calulat
th
15.
that
imagin
th
b
Now,
c
Rpat
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n
=
tak
100,
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a
th
fair
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Proailitis
look
many
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you
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of
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a
of
of
to
an
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and
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toss
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50
writ
and
50
tims
using
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rsults
Hads
it
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atually
it
that
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down.
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outoms.
rording
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is
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lt
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numr
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,
so
you
a
run
st
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lngths
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would
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of
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will
thy
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numr
what
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data.
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random.
in
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of
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tossing
run
dsri
random
of
000
of
th
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of
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hav
imaginary
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proaility
would
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non-random
st
word
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tossing
numr
st
whih
of
ar
whthr
and
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random’?
outom
tossing
longst
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random
Th
th
your
th
data.
In
ral
simulation
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in
you
rst
until
1000,
whthr
For
a
th
at
han
imagin
this
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fruit
at
4
Prtnd
3
not
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undrstand
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6
hoir
20
randomnss
pi
2
5
to
that
Exploration
4
a
Randomness
●
1
that
15
proaility
of
proaility
P R O B A B I L I T Y
fmal
ii
b
th
A N D
of
is
th
losst
to
proportion
half
of
th
rsults?
tims
th
vnt
trials.
happn
do
Hads
not
in
th
show
long
th
run.
Somtims
rgularity
that
vnts
ours
may
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rptitions.
idas
proaility.
rordth
‘rolling
a
lad
If
to
you
numr
6’.
As
th
tak
of
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a
6s,
fair
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di
and
an
of
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roll
it
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trials
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100
th
inrass,
tims
or
and
1000
xprimntal
this
thortial
tims
and
proaility
proaility
gts
of
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to
th
1
thortial
proaility
valu
of
6
4.4 What are the chances?
15 3
Practice
1
Amlia
7
wants
xprimnt.
2
to
pik
Whih
a
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th
rst
b
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th
last
c
Putting
Jo,
Amy
Aftr
did
and
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not
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of
th
roll
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5
5
Sam
6
at
studnts
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to
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of
play
Jo
all
a
all.
randomnss
and
a
will
th
6
gam
Jo
th
hr
a
lass
truly
for
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proaility
random
sltion?
room.
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into
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twi,
thinks
from
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into
studnts
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Sam
random
walk
on
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rolld
at
mthods
studnts
nams
rounds,
5
of
Amy
and
proaility
to
a
hat,
you
nd
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Amy
and
a
6
must
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5
throw
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how
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a
6
tims,
hating.
ths
out.
to
Us
rsults
start.
and
ar
Sam
th
du
to
han.
3
Max
ips
a
a
Explain
b
H
oin
10
tims
whthr
or
and
not
gts
this
7
Tails.
shows
his
oin
is
iasd.
Biasd
kps
ipping
th
oin
and
ounting
th
Tails.
Th
tal
shows
his
rsults.
Number
Do
4
a
you
52
500
265
1000
580
If
and
wins
a
Hadly
and
of
if
3,
Eliott
throws
a
Draw
a
b
Calulat
tr
i
Ptr
ii
Eliott
iii
Ptr
15 4
a
h
h
diagram
th
wins
wins
wins
on
on
whthr
his
to
this
or
wins;
if
outom.
a
If
th
sum
wins
Is
sum
multipl
of
if
4
th
of
thn
if
if
thy
oth
rolld
a
two
is
odd.
fair?
di
Morgan
is
a
wins.
6.
h
h
throws
throws
this
a
a
Tail,
Tail
it
it
is
is
gam.
turn
turn.
think
4 Representation
this
is
a
fair
Eliott’s
Ptr’s
turn
you
sors.
sum
gam
th
that:
third
thir
th
gam.
rprsnt
rst
not
is
thn
fair.
oin
wins;
rst
his
a
th
ruls.
it
di,
Morgan
again.
happn
proaility
on
is
answr.
6-sidd
and
th
If
roll
gam
Had
a
your
rprsnt
playing
Had
roll
vn,
wins.
would
ar
a
to
thy
this
what
is
Justify
hang
Hadly
Suggst
throws
ah
spa
if
fair?
sum
multipl,
ii
Disuss
th
Morgan
Dtrmin
Ptr
is
Morgan
i
and
oin
sampl
nithr
Eliott
c
th
Hadly
Ptr
of Tails
100
multipl
5
Number
24
Hadly
If
ips
50
think
Draw
b
of
gam.
turn.
turn.
If
mans
unfair.
S TAT I S T I C S
Reect
Dsign
a
●
ah
●
on
and
gam
for
playr
playr
discuss
two
has
has
an
a
gratr
P R O B A B I L I T Y
4
playrs
qual
A N D
using
han
han
di,
of
of
oins
or
spinnrs,
so
that:
winning
winning.
Summary
Th
sampl
spa
thomplt
an
st
xprimnt.
S,
of
It
is
all
an
a
rprsntation
possil

a
of
outoms
list,
a
tal
or
P(A)
from
rprsnts
P(A′)
is
th
th
proaility
proaility
of
A
of
not
vnt
A
ourring.
ouring.
a
P(A)
+
P(A′)
=
1
diagram.
Th
A
singl
event
is
a
sust
of
th
possil
proaility
proportion
listd
in
th
sampl
1
Miloš
is
taking
On
two
systm
failing
numr
two
ours
grads)
of
A,
summr
B,
grad).
is
‘pass/fail’
whil
C,
F
lasss
th
(with
Assum
othr
F
that
th
(thos
has
ing
th
at
a
th
loal
ar
3
th
At
th
of
ours
is
shool
12
ans
grading
rahs
only
and
proaility
of
vnt
th
happning
vnt
would
is
th
our
in
a
trials.
for
Write
b
Use
th
your
and
th
hr
on
10
oolr
frind
of
ans
to
th
of
gra
oolrs
soda.
a
ontains
Rhona
drink
for
hrslf
Maro.
of
a
tr
diagram
and
calculate
th
qual.
proaility
a
pini,
jui
into
on
Draw
ah
an
tims
practice
ollg.
only
of
spa.
larg
Mixed
of
outoms
sampl
spa
diagrams
to
as
a
nd
list
th
and
as
a
tal.
proaility
that:
a
sh
gras
two
juis
b
sh
gras
two
drinks
that
ar
th
c
sh
gras
two
drinks
that
ar
dirnt
d
nithr
sam
that:
i
Miloš
passs
oth
lasss
ii
Miloš
passs
xatly
on
4
lass
Olivia
On
iii
Miloš
fails
oth
Thr
ar
andO.
a
four
Ths
Rhsus
Write
ar
fator,
Forxampl,
a
main
th
lood
paird
whih
your
with
is
lood
sampl
typs:
B,
AB
somthing
ithr
typ
spa
A,
for
‘+’
or
ould
th
‘

6.
alld
that
ar
othr
By
all
lood
proaility
b
typ
typs
that
AB
di
O
d
a
lood
e
a
‘positiv’
has
of
or
othrwis:
di
sids
at
th
and
sam
thr
tim.
lak
a
Find
a
th
tr
sids
numrd
diagram,
tal
from
sids.
of
1
how
many
dirnt
dirnt
ominations
sh
an
possil
roll.
Calculate
proaility
lood
ar
you
qually
likly,
what
is
th
c
th
d
an
Calculate
rd
hav:
and
or
vn
that
sh
will
roll
a
th
that
sh
will
roll
a
lak
Calculate
proaility
and
th
lss
that
sh
will
roll
a
numr.
a
5.
proaility
than
2.
lood
typ
to
outoms
B+
numr
typ
rd
jui.
possil.
lood
c
6-sidd
thr
mans
rd
If
two
has
a
’.
b
typs
rolls
di
gts
lasss.
Th
2
prson
othr
lood
than
A
or
B
typ?
4.4 What are the chances?
15 5
5
Ann
has
a
whistls
ag
and
ontaining
1
grn
3
lu
whistls,
4
rd
b
Calculate
whistl.
from
Simon
3rd
Th
Ann
has
a
ag
ontaining
2
lu
whistls
8
ar
hooss
a
idntial
whistl
at
xpt
random
for
th
from
olor.
hooss
a
whistl
at
random
hr
from
ag
Rpat
Q7,
Calculate
Draw
a
tr
diagram
information
ofah
of
and
th
to
write
vnts
rprsnt
down
on
th
A
typial
of
of
120
ut
how
this
th
tnagr
tim
th
itm
proailitis
should
2000
studnts,
rvald
proaility
ranhs
takn
is
rplad.
hang.
onsum
th
aloris
thr-fths
pr
of
day.
whih
A
survy
wr
mal,
following:
th
●
tr
on
ar
hisag.
this
th
that
alulator
ag.
approximatly
a
on
and
9
Simon
th
proaility
and
and
whistls.
whistls
th
protrator
80
studnts
at
mor
than
th
rommndd
diagram.
amount.
b
Calculate
th
proaility
that
oth
Ann
and
●
Simon
will
hoos
a
lu
Half
of
th
girls
rommndd
c
Calculate
hosn
th
y
proaility
Ann
will

a
hosn
y
that
th
whistl
dirnt
olor
to
A
rnt
high
study
amounts
of
of
●
sugar
and
6
th
Fiv-sixths
of
th
oys
at
mor
than
th
amount.
Simon.
24
sodas
ain,
rvald
12
hav
that
high
8
hav
hav
Th
sam
numr
rommndd
of
oys
amount
of
as
girls
at
th
aloris.
amounts
a
of
than
th
●
6
lss
amount.
rommndd
on
at
whistl.
Complt
a
two-way
tal
to
rprsnt
this
oth.
information.
a
Draw
a
Vnn
diagram
to
rprsnt
this
Calculate
th
proaility
that
a
studnt
sltd
information.
What
is
random
th
proaility
from
th
group
b
is
high
in
sugar
ut
c
is
high
in
ain
d
is
not
that
in
not
a
soda
th
in
pikd
at
random:
b
at
c
is
at
th
rommndd
amount
of
aloris
study:
mal
and
at
lss
than
th
rommndd
ain
amount
only
d
7
A
ag
high
ontains
protrators
at
random
takn
in
at
four
(P ).
and
ain
itm
is
replaced.
(C
takn
A
at
mor
givn
sugar?
alulators
On
not
or
)
and
from
sond
six
th
itm
random.
10
ag
is
thn
In
a
th
group
rst
holstrol
prssur
Complt
th
proailitis
tr
in
diagram
th
spas
y
C
P
C
P
P
4 Representation
popl,
oth;
28
that
high
23
hav
a
amount
fmal.
10
ar
lood
popl
high
prson
halthy
and
prssur,
hav
high
holstrol.
sltd
at
high
lood
Find
th
random:
writing
providd.
C
15 6
50
rommndd
ar
ithr
or
and
th
thy
of
hav
proaility
a
than
that
a
has
high
lood
prssur
b
has
high
lood
prssur
and
c
has
high
lood
prssur
or
d
has
high
holstrol
only.
high
high
holstrol
holstrol
S TAT I S T I C S
Review
in
A N D
P R O B A B I L I T Y
context
Identities and relationships
Problem
on
solving
if
1
Th
hart
diats
valus.
shows
for
th
risk
dirnt
BMI
is
a
of
ody
way
of
dvloping
mass
indx
masuring
Typ
fat
in
th
ody.
For
adults,
a
th
(gn)
parnts
18.5
and
ar
would
riv
ah
MM
and
parnt.
Mm,
For
thn
instan,
‘M’
‘m’
an
‘M’
from
th
rst
th
hild
parnt
and
an
(BMI)
th
or
from
th
sond.
amount
halthy
BMI
Copy
and
omplt
th
sampl
spa
for
th
is
dirnt
twn
from
2
a
of
alll
outoms.
25.
100
Mother
93.2
gnipoleved
MM
80
fo
setebaid
M
Mm
M
M
mm
m
m
m
60
54.0
ytilibaborP
2
M
40.3
epyT
40
MM
27.6
M
20
15.8
8.1
1.0
5.0
4.3
2.9
M
0
5
3
≥
3
.
4
9
9
3
.
2
3
.
0
9
9
2
.
8
2
.
6
9
2
.
4
9
9
2
>
2
.
2
1
.
0
9
9
Father
Mm
-
-
-
-
-
-
-
-
3
3
3
1
2
9
2
7
2
5
2
3
2
1
1
9
m
2
Body
mass
index
(kg/m
)
m
a
An
adult
has
BMI
28.
What
is
thir
risk
of
mm
dvloping
Typ
2
diats?
m
b
Describe
hangs
how
as
th
BMI
risk
of
dvloping
diats
b
inrass.
If
a
hild
inhrit
c
An
adult
has
BMI
31.
H
loss
wight
BMI
risk
of
26.
Typ
How
2
many
diats
tims
smallr
is
orn
Write
som
dvloping
to
2
justify
Vinnt
his
Typ
your
knows
hildrn
family.
advi
for
c
2
diats.
th
Use
risk
that
thr
h
a
Calculate
proailitis
has
is
a
15%
thr
han
that
runs
hildrn.
of
in
Find
d
his
all
b
at
of
his
two
of
hildrn
having
th
his
Proaility
liklihood
that
of
his
hildrn
having
hildrn
having
th
an
of

usd
hildrn
in
gntis
inhriting
a
hav
th
th
proaility
gn
that
th
hild
will
(m).
parnts,
oth
of
whom
ar
Mm,
th
proaility
myopia
that
th
rst
that
only
that
nithr
hild
(mm).
th
proaility
on
hild
myopia.
Calculate
th
proaility
hild
disas
hav
myopia.
disas.
to
prdit
ondition
th
Determine
hav
to
whih
hav
in
gns
ordr
for
th
th
parnts
would
proaility
that
1
from
thir
rst
hild
has
myopia
to

2
thir
parnts.
rprsntd
(myopia)
is
Suppos
y
‘M’
th
whil
rprsntd
Reect
and
How
you
hav
gn
that
y
for
of
‘m’.
normal
short
sight
is
sightdnss
Ahild
inhrits
discuss
xplord
th
statmnt
of
inquiry?
Giv
spi
xampls.
Statement of Inquiry:
Undrstanding
using
a
disas
g
3
hav
Calculate
will
non
proaility
myopia.
th
f
c
hav
of:
thr
last
thy
hildrn.
Calculate
will
a
will
myopia
two
two
will
e
proaility
th
Suppos
hav
disas
th
whn
of
ommnts.
inhriting
Suppos
rduing
alulat
only
his
now?
arry
d
‘mm’,
myopia
and
hild
has
inhrits
logial
halth
and
making
rprsntations
and
halthir
hois
rsult
from
systms.
4.4 What are the chances?
157
5
Simplication
The process of reducing to a less complicated form
go
Programming
to
pen
In
computer
x:
programming,
a
block
is
a
section
of
code
grouped
together.
down
▼
Experienced
programmers
write
code
in
direction
for
at
least
three
reasons:
each
block
can
it
makes
it
simpler
for
others
4
to
move
understand,
perform
just
one
simple
task,
the
act
of
is
made
working
simpler.
through
(The
a
term
‘debugging’
computer
program
here
errors
that
cause
the
program
to
fail
or
to
identify
give
degrees
90
means
and
pen
remove
steps
320
and
turn
debugging
90
in
repeat
blocks
-160
that
point
is
y:
-160
code
up
incorrect
to
results.
It
is
much
easier
to
do
this
when
the
code
has
been
to
written
in
blocks
that
can
be
tested
independently.)
▼
point
Get
a
code
part
sheet
here
of
a
of
graph
written
graphic
paper
using
‘proof’
and
see
Scratch
of
the
is
if
you
gure
designed
innite
to
series
out
do.
sum
what
The
the
result
below,
is
in
repeat
go
to
direction
8
will
learn
about
in
IB
DP
q
x:
4
down
n
move
1
1
+
2
+
4
8
+
...
=
16
go
Scratch is developed by the Lifelong Kindergarten
Group at the MIT Media Lab. See http://scratch.mit.edu
Smar tphones
replaced
a
camera,
clock,
●
the
a
(and
need
the
for
laptop,
applications
other
diary,
devices
maps,
that
and
run
on
them)
equipment,
ashlight,
have
such
calendar,
alarm,etc.
Suggest
our
made
15 8
some
lives,
life
but
ways
also
more
that
smartphones
suggest
some
complicated.
up
1
to
turn
Smartphones
steps
1
pen
+
q
Mathematics.
pen
1
y:
which
repeat
you
90
ways
have
that
simplied
they’ve
as:
q
x:
90
set
to
set
to
y:
q
degrees
Simplied
In
the
writing
making
the
literacy,
learn,
of
Chinese
the
written
simplied
because
characters
The
table
and
three
to
it
a
Chinese
language
Chinese
has
fewer
Roman
here
shows
hard
was
to
many
learn.
introduced.
strokes
script,
the
language,
per
based
Pinyin,
on
traditional
Beginning
Simplied
character.
in
characters
1949,
Chinese
Pinyin
is
the
in
is
are
an
very
effort
probably
conversion
complex,
to
increase
easier
of
Chinese
pronunciation.
Simplied
and
Traditional
characters
for
three
Simplied
Traditional
character(s)
character(s)
数学
數學
Pinyin
Mathematics
shù
To
count
shù
数
數
To
learn
xué
学
學
书
書
马
馬
谢
謝
thank
The
of
rst
xiè
near
one
of
a
the
the
village
Beijing,
of
xué
character
Cuandixia,
name
nouns
verbs.
English
To
to
is
most
complicated
traditional
characters,
up
of
30
Chinese
made
strokes.
15 9
Are
you
saying
I’m
irrational?
5.1
Objectives
●
Simplifying
Inquiry
irrational
numerical
●
expressions
F
MROF
●
Approximating
●
Applying
●
Performing
rules
expressions
questions
What
is
the
and
●
What
a
●
How
do
●
How
are
an
between
irrational
a
rational
number?
radicals
of
radicals
operations
that
on
contain
to
simplify
radicals
to
is
radical
(surd)?
them
you
approximate
a
radical?
simplify
radicals
C
the
●
rules
How
is
the
for
●
D
Can
rules
radicals
related
terms
radicals
in
to
algebra?
similar
to
fractions?
irrational
form
of
combining
simplifying
simplifying
ATL
dierence
number
rational
numbers
be
combined
to
numbers?
Communication
Draw
reasonable
conclusions
and
generalizations
5.1
10.3
E5.1
Conceptual understanding:
Forms
can
16 0
be
changed
through
simplication.
N U M B E R
Y
ou
●
should
identify
already
radicals
and
know
how
understand
to:
1
Simplify
each
expression.
Muhmm
what
they
in
represent
a
9 + 16
b
9 +
16
Mus
(.
c
3 ×
3
d
11÷
l-Khwrizmi
780–.
850
ad)
11
rfrr
to
rtionl
1
e
3 ×
6 ×
8
numrs
f
s
audible,
9
n
irrtionl
numrs
s
inaudible.
Reviewing
F
radicals
l
●
Wht
is
th
irn
twn

rtionl
numr
n
to
th
‘
’
Wht
●
How
is
o
(smm,
numr?
mning
●

ril
you
(sur)?
or
pproximt

‘f ’
‘um’)
irrtionl
ril?
whih
square
root
(lso
known
s

radical
or
surd )
of

positiv
numr
x
is
ws
Any
Th
of
In
x,
whih,
positiv
gnrl,
For
numr
principal
n
is
square
writtn
‘th
writtn
Hr
r
som
= 5,
π
1
hr
itslf,
squr

givs
roots:
positiv
th
on
originl
positiv
numr
x
is
th
numr
n
on
positive
s
‘surus’.
x.
ngtiv.
squr
root
.
ut
mns
th
th
squr
25
s
of
positive
root
of
squr
25
n
root.
lso

5.
This
is
5
wht
7.262626…
r
th
of
y
1
som
w
ll
of
wht
twn

numrs’:
18.2
w
17.41002368…
irns
‘rtionl
3
xmpls
2.39841…
Stt
two
root
xmpls
1.45
An
hs
root’
own
Exploration
multipli
s
squr
25
xmpl,
normlly
whn
thn
into

Ltin
numr
for
numr,
trnslt
A
ltr
Ari
n
wor
irrtionl
This
(surds)
ll
‘irrtionl
numrs’:
0.83126674…
‘rtionl
numr’
n
n
‘irrtionl
your
rsoning.
numr’.
2
Dn
3
How
A
h
woul
rational
of
thm
you
in
your
lssify
number
is


own
wors.
numr
numr
tht
lik
n
?

Explin
writtn
s

frtion
whr
W
hv
nms
n
.
For
xmpl:
1.2,
−0.3,
n
0
r
fw
irrational
spil
irrtionls,
numrs.
π,
An
number
is

numr
tht
nnot

writtn
s

ϕ,
n
imls.
numrs
n
cannot
π
r

rprsnt
xmpls
of
s
trminting
irrtionl
or
lik
Eulr’s
frtion.
numr,
Irrtionl
only
ll

rtionl
givn
to
.
rpting
numrs.
5.1 Are you saying I’m irrational?
1 61
Hr
●
r
Th
squr
thir
r
●
som
importnt
roots
vlus
r
rtionl
Th
squr
numrs
of
fts
out
positiv
whol
squr
squr
numrs.
roots:
numrs
4
=
2,
r
rtionl
= 5,
25
numrs
= 10 ,
100
n
us
2025
=
45
Tip
numrs.
roots
2 ,
of
positiv
247
24 ,
non-squr
n
r
1000
numrs
r
irrtionl
irrtionl.
us
thir
Th
trms
n
‘sur’

writtn
s

frtion.
A
ril
(or

sur)
is
nothr
wor
for
to
rfr
root
of

numr
tht
hs
n
irrtionl
Som
irrtionl
squr
root,
numrs
yt
thy
r
(suh
still
π,
s
for
irrtionl
xmpl)
n’t

writtn
s

numrs.
You
ithr
oth)
my
wor
us
and
●
Try
to
●
Is
possil
it
n
discuss
th
to
vlu
tk
in
ooks.
1
of
th
s
(or
mthmtis
Reect
n
squr
vlu.
root.
●
oth
to
th
irrtionl
squr
r
vlus
us
nnot
‘ril’
Th
.
squr
root
of

ngtiv
numr?
If
not,
why
not?
ATL
Exploration
In
this
For
1
xplortion,
th
In

2
purpos
opy
of
of
this
you
will
this
xplortion,
tl,
stimt
omplt
=

squr
you
th
n
vlus
root
without
ignor
of
just
using
ngtiv
the

lultor.
squr
rational
squr
roots.
roots.
Tip
=
1,
4,
th
9,
n
rst
fw
numrs.
2
Look
t
th
n
vnly
3
,
sp
Squr
your
Commnt
stimt
4
Rpt
3
n
squr
6
Using
16 2
1
th
th
your
roots
1
tht
r
r
similr
still
lnk.
of
1
n
stimt
from
your
n
2.
th
stp
2
r
Assuming
vlus
using
stimts.

in
tht
of
twn
thy
n
r
.
lultor.
Cn
you
n

ttr
pl?
roots
ll
vnly
tht
squr
vnly
or
2,
vlus
of
rsoning,
5 Simplication

in

2
n
twn
stimt
3,
n
pirs
th
of
twn
onsutiv
vlu
of
h
on.
lultor.
stimts.
giv
squr
suggst
twn
n
using
your
sp
lrg
r
roots
sp,
vlus
ury
roots
r
twn
n
ury
squr
th
tht
oth
stimt
on
squr
r
iml
Assum
Commnt
roots
twn
numrs
Squr
smll
thy
stimt
on
to
for
4.
whol
5
squr
so
you
mor
Dos
ssuming
urt
tht
stimts
th
for
roots?
mtho
for
stimting
ny
squr
root.
16
r
squr
N U M B E R
Practice
1
Without
of
2
ths
1
using
writ
own
or
stimt
th
squr
121
b
64
c
15
d
12
e
27
f
50
using

lultor,
stimt
th
vlu
of
th
prinipl
a
24
b
55
c
82
d
90
e
56
f
99
Problem
By
ths
is
2
lss
vlus
thn
15
(<)
of
or
8
irrtionl,
iml.
you
th
11
7
d
of
h
squr
roots.
ny
rils
grtr
thn
or
othrwis,
(>),
shoul
trmin
rpl
th
whih
to
omplt
sttmnts.
7
a
roots
solving
stimting
inqulity,
if
lultor,
a
Without
3

numrs.
A
so
it
nnot
lultor
2
wrot
n
orrt
to
b
3 +
e
5 ×

5
3
4
rprsnt
pproximt

lrg
c
38
2
numr
×
y
to
of
6


23
4
f
trminting
iml,
iml
×
50
s
it
10
25 +
iml
suh
pls,
1+
or

rurring
1.41421356.
oul
only
15
Evn
vr
Numrs
n
n
pproximtion.
Th
only
wy
to
writ
th
exact
vlu
of
th
squr
root
of
lik

2

rprsnt
is
s

trminting
4
2.
Unlik
th
iml
form
of,
for
xmpl,
=
0.363636...
whih
is
iml,
in
this
11
rurring,
th
iml
form
of
2
os
notrur.
s
0.375.
numrs
Hippsus
irrtionl
intgrs
of
Mtpontum,
numrs.
(i..
s

H

followr
show
rtionl
of
tht
numr),
Pythgors,
nnot
n
tht

is
si
writtn
thrfor
it
to
s
hv
th
must

isovr
rtio
of
n
s
two

isovry
muh
to
tht
h
of
th
irrtionl
sntn
numrs
Hippsus
to
upst
th
y
n
trri
rowning
t
Pythgors
s
s
rprsnt
rurring
s
This
suh
imls,
irrtionl.
Othr
in
innit
this
0.26666666…
so
(oring
lgn).
Exploration
Consir
th
3
following
list
of
numrs:
Tip
1
Without
using

lultor,
orr
th
numrs
from
smllst
to
lrgst.
Your
strtgy
involv
2
Us

lultor
3
Suggst
to
hk
your
nswr.
nomintors

strtgy
tht
osn’t
involv
using

lultor
to
ompr:
squring
Dos
a
b


frtion
ril
to
to


whol
whol

numr
lwys
om
numr
whn
squr

ril
to

n
numrs.
numr
lrgr
c
oul
ommon
you
it?
frtion.
Continued
on
next
page
5.1 Are you saying I’m irrational?
16 3
4
Without
pirs
of
using

lultor,
numrs.
a
Rpl
5
d
Reect
and
discuss
us
your
th
y
strtgis
<,
>
or
to
ompr
th
following
=
b
c
e
f
6
Th
2
squr
sign,
Th
only
wy
to
writ
th
exact
vlu
of

ril
numr
is
y
root
ll
ril
sign.
sign.
Any
●
is
using
th
th
,
Wht
ny
r
th
possil
vntgs
of
using
suh
nottion
for
rils?
Ar
thr
trm
it
isvntgs?
is
numr
or
unrnth
ll
th
rin.
●
How
n
os
this
xprssion
irntly
whn
pproximt
How
C
●
t
to
thy

How
r
How
in
is
th
ril
r
wy
you
prform
numrs?
writtn
using
Do
th
oprtions
rils
sign
on
hv
thn
whn
thy
r
iml?
radicals
trms
●
nottion
ontining
th
ruls
behave
of
rils
rlt
to
th
ruls
for
omining
lgr?
simplifying
rils
similr
to
simplifying
frtions?
Aoring
to
th
ATL
Exploration
Ntionl
4
n
You
r
going
to
xplor
th
thr
si
ruls
of
Oni
Atmosphri
rils.
Aministrtion,
Rule
sp
1
pr
1
Evlut
ths
s
a
b
c
d
e
f
son)
(in

gnrl
rul
from
wht
you
foun
in
stp

1
th
pth
mtrs)
on,
th
th
=
,
2
whr
3
s
d
of
using
formul
Rule
t
tsunami
movs is trmin
y
Du
th
mtrs
mounts.
whih
2
(in
Us

lultor
to
vlut
ths
mounts.
g
(th
lrtion
u
to
2
grvity)
4
a
b
c
d
e
f
Du

gnrl
rul
from
wht
you
foun
in
stp
3
Continued
16 4
5 Simplication
on
next
page
is
9.8
m/s
N U M B E R
Rule
5
6
Th
n
3
Us

lultor
vlut
ths
ril
xprssions.
a
b
c
d
e
f
Du
ruls
of

gnrl
rils
iviing
Example
a
to
rul
tht
from
you
wht
you
isovr
foun
in
in
stp
5
Explortion
4
rlt
to
multiplying
rils.
1
Simplify
b
Simplify
a
When multiplying, the order doesn’t matter (associative property).
(Rule 2)
Leave the nal answer without the × sign.
b
Rewrite a division as a fraction.
Recall the rules of fractions:
(Rule 3)
Rules
For
of
ny
Rule
1
radicals
rl
numrs
Rule
:
2
Rule
3
b
≠
0
5.1 Are you saying I’m irrational?
16 5
Practice
1
Simplify
2
h
xprssion.
18
26
a
×
5
3
b
21 × 21
5
7
c
×
11
d
2
3
5
42
e
Fill
in
th
×
2
35
g
h
6
Problem
2
12
f
75
4
20
×
8
solving
lnk
18
to
mk
th
xprssions
qul.
6
=
9
3
×
ATL
Exploration
1
Simplify
2
Whn

th
5
frtion
frtion
is
.
qul
to
1,
trmin
wht
this
tlls
you
out
th
numrtor
n
th
nomintor.
3
Hn,
writ
own
(s
n
qulity)
wht
you
n
u
from
.
h
1
4
Using
th
5
Suppos
Pythgorn
thorm,
n
th
hypotnus
of
this
tringl.
1
lrgr
you
lign
right-ngl
thr
tringl
suh
lik
tringls
long
th
hypotnus
h
1
h
h
1
3
1
1
h
h
1
1
1
Writ
own
6
Now,
using
7
Wht
n
n
xprssion
th
you
and
sm
for
tringl,
u
from
discuss
1
th
us
th
lngth
th
of
th
hypotnus
Pythgorn
prvious
two
stps?
your
nings
3
of
thorm
Drw

this
to
lrgr
n
th
rsonl
3
√
in
stp
3
to
Explortion
5.
2
1
your
18
nings
in
stp
7
of
1
Ar
●
thy
How
quivlnt?
os
this
igrm
3
illustrt
your
nings?
√
2
1
1
√
1
1
3
16 6
nw,
h
1
Compr

1
1
●
mk
this:
1
Reect
to
5 Simplication
2
tringl.
lngth
of
th
onlusion
hypotnus.
or
gnrliztion.
N U M B E R
In
Explortion
simplies
5,
to
you
Exploration
Blow
r
foun
tht
som
rils
1
Explin
th
2
Explin
how
3
Dtrmin
in
form
most
4
will
th
Explin
Explortion
Example
Method
pross
how
6
illustrt
Simplify
A
wy
of
sying
this
is
tht
18
6
n
thir
Unsimplied
is
18 .
=
.
you
for
mny
you
n

Simplied

simplify
wys
you
form.
form
ril.
.
n
nswr
qustion
2.
Stt
whih
simpli.
know
monstrt
it
form
simplifying
woul
how
simpli
whn
on
irnt

ril
mtho
of
can

simpli.
simplifying
rils.
Exmpl
2
mtho.
2
.
1:
9
is
a
square
number
factor
of
18.
(Rule 2)
Write
Method
the
nal
answer
without
the
×
sign.
2:
The
prime
factorization
of
18
is
2
×
3
×
3.
(Rule 2)
(Rule 1)
5.1 Are you saying I’m irrational?
167
Reect
●
Whn
is

●
it
and
simplifying
importnt
squr
root
Compr
r
●
discuss
th
y
is
Somtims
Example
squr
root
ftors
using
ftors
mthos
using
tht
r
tht
us
r
in
Mtho
squr
not
1
from
Exmpl
numrs?
squr
Exmpl
2.
Cn
you
2,
why
simplify
numrs?
Woul
you
sy
tht
thy
mthos?
simplifying
th
Dtrmin

n
two
quivlnt
How
to
4
squr
rils
root
lik
nnot

simplifying
frtions?
simpli.
3
whthr
or
not
a
ths
rils
n

simpli.
b
c
15
has
factors
1,
3,
5,
and
15.
a
Only
You
could
However,
No,
nnot

rewrite
3
and
5
15
1
as
are
is
a
not
a
square
product
square
factor.
of
3and
5.
numbers.
simpli.
45
has
factors
1,
3,
5,
9,
15
and
45.
b
9
Ys,
n

is
the
largest
square
factor.
simpli.
30
has
factors
1,
2,
3,
5,
6,
c
Only
No,
nnot
Thr
r
two

simpli
situtions
whn

squr
●
th
squr
root
of

prim
●
th
squr
root
of

omposit
thn
Th
1
nnot
onition
squr
ftor
Practice
Exprss
to

root
numr
nnot
nnot
numr
tht
simpli:
simpli
hs
no
squr
ftors
othr
thn

squr
root
is
tht
th
numr
hs
t
lst
on
1).
3
h
ril
in
its
simplst
form.
1
24
2
32
3
72
4
6
960
7
675
8
864
9
16 8


simpli.
simplify
(othr
1
5 Simplication
125
991
5
10
135
992
is
a
square
factor.
10,
15,
30.
N U M B E R
Problem
solving
4
6
99 a
11
200b
12
8
2
c
288 x
13
3
y
ATL
Exploration
A
7
vinculum
horizontl
Whn
ing
n
sutrting
rils,
th
sm
ruls
pply
s
whn
ing
or
sutrting
whol
numrs,
frtions
or
Writ
n
a
own
thn
h
n
Thr
th
twos
of
th
following
sttmnts
in
vrils.
mthmtil
nottion,
four
for
grouping
things
togthr.
nswr.
plus
twos
mks
In
twos.
1637,
Rné
Dsrts
b
Two
sixths
plus
thr
sixths
mks
Fiv
x’s
d
Thr
plus
six
x’s
to
sign
squr
roots
of
two
plus
v
squr
roots
of
squr
roots
of
with
Writ
own
nottion
h
n
of
th
trmin
thy
xprssions
n

furthr
using
mthmtil
w
ril
us
Thr
b
On
c
Six
d
Two
twos
plus
four
rt
symol
toy.
simpli.
S
a
to
two.
following
if
th
two
th
2
th
ril
vinulum
mks
th
omin
Grmn
x’s.
mks
ws
sixths.
rst
c
us
mthmtil
nottion
1

you
in
r
is
lin
to
vs.
how
it
is
us
istinguish
from
3
Bs
thir
x’s
plus
plus
v
squr
on
thr
stps
y’s.
roots
1
qurtrs.
of
n
two
2,
plus
u
two

squr
rul
for
roots
ing
of
thr.
n
sutrting
squrroots.
4
Compr
thy
5
Simplify
a
n
similr
h
Without
ontrst
n
of
in
ths
using

ths
wht
four
wys
four
xprssions.
r
thy
Stt
in
wht
wys
r
irnt.
xprssions.
lultor,
trmin
whih
of
th
xprssions
rqul.
b
Us
c
Of
your
th
lultor
xprssions
itssimplst
6
Eh
Du
of

ths
rul
to
vrify
tht
r
your
qul,
nswr.
suggst
whih
on
is
writtn
in
form.
xprssions
for
hs
n
ing/sutrting
simpli.
ril
xprssions.
5.1 Are you saying I’m irrational?
16 9
You
th
●
n
2a
(2
●

sm
+
+
2a
3a
Th
n
3)a
+
hv
3b
or
sutrt

For
simpli

ommon
is
tru
●
2
5 + 3
●
2
5
5
+ 3
Example
two
numrs
only
whn
you
r
ing
multipls
of
xmpl:
to
5a,
sin
a
n

ftor
out
to
giv
5a
nnot
no
sm
Whr
or
mount.
for
= 5
7
simpli
ftors.
squr
sin
Thus,
a
2a
n
+
3b
b
r
is
not
writtn
lik
in
trms,
its
tht
simplest
is,
a
n
b
form
roots:
5
nnot

simpli.
4
possil,
simplify
a
ths
xprssions
b
tht
involv
squr
c
roots.
d
a
Just
b
Convert
to
like
equivalent
4x
+
2x
square
=
roots.
Simplify
Substitute
Adding
only
There
c
for
terms
when
are
no
involving
they
like
in
have
terms,
the
square
the
so
original
same
this
is
.
expression.
roots
is
square
the
6x
possible
root.
simplest
form.
The
largest
square
factor
of
48
is
16.
The
largest
square
factor
of
32
is
16.
d
This
Reect
●
How
●
In
and
os
prt
d
discuss
simplition
of
Exmpl
4,
5
hlp
why
you
is
it

or
ttr
sutrt
to
unlik
rils?
simplify
to
?
●
How
is
ing
sutrting
●
How
17 0
is
n
sutrting
squr
roots
similr
to
ing
n
polynomils?
simplifying
squr
roots
5 Simplication
similr
to
simplifying
frtions?
cannot
be
simplied
any
further.
N U M B E R
Rule
4
of
For
radicals
:
Similrly,
you
Practice
Simplify
1
7
3
12
9
lso
sutrt
rils
th
sm
wy:
xprssions.
3
a + 3
in
4
ths
3
n
17
2
− 9
a + 3
4
2
6
2
+ 16
pq
17
+ 4
pq
Th
27
5
12
12
+
prio
sons
7
2
75
54
9
5
−
27
20
+
45
Problem
8 +
8
−
24
12
10
128 +
+
T
in
48
32
+
48 +
18
75 +
162
of

simpl
pnulum
of
L
in
givn
T
=
ft
is
lngth
.
solving
2
11
4
12 x
+ 3
27 x
3x
12
48
3
13
27 x
Anothr
3x
wy
Reect
Until
to
you
nomintor
You
n
Thr
is
think
r
numr
simplify
and
now,
two
of
2
14
discuss
hv

s
1
possil
iml
(rtionl
pls
tht
÷
rpt
●
How
iult
is
●
How
iult
woul
to
or
of
th
ivi

10
innit
thos
y
to
frtions
s
iml
n
n
with
s
(irrtionl
it
rtionliz
suh
n
typs
pls
it
4
to
lt
numr,
numrs),
on’t
is
+ 8 y
− 5
9x
+ 18 y
nomintor.
6
proly
whol
of
rpt
rils
4 x
or
÷
whr
th
.
3.
numrs:
numr
with
only
n
thos
of
with
iml
innit

nit
pls
numr
of
tht
iml
numrs).
rtionl
ivi
y
numrs?
n
irrtionl
numr?
5.1 Are you saying I’m irrational?
171
y
Exploration
Consir
th
frtion
1
Dsri
2
Simplify
3
Dtrmin
4
Simplify
5
Compr
this
8
th
.
mount
tht
this
frtion
rprsnts.
.
Tip
whthr
×
is
qul
to
.
For
×
4,
us
th
rul
your
frtion
stp
nswr
to
.
Is
it
now
sir
to
sri
th
mount
tht
rprsnts?
ATL
Reect
●
and
Prforming
hs

n
rtionl
nomintor.
●
oprtion
Is
this
rtionlizing
If
xplin
so,
5
b
of
>
7
so
nomintor
Dos
Rule
For
discuss
tht
similr
th

frtion
inst
to
is
simplifying
nomintor
with
known
s
n
mk
it

ril
in
th
rtionlizing
irrtionl
sir
to
nomintor
th
numr?
ompr
numrs?
why
.
radicals
0:
Example
5
Simplify:
a
b
a
To
b
17 2
5 Simplication
simplify
multiply
by
N U M B E R
Somtims,
●
Simplify
●
Us
you
th
n
3:
Rtionliz
Example
th
n
mthos
th
to
simplify
nomintor
ril
xprssions:
sprtly
a
to
=
b
●
irnt
numrtor
a
Rul
us
simplify
th
frtion
b
nomintor
6
Simplify:
a
b
a
Simplify
the
denominator
Rationalize
Simplify
the
numerator
the
and
and
cancel.
denominator.
denominator
b
separately,
Practice
Simplify
th
then
cancel.
5
following
nomintor
in
your
xprssions.
nl
Thr
shoul
45
40
1
in
th
6
7
5
12
14
10
45
11
12
18
28
27
2
8
2
5
9
8
10
6
3
1
4
3
25
15
Problem
ril
3
5
9
no
1
2
10
5

nswr.
150
solving
3
3 xy
108 x
13
15
3
3x
8
5
14
15 x
y
16
10 xy
32 x
5.1 Are you saying I’m irrational?
17 3
D
How
●
Cn
Irrtionl
of
radicals
irrtionl
numrs
Pythgors.
ook
The
B:
Dsri
To
are
nd
a
Euli
isovr
it
wsn’t
prov
Invstigting
pttr ns
general
similar.
consistent
wr

th
omin
roun
until
to
520
roun
xistn
of
form
bce,
300
rtionl
uring
bce
tht
irrtionl
th
numrs?
tim
th
Grk
numrs
in
his
Elements.
Objective
ii.
numrs
Howvr,
mthmtiin
behave
If
rule,
the
with
s
rule
your
pttr ns
gnrl
make
sure
applies
r uls
that
only
to
onsistnt
the
rule
some
with
applies
specic
to
cases
nings
all
the
but
specic
not
others,
cases
then
which
it
is
not
ndings.
Exploration
Som
9
irrtionl
numrs
Som
(ut
not
ll)
squr
roots
r
irrtionl
ntur
is
π
irrtionl
(=
n
is
Hr
is

list
of
irrtionl
in
numrs:
of
s

(=
2.718…)
pirs
suh

trms
r
thir
sum
is
rtionl
b
thir
sum
is
irrtionl
c
thir
prout
is
rtionl
d
thir
prout
is
irrtionl.
Suggst
vlus
th
lngth
a
its
primtr
is
rtionl
b
its
primtr
is
irrtionl
c
oth
its
primtr
Dtrmin
3
for
Th
r
Grk
tims
lttr
rltionship.
Th
Goln
musi,
ll
simpli
thr
Th
if
rtio
pinting,
1 74
ths
form
r
its
r
is
is
pprs
ϕ
is
not
just
sign
in
n
5 Simplication
is

rtngl
suh
tht:
irrtionl
is
rtionl
irrtionl.
r
possil.
Explin
to
th
givn
of
r
.
ttr
rprsnts
of
ntur,
its
n
r
of
(ϕ)
vlu
n
with
omintions
whn
phi
Th
n
n
us
whthr
thn
‘Goln
y:
Justify
or
your
not
nswr.
you
think
.
rtio’,

gomtri
ϕ
mthmtis,
mny
othr
ut
rs.
in
rhittur,
rils.
prt
numrs
of

st
known
trnsnntl
tht:
a
xprss
of
numrs.
2
in
s
rtionl
Thy
Fin
suh
3.14159…)
nnot
1
foun
numrs:
N U M B E R
Summary
●
A
of
square

root
positiv
(lso
known
numr
x
is

s

radical
numr
or
surd)
Rules
For
whn
multipli
numr
roots:
●
Th
x
is
A
Any
on
th
itslf,
positiv
positiv
principal
s
●
x.
y
n
square
positive
squr
givs
th
numr
on
of
radicals
whih,
ny
rl
numrs
rational
s
number

c
≥
0:
hs
two
squr
Rule
1
Rule
2
Rule
3
Rule
4
ngtiv.
root
of

root
of
x,
is

frtion
positiv
n
is
numr
b
≠
0
writtn
numr
tht
n
whr
For
c
xmpl,
n
n
0
r
ll
1.2,
rtionl
irrational
writtn
nnot

rpting
s
,
Mixed
Compr
y
is
frtion.
rprsnt

numr
Irrtionl
s
imls.
irrtionl
n
lso
sm
wy:
For
>
sutrt
rils
in
th
numrs.
number

0:
.
Rule
An
≥

You
For
of
b,
.
writtn
●
a,
originl
tht
b
0:
numrs
trminting
n
5
nnot
(pi)
or
r
xmpls
numrs.
practice
th
following
using
<,
>
or
pirs
of
vlus
n
Simplify
rpl
to
=
ths
rtionliz
xprssions
th
ompltly.
nomintor
whn
Rmmr
nssry.
9
1
20
2
13
12
14
15
125
16
17
10
19
xy
18
6
3
6
3
4
7
6
Without
using

lultor,
estimate
th
vlu
×
360
prinipl
squr
roots
orrt
to
18
3 ×
12
of
3
th
121x
7
1.p.
98
5
×
x
y
20
2
45
5
6
6
15
17
21
22
24
3x
54
7
8
12
10
23
Rtionliz
th
nomintor
for
h
24
12
13
2
13
12
ril
xprssion:
1
25
5
27
2
8
3
50
26
6
4 x
+ 3
9x
6
10
9
2
20t
+ 4
45t
3
28
3
200 a
+
128a
−
8a
15
11
12
4
8
75
Reect
and
How
hv
you
Giv
spi
discuss
xplor
th
sttmnt
of
onptul
unrstning?
xmpls.
Conceptual understanding:
Forms
n

hng
through
simplition.
5.1 Are you saying I’m irrational?
175
6
Quantity
An amount or number
How
many
Suppose
of
sh
this?
all
not
the
want
lake.
could
dead
accurate
One
a
You
the
and
you
in
know
the
that
but
it’s
estimate
100
sh,
tag
release
them
back
that
these
sh
all
evenly
Later,
that,
and
you
of
5
previously.
You
tagged
in
sh
batches
of
capture
all
you’d
then
batches.
2000
5
100
there
sh
in
are
the
the
sh
an
and
either!
is
called
then
Assume
themselves
again
that
that
So
how
you
to
per
If
for
and
batch
lake.
nd
tagged
are
100
many
need
you
to
assume
every
catch
estimated
the
you
there
sh?
sh
have
sh
lake.
sh
would
tagged
an
practical,
quantity
ones
tagged
would
At
100
lake.
sh
100
catch
you
mean
100
count
you
throughout
know
the
the
distribute
are
give
about
method:
into
capture
go
and
very
them,
randomly
these,
of
the
capture-recapture
Capture
not
quantity
you
lake
would
considerate
to
the
would
drain
count,
way
to
How
sh;
too
sh?
100,
100/5
that
20
×
=
20
would
100
=
lake.
●
How
in
many
the
lake
sh
if
would
you
you
estimate
recaptured
11
are
tagged
sh?
●
●
What
are
being
made
What
none
the
in
assumptions
this
conclusion(s)
of
tagged?
17 6
all
the
sh
that
are
method?
could
you
you
caught
draw
later
on
if
were
Skating
In
the
by
Olympics,
gure
skating
single
number
complicated
numbers
how
do
routines
from
than
the
boil
each
you
complex
down
judge?
might
to
It’s
The
just
a
nal
which
more
think.
score
is
randomly
awarded
by
7
7scores,
the
accumulate
points
in
a
to
prevent
competition
and
judges
also
to
from
make
the
9
highest
and
using
the
the
computer
scores
judges.
and
a
Of
lowest
those
scores
remaining
5
system
areadded
designed
selects
of
aredisregarded
Skaters
tallied
xing
scoring
up
for
the
nal
score.
a
less
subjective.
First,
skaters
move
they
get
a
basic
execute,
score
for
regardless
every
of
how
its
executed.
The
judges
this
basic
then
score,
add
to
based
or
on
subtract
how
from
well
it
was
performed.
Skaters
are
transition,
skating
it
execution,
hot
possible
Scoville
The
to
give
various
Bell
heat
a
on
choreography,
interpretation
and
a
is
the
Bell
a
a
of
to
diluted
trained
naturally
of
scale
method
then
good
types
hot?
create
and
peppers
is
to
panel
method
does
is
devised
isdissolved
given
judged
skills.
How
Is
also
barely
Carolina
do
in
just
a
of
In
that.In
1912,
who
rate
given
amount
the
scale,
sugar
the
that
the
American
Scoville’s
solutionof
imprecise,
of
it
is
method,
water.
heat
level
based
heat
pharmacist
(and
driedpepper
Varying
in
on
concentrations
Scoville
human
pain!)
Wilbur
are
HeatUnits.
subjectivity,
experienced
but
when
it
eating
peppers.
register
Reaper,
on
weighing
pepper
0
spiciness?
tasters
estimation
chili
of
Bird’s
100
000
–
in
with
at
eye
350
a
2.2
rating
million
Red
000
of
000
The
Scoville
Savina
350
0.
–
current
000
of
units.
habanero
580
champion
Ghost
over
1
pepper
000
000
17 7
Can
I
exchange
this
please?
6.1
Global
context:
Objectives
●
Converting
Globalization
Inquiry
between
dierent
currencies
and
sustainability
questions
●
Why
●
What
●
When
do
we
need
to
convert
currencies?
F
●
Recognizing
●
Solving
●
conversions
word
problems
SPIHSNOITA LER
Understanding
and
the
commission
in
real-life
involving
meaning
of
contexts
Calculating
ATL
exchange
rate?
dierent
C
rates
are
currencies
quantities
in
dierent
equivalent?
charges
commission
Information
Access
an
currencies
●
●
is
information
charges
How
do
systems
inuence
communities?
D
literacy
to
be
informed
and
inform
others
6.1
7.1
Statement of Inquiry:
Quantities
and
measurements
illustrate
the
relationships
between
7.2
human-made
17 8
systems
and
communities.
N U M B E R
Y
ou
●
should
calculate
already
the
know
percentage
of
how
an
1
amount
to:
Find:
a
c
●
calculate
percentage
change
2
25%
20%
of
of
386
78
kg
Ω
b
1%
of
83
mm
d
8%
of
75
liters
The cost of a package of noodles
increases from $1.75 to $2.15. What
is the percentage increase in cost?
Introduction
F
●
Why
●
do
What
is
we
an
need
to
to
currency
convert
exchange
conversion
currencies?
The
rate?
of
The
number
of
people
travelling
around
ocial
monetary
the
world
is
constantly
a
country
there
is
currencies
no
to
Example
Convert
single
global
understand
currency,
the
you
purchasing
need
power
to
of
be
a
able
to
convert
particular
is
increasing.
called
As
standard
the
currency
between
currency.
1
1500
Swiss
1
CHF
=
1.20
EUR.
1
CHF
=
1.20
EUR
Francs
(CHF)
into
Euros
(EUR).
The
exchange
rate
is
Write out the exchange rate.
Method 1:
Think: ‘What mathematical operation do I
1500
×
1
CHF
=
1500
×
1.20
=
1800
EUR
EUR
need to perform to get from 1 to 1500?’
Multiply both sides by 1500.
Method 2:
Multiply by the exchange rate as a
1500
CHF
fraction, to cancel the Swiss Francs.
Example
Convert
2
800
Euros
into
Swiss
Francs.
The
exchange
rate
is
1
CHF
=
1.20
EUR.
Method 1:
1
CHF
=
1.20
CHF
=
1
CHF
⇒
800
EUR
=
=
EUR
EUR
800
666.67
EUR
Write out the exchange rate.
Express the exchange rate in terms of Euros.
Multiply both sides by 800.
CHF
Method 2:
Multiply by an equivalent fraction.
6.1 Can I exchange this please?
17 9
Reect
and
discuss
In
1
Switzerland,
the
●
Divide
both
sides
of
1
CHF
=
1.20
EUR
by
1.20.
What
have
you
is
●
What
do
your
results
mean
in
the
context
of
currency
smallest
0.05
for
exchange?
of
a
CHF,
rate
terms
of
of
exchange,
another
Objective:
ii.
apply
In
this
A.
the
Knowing
selected
practice
or
set
exchange
rate,
gives
the
value
of
one
currency
in
currency.
666.67
you
and
understanding
mathematics
will
use
the
successfully
skill
of
when
converting
solving
currencies
to
problems
solve
the
following
problems.
Practice
1
1
US
2
1
CHF
3
Dollar
=
1.02
=
0.90
Euros
(EUR).
1200
CHF
into
USD.
b
Convert
1734
USD
into
CHF.
The
exchange
a
is
You
1
Using
The
to
will
the
table
from
=
0.52
convert
exchange
600USD
b
rate
CAD
want
rate
be
same
below
is
Canadian
GBP.
600
1
US
USD
worth
in
Chinese
Dollars
=
some
Peso
have
how
many
Find
18 0
EUR.
GBP
(USD)
AUD.
into
into
into
Great
Australian
Calculate
British
Pounds
Dollars
(AUD).
CAD.
how
much
your
convert
500
rates
AUD
for
the
into
USD.
Japanese
Yen
(JPY).
JPY
0 .053
how
to
Krone
1250
has
how
you
855
Yen
will
Yen
Mexican
nearest
he
exchange
Give
Pesos
will
Pesos
Peso.
6 Quantity
to
receive.
Mexican
many
many
the
0 .066
Japanese
Yuan
friend
correct
into
0.0076
Calculate
c
USD
0.12
Yuan
You
Your
(CAD)
870
exchange
1
Norwegian
b
1.26
rate,
Euro
a
750
AUD.
exchange
shows
Dollars
Convert
Currency
Mexican
Convert
USD.
Convert
The
5
(USD)
a
(GBP)
4
1
to
for
your
Chinese
answer
exchange
for
Yuan.
to
the
Calculate
nearest
Japanese
Yuan.
Yen.
receive.
there
are
to
the
Euro.
Give
your
answer
may
so
conversion
Example
A
coin
found?
be
(as
2)
in
you
given
666.65
CHF
666.70
CHF.
or
N U M B E R
Problem
6
a
The
solving
exchange
rate
from
Turkish
Lira
(TRY)
to
Euros
(EUR)
is
On
1
TRY
=
0.25
EUR.
What
is
the
value
of
1
Euro
in
Turkish
January
1999,
b
Victor
receives
500
Swiss
Francs
(CHF)
for
400
this
information
to
nd
the
value
of
1
the
concept
EUR.
of
Use
Euro
in
Swiss
the
Euro
7
Find
Use
to
the
the
value
of
exchange
1
Turkish
rate
table
Lira
here
in
for
Swiss
this
coins
Francs.
question.
Give
answers
and
notes
recognized
legal
tender
BRL
CRC
1
However,
members
1 USD
1
1 BRL
0.3976
2.580
1
0.001865
207.8
(EU)
0.004812
have
into
In
Brazilian
Brazil,
Costa
In
Andrew
the
table
trip
Real
spent
Rica
answers
a
he
in
350
US.
He
converted
Calculate
BRL.
(CRC)
spent
his
the
(BRL).
Colón
converted
that
Give
he
Rican
Costa
He
Copy
started
Then
did
he
200 000
remaining
Euro
how
he
CRC.
below
correct
and
to
3
1
BGN
1
CLP
adopted
as
their
BRL
Costa
his
he
into
US
Dollars
he
Rica.
returned
(USD)
received.
How
remaining
USD.
currency.
to
many
BRL?
the
Calculate
US.
the
amount
nd
the
exchange
rates
for
the
three
currencies.
s.f.
AOA
AOA
to
for
Then
back
750
many
went
receive
money
Union
received.
(Angolan
1
all
the
1
common
Andrew
not
536.1
the
1 CRC
8
from
2001.
of
European
c
BGN
Kwanzy)
CLP
(Bulgarian
1
Lev)
(Chilean
0.0170
Peso)
6.03
ATL
Exploration
1
1
€100.
You
start
value
2
You
of
Convert
amount
4
with
€100
spend
original
3
as
correct
4s.f.
USD
b
and
were
January
a
was
Francs.
introduced
c
1
Lira?
Repeat
in
75%
of
amount
the
you
the
have
steps
your
a
currency
you
left
is
converted
have
25%
for:
£
(Japanese
converter
site
or
app
to
nd
the
(USD).
Calculate
above
¥
(Venezuelan
Use
Dollars
left.
money
Dollar/AUD);
Bs
US
left
of
€100,
how
back
your
(Great
so
much
into
initial
British
Yen/JPY);
$
you
have
money
Euros.
only
you
25%
have
Decide
left
of
your
in
USD.
whether
the
€100.
Pounds/GBP);
(Mexican
$
(Australian
Peso/MXN)
and
Bolivar/VEF).
6.1 Can I exchange this please?
181
Financial
C
●
When
are
Exploration
Claudia
2000
1
lives
Euros
in
into
Calculate
Before
1EUR
=
Germany.
many
Calculate
3
Determine
4
Calculate
dierent
currencies
equivalent?
is
planning
at
Canadian
a
sells
holiday
of
Dollars
Germany,
She
a
rate
the
1
she
Claudia
Dollars
to
EUR
Canada.
=
1.39
She
exchanges
CAD.
receives.
becomes
back
to
ill
the
and
bank
has
at
a
to
rate
of
CAD.
how
many
how
Euros
much
Claudia’s
Selling
in
Dollars
leaves
holiday.
1.48
2
She
Canadian
even
cancelher
quantities
2
how
she
mathematics
(Going
money
loss
on
she
as
a
receives.
the
bank
makes
percentage
holiday)
of
from
her
the
original
Buying
(On
two
transactions.
2000
return
EUR.
from
holiday)
The
If
you
have
country
to
is
a
currency
(HC)
foreign
selling
The
bank’s
CAD
is
=
(FC)
a
then
fee
buy
change
the
If
it
that
bank
have
you
home
another.
The
foreign
want
to
currency
buying
foreign-exchange
into
and
you
and
currency.
currency
they
to
home
your
currency
change
(HC)
foreign
providers
commission
then
to
the
your
bank
is
and
rates
from
the
view
of
are
point
the
of
bank.
currency.
charge
is
it
buying
selling
(FC)
for
charged
in
the
sell.
3
0.73
5000
=
your
want
foreign
exchange
CAD
Convert
one
that
Example
1
the
commission
currency
is1
you
currency
you
exchanging
A
and
from
rate
USD.
CAD
0.73
to
from
The
Canadian
bank
USD,
Dollars
charges
taking
into
1%
(CAD)
US
Dollars
(USD)
commission.
account
USD
to
the
bank’s
commission.
Write out the exchange rate.
First multiply both sides by 5000
5000
×
1
CAD
=
5000
×
0.73
USD
to work out the conversion.
5000
CAD
=
3650
USD
Find 1% of the conversion to work
3650
USD
×
0.01
=
36.5
USD
out the cost of the commission.
Subtract the cost of the commission from the conversion
3650
USD
−
36.5
USD
=
3613.50
USD
to nd out how much you would receive.
18 2
6 Quantity
N U M B E R
Practice
1
1
US
2
Dollar
a
Convert
b
A
(USD)
840
=
USD
commission
of
0.90
into
1%
Euros
(EUR).
EUR.
is
charged
for
the
conversion.
Find
the
cost
of
the
More
commission
in
than
worth
c
Calculate
the
nal
number
of
Euros
received.
1
CHF
=
1.02
USD.
A
commission
of
1%
is
3
The
3500
exchange
Rand
(RND)
Convert
800
You
want
bank.
charge
of
3
Your
S
The
A
be
the
answer
the
For
are
rates
this
1
going
each
of
at
the
Seren
GBP
2
In
fountain
the
Pounds
The
(GBP)
bank
into
charges
South
2%
every
in
day.
African
commission.
is
least
by
(GBP)
=
1.30
Francs
some
of
in
into
CHF.
Swiss
There
Francs
is
also
(CHF)
a
xed
at
your
bank
GBP
Swiss
GBP
could
and
received
Francs
of
buy
with
430
133
CHF.
GBP.
Calculate
how
EUR
per
Euros
amount
received
in
exchange
for
B
British
B
into
charged
320
you
exchanged.
terms
from
1%
GBP
transaction.
she
of
in
is
1
GBP
=
1.27
EUR.
transaction.
spending
GBP
money
he
needs
by
two
to
for
his
holiday.
convert.
Give
your
Pound.
3
compare
of
the
the
next
her
the
journey
two
stage
(AUD)
she
many
to
started
Dollars
S
nearest
stage
oered
for
RND.
Pounds
1
Swiss
minimum
the
is
each
number
Exploration
You
British
rate
Pounds
rate
wants
to
British
21.33
are
the
RND.
many
Express
commission
Calculate
into
=
exchanged
British
exchange
Nathan
Great
for
how
friend
Pounds.
5
from
GBP
change
GBP
b
Let
USD.
exchange
Calculate
c
into
into
solving
a
many
1
GBP
to
The
rate
is
Problem
4
CHF
coins
charged.
Rome
Convert
in
thrown
Trevi
2
$2500
EUR.
trip
the
in
oered
below,
banks.
of
into
rates
work
Decide
on
out
the
banks:
Bank
1
and
Bank
the
conversion
using
best
conversion
and
2.
the
then
use
journey.
Australia.
Great
British
She
converted
Pounds
1600
(GBP).
Australian
Calculate
how
many
received.
UK
she
Euros
spent
she
520
GBP.
received
for
Then
her
she
went
remaining
to
France.
Find
how
GBP.
Continued
on
next
page
6.1 Can I exchange this please?
18 3
3
In
France,
Seren
Sheconverted
amount
that
spent
her
190
EUR.
remaining
Seren
Then
money
GBP
GBP
EUR
EUR
sells
buys
sells
buys
0.485
0.515
0.62
0.66
1.271
1.299
1.94
2.06
1
EUR
1.465
1.565
Bank
AUD
GBP
2.0
1
EUR
1.553
that
1
a
have
currency
commission)
GBP
buys/sells
EUR
buys/sells
0.668
0.793
2
dierent
conversion
buying
sites
and
only
selling
oer
a
rates?
single
conversion
rate.
this?
is
the
dierence
in
nancial
currency
might
Practice
1%
1.323
discuss
banks
work
How
0.787
0.52
and
foreign
●
buys/sells
1
What
0.753
(charges
AUD
Whyis
●
2
1
Many
1
AUD
GBP
●
the
buys
1
do
Australia.
Calculate
sells
AUD
Why
to
AUD.
AUD
1
●
returned
into
received.
Bank
Reect
she
back
you
to
between
sites
institutions
go
decide
on
that
and
provide
those
information
aimed
at
people
for
people
needing
a
holiday?
which
bank
or
foreign
exchange
center
to
use?
3
Dafydd
is
going
on
holiday.
He
wants
to
exchange
700
USD
Buy
for
AUD.
Use
the
table
to
calculate
how
many
AUD
he
1
USD
receive.
b
Bernice
980
USD
2
The
ve
has
AUD
she
table
returned
back
will
shows
other
into
from
USD.
holiday.
Use
the
to
to
exchange
calculate
how
a
bank’s
exchange
rate
between
1
Danish
(ALL)
France
Mexico
Buy
DKK
19.25
1.17
1.28
0.13
0.15
(CAD)
0.19
0.22
(MXN)
2.21
2.24
(HKD)
(EUR)
Canada
DKK
18.75
Kong
many
Krone
currencies.
Albania
18 4
wants
receive.
Sell
Hong
She
table
6 Quantity
AUD
Sell
AUD
will
(DKK)
and
1.395
1.379
N U M B E R
a
Your
into
they
b
friends
receive
Your
they
A
they
are
table
below
bank
charges
exchange
in
to
two
exchange
travel
so
CAD
Mexico,
Pesos
decimal
1500
they
they
into
CAD
and
into
lost
returning
DKK
their
on
the
shows
a
part
1%
of
a
currency
commission
after
conversion
the
GBP
USD
CAD
USD
0.65
1
q
1
CAD
0.53
0.81
1
calculations
Calculate
the
this
the
taking
He
UK
Christine
Shillings
Denmark
number
of
their
back
holiday.
into
CAD.
lives
1
buys
Bank
2
has
1
an
from
to
In
the
give
USA
exchange
spends
whether
at
a
bank.
your
answers
correct
to
2
d.p.
890
GBP.
Canada
he
has
to
1500
he
He
plans
enough
the
UK
USD.
then
to
CAD
and
How
then
many
exchanges
stay
for
in
a
a
his
hotel
hotel
home
GBP
he
via
receive?
remaining
for
room
again
will
three
nights.
charged
at
night.
in
(TZS)
question
chart
q
trip
CAD.
per
Bank
a
wants
Stefan
into
CAD
to
the
of:
ii
Determine
45
in
value
p
money
Pesos
conversion.
1
In
DKK
transaction.
1.90
is
before
money
p
Stefan
600
Mexican
Calculate
1
all
of
places.
exchange
have
on
Krone.
GBP
Canada.
c
they
number
1
i
b
the
solving
The
a
to
many
The
For
MXN
remaining
correct
family
leaving,
Calculate
DKK.
300
their
how
Problem
4
spend
Before
(MXN).
600
unable
Calculate
Mexico.
receive,
Canadian
They
3
for
friends
to
Pesos
exchange
DKK
c
go
Mexican
Tanzania.
into
TZS
US
for
She
Dollars
0.0005
exchange
rate
wants
USD,
of
to
exchange
350 000
Tanzanian
(USD).
1
and
TZS
=
sells
1
TZS
0.00051
for
USD,
0.0004
and
USD.
charges
a
1%
commission.
At
which
bank
would
you
advise
Christine
to
exchange
her
money?
6.1 Can I exchange this please?
18 5
Currency
D
●
How
do
as
a
systems
quantity
inuence
communities?
ATL
Exploration
1
Compare
the
4
DETECT
currencies
for
each
group
of
countries
Group
in
the
tables.
grouped
in
Decide
this
why
the
countries
have
1
Dene
2
Examine
the
3
Track
task
A
been
France
way.
information
Germany
2
Research
the
bitcoin
and
answer
these
down
questions.
Spain
sources
a
What
is
the
bitcoin
and
how
does
it
and
nd
Greece
work?
information
in
Italy
those
b
Who
created
the
bitcoin?
Why
was
the
sources
bitcoin
4
Extract
developed?
meaningful
Group
c
Is
the
bitcoin
secure?
Is
it
B
legal?
information
Denmark
5
3
Exchange
rates
are
constantly
Create
United
Research
ve
currencies
and
analyze
change
in
the
last
year.
Group
b
Determine
which
about
this
have
changed
the
most.
C
Think
Switzerland
why
might
be.
Sweden
Serbia
Reect
Single
and
discuss
3
currency:
●
Why
have
●
Why
do
●
What
some
some
are
the
European
EU
countries
countries
advantages
opt
and
out
decided
of
the
not
to
join
the
EU?
Euro?
disadvantages
of
a
single
European
currency?
New
currencies:
●
Does
the
●
Does
every
extent
does
Exchange
●
Of
17
What
the
of
bitcoin
person
most
them
do
letters
use
you
are
capital
future?
inuence
right
equal
traded
a
to
create
access
world
exchange
currencies,
strikethrough
think
used
the
in
the
rather
their
or
symbol.
struck-through
than
just
letters?
18 6
the
have
to
rates:
double-strikethrough
Why
a
have
everyone
factors
20
have
6 Quantity
plain
inform
others
the
6
percentage
product
Kingdom
to
a
a
changing.
rate?
their
these
own
currency?
opportunities?
To
what
Task
reection
N U M B E R
Summary
A
rate
value
The
of
of
exchange,
one
commission
providers
Mixed
1
1
EUR
or
currency
charge
is
a
for
exchange
in
terms
fee
that
rate,
of
gives
another
the
currency.
foreign-exchange
exchanging
one
currency
into
another.
currency
The
buying
view
of
the
=
1.10
Convert
6
USD.
The
950
EUR
into
1
buy
and
selling
who
rates
is
is
charged
in
the
sell.
are
buying
from
or
the
point
selling
of
currencies.
exchange
Convert
2750
USD
Mexican
Peso
into
(MXN)
Baht
rate
from
(THB)
is
US
1
Dollars
USD
=
(USD)
33.45
THB.
USD.
=
the
answers
to
the
following
correct
EUR.
to
2
and
bank,
Thai
Give
b
commission
they
practice
to
a
The
that
0.055
US
4
s.f.
Dollars
a
Find
the
value
b
Calculate
of
115
US
Dollars
in
THB.
(USD).
a
Convert
b
A
4400
MXN
into
c
commission
of
1%
is
c
the
cost
Calculate
of
the
the
nal
Alexis
receives
commission
number
of
in
of
USD.
USD
d
received.
1
for
THB
1
CHF
=
charged.
0.72
GBP.
Convert
A
commission
1600
CHF
into
of
2%
7
4
RUB
A
bank
a
GBP
Sell
107.30
Jasmine
She
wants
Russian
is
to
going
Rubles
for
holiday.
exchange
buys
how
1400
GBP
for
a
Find
b
Frida
RUB
she
will
Kali
She
has
wants
Calculate
5
The
table
returned
to
change
how
many
below
conversion
from
chart.
holiday.
1420
RUB
GBP
show
part
Find
the
she
of
GBP
AUD
670
New
THB.
value
1
of
Zealand
Dollars
Calculate
the
Australian
(EUR)
Frida
how
has
US
the
Dollar
Dollar
and
a
back
will
to
GBP.
of
c
to
1
sells
to
Euros
cancel
back
AUD
value
in
New
(AUD)
1
AUD
exchange
=
Frida
her
later
0.73
8
p
and
q
Ed
EUR
GBP
AUD
had
Swiss
1
p
1.53
1.27
1
1.92
q
0.52
1
2%
a
b
lost
to
on
for
800
AUD
trip
and
when
EUR,
will
receive.
changes
the
rates
sells
1
are
1
how
many
AUD
=
Australian
receives.
how
the
change
Francs
was
Find
she
Determine
has
receive.
currency
value
wants
many
EUR’.
Dollars
rate
EUR
600
receive.
0.69
b
USD.
NZD.
the
hermoney
(RUB).
many
in
EUR.
‘buys
Calculate
THB
RUB
105.80
on
1
Dollars.
0.72Euros
0.70EUR.
1
of
is
GBP.
for
Buy
14
in
Calculate
Zealand
3
value
charged.
(NZD)
Find
the
USD.
many
=
Dollars
she
transaction.
British
(CHF)
GBP
Australian
at
1.4
a
Pounds
bank.
CHF.
(GBP)
The
The
into
exchange
bank
charged
commission.
Determine
how
bought
with
Ed
100
has
Find
how
many
200
CHF
many
Swiss
Francs
Ed
GBP.
of
his
British
initial
amount
Pounds
6.1 Can I exchange this please?
he
left.
could
18 7
buy.
9
Santiago
67 000
lives
in
Mexican
Mexico.
Pesos
He
wants
(MXN)
into
to
change
Euros
a
Calculate
Pavla
leaves
Zealand.
A
bank
buys
1
MXN
for
0.051
EUR
and
MXN
for
0.0531
exchange
1MXN
=
of
THB
Pavla
buys.
has
20
and
000
travels
THB
to
and
New
uses
buy
New
Zealand
Dollars
(NZD).
these
The
EUR.
center
has
an
exchange
0.054
EUR
and
rate
is
24
450
THB
=
1000
NZD.
rate
b
of
Thailand
She
exchange
An
number
sells
to
1
the
(EUR).
charges
Calculate
the
total
number
of
New
Zealand
a
Dollars
Pavla
receives.
2%commission.
c
Find
the
EUR
dierence
he
would
exchange
between
receive
the
from
number
the
bank
Find
an
New
Zealand
Give
your
Pavla
She
travels
from
changes
Baht.
The
Review
The
in
He
is
rate
and
He
into
is
Republic
Koruna
1
CZK
to
bank
a
and
b
of
will
rate
is
of
Determine
Let
1330
s
be
Jim’s
and
be
=
company
starting
a
exchanging
Dollars
1
GBP
=
10
GBP
for
how
to
the
2.07
many
are
in
British
2
the
company
Andy,
to
in
(SGD).
SGD,
each
SGD
b
of
GBP
.
of
the
15 000
account.
to
and
Jim
SGD
there
is
s
in
in
used
could
used
Jelena
10
The
buy
second
He
rate
year,
from
has
can
two
use
Jim
ways
the
of
=
SGD,
for
each
Singaporean
2%
=
for
0.49
and
which
to
How
you
have
and
human-made
18 8
3000
his
USD
to
USD
3000
to
the
USD
buy
buy
to
Swiss
USD
Francs.
Brazil
buy
exchange
conversion
Reals.
and
at
Swedish
rates
per
ve-year
2010
CHF
1
Krone.
USD
at
intervals.
2015
0.97
CHF
0.85
CHF
2.50
BRL
1.52
BRL
3.29
BRL
7.25
SEK
6.40
SEK
9.30
SEK
UK
bank
(the
Ignoring
any
there
is
a
bank
bank
buying
and
selling
costs:
Calculate
(the
or
he
exchange
and
there
is
a
can
rate
bank
chosen
the
better
discuss
explored
the
Calculate
and
of
illustrate
inquiry?
the
person
received
2005.
amount
in
USD
each
person
receive
if
they
chose
to
sell
in
2010.
charge
c
communities.
6 Quantity
the
Calculate
if
the
amount
they
chose
to
USD
in
Give
specic
examples.
relationships
between
each
to
option.
statement
each
in
use
3
measurements
systems
amount
currency
is
transaction).
is
the
charge
Statement of Inquiry:
Quantities
3000
to
his
back
and
had
converting
British
transaction),
bank
GBP,
each
Reect
each
b
receive
Decide
3000
shows
of
would
SGD
of
=
is
2.06
GBP
Jelena
2005
of
needs
Singapore
and
his
table
time
in
terms
b
1
NZD
2005.
his
used
their
the
1
a
a
GBP
of
form
Pounds
received
Express
SGD
He
GBP
.
exchange
1
the
d.p.
The
transaction.
SGD
Neil
invest
the
number
for
end
transfer
bank
in
2
1.34THB.
1.32
At
to
Thai
GBP.
the
exchange
c
answer
correct
sustainability
Singaporean
charge
with
between
Koruna.
Thailand.
(CZK)
Neil
exchange
rate
Czech
the
Andy
(GBP)
and
context
expanding
Singapore.
Czech
Czech
exchange
headquarters
UK.
the
29 000
Globalization
1
exchange
Dollars
center.
xCZK,
10
approximate
of
2015.
person
convert
it
would
in
City
skylines
6. 2
Global
context:
Objectives
●
Constructing
relative
Globalization
Inquiry
and
interpreting
frequency
histograms
frequency
with
equal
●
and
F
class
and
sustainability
questions
What
are
chart
and
How
do
the
a
dierences
between
a
bar
histogram?
widths
●
●
and
interpreting
histograms
with
unequal
Describing
distributions
frequency
class
density
C
you
distribution
accurately
from
a
analyze
a
data
histogram?
widths
●
How
can
●
How
do
real
data
ever
be
misleading?
D
individuals
stand
out
in
a
crowd?
ATL
Critical
Revise
thinking
understanding
based
on
new
information
and
evidence
Statement of Inquiry:
4.1
How
to
quantities
establish
trends
in
a
are
represented
underlying
can
help
relationships
and
population.
4.2
6.2
18 9
SPIHSNOITA LER
●
Constructing
Y
ou
●
should
represent
discrete
already
continuous
data
frequency
in
a
know
how
and
1
to:
The
grouped
heights
(rounded
to
measures
grouped
of
and
central
dispersion
frequency
2
from
5.8,
2.2,
4.9,
3.0,
4.7,
5.3,
2.6,
4.5,
3.7,
2.3,
5.4,
5.7,
3.5,
2.1,
and
3
your
modal
contains
Before
char ts
now,
you
quantitative
are
the
may
discrete
representing
frequency
for
this
frequency
class
the
4.1
and
the
table
data.
table,
class
median.
estimates
for
the
mean
and
range.
as
qualitative
or
quantitative
data:
shoe
c
What
2.8,
widths
grouped
that
b
●
2.5,
grouped
the
a
Bar
3.6,
class
nd
Classify
data
4.2,
a
equal
From
a
table
qualitative
quantitative
F
are:
2.6,
the
recognize
seedlings
5.2,
Calculate
●
d.p.)
24
3.1,
with
tendency
of
2.4,
Construct
nd
1
cm
table
4.0,
●
in
and
data.
quantitative
drawn
is
shoe
a
between
bar
However,
data
shoe
color
price
frequency
dierences
have
size
charts
strictly
a
bar
for
histograms
chart
both
and
a
histogram?
qualitative
speaking,
a
chart
and
with
bars
histogram.
A
bar
char t
y
A
bar
chart
represents
qualitative
data.
30
All
●
There
the
bars
are
have
spaces
●
The
●
Frequency
height
of
the
equal
between
bar
25
width.
the
ycneuqerF
●
bars.
represents
the
20
15
10
frequency.
5
is
on
the
vertical
axis.
0
x
Cat
A
histogram
represents
quantitative
discrete
data
and
all
Dog
A
Rat
Parrot
Snake
histogram
y
continuous
data.
8
7
●
●
bars
There
The
may
are
area
no
of
or
may
spaces
the
bar
not
be
of
between
equal
the
represents
the
width.
6
ycneuqerF
●
The
bars.
5
4
3
frequency.
2
1
●
Frequency,
relative
frequency
or
frequency
density
are
0
on
the
vertical
axis.
x
1.10
1.20
1.30
1.40
Height
19 0
6 Quantity
1.50
(m)
1.60
1.70
1.80
S TAT I S T I C S
A N D
P R O B A B I L I T Y
ATL
Reect
and
discuss
The
1
word
originates
Decide
which
of
these
graphs
are
bar
charts,
which
are
histograms
are
neither.
Justify
your
words:
histos
answers.
meaning
upright’
Favourite
from
and
Greek
which
histogram
colors
Pupil
to
teacher
‘stands
and
ratio
y
gram
y
meaning
‘a
16
7
drawing’.
14
6
ycneuqerF
12
5
egatnecreP
4
3
2
10
8
6
1
4
0
x
Blue
Red
Green
Orange
2
Color
0
x
<18
18
19
20
21
22
23
24
25
26
27 Over
27
Pupil/Teacher
ratio
Lengths
Mid-term
examination
of
leaves
scores
y
y
25
12
25
10
ycneuqerF
ycneuqerF
8
6
4
2
20
15
10
As
the
scores
not
0
24 26 28 30 32 34 36 38 40 42 44
x
0
x
40
60
Score
80
Length
100
120
140
the
start
zigzag
values
histograms
for
continuous
one
table
Weight,
some
not
on
the
axis.
1
shows
week.
symbol
data
scores
Example
zero,
that
are
included
Frequency
at
(mm)
indicates
The
do
5
Draw
x
the
a
(grams)
19.5
<
x
≤
20
weights
histogram
in
grams
to
Freq
3
of
50
represent
baby
the
chicks
hatched
at
a
farm
in
data.
Weight,
x
(grams)
22.5
<
x
≤
23
Freq
4
20
<
x
≤
20.5
5
23
<
x
≤
23.5
6
20.5
<
x
≤
21
1
23.5
<
x
≤
24
5
21
<
x
≤
21.5
6
24
<
x
≤
24.5
5
21.5
<
x
≤
22
2
24.5
<
x
≤
25
6
22
<
x
≤
22.5
6
25
<
x
≤
25.5
1
Continued
on
next
page
6.2 City skylines
19 1
y
7
Use
your
frequency
table
to
draw
6
the
histogram
on
graph
paper.
5
ycneuqerF
4
3
Plot
the
class
boundaries
on
2
thex-axis
and
the
frequency
on
1
the
0
19
20
21
22
23
Weight
24
25
y-axis.
Draw
the
bars
with
x
26
nogapsbetween
them.
(g)
7
6
5
ycneuqerF
4
3
Use a GDC to check
2
your histogram.
1
0
19.5
20.5
21.5
22.5
Weight
In
to
Example
one
1,
the
decimal
23.5
weights
place,
24.5
25.5
(g)
so
a
are
continuous
histogram
is
data,
the
even
correct
though
they
are
rounded
representation.
The
are
class
intervals
sometimes
ATL
Reect
Ages
are
nearest
If
a
and
rounded
meter,
woman
continuous
The
discuss
table
is
its
dierently
height
40
years
gives
the
●
What
●
How
would
are
1
The
most
her
age
centimeter,
Height
of
h
<
If
a
r
tree
is
10
meters
tall
to
the
10.5.
satises
40
≤
a
<
41,
where
in
an
a
is
a
orchestra.
26–33
34–41
42–50
5
8
10
7
draw
class
you
bars
on
boundaries
write
the
class
of
the
are
chicks
recorded
(cm)
from
a
histogram
for
each
interval,
in
Example
this
to
show
these
ages?
class?
using
≤
and
≤
x
<
2.5
1
≤
x
<
3.5
3
3.5
≤
x
<
4.5
7
4.5
≤
x
<
5.5
14
5.5
≤
x
<
6.5
18
6.5
≤
x
<
7.5
5
7.5
≤
x
<
8.5
2
frequency
grouped
1,
measured
frequency
Frequency
2.5
19 2
a
musicians
1.5
a
data.
≤
<,
for
the
class
1
heights
Draw
9.5
18–25
you
the
would
Practice
to
satises
old,
ages
Frequency
How
h
called
quantity.
Age
●
2
histogram
6 Quantity
to
represent
the
data.
to
the
table:
nearest
18–25?
bins.
S TAT I S T I C S
2
The
ages
of
100
shoppers
Age
14
≤
Number of users
Draw
a
3
The
the
<
24
24
≤
x
<
for
a
34
survey
34
≤
30
histogram
to
represent
x
are
<
given
44
in
44
≤
15
the
the
x
<
P R O B A B I L I T Y
table.
54
54
8
≤
x
<
64
6
data.
solving
incomplete
masses
x
chosen
41
frequency
Problem
randomly
A N D
(to
table
the
and
frequency
histogram
nearest
kilogram)
of
some
give
some
female
information
polar
about
bears.
y
16
14
12
ycneuqerF
10
8
6
4
Class
Class
inter val
boundaries
Frequency
80–119
79.5–119.5
15
120–159
2
160–199
70
90
110
130
150
Mass
a
Use
b
the
You
histogram
Complete
Frequency
can
200–239
x
0
the
to
190
210
230
8
250
(kg)
complete
the
table.
histogram.
histograms
represent
170
for
discrete
ungrouped
data
discrete
Eggs
data
hatched
in
incubators
y
with
a
ver tical
line
graph.
30
This
type
discrete
of
graph
data.
A
is
not
single
suitable
ver tical
for
line
grouped
to
30
represent
22
but
categor y
1–3
separate
eggs
lines
at
could
0,
1,
2
be
and
misleading,
3
could
be
confusing.
ycneuqerF
the
20
18
20
15
12
10
10
8
4
2
1
0
0
0
1
0
2
3
4
5
Number
A
histogram
for
grouped
discrete
data
For
to
represent
discrete
from
upper
lower
data,
the
1–3,
point
boundar y
boundar y
of
of
group.
4–6,
7–9,
halfway
one
the
etc.,
plot
between
class
eggs
8
9
10
11
12
hatched
hatched
in
incubators
y
inter val
the
the
and
the
next.
100
ycneuqerF
bars
each
of
7
uses
Eggs
bars
6
80
60
40
20
For
4–6,
Star t
the
plot
the
rst
same
the
bar
width.
bar
at
between
0.5,
to
3.5
make
and
all
6.5.
the
bars
0
x
1
2
3
4
5
Number
6
of
7
eggs
8
9
10
11
12
13
hatched
6.2 City skylines
19 3
13
x
A
histogram
1,
2,
1.5
3,
to
4
…
2.5,
class
ungrouped
and
plot
bars
so
from
discrete
0.5
to
y
data
60
1.5,
50
on.
40
ycneuqerF
You
for
uses
the
intervals
30
20
10
of
cumulative
frequency
curves
in
0
x
1
a
similar
way.
Practice
1
The
Number
table
shows
of
the
numbers
people
Frequency
a
Classify
b
Represent
There
the
were
unhatched
3
data
the
25
as
data
trays
eggs.
18
17
13
9
21
2
5
12
1
14
11
13
15
18
21
14
15
9
12
14
15
16
24
in
grouped
b
Construct
a
frequency
lengths
cm.
of
of
30
The
Swiss
table
the
Determine
b
Draw
taxi
the
8
11
6
4
2
2
or
continuous,
frequency
in
an
of
incubator.
cheese
shows
for
plant
the
that
table
histogram
or
ungrouped.
histogram.
eggs
frequency
grouped
Each
tray
hatched
for
the
this
leaves
started
from
each
with
tray
24
were:
data.
data.
were
measured
to
the
results.
of
≤
x
<
15
15
≤
whether
records
a
x
<
20
20
≤
x
<
25
25
≤
x
<
30
cm)
frequency
driver
course
cars.
6
3
a
A
33
5
Frequency
4
in
leaf
nearest
a
travelling
4
10
(to
people
3
eggs
a
Length
orders
2
numbers
Construct
nearest
of
4
1
a
a
The
of
discrete
of
The
12
5
3
2
Number
2
2
long
this
is
8
discrete
histogram
the
to
distances
weekend
or
12
continuous
represent
of
his
the
7
data.
data.
journeys,
to
the
nearest
km,
shift.
over
For
9
Distance
(km)
1
5
6
grouped
continuous
10
11
15
16
20
21
25
12,
etc.
13
are
the
intervals.
Frequency
10
6
8
5
Write
down
b
Determine
the
the
modal
class
class.
boundaries
for
each
Draw
19 4
a
frequency
histogram
6 Quantity
for
this
boundaries
are
8.5
8.5
<
x
class.
so
c
data.
class
The
2
class
a
data,
16
on.
12.5
≤
or
12.5,
and
S TAT I S T I C S
Problem
5
This
A N D
P R O B A B I L I T Y
solving
histogram
has
no
title
or
axis
y
labels.
ATL
a
Determine
whether
it
represents
discrete
35
or
continuous
data.
30
b
Suggest
what
the
c
Do
think
data
could
be.
25
you
that
the
data
represented
20
here
contains
any
outliers?
If
not,
why
not?
15
10
5
0
6
The
histogram
that
Macey
day
in
shows
spent
the
doing
number
of
homework
hours
1
2
3
4
5
6
4
5
6
7
x
y
each
9
June.
8
a
Use
the
information
histogram
to
in
construct
the
a
frequency
7
grouped
6
frequency
Use
an
your
frequency
estimate
for
the
table
mean
to
ycneuqerF
b
table.
calculate
time
that
5
4
3
Macey
spent
doing
homework.
2
1
0
1
2
3
T ime
Distributions
C
●
How
do
you
in
x
(hours)
histograms
accurately
analyze
a
data
distribution
from
a
histogram?
Examining
data
set
To
it
a
histogram
can
help
you
visualize
certain
characteristics
of
the
represents.
comment
on
the
distribution
of
Center
measure
of
central
Spread
measure
of
dispersion
Outliers
extreme
Shape
unimodal,
data
data,
describe:
tendency
values
bimodal
that
or
don’t
t
the
pattern
multimodal
6.2 City skylines
19 5
Example
This
2
frequency
caught
in
the
histogram
Kispiox
shows
River,
the
British
lengths
in
centimeters
Colombia,
of
57
sh
Canada.
y
11
10
9
8
7
ycneuqerF
6
5
4
3
2
1
0
10
20
30
40
50
60
70
Length
a
80
90
100
110
120
130
x
(cm)
Find:
i
the
modal
ii
the
class
iii
an
class
interval
estimate
b
Describe
a
i
The
ii
57
the
the
contains
the
median
range.
distribution.
modal
data
for
that
class
is
60
≤
x
<
70.
items
Count
The
median
The
class
=
=
29th
item
of
each
iii
b
Estimate
The
is
median
also
the
caught
There
Most
90
≤
are
x
<
for
≤
x
the
lies
in
modal
are
sh
60
contains
range
the
have
=
120
class
The
spread
outliers.
caught
100
70
class.
widely
no
<
60
≤
20
x
<
20
60
≤
=
100
70
of
cm
x
bar,
6 Quantity
of
data
items
in
until
you
reach
the
29th.
<
and
the
to
cm
this
sh
120
is
70
cm.
bimodal.
cm
or
To
describe
measure
and
a
State
of
a
distribution,
central
measure
if
there
describe
19 6
number
median.
distribution
length
cm.
–
lengths
from
The
the
the
data.
the
is
of
use
tendency
dispersion
an
outlier,
shape.
a
(center)
(spread).
and
S TAT I S T I C S
These
same
three
50
histograms
eggs
ungrouped
measured
data,
all
show
one
appears
to
the
week
be
a
weights,
before
multimodal
Ungrouped
to
the
hatching.
nearest
The
gram,
rst
of
A N D
P R O B A B I L I T Y
the
histogram,
for
distribution.
data
y
ycneuqerF
6
4
2
x
0
35
40
45
50
55
Weight
60
65
70
75
(g)
4
class
inter vals
y
This
appears
to
be
a
unimodal
distribution
24
43
≤
w
<
53)
that
is
not
ycneuqerF
(at
symmetrical.
18
12
6
0
x
35
40
45
50
55
60
Weight
10
This
that
appears
is
fairly
to
be
a
bimodal
class
65
70
75
(g)
inter vals
y
distribution
15
symmetrical.
12
ycneuqerF
9
6
3
0
x
35
40
50
45
Weight
Reect
●
How
●
What
and
can
are
discuss
changing
the
the
eects
of
60
55
65
70
75
(g)
3
class
intervals
having
too
change
few
the
classes?
shape
What
of
a
about
distribution?
too
many
classes?
●
How
of
many
the
data
Practice
1
is
set
intervals
when
should
deciding
on
you
the
use?
Do
number
you
of
need
to
consider
the
size
classes?
3
Determine
i
class
whether
each
histogram:
symmetrical
ii
is
a
unimodal,
bimodal
or
multimodal
iii
b
contains
outliers
c
y
y
16
y
140
12
fo
8
rebmuN
ycneuqerF
10
6
4
120
100
ycneuqerF
slaudividni
14
80
60
40
20
2
032
022
012
percentage
002
fat
x
x
50
Body
Body
091
45
081
40
071
35
061
30
051
25
041
x
20
031
0
0
mass
(g)
Spanish
6.2 City skylines
test
score
19 7
2
The
26
histogram
black
lengths
y
of
10
b
Find
down
the
the
class
modal
interval
class.
that
8
contains
ycneuqerF
Write
median.
c
Calculate
d
By
rst
an
the
estimate
constructing
frequency
for
the
rats.
a
the
shows
table,
mean
for
a
the
range.
6
4
grouped
calculate
an
2
estimate
length.
0
x
28
e
Describe
the
Problem
3
The
to
44
40
Length
48
52
(cm)
solving
histogram
money,
36
32
distribution.
the
shows
nearest
the
amounts
$10,
y
of
18
some
16
families
spend
on
food
each
week.
14
Write
down
b
Calculate
amount
the
an
total
number
estimate
spent
on
for
the
of
families.
12
ycneuqerF
a
mean
food.
10
8
6
c
Comment
on
the
distribution
of
the
data.
4
2
0
x
145
155
Amount
4
5
Imogen
a
Explain
b
Sketch
Here
are
nature
why
a
bar
the
she
chart
in
in
her
should
weights
reserve
the
to
class
use
predict
(to
the
Snowy
a
to
bar
what
nearest
tell
her
chart
the
to
their
show
whole
kg)
of
50
35
36
24
29
29
29
34
38
35
35
35
36
60
35
50
34
48
41
41
51
42
35
36
32
61
30
40
41
19
33
34
17
35
35
38
35
42
20
29
50
33
37
28
49
58
45
40
b
Describe
order
data
frequency
the
to
make
points,
you
histogram
fair
the
comparisons
use
a
total
for
this
relative
between
frequency
frequency
axis.
19 8
175
($)
month.
results.
would
emus
look
in
an
like.
Australian
data.
histograms
can
of
food
distribution.
frequency
proportion
vertical
a
the
165
on
Mountains.
24
Construct
birthday
distribution
19
a
the
everyone
33
Relative
In
asked
spent
6 Quantity
in
data
sets
histogram.
each
class
with
dierent
Relative
interval.
It
numbers
frequency
is
plotted
of
shows
on
the
S TAT I S T I C S
Objective:
v.
justify
real-life
This
in
Applying
whether
a
mathematics
solution
makes
in
real-life
sense
in
the
P R O B A B I L I T Y
contexts
context
of
the
authentic
situation
exploration
an
the
D.
A N D
authentic
comparison
encourages
real-life
of
data
students
context.
makes
The
sense
to
see
mathematics
students
in
this
should
specic
be
as
a
able
tool
to
for
solving
justify
problems
whether
or
not
case.
ATL
Exploration
The
frequency
Kalamazoo
show
the
Sable
10
<
x ≤
20
2
20
<
x ≤
30
52
30
<
x ≤
40
93
40
<
x ≤
50
30
(cm)
<
x ≤
60
25
60
<
x ≤
70
33
70
<
x ≤
80
30
80
<
x ≤
90
30
90
<
x ≤
100
40
100
<
x ≤
110
25
110
<
x ≤
120
30
120
<
x ≤
130
5
Draw
both,
sh
x ≤
140
Copy
b
Find
the
Au
Sable
and
the
each
the
River
Length,
x
10
<
x ≤
20
0
20
<
x ≤
30
8
30
<
x ≤
40
15
40
<
x ≤
50
30
50
<
x ≤
60
10
60
<
x ≤
70
11
70
<
x ≤
80
10
80
<
x ≤
90
7
90
<
x ≤
100
7
(cm)
Frequency
100
<
x ≤
110
8
110
<
x ≤
120
1
120
<
x ≤
130
2
130
<
x ≤
140
4
Total
frequency
with
a
in
7
frequency
a
caught
Frequency
50
Total
of
Kalamazoo
x
<
lengths
River
Length,
130
2
tables
rivers.
Au
1
1
histogram
y-axis
table,
total
from
0
adding
a
frequency
for
to
each
set.
Use
the
same
scale
for
100.
third
of
data
frequency
each
column
data
for
‘Relative
frequency’.
set.
c
Round
d
Draw
the
a
values
relative
boundaries
3
Look
a
of
b
at
sh
of
in
tendency?
the
set
places
histogram
and
is
class
with
frequency
two
relative
n
What
the
should
rivers?
histograms
frequency
data
relative
of
the
central
a
x-axis
histograms
lengths
in
decimal
and
for
add
each
to
your
data
frequency
tables.
set.
on
the
Plot
the
class
y-axis.
histograms.
pair
pair
belonging
2
frequency
the
Which
Relative
For
your
Which
on
to
you
the
proportion
use
Explain
should
about
you
to
the
distribution
fully.
use
to
measures
(or
compare
nd
of
the
measures
of
dispersion?
percentage)
of
the
data
set
interval.
members,
a
class
interval
with
frequency
f
has
.
6.2 City skylines
19 9
Practice
4
Tip
1
The
tables
show
the
length
of
time
some
men
and
women
spent
on
their
ATL
mobile
phones
Time
one
spent
minutes
day.
in
When
Frequency
Time
(men)
spent
minutes
in
Frequency
two
use
(women)
descriptive
words
0
≤
x
<
15
5
0
≤
x
<
15
comparing
distributions,
such
wider,
15
≤
x
<
30
8
15
≤
x
<
30
5
30
≤
x
<
45
10
30
≤
x
<
45
7
45
≤
x
<
60
5
45
≤
x
<
60
14
60
≤
x
<
75
2
60
≤
x
<
75
20
more
Explain
these
2
why
data
b
Calculate
c
Draw
d
Describe
e
Compare the
The
the
tables
Mass,
length
the
a
relative
m
frequencies
for
histograms
of
time
masses
Siberian
(kg)
frequency
histogram
to
compare
each
for
class
the
interval.
two
sets
of
data.
of
spent
on
the
some
male
phone
and
husky
by
female
Female
Frequency
the
Mass,
m
men
and
Siberian
Siberian
(kg)
women.
huskies.
husky
Frequency
17
≤
m
<
19
3
17
≤
m
<
19
5
19
≤
m
<
21
6
19
≤
m
<
21
8
21
≤
m
<
23
6
21
≤
m
<
23
4
23
≤
m
<
25
11
23
≤
m
<
25
2
25
≤
m
<
27
9
25
≤
m
<
27
0
b
Draw
c
Comment
the
relative
relative
Siberian
Problem
a
use
distribution.
show
Calculate
at
to
frequency
each
a
The
need
relative
relative
Male
3
you
distributions.
on
frequency
frequency
the
of
each
histograms
distribution
of
the
for
class
each
masses
interval
data
of
the
as
a
percentage.
set.
male
and
female
huskies.
solving
histogram
shows
community
the
relative
fundraising
frequency
event.
There
of
were
items
32
sold
items
at
in
dierent
total.
y
0.25
ycneuqerf
0.1875
0.125
evitaleR
0.0625
x
0
1
6
11
16
Selling
Calculate
20 0
how
many
items
6 Quantity
sold
for
less
price
21
26
31
(dollars)
than
16
dollars.
36
prices
narrower,
varied,
varied.
a
as:
4
less
S TAT I S T I C S
Histograms
D
This
●
How
can
●
How
do
stem-and-leaf
gram,
of
stem
0
sh
real
data
ever
individuals
diagram
caught
with
in
a
unequal
be
class
A N D
P R O B A B I L I T Y
widths
misleading?
stand
out
in
a
crowd?
represents
the
weights,
measured
to
the
nearest
river.
leaf
8
8
9
2
1
2
2
2
3
4
8
3
2
3
5
5
5
6
7
7
4
1
2
2
3
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
5
6
6
6
6
7
7
7
7
8
8
8
9
9
9
9
9
1
1
1
2
2
3
4
4
4
5
5
5
6
6
6
6
6
7
7
7
7
7
7
8
8
8
8
8
8
8
8
9
9
9
9
9
9
9
0
0
0
0
0
0
1
1
1
2
2
2
3
3
3
4
4
4
4
4
4
5
5
6
6
7
7
8
8
8
8
9
9
9
0
0
1
2
4
5
5
5
7
8
1
5
6
7
8
9
Key :
Both
This
into
This
2
these
|
5
histograms
histogram
small
means
class
grouping
25
g
show
represents
intervals
has
the
the
of
resulted
weights,
data
equal
in
to
the
nearest
of
the
same
136
sh.
grouped
50
width.
three
gram,
45
class
40
intervals
containing
no
data.
35
ycneuqerF
30
25
20
15
10
5
0
10
This
into
histogram
fewer,
represents
wider
classes
the
of
data
equal
20
30
40
50
60
Weight
(g)
40
60
70
80
90 100
70
80
90 100
grouped
width.
80
70
60
ycneuqerF
50
40
30
20
10
0
10
20
30
50
Weight
(g)
6.2 City skylines
2 01
Reect
and
Why
using
does
comment
Both
are
of
the
is
small
shape
histograms
when
concentrated
histogram
where
Using
a
the
appropriate
data
the
on
discuss
the
with
data
unequal
river
could
Weight
number
of
are
data
a
accurate
is
very
classes
class
widths,
like
wide
classes
make
representations
spread
You
where
concentrated
look
of
it
dicult
to
distribution?
together.
wide
is
4
can
the
out,
and
combine
data
is
of
the
narrow
these
spread
data.
classes
ideas
out,
Wide
to
and
help
create
classes
when
a
narrow
the
better
classes
together.
the
grouped
frequency
table
for
the
sh
caught
in
this:
(g)
Frequency
0
<
w
≤
30
10
30
<
w
≤
40
8
40
<
w
≤
45
20
45
<
w
≤
50
16
50
<
w
≤
55
12
55
<
w
≤
57
11
57
<
w
≤
59
15
59
<
w
≤
61
9
61
<
w
≤
65
14
65
<
w
≤
70
11
70
<
w
≤
100
10
Class
width
30
g
Class
width
10
g
Class
A
frequency
histogram
to
Class
represent
this
frequency
table
looks
like
this:
25
ycneuqerF
20
15
10
5
0
10
20
30
40
50
Weight
Reect
●
Do
●
Does
and
you
it
think
obey
proportional
Because
classes
the
have
20 2
discuss
this
the
to
height
ended
of
up
60
70
80
90 100
(g)
5
histogram
principle
the
represents
that
in
a
the
data
histogram,
well?
the
area
of
each
bar
is
frequency?
each
bar
looking
6 Quantity
is
width
its
frequency,
much
too
large.
the
bars
representing
wider
5
width
g
2
g
S TAT I S T I C S
Frequency
density
When
is
data
proportional
In
a
For
for
the
sh
0
<
density
≤
intervals
the
data
data,
30
histogram,
class
density
weight
w
class
of
so
unequal
that
the
widths,
area
of
bar
to
frequency
each
bar
is
has
the
area
of
each
represent
is
plotted
the
rst
frequency
this
bar
is
equal
to
the
interval.
on
the
vertical
axis.
class
height
10.
Area
The
a
frequency.
that
frequency
interval
in
represents
the
frequency
frequency
The
to
P R O B A B I L I T Y
histograms
grouped
densityhistogram
A N D
class
=
frequency
=
10
(frequency
interval
density)
has
width
What
is
30
the
and
area
height
of
10.
this
rectangle?
class
For
any
rectangle,
Rearranging
area
=
height
frequency
=
frequency
this
Example
The
in
to
the
frequency
Weight,
gives
nearest
x
(g)
Freq
≤
x
<
19
14
19
≤
x
<
21
10
21
≤
x <
22.5
22.5
≤
x
<
25
10
25
≤
x
<
28
7
a
Weight,
the
30
so:
density
×
denition
class
of
width.
frequency
density:
(g)
gram,
of
50
Blue
Heron
chicks
are
given
9
frequency
x
whole
table.
14
Construct
width,
=
3
weights,
this
formula
×
width
density
Freq
histogram
Class
width
for
this
data.
Add
Frequency
14
≤
x
<
19
14
5
2.8
19
≤
x
<
21
10
2
5
21
≤
x
<
22.5
9
1.5
6
22.5
≤
x
<
25
10
2.5
4
25
≤
x
<
28
7
3
2.3
columns
frequency
density
for
class
density
to
width
the
=
Round
degree
Continued
on
next
decimals
of
to
and
table.
=
a
2.8
suitable
accuracy.
page
6.2 City skylines
20 3
Weights
of
Blue
Heron
chicks
y
6
Draw
with
the
histogram
weight
on
the
5
horizontal
axis
frequency
density
ytisned
the
vertical
ycneuqerF
The
rst
from
2
14
height
the
bars
bar
to
of
0
x
14
15
16
17
18
19
20
21
Weight
Reect
the
What
●
How
modal
would
you
class?
work
between
weighed
mode
of
of
two
a
data
data,
each
chicks
chicks,
all
chick
would
with
continuous
–
in
case,
The
other
that
is
modal
Practice
The
is
as
data,
class
is
most
you
class
g?
28
the
to
the
weights
with
with
the
the
are
weight.
would
interested
are
they
weight
nearest
there
really
number
estimate
occurring
chick
same
so
the
an
Why
commonly
are
the
most
greatest
greatest
data
g,
You
be
number
estimates?
point.
if
very
would
With
you
had
likely
the
that
probably
have
mode.
where
closely
the
even
it’s
no
is
chicks
only
data,
0.1
in
of
for
the
clustered
frequency
frequency
data
is
most
together.
density:
21 ≤
x
<
In
22.5.
density.
5
x
in
centimeters,
(cm)
≤
x
<
55
13
55
≤
x
<
60
20
60
≤
x
<
65
26
65
≤
x
<
72
19
a
Copy
and
b
Construct
complete
a
of
some
Frequency
45
20 4
24
correct
interval
the
27
for
about
Heron
weights,
where
class
26
this?
estimate
and
exactly
what
words,
g
is
What
Blue
recorded
have
an
g?
22
the
25
(g)
Why
20
the
dierent
the
lengths,
Length,
set
such
than
24
3:
out
who
dense
1
the
23
6
Example
less
weight
this
is
in
whoweighed
continuous
With
discuss
histogram
●
50
and
22
the
frequency
Class
vipers
width
are
recorded
10
histogram
for
in
Frequency
1.3
table.
density
6 Quantity
horned
this
data.
the
table.
density
goes
19
and
2.8.
with
between
no
axis.
3
1
The
on
4
a
In
and
no
them.
has
Draw
gaps
S TAT I S T I C S
Problem
2
The
ages
A N D
solving
of
people
visiting
the
Eiel
Tower
are
recorded
in
the
table.
In
Ages,
x
P R O B A B I L I T Y
(years)
Frequency
2015
were
visitors
0
≤
x
<
15
15
15
≤
x
<
25
28
25
≤
x
<
40
30
Tower.
deck
is
above
the
40
≤
x
<
60
there
6.91
to
≤
x
<
The
Construct
b
From
and
the
80
i
Explain
ii
State
3
State
The
your
at
The
ground,
highest
you
for
medical
gives
5
<
L
≤
7
15
7
<
L
≤
9
24
13
19
13
<
L
≤
16
9
16
<
L
≤
20
6
b
Calculate
The
100
table
km
Time,
a
frequency
an
10
charity
t
estimate
and
shows
15
the
b
State
c
Calculate
the
and
2
a
data.
were
80
ten
and
visitors
100
years
between
60
old.
made.
reasonable
assumption.
80
patients
about
the
in
total
lengths
during
of
morning
these
for
the
histogram
number
of
for
the
information.
patients
who
saw
their
doctor
for
minutes.
taken
by
members
of
a
cycling
club
to
nish
a
110
<
t
≤
130
130
6
frequency
modal
an
<
t
≤
145
145
30
density
<
t
≤
160
160
45
histogram
for
the
<
t
≤
55
185
185
<
t
≤
195
28
195
<
t
≤
16
information.
class.
estimate
for
the
number
of
cyclists
who
took
between
1
1
2
a
see
density
times
Frequency
Construct
there
between
has
this
ride.
(min)
a
is
for
Frequency
7
Construct
Union.
minutes.
5
a
the
data.
center
≤
≤
this
information
L
L
that
Voleta
this
<
<
visitors
think
class
(min)
between
4
in
ten
0
9
in
answer.
table
appointments
L
a
and
histogram
estimates
assumption
modal
doctors
Length,
old,
the
density
Voleta
whether
the
surgery.
frequency
years
upper
meters
20
graph,
Justify
c
a
Eiel
42
100
a
the
276
European
60
million
3
hours
to
complete
the
ride.
2
6.2 City skylines
20 5
250
5
The
histogram
sports
represents
the
distances
students
threw
the
javelin
on
school
day.
y
In
javelin
the
11
distance
of
throw
rounded
the
10
is
9
down
to
the
nearest
ytisned
8
centimeter.
7
ycneuqerF
6
5
4
3
2
1
0
x
5
10
15
20
25
30
35
Distance
6
a
Construct
b
Find
c
Calculate
d
Find
This
the
the
a
frequency
number
an
between
Find
110
the
interval
for
and
number
of
125
the
that
represents
cm
for
the
45
50
55
60
65
distribution.
throws.
estimate
class
histogram
of
table
40
(meters)
mean
distance.
contains
children’s
the
median.
heights.
Fifteen
children
measured
cm.
children
in
the
sample.
y
ytisned
5
4
ycneuqerF
3
2
1
0
110
120
130
140
Height
7
In
of
a
histogram,
50.
the
A
second
height
of
a
bar
bar
this
of
is
4
width
cm
2
wide
cm
and
and
150
160
170
x
(cm)
height
represents
5
a
cm
represents
frequency
of
a
frequency
50.
Calculate
bar.
Summary
A
●
●
●
bar
chart
All
the
There
The
represents
bars
are
have
spaces
height
of
qualitative
equal
bar
●
The
●
There
●
The
●
Frequency,
bars
are
the
A
Frequency
histogram
is
on
the
represents
the
all
represents
continuous
area
vertical
spaces
not
be
of
between
equal
the
width.
bars.
of
the
are
relative
on
the
quantitative
discrete
represents
the
frequency.
frequency
vertical
or
frequency
axis.
data
modal
class
is
data.
6 Quantity
bar
axis.
frequency
20 6
may
frequency.
The
and
no
or
bars.
density
●
may
width.
between
the
data.
density.
the
class
with
the
greatest
S TAT I S T I C S
To
comment
on
the
distribution
of
data,
describe:
For
a
with
Center
measure
of
central
Spread
measure
of
dispersion
a
data
with
frequency
baris
extreme
set
f
n
has
members,
relative
a
P R O B A B I L I T Y
class
interval
frequency
.
tendency
In
Outliers
data
frequency
A N D
values
that
don’t
t
equal
to
density
the
histogram,
frequency
for
the
that
area
class
of
each
interval.
the
The
frequency
density
is
plotted
on
the
vertical
axis.
pattern
Shape
unimodal,
Relative
frequency
is
bimodal
the
or
multimodal.
proportion
(or
percentage)
The
of
the
data
set
belonging
in
the
class
modal
frequency
Mixed
1
The
class
is
the
class
with
the
greatest
interval.
density.
practice
table
shows
the
heights
of
24
4
students.
Fifty
students
eating
Height,
x
(cm)
Frequency
The
out
results
frequency
140
≤
x
<
150
6
150
≤
x
<
160
7
in
recorded
one
are
how
month
shown
(to
in
much
the
the
they
nearest
partially
spent
on
dollar).
complete
table:
Amount spent
Class
boundaries
Frequency
(dollars)
160
≤
x
<
170
6
170
≤
x
<
180
4
180
a
≤
<
Explain
used
b
2
x
to
190
why
restaurant
from
one
8pm
week.
to
the
frequency
set
9pm
the
on
table
histogram
of
frequency
The
21–30
15
31–40
0
41–50
4
51–60
3
61–70
1
histogram.
of
consecutive
the
pizzas
nights
served
during
a
results.
Draw
to
of
frequency
weights
nearest
13
be
5
6
8
9
10
3
8
12
4
5
histogram
for
this
of
50
hoglets
are
recorded
to
the
clearly
this
b
State
the
modal
c
State
the
class
d
Describe
label
a
frequency
histogram
information.
class.
the
that
contains
the
median.
distribution.
The
in
a
table
shows
the
age
distribution
of
teachers
school.
gram.
Age,
93
88
105
90
92
89
90
88
108
86
103
96
94
100
104
98
100
120
115
94
84
130
110
125
115
112
105
129
118
105
95
124
112
96
114
128
132
95
129
122
120
85
93
108
105
88
95
105
123
97
a
and
represent
data.
5
3
14
data.
number
shows
can
0.50–10.50
10.50–20.50
ser ved
Frequency
Draw a
this
records
The
Number
pizzas
a
represent
Construct
A
1
1–10
11–20
Construct
a
grouped
frequency
table
x
(years)
Number
of
20
<
x
≤
30
6
30
<
x
≤
40
4
40
<
x
≤
50
3
50
<
x
≤
60
2
60
<
x
≤
70
2
teachers
for
a
Calculate
an
b
Construct
c
Describe
estimate
for
the
mean
age.
thisdata.
b
Construct
a
histogram
to
represent
this
a
histogram
to
represent
this
data.
data.
the
distribution.
6.2 City skylines
207
6
The
and
heights
Peru
of
are
14-year-old
given
in
the
students
from
frequency
Sri
tables
Lanka
8
below.
In
New
old
The
Heights
of
students
(measured
Heights
to
the
in
Sri
Lanka
nearest
cm)
130–139
11
140–149
14
150–159
8
160–169
6
(measured
to
the
in
Peru
nearest
x
Explain
cm)
a
22
130–139
41
140–149
32
150–159
6
160–169
6
it
is
necessary
frequency
gives
were
asked
their
rst
the
how
child.
results.
Frequency
<
x
≤
20
6
20
<
x
≤
22
14
22
<
x
≤
25
19
25
<
x
≤
30
21
30
<
x
≤
40
30
Determine
the
to
Construct
c
Calculate
histogram
to
type
of
histogram
would
representation.
the
an
determine
construct
which
best
b
histogram.
estimate
the
modal
for
the
mean
and
class.
solving
a
compare
The
histogram
shows
the
lengths
of
a
sample
these
of
data
had
15
give
9
relative
mothers
they
Frequency
120–129
why
table
(years)
Problem
a
90
when
frequency
Age,
3
students
were
Frequency
120–129
of
Zealand,
they
European
otters.
sets.
y
b
Comment
of
on
14-year-old
the
distributions
students
in
Sri
of
the
Lanka
heights
and
Peru.
10
7
The
frequency
nearest
kg)
of
distribution
35
parcels
is
for
the
given
masses
in
the
(to
the
9
table.
8
7
(kg)
6–8
Frequency
9–11
4
12–17
6
18–20
10
21–29
3
ycneuqerF
Mass
6
5
4
12
3
a
Determine
b
Explain
the
class
boundaries
for
each
class.
2
1
why
histogram
you
for
this
cannot
draw
a
frequency
0
data.
x
50
60
70
Length
c
Copy
and
complete
the
frequency
80
90
100
(cm)
density
table.
a
Find
the
class
interval
that
contains
the
median.
Mass
Frequency
Class
Frequency
width
density
b
(kg)
d
6–8
4
3
9–11
6
3
12–17
10
6
18–20
3
3
21–29
12
9
Construct
20 8
a
frequency
density
6 Quantity
histogram.
Calculate
an
estimate
for
the
mean
length.
S TAT I S T I C S
10
The
histogram
some
shows
customers
at
a
the
waiting
times
for
a
State
the
modal
b
Calculate
A N D
P R O B A B I L I T Y
class.
supermarket.
an
estimate
for
the
mean
waiting
time.
ytisned
2
ycneuqerF
Find
d
Find
the
an
total
who
i
x
20
60
100
Waiting
Review
in
estimate
for
of
the
customers.
Twenty-ve
140
time
180
220
had
to
number
of
customers
between
shown
in
and
women
children
the
60
and
90
seconds
260
ii
(seconds)
longer
than
2
minutes.
sustainability
from
Mexico
were
asked
Number
many
wait:
context
Globalization
how
number
1
0
1
c
they
frequency
Number
of
had.
The
results
per
children
Frequency
histogram.
children
of
0
1
2
3
4
5
4
8
5
4
5
2
are
woman
Use
the
and
compare
results
from
parts
a
and
b
to
describe
y
the
distribution
of
the
numbers
of
10
children
per
woman
in
Australia
and
Mexico.
9
8
2
The
amount
of
land
covered
by
rainforest
in
the
7
ycneuqerF
Amazon
6
5
region
the
past
the
percentage
has
decades.
been
This
rapidly
frequency
declining
table
over
show
4
of
deforestation
for
20
South
3
American
countries.
2
1
0
1
2
Number
a
Show
that
woman
b
A
is
group
asked
given
the
25
4
0.3%
1.6%
2.1%
0.3%
2.7%
2.2%
3.5%
2.0%
9.2%
3.2%
0%
3.0%
3.3%
4.0%
0.6%
0.5%
4.0%
1.7%
2.6%
5
children
number
women
same
this
Reect
and
How
you
have
mean
of
3
of
children
per
1.48.
of
the
in
0.4%
x
0
from
question.
Australia
The
results
were
a
Sketch
a
frequency
b
Comment
c
Determine
histogram
for
this
data.
are
on
the
distribution.
table.
discuss
explored
the
any
data
that
could
be
outliers.
6
statement
of
inquiry?
Give
specic
examples.
Statement of Inquiry:
How
and
quantities
trends
in
a
are
represented
can
help
to
establish
underlying
relationships
population.
6.2 City skylines
20 9
7
Measurement
A method of determining quantity, capacity or dimension
using a dened unit
Is
10
Would
always
you
than
100?
rather
●
wait
●
pay
●
walk
10
●
have
100
●
carry
Why
less
10
minutes
100
cents
km
10
g
or
or
or
100
or
we
100
seconds
Qua
$10
10
liters
do
or
ntity
+
unit
=
mea
kg
of
100
ml
devised
of
water?
standard
units?
systems
of
measurement
compare
quantities.
and
UK
were
based
on
body
based
on
hands,
forearms)
or
other
b etween
3
miles
of
units
dened
varied
the
for
worldwide,
own
make
such
1
as
system
all
time’.
to
was
pounds
some
more
example:
be
a
and
precisely,
country.
to
Metric
although
for
inch.
feet,
country
The
system
units
are
countries
idea
‘for
all
understood
still
use
their
systems.
What
are
the
non-standard
or
●
available,
measurements
from
metric
people,
●
readily
barleycorns
Standard
but
were
disadvantages
units
such
as
of
using
hands,
paces
barleycorns?
What
are
standard
21 0
the
advantages
units?
Are
of
there
using
any
size
7
is
1
barleycorn
objects
(one-third
that
are
barleycorn
s.
parts
size
(thumbs,
sizes
The
Early
dierence
measurements
shoe
to
still
and
t
gold
US
describe
men
meters
need
Humans
sure
disadvantages?
of
an
inch).
6
and
How
big
Although
units,
we
most
is
it?
have
people
sophisticated
are
not
very
interpreting
measurements.
●
sofa
Would
a
t
family
As
in
big
Media
such
two
as
reports
as
‘an
the
gives
of
A
a
blue
cm
comparisons
the
×
visualizing
80
cm
×
70
standard
or
cm
size
or
A
blue
and
of
‘an
whale
weigh
as
can
be
much
up
as
to
200
30m
long
tonnes.
area
Belgium.’
these
you
blue
and
of
160
at
and
whale
use
of
good
tools
car?
pitches’
size
Which
measuring
a
area
football
twice
●
a
measuring
the
two
best
comparisons
idea
of
the
size
whale?
whale
weigh
as
can
be
much
as
as
long
33
as
9
family
cars
elephants.
21 1
Y
es,
I’m
absolutely
positive
7.1
Global
context:
Objectives
●
Knowing
value
●
of
a
value
of
a
Inquiry
dierent
denitions
for
the
absolute
the
and
sustainability
questions
●
What
●
How
is
an
absolute
value?
F
number
Understanding
Globalization
properties
of
the
absolute
C
number
value
are
the
properties
similar
to
other
of
absolute
properties
in
SPIHSNOITA LER
mathematics?
●
How
can
error
estimating
●
How
●
Can
can
or
be
measured
when
approximating?
error
best
be
measured?
D
you
ever
know
the
exact
value
of
ameasurement?
ATL
Critical
Draw
thinking
reasonable
conclusions
and
generalizations
6.1
Statement of Inquiry:
Quantities
and
human-made
measurements
systems
and
illustrate
the
relationships
between
communities.
7.1
7.2
21 2
N U M B E R
Y
ou
●
should
calculate
already
with
negative
know
how
integers
1
to:
Work
a
c
●
nd
the
square
root
of
a
number
2
out
4
these
calculations.
6
5
×
Work
3
out
these
b
3
+
7
d
18
÷
−2
expressions.
2
a
b
121
7
1
c
d
0.01
4
●
substitute
numbers
positive
into
and
negative
3
expressions
When
a
a
+
a
=
4
and
b
b
=
−2,
b
a
d
b
2
c
Introduction
F
●
What
is
an
to
absolute
2
a
e
nd:
b
f
a
absolute
ab
value
value?
Activity
°C
40
State
the
temperature
dierence
between:
1
and
30
noon
sunrise
noon
20
2
sunrise
and
sunset
3
sunrise
and
night
4
noon
To
and
calculate
the
other.
choose
Here
1
is
the
the
Reect
When
use
we
the
sunrise
sunrise
sunrise
noon
and
is
is
is
is
the
−
10°
−
10°
night,
8°
−
10°
sunset,
15°
−
and
calculate
temperature
=
b,
a
you
b
or
subtract
b
a?
Or
one
did
from
you
dierences:
temperature
rose
by
17°,
17°
the
=
=
a
answer?
temperature
rose
by
5°,
5°
the
temperature
dropped
by
2°,
−2°
the
27°
quantities
always
the
sunset,
15°
and
you
noon,
27°
and
two
positive
discuss
about
positive
did
a
and
and
dierence
talk
between
calculating
dierence
Between
so
of
dierence
Between
so
4
the
–10
gave
dierence
Between
so
3
way
the
night
Activity,
that
sunrise
0
dierence
this
Between
10
sunset
one
one
so
2
In
the
a
sunset
=
temperature
dropped
by
12°,
−12°
1
dierences
in
real-life
situations,
why
do
we
usually
dierences?
7.1 Yes, I’m absolutely positive
213
After
who
playing
had
the
dierence
7
points’.
unusual
The
by
The
for
went
we
The
the
losers
it
has
absolute
The
say
If
up.
‘We
‘We
lost
had
implied
scored
the
The
by
context.
points
temperature
numbers
may
points’.
tells
gives
say
It
you
you
‘We
would
the
won
be
by
highly
points’.
the
7
7
won?’
much?’
winners
7
in
‘Who
how
less
When
than
dropped
used
are
a
the
by
always
team
other
20°,
it
loses
team,
obviously
positive,
because
value.
value
that
scores.
say
is
question
‘By
obviously
absolute
between
to
more.
andnot
the
may
the
Asking
two
dierence
7points
down
use
asking
score.
someone
negative
not
game,
between
7points,
and
a
highest
of
number
a
number
and
0.
is
We
the
distance
write
to
on
mean
the
number
‘the
line
absolute
value
Tip
of
x’.
On some calculators
5
5
the
x
–5
–4
–3
–2
–1
0
1
2
3
4
absolute
function
5
abs(x).
For
Using
example,
absolute
,
value
and
.
notation,
the
temperature
°C
dierences
you
calculated
in
the
Activity
are:
40
1
Between
sunrise
and
noon:
30
noon
27° − 10°
=
17°
= 17°
20
2
Between
sunrise
and
sunset
sunset:
10
15° − 10°
=
5°
sunrise
= 5°
night
0
3
Between
8° − 10°
4
Between
sunrise
=
−2°
noon
15° − 27°
=
=
−12°
Let
x
a
=
a
=
a
if
a
≥
= 12°
1
0,
and
e
2
what
State
your
Give
the
–10
sunset:
12
Describe
night:
2°
and
Exploration
1
and
you
ndings
positive
x
=
−a
b
a
=
0
c
f
a
=
3.14159
g
notice
as
a
if
a
about
general
square
<
root
0.
Find
your
a
a
x
=
=
for
each
value
−5
−3.14159
h
answers.
rule.
of:
b
c
d
e
f
g
h
State
what
your
21 4
you
ndings
notice
as
a
about
general
7 Measurement
your
rule.
answers.
a
d
a
Describe
of
a
=
−0.003
value
button
is
N U M B E R
Denition
of
absolute
The
value
is
1
rst
an
piecewise
The
denition
example
of
function
dierent
has
the
values
Reect
●
Why
and
can’t
number,
●
Can
number
●
Can
1
Write
surd
take
the
take
the
,
take
if
the
∈
value
or
root
of
the
square
root
of
a
negative
?
of
the
absolute
value
of
any
real
ℝ)?
absolute
value
of
any
real
number?
1
down
when
the
value
of
the
following.
Leave
you
answer
as
a
simplied
possible.
35
a
absolute
square
a
x
2
example:
(
you
Practice
you
for
you
discuss
dierent
of
35
b
234
c
234
d
4
e
2.8
f
5.6
g
h
0
8
2
2
i
3
10
j
k
36
l
4
5
3
25
m
2
Find
the
a
4 + 9
e
5.1
32
n
value
−
o
−
24
−
p
( −10 )
of:
b
−2 + 5
c
3 × 6
g
12
4
d
9
9
3
7.2
f
−7.5
×
2
× 10
h
5
−3
i
+
8
Problem
3
4
−3
j
Write
two
numbers
b
Write
two
calculations
a
highway,
highway
between
a
same
safely
b
The
is
average
for
there
driving
of
relative
speeds
−4
k
×
−2
−4 × −2
l
have
an
whose
too
fast
should
answers
or
be
absolute
too
no
value
have
slow
more
can
an
of
absolute
cause
than
a
4.
20
value
accidents.
km/h
of
For
6.
safe
dierence
speeds.
section
travelling
that
driving
driving,
cars’
Gauthier
8
solving
a
On
−
at
to
each
speed
the
at
97
km/h.
highway.
on
to
is
driving
whether
or
at
not
113
km/h
they
are
on
the
driving
other.
a
average
others
Nathan
Determine
be
stretch
speed,
driving
of
highway
determine
safely
is
the
110
km/h.
minimum
relatively
to
Louis
If
Louis
and
on
is
maximum
this
stretch
ofhighway.
7.1 Yes, I’m absolutely positive
a
denition
2
for
a
function.
21 5
Proper ties
and
applications
C
The
of
absolute
value
is
absolute
also
called
modulus
●
How
are
the
properties
of
absolute
value
similar
to
or
●
How
can
Objective:
B.
ii.
patter ns
In
describe
allows
mathematics?
error
be
Investigating
Exploration
table
in
2
you
you
to
as
measured
will
be
estimating
or
approximating?
patter ns
general
observe
when
r ules
looking
for
consistent
patterns.
with
ndings
Organizing
the
information
in
a
trends.
ATL
Exploration
1
Determine
a
0.005
d
=
a
=
down
Write
3
11,
a
d
a
=
a
=
your
=
what
ndings
b
− 3,
=
b
you
each
down
absolute
2.625
d
general
=
pair
− 6,
b
value.
1.001
h
property.
of
=
values
a
4
and
b
c
f
as
pair
− 30
your
a
each
a
b
a
of
a
=
general
values
− 2,
b
property.
of
=
a
and
b
4
c
e
you
an
notice.
10
what
be
e
ndings
for
=
as
for
− 15
your
and
3,
c
b
=
could
g
7
b
down
Describe
Write
b
− 0.2,
Find
numbers
392
and
Describe
real
f
Find
a
these
b
Write
a
of
5
e
2
2
which
f
a
a
=
=
1,
b
=
− 0.2,
− 5
b
=
− 0.8
notice.
ndings
as
a
general
property.
2
4
Find
a
a
=
and
a
for
5
b
d
following
a
=
values
of
− 3
a:
c
e
Describe
Write
5
the
Copy
what
down
and
a
you
notice.
general
complete
rule
this
consistent
with
your
ndings.
table:
a
2
2
3
Describe
any
generalizations
that
you
notice.
Continued
21 6
7 Measurement
the
the
other
numerical
properties
value
on
next
page
value.
N U M B E R
Write
help
6
down
your
Copy
and
complete
a
b
8
3
6
a
general
rule
using
the
last
three
Draw
conclusions
the
any
this
table:
symbols
of
from
≥,
≤
absolute
numbers
a,
b
∈
the
and
=
table.
to
complete
these
general
properties:
value
ℝ:
1
2
3
4
5
6
7
8
Practice
2
Substitute
the
1
=
=
7,
to
7
Proper ties
a
columns
9
5
For
as
4
1
Use
ndings
you.
b
values
for
a
and
b
a
to
nd
the
value
of
each
expression.
− 4:
a
a + 2b
b
d
−2 a − b
e
2b
c
−2 a + b
3
2
a
=
−5,
a
a +
d
3a
b
=
a + b
3b
3a
f
b
3:
b
−a +
a − b
− b
c
a +
a
+ b
+
b
4
a
e
3b
b
f
a
7.1 Yes, I’m absolutely positive
21 7
3
a
=
1,
b
=
−1:
a
a +
a
a
+ b
+
b
−b
b
−
b
− a −
a
+
b
a
+
b
a
+
b
c
a
b
3
d
e
b
a
Problem
4
Determine
Any
For
and
the
values
you
height
of
accuracy
that
Would
make
it
centimeter?
Every
an
more
but
is
Or
accurate
not
as
b
such
a
for
as
sense
than
accurate
a
in
to
make
it
weighing
it
the
to
idea
of
positive
certain
the
number.
degree
two
You
of
cities
accuracy.
in
choose
actual
between
a
kilometers,
degree
distance
cities
to
the
or
of
height.
nearest
kilometer?
only
to
the
nearest
the
a
a
between
nearest
object
to
to
distance
the
is
− 3
centimeters.
an
accurate
an
b
made
and
the
to
−
distance
give
is
weigh
weighing
as
to
height
is
the
meters
calculate
you
can
8 − b
length,
enough’
person’s
You
which
measure
person
‘good
a
value.
of
might
measurement
exact
b
b
solving
measurement,
example,
11
a
f
a
to
a
certain
nearest
gram
nearest
or
degree,
milligram,
to
the
so
it
and
nearest
is
not
this
is
centigram,
microgram.
18
18
17
17
16
length
The
true
perfect
value
the
a
describes
following
possible
21 8
17.4
cm
length
measurement
measurement.
Accuracy
In
of
=
This
how
is,
close
exploration,
measurement
as
the
by
a
is
its
the
value
nature,
true
7 Measurement
and
value.
17.42
that
not
measurement
example,
=
cm
would
really
is
to
practice
obtained
possible
the
we
be
true
use
to
by
a
obtain.
value.
the
most
accurate
N U M B E R
ATL
Exploration
1
Measure
your
centimeter.
2
Now
as
Write
height,
Copy
measure
you
3
can,
to
them
in
the
the
your
the
the
length
table
and
height,
nearest
of
a
piece
write
the
your
length
millimeter.
second
column
of
of
nearest
paper,
your
to
cm
piece
The
of
of
in
paper,
these
to
width
the
and
be
the
of
rst
the
an
to
the
nearest
column.
width
true
eraser
of
values
an
for
eraser
these
as
accurately
measurements.
True
value
accurate
the
–
most
measurement
nearest
Absolute
Relative
Percentage
error
error
error
mm)
height
length
of
a
piece
paper
The
3
the
table.
(to
Your
and
measurements
Consider
Measurement
the
a
of
width
Calculate
The
of
the
an
absolute
absolute
error
in
a
eraser
error,
error
Δx,
for
of
a
each
measurement
measurement
is
the
using
actual
this
denition:
amount
of
measurement.
Δ x
Write
the
4
Compare
5
Calculate
The
size
absolute
the
the
absolute
relative
relative
of
the
errors
error
thing
in
your
errors.
error
is
Write
for
the
being
table.
each
error
in
down
what
you
measurement
a
notice.
using
measurement
this
denition:
compared
to
the
measured.
Δ
Relative
6
Write
the
7
Calculate
error
=
relative
the
percentage
as
percentage.
8
Write
the
9
Compare
error
percentage
the
in
percentage
The
a
errors
your
error
is
the
errors
percentage
in
table.
for
each
relative
the
errors.
measurement
error
of
a
using
this
measurement
denition:
written
table.
Write
down
what
you
notice.
7.1 Yes, I’m absolutely positive
21 9
Example
Frederico
and
says
1
measured
that
a
Find
b
Explain
a
An
Let
x
=
why
of
Relative
has
true
there
5%
true
=
height
Frederico
Frederico’s
error
his
to
be
made
132
an
cm.
error
His
of
doctor
5%
in
his
then
measures
him
measurement.
height.
are
two
means
a
possible
relative
values
error
of
for
his
true
height.
0.05.
Write the percentage error as a relative error.
value
absolute
error
error
=
Substitute into the denition.
Solve for x
0.05x
=
132
x
or
1.05x
=
132
x
=
125.714
0.05x
=
x
132
0.95x
=
−132
x
=
138.947
Use the same level of accuracy as
Frederico’s
true
height
is
either
126
cm
or
139
cm.
in the question.
b
Absolute
error
measurement
we
would
Practice
does
is
need
not
above
more
tell
or
us
whether
below
his
information
Frederico’s
true
to
height
know
–
which.
3
‘To
1
Elias
estimated
23people
2
Breanna
240.5
that
attended.
measured
ml,
240.9
20
Find
the
ml,
people
the
would
percentage
capacity
240.2
attend
ml,
of
a
239.5
error
mug
ml,
an
ve
evening
in
his
In
fact,
Calculate
the
mean
value
for
the
The
tr ue
capacity
of
the
mug
is
of
the
Her
results
mean
3
Markéta
was
4
240
Amelia
Find
5
you
‘To
human;
err
to
A
made
two
factory
2%.
Find
2 20
Find
a
this
250
the
an
is
g
an
foul
capacity
.
240.1
pack
error
correct
produces
range
acceptable
of
percentage
18%
possible
the
is
really
ml.
Calculate
the
percentage
up
you
need
error
computer.’
result.
think
bought
g.
1711
were:
Paul
Do
cereal
of
in
butter.
percentage
She
dierence
a
error?
weighed
in
mass.
calculation.
Her
it
to
Take
Explain.
nd
250
incorrect
its
g
actual
as
the
answer
mass
true
is
value.
$27.50.
answers.
bars
possible
with
mass
masses
7 Measurement
to
ml.
a
c
human;
divine.’
Pope,
things
b
is
Alexander
but
a
err
forgive,
estimate.
times.
241.2
lecture.
of
58
g,
cereal
and
percentage
bars
from
this
error
up
factory.
to
Ehrlich,
1969
N U M B E R
Problem
6
Huýnh
fails
500
7
to
g
Find
●
30
kg
measured
How
can
How
do
value
and
Which
type
Justify
your
Absolute
value
without
Think
to
a
websites
and
has
the
the
be
error
before
12%
or
___ g
largest
an
Scales
will
absolute
error
error
of
of
experiment.
more.
statement:
percentage
best
lab
of
than
percentage
Huýnh
that
fail
A
scale
uses
a
measure
the
500
g
a
test.
error.
1%
10%
values
useful?
measured?
inuence
communities?
error
measurement.
in
a
tool
or
from
your
These
errors
accuracy
in
either
come
reading
it.
3
(absolute,
is
the
absolute
measured
an
positive
births
this
more
measuring
the
relative
when
error.
or
claim
it
When
negative
to
give
value
value.
absolute
situations
Some
a
or
absolute
error
the
error
or
percentage)
is
the
best
to
use?
arguments.
the
describe
has
of
taking
of
a
discuss
error
and
that
systems
of
science
Complete
with
measures
accuracy
the
___ g
with
are
●
in
percentage
than
●
the
a
scale.
less
measured
Reect
●
each
as
scales
has
measurement
kg
Absolute
●
the
it
When
D
●
if
test
the
500
all
test
mass
●
from
tests
the
mass
solving
of
Could
the
you
dierence
measure
between
the
the
absolute
true
error
value?
would
is
it
be
better
important
not
to
to
use
know
the
absolute
whether
the
value
error
value?
the
true
world
population,
plus
the
number
of
The
deaths
so
far
that
year,
so
naturally
this
population
number
world’s
is
population
changing
constantly.
If
the
population
gure
varied
between
7 379 930 858
is
7 379 930 871
over
a
one-minute
for
population?
period,
what
would
be
a
good
47
people
world
Could
anyone
ever
know
the
exact
value
of
kilometer.
the
The
world
most
densely
population?
populated
is
Reect
and
discuss
4
Why
do
we
often
use
people
approximations
instead
of
exact
amounts?
Is
per
appropriate
to
use
If
you
you
0.01
●
Is
it
measure
can
still
mm.
a
length
make
Can
important
you
to
approximations?
measurement
a
know
How
help
you
the
is
approximations?
to
more
ever
16 205
populated
Mongolia
●
at
square
it
least
always
country
Monaco
kilometer;
●
per
approximation
square
the
density
and
the
nearest
exact
know
exact
can
and
measurement
the
exact
values,
knowing
decide
cm
or
length
can
the
whether
to
of
you
size
an
then
of
to
the
the
nearest
exact
0.1
mm,
mm
square
with
2
or
something?
always
the
use
error
in
approximation
a
is
acceptable?
7.1 Yes, I’m absolutely positive
per
kilometer.
2 21
Summary
The
on
absolute
the
value
number
means
‘the
line
of
a
number
between
absolute
value
5
is
that
of
the
distance
number
and
0.
x’.
–4
–3
–2
–1
0
1
2
true
that
would
to
3
4
the
The
example,
Properties
of
any
a
measurement
describes
by
how
a
is
the
perfect
close
a
value
measurement.
measurement
is
absolute
Δx,
error,
Δx
amount
of
error
of
in
a
a
measurement
is
the
measurement.
=
value
The
For
of
obtained
5
and
absolute
be
truevalue.
actual
For
value
Accuracy
5
x
–5
The
relative
error
is
the
error
in
a
measurement
ℝ:
numbers
compared
to
the
size
of
the
thing
being
measured.
Δ
1
2
3
4
Relative
The
error
percentage
measurement
5
6
7
8
Mixed
1
Find
a
−
error
is
written
the
as
a
relative
error
of
a
percentage.
practice
the
value
of
these
expressions.
−
b
33
c
=
2
7.8
(
f
bought
3
measured
it,
of
Find
3
5
e
Soraya
she
meters
found
of
ribbon.
there
When
were
3.15
she
meters
4 − 6.5
12.3
d
−2.3 + 8.05
4
5
ribbon.
meter
Moa
the
percentage
error
in
the
measurement.
charges
€10
to
iron
8
shirts.
On
average,
2)
9
she
g
27
Find
the
−
h
−
81
irons
value
of
these
shirts
a
Determine
b
Determine
c
On
20
2
8
per
how
hour.
much
she
charges
to
iron
shirts.
expressions.
her
average
hourly
rate.
2
a
4
×
b
−5
−
− ( −6 )
Monday,
and
c
d
i
12
ironed
She
20
shirts
charged
the
in
2
hours
customer
€25.
Determine
ii
−
Moa’s
hourly
rate
for
these
shirts.
6
2
1
9
−
12
8
the
percentage
her
average
her
hourly
hourly
dierence
rate
(in
between
part
b)
and
5
15
56
g
Find
144
×
f
×
5
×
−3
d
On
Friday,
rate
she
for
took
3
these
hours
shirts.
and
12
minutes
h
7.2
4
8
−
6.25
×
7 − 3
to
iron
20
shirts.
dierence
3
Moa
minutes.
27
6
e
15
The
exact
length
of
a
bent
stick
is
22.2
cm.
length
his
to
it
with
a
straight
be
21.5
cm.
Find
ruler
the
and
estimated
percentage
between
the
her
percentage
average
error
part
b)
and
her
hourly
rate
7 Measurement
for
its
of
Which
rate
is
the
Which
one
is
an
exact
rate?
estimate.
2 2 2
hourly
rate
Daniel
(in
measured
Find
estimate?
these
shirts.
N U M B E R
Review
in
context
Globalization
1
The
most
recent
and
sustainability
census
in
China
was
in
July2015
2
UK
and
the
population
government
was
statistics
In
two
be
1 376 049 000
separate
a
Calculate
reported
to
be
articles,
in
1 376 048 943.
this
gure
population
was
a
quoted
million
1.38
absolute
error
in
each
of
compared
to
the
ocial
Calculate
the
Discuss
percentage
whether
it
is
Find
the
Find
the
Government
reliability
gure
of
may
Calculate
e
sources
the
be
better
the
way
to
suggest
it
is
use
absolute
this
is
possible
that,
due
measured,
inaccurate
what
Calculate
Discuss
to
what
report
Here
are
In
by
as
plus
an
range
or
the
the
given
might
be
a
population
the
reliable
of
populations
in
States
in
population.
half
a
error
in
the
headline
million.
17.7%
over.
of
d
error.
e
population.
gure
some
2013
the
the
population
percentage
were
was
aged
65
17.4%.
Find
the
aged
65
actual
and
change
to
over
in
the
between
number
2013
and
of
people
2014.
1.8%.
Is
this
gure
Suggest
an
how
increase
these
or
gures
healthcare
in
the
a
decrease?
could
be
useful
for
UK.
use
China.
of
In
3
Tobias
went
grocery
roommate
shopping
Felix.
They
for
himself
usually
split
and
the
cost
countries,
groceries
in
half,
and
since
they
never
have
change,
they
usually
round
the
amount
paid
before
paid
exactly
splitting
it
in
half.
Today,
Tobias
80 688 545
€95.74,
and
he
rounded
the
amount
30 331 007
Australia
to
an
even
€100
before
telling
Felix
that
he
23 968 973
owed
Monaco
Tobias
€50.
37 731
a
gures
could
also
be
inaccurate
by
Calculate
absolute
errors.
Use
these
to
the
percentage
error
between
the
1.8%,
true
the
change
the
up
nd
rose
321 442 019
Malaysia
the
population
2015:
Germany
If
UK
64 510 376.
percentage
of
2014,
exact
United
the
ocial
minus
absolute
for
to
the
of
also
2014
or
his
g
to
exact
planning
f
and
errors.
and
d
year
errors.
c
percentage
2013
64 106 779
gure
c
a
gure.
b
b
in
these
a
gures
half
billion.
from
the
by
to
Between
and
grows
ocial
amount
Tobias
paid
and
the
rounded
give
amount.
reliable
values
to
quote
for
these
populations,
b
and
justify
why
the
values
are
Discuss
whether
Felix
pay
c
to
Calculate
have
and
and
How
you
specic
exact
reimbursed
what
error
he
explored
the
statement
of
is
reasonable
for
amount
Tobias.
Then
between
should
Felix
what
have
should
calculate
Felix
did
Discuss
paid.
what
you
errors.
notice
about
Would
it
the
have
two
made
sense
examples.
to
just
calculate
absolute
errors
in
this
case?
Statement of Inquiry:
Quantities
and
human-made
the
pay
inquiry?
percentage
Give
it
discuss
d
have
not
€50.
the
percentage
Reect
or
reliable.
measurements
systems
and
illustrate
the
relationships
between
communities.
7.1 Yes, I’m absolutely positive
2 23
How
do
they
measure
up?
7.2
Global
context:
Objectives
●
Converting
metric
SPIHSNOITA LER
●
Converting
●
Using
●
Solving
units
Inquiry
between
units
of
area
metric
and
between
metric
correctly
problems
units,
in
and
F
imperial
problem
involving
Deciding
is
ATL
Use
●
including
volume
units
●
compound
the
answer
to
a
●
problem
sustainability
questions
What
are
How
do
volume
D
if
and
the
dierent
systems
of
measurement?
C
solving
measures
●
Globalization
Should
of
●
we
into
we
convert
measures
dierent
adopt
a
of
area
and
units?
single
global
system
measurement?
How
do
systems
inuence
communities?
reasonable
Communication
intercultural
understanding
to
interpret
communication
6.1
Statement of Inquiry:
Quantities
and
human-made
measurements
systems
and
illustrate
the
relationships
between
communities.
7.1
7.2
2 24
N U M B E R
Y
ou
●
should
recognize
and
already
units
for
know
length,
how
mass
1
to:
Group
volume
these
lengths,
measurements
masses
Which
is
the
2
kg
34
9
km
and
odd
mm
one
15
into
volumes.
out?
g
24
s
5
ml
3
●
write
powers
of
ten
in
full
2
13
Write
in
miles
down
these
3
multiply
of
and
divide
by
powers
3
of
cm
ten
b
2
10
c
10
Calculate.
ten
2
nd
the
area
of
a
square
4
280
nd
of
a
the
sur face
area
and
volume
×
0.001
Calculate
side
●
8.4
powers
6
10
a
●
liters
full.
a
●
6
5
the
length
b
area
12
34 500
of
a
÷
10
square
with
cm.
A cuboid measures 3 m × 4 m × 5 m.
cuboid
Calculate
its
sur face
area
and
volume.
Units
F
●
for
What
are
Exploration
1
Add
of.
●
●
to
this
Group
the
dierent
systems
by
writing
similar
units
down
all
measurement?
per
the
of
measurement
you
can
think
Think
●
hour
units
together.
(kg)
kilometers
of
1
list
kilograms
measurement
minutes
●
(km/h)
degrees
you
(min)
of
use
your
list
with
others.
Did
anyone
(°)
group
units
Add
●
to
this
list
all
the
●
length
types
of
measurement
●
volume
you
can
think
length
and
distance
the
same
measurement?
What
about
travelling,
is
one
about
words
how
volume
capacity?
Compare
it
etc.
volume
and
3
and
cooking,
mass
the
and
in
of.
Think
Are
in
dierently?
when
2
units
mathematics
science,
Compare
the
used
your
for.
two
Are
lists.
there
Match
any
each
types
of
unit
to
the
type
measurement
of
that
measurement
have
●
Does
●
Why
each
●
Can
and
every
are
used.
(1D,
only
unit?
Reect
capacity
discuss
units
often
measure
have
dierent
more
what
3D)
space
does
refer?
1
measurement
there
To
2D,
are
than
units?
units
one
Explain
for
a
type
single
of
why
type
this
of
is
the
case.
measurement?
measurement?
Explain
fully.
7.2 How do they measure up?
2 25
Practice
1
Suggest
1
sensible
units
for
these
specic
measurements.
The
a
A
matchbox
measures
40
×
25
×
‘A’
series
12
of
b
The
length
of
an
A4
sheet
of
paper
is
297
c
The
length
of
an
A4
sheet
of
paper
is
29.7
paper
used
sizes
Only
the
Canada
d
A
shoebox
has
a
capacity
of
to
e
A
shipping
f
The
oor
g
The
length
container
measures
US
do
currently
6307
6.06
×
2.44
×
the
of
a
house
is
standard
2.59
Mexico,
250
in
a
swimming
pool
is
Select
appropriate
units
for
Chile
50
and
2
practice
Venezuela,
Colombia,
of
and
not
conform
ISO
although
area
is
worldwide.
these.
the
favour
Philippines
the
US
letter
format.
a
The
weight
b
The
dimensions
c
The
area
d
The
speed
e
The
height
f
The
dimensions
g
The
speed
Systems
of
of
of
a
a
farm
of
wall
of
an
of
of
a
a
a
tractor
tennis
in
a
court
house
ambulance
girae
of
a
piece
javelin,
of
when
land
thrown
by
an
athlete
The
measurement
International
System
The
two
most
widely
used
systems
of
measurement
are
the
metric
system
of
Units
and
(abbreviated
the
imperial
SI
system.
from
the
French
‘Le
Système International
A
system
units.
All
of
measurement
other
units
in
is
the
a
system
system
are
of
measures
derived
based
from
the
on
base
a
set
of
base
units.
d’Unités’)
ocially
came
being
the
The
metric
into
11th
Conference
system
Weights
The
metric
system
is
used
widely
in
Europe
and
most
of
the
rest
of
the
world.
October
●
International
meter
(m)
for
System
of
measuring
Units
(or
SI)
denes
seven
base
length
kilogram
●
second
(kg)
for
measuring
The
French
the
for
measuring
time
ampere
(A)
for
measuring
the
Kelvin
●
mole
(K)
(mol)
for
for
measuring
measuring
electric
even
candela
(cd)
for
amounts
the
of
a
intensity
substance
10
days
10
hours
of
with
in
a
in
week,
a
day,
in
an
light.
in
a
and
100seconds
minute.
returned
to
They
traditional
timekeeping
7 Measurement
to
time
temperature
hour
2 2 6
1793
current
thermodynamic
measuring
In
tried
100minutes
●
system
French
decimalize
●
in
brought
metric
Revolution.
they
●
Paris
mass
after
(s)
in
1960.
units:
in
●
on
and
Measures
The
at
General
in
1806.
N U M B E R
All
other
the
unit
units
are
derived
from
these
and
are
called
derived
units.
For
example,
In
of
speed,
in
terms
of
the
base
units,
is
some
books,
m/s
m/s.
1
is
The
SI
which
(or
act
metric
as
system)
multipliers.
allows
The
other
unit
units
prexes
to
be
specify
created
a
by
power
of
using
written
ms
prexes
ten:
3
●
a
kilometer
is
10
times
larger
than
a
meter
(standard
unit),
3
1
km
=
1
×
10
m
a
nanosecond
5
nanoseconds
=
1000
m
−9
●
is
10
times
smaller
than
a
second
(standard
unit),
−9
Dividing
●
a
by
gram
the
=
5
factor
(standard
×
10
seconds
allows
unit)
you
is
to
=
0.000 000 005
convert
smaller
than
a
a
seconds.
standard
unit
into
another
unit:
kilogram,
3
500
●
a
g
=
meter
500
g
÷
10
(standard
=
0.5
unit)
is
kg
larger
than
a
micrometer,
−6
1
m
=
1
m
÷
10
Prex
=
1 000 000
Letter
m
Power
Some
derived
units
12
tera-
T
10
are
a
9
giga-
G
10
mega-
M
10
named
person
after
closely
associated
with
6
them.
For
example,
3
kilo-
k
10
deci-
d
10
centi-
c
10
milli-
m
10
micro-
μ
10
nano-
n
10
the
1
Newton
ameasure
(N)
of
is
force.
2
1
2
N
=
1
kg
m/s
3
6
9
Practice
1
Find
2
these
a
3.5
c
257
conversions.
megatonnes
mm
to
to
tonnes
meters
b
8
gigabytes
d
6.5
dl
to
to
bytes
liters
‘How
many
millimeters
2
How
meter’
a
c
are
in
a
many:
nanoseconds
watts
are
in
a
are
in
a
second
kilowatt
b
d
micrometers
centiliters
are
are
in
in
a
a
meter
is
the
same
as
‘Convert
to
millimeters’.
meters
liter?
7.2 How do they measure up?
2 27
3
Find
these
conversions.
Tip
a
2
m
to
c
2500
μm
watts
to
microwatts
b
3.2
g
to
mg
d
0.5
seconds
to
nanoseconds
Check
that
answer
4
Find
these
conversions.
sense.
a
285
cm
=
_____
m
b
923.5
mg
=
_____
Should
converted
g
smaller
c
4358
e
7.263
g
18 655
i
0.05
your
makes
m
=
_____
km
d
4358
m
cg
f
12.45
l
h
560
j
380 506
=
_____
=
______
l
=
_____
or
be
larger
mm
than
kg
the
unit
the
original
cl
unit?
ml
km
=
=
_____
Problem
5
A
5
human
μm
_____
cm
mm
=
_____
mg
=
cm
_____
kg
solving
ovum
across.
has
diameter
Calculate
how
0.1
mm.
many
The
times
head
larger
of
the
a
human
ovum
is
sperm
is
compared
roughly
to
the
sperm.
6
The
1
cells
μm.
of
How
microscope
the
bacterium
many
of
Staphylococcus
these
cells
could
t
aureus
on
a
are
spheres
rectangular
75
with
mm
diameter
by
25
mm
slide?
1
μm
On
cell
the
same
1
area
length
The
imperial
imperial
other
1
system
is
now
chiey
used
in
the
US
(US
customary
and
many
systems
measures
paces
μm.
of
measurement
had
their
own
standard
on
are
which
were
based.
Small
distances
were
counted
in
number
(yards)
and
larger
distances
in
miles
(1000
paces).
Capacity
and
using
kitchen
Eventually
a
items,
such
standard
was
as
cups,
set
so
because
that
all
it
was
mainly
measurements
used
were
units,
in
the
they
both
from
everyone.
same
Reect
●
What
as
●
and
diculties
paces
What
discuss
and
types
might
SI
(or
measures
of
For
metric)
are
example,
problems
2 2 8
arisen
in
measuring
with
units
such
yard
in
≡
might
have
occurred
from
the
use
of
units?
standards
dened
1
have
cups?
non-standardized
The
2
are
terms
0.9144
of
now
that
meters,
7 Measurement
accepted
worldwide
and
all
other
standard.
where
≡
means
‘is
equivalent
to’.
are
same
derived
for
US
was
but
cooking.
between
of
customary
measured
a
side
units).
imperial
feet
as
with
dierences
the
each
the
system
There
Originally,
up
μm
square
The
slide,
takes
system.
the
N U M B E R
Some
conversions
marked
and
*
are
Pound
for
exact;
metric
all
dened
km
0.625
1
m
1
kg
1
liter
1.76
1
liter
2.113
39.4
2.205
1999,
metric
the
Mars
units
Mars,
for
mile
be
are
shown
exact
by
in
the
the
tables.
Those
International
Yard
pints
mile
1.609
km
1
foot
30.48
1
inch
2.54
1
ounce
(oz)
1
gallon
(UK)
4.546
liters
1
gallon
(US)
3.785
liters
cm
(*)
Orbiter
contractor
such
it
crucial
is
cm
(*)
(lbs)
28.35
g
(US)
Climate
where
1
(UK)
pints
a
Metric
inches
pounds
while
measurements
to
units
to
Imperial
Imperial
1
close
imperial
are
Agreement.
Metric
In
and
others
was
used
lost
imperial
calculations
thought
because
to
have
units.
caused
been
the
NASA
Failing
the
to
probe
destroyed
by
team
used
convert
to
come
the
the
too
planet’s
atmosphere.
Example
Find
a
8
b
these
UK
13
1
conversions.
gallons
liters
to
to
UK
liters
gallons
× 4.5
1
a
UK
gallon
≃
4.5
liters
1 UK gallon
8
×
4.5
=
36
To
liters
1
b
13
÷
4.5
=
2.888…
≈
2.9
convert
UK
gallons
(1
4.5 liters
gallons
gallon
÷
Use
UK
≃
UK
the
≃
4.5
to
liters,
multiply
by
4.5.
liters
4.5
inverse
operation.
d.p.)
To convert liters to UK gallons, divide by 4.5.
Round
Objective:
iv.
In
justify
Practice
3,
Practice
1
Find
D.
the
Applying
degree
justify
of
the
mathematics
accuracy
degree
of
of
a
in
real-life
to
a
sensible
degree
of
accuracy.
contexts
solution
accuracy
of
each
of
your
solutions.
3
these
conversions.
a
1
inch≃
___
c
1
pound
≃
e
1
UK
pint
m
____
≃
kg
____
liters
b
1
g
≃
d
1
liter
f
1
cm
____
≃
=
ounces
____
___
gallons
(UK)
feet
7.2 How do they measure up?
2 2 9
2
Find
how
many:
a
kilometers
c
pounds
is
50
miles
b
miles
is
50
kilometers
d
centimeters
For
is
70
kilograms
is
6
2h,
convert
e
centimeters
is
g
UK
2
pints
is
Problem
3
4
Determine
58
miles
c
50
cl
e
6
In
This
in
model
or
is
of
intended
a
By
or
1983,
due
of
a
to
fuel
a
for
One
the
b
of
Find
●
each
oz
f
6
ft
members
stage
the
of
megalith
5
UK
pints
is
1
a
ctional
(large
kg
mile
or
or
rock
standing
pair.
or
60
100
180
lbs
g
cm
group
asked
stone).
The
height
to
would
that
an
appropriate
have
measure
they
been
of
for
an
model
18-inch
was
measure,
determine
members
should
eective.
length
intended
metric
the
to
use
of
fuel
band
the
model
on
a
Canada
of
of
fuel
between
journey.
jet
have
fuel
been
0.803
has
a
many
How
has
a
per
mass
and
a
1.77
your
liters
short
convert
kg.
conversion
fuel
of
live
in
The
fuel
through
plane
have
stage.
of
in
its
journey
needed
already
Calculate
answer
pounds).
of
tell
liters
0.803
your
answer
halfway
kilograms.
7682
of
Give
Give
we
out
were
liter,
of
units
do
pounds
loaded.
kg
ran
There
mass
loaded.
how
ight
how
in
the
much
22 300
kg
tanks.
fuel
liters.
1.77
Calculate
was
how
used.
much
(One
fuel
liter
was
liters.
the
plane
was.
us
measures
of
area
and
volume
into
dierent
units?
ATL
Exploration
1
The
diagram
length
100
2
shows
a
square
with
side
length
1
m
and
a
square
with
mc
1
001
m
a
Explain
b
Calculate
why
the
these
area
two
of
m
squares
each
100
have
cm
the
same
area.
square.
Continued
23 0
side
cm.
1
7 Measurement
miles
kilometers.
show.
imperial
Air
What
C
for
4
given
the
actually
c
is
solving
liter
jet
largest
d
model
mix-up
Instead
is
pint
Tap,
what
an
should
centimeters
25
their
for,
for
h
b
Stonehenge
Problem
In
measurement
cm
Spinal
Decide
liters
km
converting
asked
5
72
UK
20
whether
b
f
solving
1
a
feet
gallons
which
a
or
3
rst
inches
on
next
page
to
N U M B E R
c
Copy
and
complete:
2
1
d
2
m
=
____
cm
Sketch a square with side length 1 cm and a square with side length 10 mm.
2
How
do
you
2
to
Sketch
a
cm
cube
a
Calculate
b
Copy
c
2
to
mm
with
the
and
=
Work
side
length
volume
of
1
each
m
and
a
How
do
you
convert
from
cube
with
side
length
100
cm.
cube.
complete:
____
out
cm
the
length
volume
10
of
a
cube
with
side
How
do
you
3
length
1
cm
and
a
cube
with
mm.
3
d
?
3
m
side
cm
?
3
1
from
2
mm
2
convert
convert
from
cm
3
to
mm
?
How
do
you
convert
from
3
mm
to
cm
?
Volume
is
generally
3
measured
Capacity
refers
to
the
amount
of
space
available
to
hold
refers
to
the
amount
of
space
actually
cm
,
something.
and
Volume
in
capacity
in
occupied.
liters.
3
1
cm
=
1
ml
3
1000
ATL
Practice
1
Find
these
20
c
5000
=
1
liters
conversions.
2
a
cm
4
cm
2
to
2
mm
b
600
d
4.5
mm
2
to
cm
For
2
cm
2
to
2
m
m
Q1f,
convert
2
to
cm
2
m
2
to
cm
then
to
2
2
e
2
2.9
Find
cm
these
2
to
mm
f
0.7
m
b
5000
d
3
f
10
mm
2
to
mm
conversions.
3
m
2
3
a
40
to
c
2 400 000
e
2.5
3
cm
mm
3
to
For
cm
Q2f,
rst
3
convert
3
cm
3
to
cm
3
liters
to
to
A
4
a
piece
1
of
decimeter
Copy
and
has
dm
volume
(dm)
m
to
liters
=
10
500
cm
Find
c
A
Find
its
volume
in
m
cm
3
=
____
cm
3
b
3
.
complete:
3
1
the
number
of
dm
3
in
1
m
.
3
5
Find
with
removal
the
an
Suggest
1
liter
truck
capacity
average
Problem
6
has
in
a
capacity
liters
depth
of
of
a
2.15
of
6
m
swimming
.
3
Find
pool
its
that
capacity
in
measures
dm
33
m
×
16
m,
m.
solving
suitable
of
cm
3
cm
foam
3
to
ml
3
3
m
3
m
dimensions
for
a
cuboid-shaped
carton
designed
to
hold
juice.
7.2 How do they measure up?
2 31
Exploration
A
cuboid
is
30
3
cm
tall,
50
cm
long
and
1
m
wide.
mc
03
1
50
m
cm
3
1
a
Use
the
lengths
in
cm
b
Use
the
lengths
in
m
c
Show
a
Calculate
b
Use
to
calculate
the
volume
of
the
cuboid
in
cm
3
that
your
two
to
calculate
results
are
the
volume
of
the
cuboid
in
m
equal.
2
2
the
surface
area
of
the
cuboid
in
cm
2
3
A
box
step
in
a
the
to
nd
shape
the
of
a
surface
cuboid
area
is
of
250
the
cm
cuboid
tall,
472
in
m
cm
long
and
545
cm
3
wide.
Calculate
Reect
In
and
Exploration
●
Which
of
the
volume
discuss
of
this
cuboid
in
m
3
3:
units
rst
the
would
cuboid?
you
use
Which
to
calculate
units
would
the
you
surface
use
to
area
give
and
the
volume
nal
answers?
3
●
Option
1:
Convert
Find
the
the
volume
dimensions
to
in
cm
3
,
and
meters,
convert
then
nd
to
the
m
.
Option
volume.
2:
Which
is
the
3
easiest
ATL
way
Practice
to
nd
the
volume
of
the
box
in
m
?
5
2
1
The
2
A
dimensions
of
a
tennis
court
are
2377
cm
×
8.23
m.
Find
its
area
2
soccer
Which
3
A
eld
measures
measure
container
has
is
0.110
easier
length
to
km
×
73
m.
Find
its
area
in
m
in
m
2
and
in
km
visualize?
9750
mm,
width
6400
mm
and
height
5640
mm.
2
a
Find
its
oor
area
b
Find
its
volume
in
m
3
Problem
in
m
solving
2
4
1
hectare
(ha)
is
the
area
of
a
square
10 000
m
2
A
square
Find
You
units
the
can
of
eld
area
use
area
232
the
and
has
of
an
the
area
eld
of
in
0.36
km
hectares.
imperial-to-metric
volume.
7 Measurement
conversion
factors
for
lengths
to
convert
N U M B E R
Example
2
3
Find
Use
the
the
1
inch
1
yard
number
of
cubic
conversions:
=
2.54
1
meters
inch
=
(m
2.54
)
in
cm,
10
1
cubic
yard
=
yards.
36
inches
cm
Write
10
=
36
yards
=
36
inches
×
10
=
360
inches
the
conversions.
then
centimeters.
inches
Convert
360
down
inches
=
2.54
10
yards
=
9.144
10
cubic
yards
cm
×
360
=
914.4
yards
m
to
inches,
Convert
3
=
10
cm
(10
10
yards
to
meters.
3
yards)
=
(9.144
m)
=
764.554…
Check
that
the
answer
makes
sense:
10
yards
3
10
cubic
yards
=
764.55
m
(to
2
d.p.)
is
shorter
than
10
m
(10
yards
=
9.144
m)
so
3
10
Practice
1
A
cubic
yards
should
be
smaller
than
(10
m)
6
rectangular
piece
of
land
a
Find
the
dimensions
b
Find
the
area
a
A
of
the
of
is
2
the
land
km
piece
in
long
of
square
and
land
3
in
km
wide.
miles.
miles.
3
2
box
measures
4
in
×
5
in
×
6
in.
Find
i
its
volume
in
cm
,
and
2
ii
its
surface
area
in
cm
3
b
Another
box
measures
3
in
×
5
in
×
8
in.
Find
i
its
volume
in
cm
,
and
2
ii
its
Write
surface
down
area
what
in
you
cm
notice
about
your
answers
to
parts
a
and
b
3
3
Find
the
number
of
cubic
centimeters
4
Find
the
number
of
square
5
A
truck
6
A
cube
(cm
)
in
5
cubic
inches.
1
yard
=
0.9144
2
meters
(m
)
in
10
square
yards.
3
has
a
capacity
of
25
m
.
Find
the
capacity
of
the
truck
in
cubic
feet.
3
has
volume
in
.
a
Find
the
side
length
of
the
b
Find
the
side
length
in
cm.
Problem
7
64
A
dressage
cube
in
inches.
solving
arena
for
horse
riding
is
a
rectangle
twice
as
long
as
it
is
wide.
2
Its
area
is
8611.13
Converting
Compound
be
.
Find
compound
measures
measured
When
ft
in
per
length
more
second
measures
in
meters.
measures
involve
meters
compound
its
are
than
and
one
unit.
density
involved,
you
in
For
example,
grams
may
per
need
to
speed
cubic
can
centimeter.
convert
both
units.
7.2 How do they measure up?
23 3
m
Density
is
the
mass
of
substance
contained
in
a
certain
volume.
The
3
density
is
of
solids
measured
Example
in
and
liquids
is
measured
in
g/cm
,
and
the
density
1
kg
density
=
1000
of
iron
is
7.87
g/cm
.
3
Find
the
density
of
iron
in
kg/m
g
Write
3
1
m
gases
3
3
The
of
g/l.
down
the
conversions
you
will
need.
3
=
1 000 000
cm
3
Method 1:
3
7.87
g/cm
convert
g/cm
Now
convert
g/m
3
to
g/m
3
×
1 000 000
=
7
870 000
3
7 870 000
First
g/m
g/m
3
3
÷
1000
=
7870
kg/m
3
to
kg/m
Method 2:
Write
the
fraction
conversion
and
factor
for
volume
as
a
multiply.
Multiplying
by
multiplying
by
is
equivalent
to
3
1,
since
1 000 000
cm
3
=
1
m
.
3
The
cm
units
Write
as
a
The
In
speed
some
Which
1
mile
=
limit
and
by
multiplying
by
time,
the
factor
is
1,
for
mass
multiply.
since
grams
equivalent
1
kg
=
to
1000
g.
cancel.
on
of
motorways
Europe,
the
in
the
speed
UK
limit
is
is
70
110
miles
per
hour
kilometers
per
(mph).
hour
(km/h).
faster?
1.609
70
mph
×
70
mph
is
234
conversion
4
parts
is
the
fraction
Multiplying
This
Example
cancel.
km
1.609
faster
=
Write
112.63
than
110
km/h
km/h.
7 Measurement
Convert
70
down
miles
the
per
conversion
hour
to
you
will
kilometers
need.
per
hour.
N U M B E R
Practice
1
7
Convert
hour
a
2
these
speeds
from
kilometers
per
hour
(km/h)
to
meters
per
(m/h).
10
km/h
Convert
b
these
54
speeds
km/h
from
c
meters
per
4.8
hour
km/h
(m/h)
d
to
meters
280
per
km/h
second
(m/s).
For
a
3600
m/h
b
43
200
m/h
c
9000
m/h
d
2160
Q2,
convert
3
Oliver
4
A
sea
is
driving
turtle
can
at
65
swim
km/h.
at
a
Find
speed
his
of
2
speed
m/s.
in
Aluminium
has
6
Oxygen
a
has
Problem
density
density
2.70
of
g/cm
Find
its
speed
in
km/h.
1.43
g/l.
Find
Find
its
its
density
density
in
in
3
Lead
has
density
11.34
g/cm
8
Gold
has
density
19.32
g/cm
9
The
speed
limit
Europe.
Are
Problem
A
11
cat
reached
Oxygen
minute,
of
per
has
in
urban
these
Find
the
mass
of
1
Find
the
mass
of
0.5
is
1
to
hour
m
of
lead.
is
the
30
mph
same?
in
the
Explain
m
US,
of
and
your
gold.
50
km/h
in
most
answer.
can
reach
top
speed
a
density
a
of
of
top
speed
12.27
1.43
of
m/s.
g/l.
48
km/h.
Determine
Enzo
says
that
The
sprinter
which
120
cl
is
of
Usain
Bolt
faster.
Tip
oxygen
has
a
grams.
Justify
whether
or
not
he
is
of
a
correct.
travelling
travel
at
110
a
speed
km.
of
Justify
25
m/s.
whether
He
or
claims
not
he
that
is
it
will
take
him
can
tell
do:
mean
Problem
1997,
the
a
car
distance
solving
world
Thrust
speed
sound
a
of
what
will
divided
In
measure
you
correct.
to
13
to
3
.
areas
speeds
a
17.16
Timeo
then
second
3
.
compound
12
meters
kg/m
Units
mass
to
solving
domestic
has
per
mg/cl.
3
10
hour
solving
7
of
per
meters
3
.
meters
m/s.
3
5
rst
m/h
SSC.
about
barrier.
land-speed
The
340
sound
meters
Explain
record
your
of
barrier
per
763.035
is
miles
reached
second.
per
when
Determine
if
hour
an
was
object
Thrust
set
moves
SSC
by
time.
by
broke
at
the
answer.
More conversions and units
D
●
Should
●
How
we
do
adopt
systems
a
single
global
inuence
system
of
measurement?
communities?
Conversions
Activity
1
Euro
(EUR)
1
USD
=
=
0.996
Swiss
1
Convert
650
2
Convert
1800
3
Calculate
an
1.10
EUR
US
Dollar
Franc
into
CHF
(USD)
(CHF)
USD.
into
USD.
approximate
value
of
900
CHF
in
EUR.
7.2 How do they measure up?
23 5
Reect
●
How
and
is
the
process
converting
and
●
a
from
currency
Quantity
as
discuss
is
of
unit.
conversion
Research
b
Which
c
What
to
units
what
systems
are
do
from
one
another?
amount
or
with
to
to
measurement?
the
similar
unit
fully.
or
do
to
conversions
Measurement
capacity
more
another
both
Explain
number.
quantity,
conversion
unit
Are
dimension
with
Explain
is
dened
using
quantity,
a
and
unit
fully.
4
natural
Determine
an
currency
more
a
as
to
standardized?
determining
Is
Exploration
1
converting
currency
conversions
dened
method
dened
of
one
4
the
and
answer
natural
of
units
natural
advantages
questions.
are.
units
of
these
are
the
used?
natural
system
of
units?
Are
there
any
disadvantages?
2
Systems
denes
man-made
unit.
a
Determine
b
Decide
Reect
●
Is
●
Why
●
What
the
is
If
a
a
units
the
a
would
prototype
natural
man-made
no
be
single
the
need
a
prototype,
measurements
discuss
there
unit
is
are
for
needs
that
compared
the
a
is,
to
an
the
object
that
prototype.
kilogram.
prototype.
5
unit
or
global
natural
system
advantages
and
unit?
of
measurement?
disadvantages
of
having
a
single
measurement?
single
chosen?
Some
what
and
of
Other
whether
day
system
●
of
a
system
Explain
unusual
●
The
Hand
●
The
Board
●
The
Shake
●
The
Foe
units
(=
4
foot
(=
of
measurement
of
measurement
inches)
(=
10
was
to
be
adopted,
which
should
be
fully.
1
for
inch
include:
measuring
×
1
foot
nanoseconds)
×
for
1
the
foot)
height
for
referring
of
a
horse
measuring
to
very
lumber
short
periods
oftime
44
(=
produced
23 6
10
when
joules)
a
star
for
goes
measuring
supernova
7 Measurement
the
enormous
amount
of
energy
N U M B E R
Summary
A
system
based
on
system
of
a
measurement
set
are
of
base
derived
is
units.
from
a
system
All
the
other
base
of
measures
units
in
the
units.
Capacity
to
hold
of
space
refers
to
the
something.
actually
amount
Volume
occupied.
of
space
refers
Both
to
available
the
amount
capacity
and
3
volume
The
two
most
common
systems
of
can
be
measured
3
the
metric
system
and
the
imperial
metric
system
allows
other
cm
using
unit
prexes,
prex
which
species
a
act
as
power
units
to
be
multipliers.
of
ten;
liters:
3
=
1
ml,
for
1000
cm
=
1
liter.
measures
such
as
speed
and
density
created
combine
by
and
system.
Compound
The
cm
measurement
1
are
in
measures
of
two
quantities.
Each
example,
Density
is
the
mass
of
substance
contained
in
a
3
the
prex
kilo-
species
10
certain
Some
common
conversions:
2
To
convert
from
m
2
to
cm
,
2
To
convert
from
mm
To
convert
from
m
to
Mixed
1
a
b
3
the
a
use
cm
cm
of
to
the
capacity
d
the
weight
cm
the
amount
f
the
distance
a
3
to
b
4.8
c
760
d
9845
e
0.01
f
400 600
kg
of
by
by
100.
1 000 000.
The
in
density
g/cm
;
,
divide
by
4
b
15
solids
and
density
of
liquids
gases
is
is
measured
measured
in
g/l.
1000.
measurement
you
4
a
a
of
in
a
swimming
storage
box
m
km
these
9500
c
0.5
d
6400
e
10
in
cm
mg
to
a
cake
Chicago
to
f
Los
Angeles
5
m
m
≃
≃
kg
___
≃
d
6
liters
e
10
f
800
UK
g
___
≃
3
to
cm
to
ml
Find
m
to
these
liters
conversions.
a
950
meters
b
36
km/h
c
10
meters
d
7200
e
54
to
to
to
hour
(m/h)
to
km/h
m/h
per
m/h
km/h
per
second
(m/s)
to
m/h
m/s
m/s
6
A
bottle
___
solving
contains
Calculate
the
from
240
number
cl
of
of
medicine.
25
ml
doses
that
can
it.
g
The
table
gives
the
heights
and
weights
of
cm
people.
feet
___
pints
≃
mm
mm
cm
100
two
8
m
2
to
Problem
7
c
to
3
conversions.
inches
2
cm
km
to
ounces
mm
3
m
to
b
2
to
cm
3
g
to
50
2
person
from
a
speed
our
conversions.
2
cm
cm
these
pool
conversions.
to
Find
poured
a
of
the
2
water
of
of
e
these
unit
walking
c
Find
divide
multiply
measure:
volume
m
,
3
,
3
mm
which
to
person’s
Find
10 000.
practice
Determine
would
2
from
by
3
to
3
convert
multiply
2
3
To
volume.
___
UK
≃
Height
Weight
190
gallons
___
Person
A
6
ft
2
Person
B
1.82
in
lbs
gallons
m
90
kg
pounds
a
Who
is
taller?
b
Who
7.2 How do they measure up?
is
heavier?
237
be
2
8
An
airline’s
SIZE:
hold
length
+
baggage
width
+
allowances
height
must
12
are:
be
A
square
275
has
area
2323.24
ft
less
Find
than
eld
the
side
length
of
the
eld
in
meters.
cm.
3
13
WEIGHT:
free
up
to
20
kg.
Each
extra
kg
The
density
of
nickel
is
8900
kg/m
(or
3
part
a
of
Jo’s
a
kg)
is
charged
suitcase
at
measures
13
18
Find
Euros.
in
×
22
take
it
as
in
×
8
the
density
whether
he
can
hold
nickel
in
g/cm
in.
Problem
Decide
of
solving
luggage.
3
14
b
Suzanne’s
bag
weighs
48
lbs.
Calculate
Platinum
she
would
have
to
pay
to
take
it
has
a
A
21 450
kg/m
of
19.32
.
g/cm
which
metal
is
denser.
baggage.
water
tank
has
length
2250
mm,
A
car
mm
and
height
3860
mm.
travels
Calculate
limit
of
the
tank
Find
number
in
is
20
50
m/s
in
km/h.
a
village
where
Determine
if
the
the
car
is
the
exceeding
capacity
at
width
speed
5800
of
as
15
9
density
density
Determine
hold
a
3
Gold
much
has
how
the
speed
limit.
liters.
16
A
top
speed
of
231.523
mph
was
set
by
a
3
10
the
of
cubic
inches
in
9
cm
Formula
11
Find
the
number
helicopter
of
2
m
in
6
square
1
1
in
=
1
yd
2.54
racing
has
in
in
2005.
maximum
A
Chinook
speed
of
87.5
m/s.
cm
yards.
Determine
Review
a
car
=
36
which
is
faster.
inches
context
Globalization
and
sustainability
3
b
Carbon
of
steel
carbon
is
steel
used
has
a
for
the
mass
main
of
cables.
about
7.85
1
cm
g.
3
Find
c
the
When
was
density
the
built,
of
carbon
bridge
yards
concrete
After
was
the
yard
=
0.9144
m
of
was
used.
original
replaced,
Calculate
inkg/m
389 000
1
cubic
steel
the
concrete
there
total
was
roadway
6.4%
quantity
less
of
deck
concrete.
concrete
in
the
3
refurbished
d
1
The
Golden
opened
a
for
Below
Find
Gate
vehicular
are
the
Bridge
some
in
trac
facts
equivalent
San
in
Francisco
A
bolt
the
of
diameter
about
the
ii
Each
of
are
measurements
the
746
two
feet
main
in
The
total
main
23 8
length
cables
is
of
tall.
cables
wire
80 000
inches
was
used
in
i
Find
the
ii
Imagine
used
miles.
7 Measurement
diameter
of
the
bolt
in
mm.
is
7 650
in
that
a
bolt
of
diameter
2.125
cm
metric
long.
iii
2.125
bridge.
mistakenly
inmm.
towers
m
1937.
units.
The
in
construction.
was
i
bridge
both
feet
used.
Calculate
the
error
N U M B E R
2
Railway
tracks
dierent
gauges
a
in
dierent
(spacing
Approximately
today
use
the
60%
countries
between
of
the
standard
have
the
rails).
world’s
gauge,
railways
which
is
1
4
ft
8
in.
Find
this
measurement
in
metric
2
units,
b
correct
Originally,
gauge
was
as
nearest
trains
transport
the
the
gauge
trains
mm.
used
trains.
for
run
a
4
ft
Half
8
passenger
more
in
an
inch
trains,
smoothly
on
curves.
the
metric
Find
length
units,
the
sizes,
iii
to
made
Find
ii
the
English
coal
added
this
the
i
for
to
in
to
of
the
the
dierence
coal-mining
nearest
gauge
in
mm.
between
the
two
gauge
mm.
Determine
whether
your
answer
in
1
part
ii
corresponds
in.
to
If
not,
nd
the
2
percentage
c
Some
countries
isexactly
in
d
1000
imperial
Until
and
error.
units,
2011,
France
Suggest
tracks
used
meter
Find
feet
used
the
Spain
and
a
gauge,
this
which
measurement
inches.
1668
standard
modied
mm
gauge,
gauge.
its
railway
engines.
Reect
and
How
you
have
the
in
Spain
why
and
use
mm.
discuss
explored
the
statement
of
inquiry?
Give
specic
examples.
Statement of Inquiry:
Quantities
and
human-made
measurements
systems
and
illustrate
the
relationships
between
communities.
7.2 How do they measure up?
23 9
Going
around
and
around
7.3
Global context: Personal and cultural expression
Objectives
●
Knowing
and
the
Inquiry
terms
chord,
SPIHSNOITA LER
Finding
the
length
●
Finding
the
angle
●
Finding
the
perimeter
●
a
Use
of
in
an
a
arc
of
sector
and
a
of
area
the
the
a
of
a
can
you
dierent
nd
parts
the
of
a
measurements
for
circle?
of
a
●
How
●
Is
are
the
parts
of
a
circle
related?
C
circle
sector
D
length
How
circle
circle
Finding
ATL
●
segment
F
●
of
arc,
sector
questions
it
advantageous
to
generate
your
own
formulae?
chord
Communication
and
interpret
a
range
of
discipline-specic
terms
and
symbols
7.3
9.1
E9.1
Statement of Inquiry:
Generalizing
construction
24 0
relationships
and
analysis
between
of
measurements
activities
for
ritual
enables
and
play.
the
G E O M E T R Y
Y
ou
●
should
nd
the
using
●
nd
C
already
circumference
=
the
2π r
area
of
a
how
circle
1
πd
or
of
know
a
circle
using
2
to:
Find
the
a
of
radius
circumference
b
of
diameter
Give
your
Find
the
2.4
5.75
answers
area
of
a
circle:
cm
of
m.
to
the
1
d.p.
two
circles
2
A
●
=
r
nd
above.
missing
lengths
right-angled
in
triangle
trigonometric
a
3
using
Find
Give
the
your
missing
answers
length
x
to
in
1
d.p.
this
triangle.
ratios
5
cm
x
30°
Par ts
F
●
How
of
can
a
circle
you
nd
the
measurements
for
the
dierent
parts
of
acircle?
ATL
Activity
1
Match
●
A
each
denition
diameter
is
a
with
chord
a
part
that
of
the
passes
circle.
F
through
E
the
center
of
a
circle.
O
C
●
A
chord
on
a
is
a
line
segment
with
its
D
endpoints
circle.
B
A
●
A
secant
two
is
a
line
that
intersects
a
circle
at
G
points.
T
●
A
tangent
only
2
Match
one
the
is
a
line
that
touches
a
circle
at
point.
following
denitions
with
A
the
parts
The
of
major
the
circle
segment
in
is
the
the
diagram.
X
larger
W
segment.
The
Z
minor
segment
is
the
smaller
Y
segment.
the
(‘Segment’
minor
usually
refers
to
segment.)
B
The
larger
arc
The
smaller
is
arc
the
is
major
the
arc.
minor
arc
7.3 Going around and around
241
A
sector
radii.
is
The
a
‘slice’
center
of
of
a
a
circle
circle
between
is
usually
two
labelled
B
radius
O,
so
the
sector
here
is
BOC.
‘Sector’
usually
minor
means
the
minor
minor
major
sector.
O
arc
sector
sector
Lorenzo
Mascheroni
radius
C
proved
that
any
you
can
with
ATL
and
Reect
and
discuss
in
a
1797
point
construct
compass
ruler
can
constructed
How
are
secants
and
chords
similar?
How
are
they
dierent?
just
a
with
compass.
was
then
that
George
Mohr,
Exploration
be
1
a
It
discovered
Danish
1
mathematician,
1
Use
a
set
mark
its
of
compasses
center.
diameter
of
the
Draw
to
a
draw
chord
a
circle.
that
is
Clearly
not
had
the
this
circle.
is
O
already
in
1672
known
as
proven
so
it
the
Mohr–Mascheroni
2
Draw
radii
from
the
center
of
the
circle
to
each
theorem.
endpoint
triangle
3
Use
a
the
you
chord.
have
protractor
sector
4
of
Draw
you
a
have
line
Determine
which
type
of
drawn.
to
measure
the
angle
of
the
drawn.
segment
Tip
from
the
center
of
the
O
circle,
a
O,
to
Explain
●
the
midpoint
why
divides
this
the
line
of
the
chord,
Use
M
segment:
triangle
in
half
M
●
divides
the
(measure
angle
to
of
check
the
that
sector
it
in
half
does).
O
b
Calculate
segment
In
a
the
circle,
center
two
of
the
angle
makes
radii
the
that
with
and
circle
a
to
the
the
chord
the
new
chord.
line
Measure
form
midpoint
an
of
to
check.
isosceles
the
triangle.
chord
forms
A
two
line
from
right-angled
triangles.
When
the
you
know
right-angled
24 2
the
angle
triangles
at
to
the
nd
center,
the
7 Measurement
you
length
can
of
the
use
trigonometry
chord.
in
one
of
symmetry.
G E O M E T R Y
Example
Find
the
1
length
of
chord
XY.
6
X
cm
30°
Y
6
cm
X
6
cm
a
Divide the isosceles triangle
15°
into two right-angled triangles.
Y
sin
15°
a
XY
is
=
=
6
=
1.5529…
1
sin
1.5529…
Practice
Give
×
your
Find
15°
×
2
=
3.11
m
(3
s.f.)
The length of the chord is twice the length of a
1
answers
the
cm
length
to
of
1
decimal
chord
place.
JK
2
Find
the
J
length
of
LM
L
6
10
chord
cm
cm
126°
135°
M
K
Problem
solving
Tip
A
3
The
sector
Find
the
AOB
length
is
of
one
third
chord
of
the
circle.
First
AB
7
of
cm
nd
the
the
angle
sector.
O
B
7.3 Going around and around
24 3
4
POQ
is
sector
Find
a
is
sector
75°.
the
of
a
circle.
Chord
radius
of
PQ
the
The
has
angle
length
8
of
the
The
cm.
Mesopotamians
used
circle.
O
a
sexagesimal
system,
base
that
60.
We
is,
use
75°
this
8
cm
today
when
measuring
time–
Q
5
ROS
is
a
sector
of
a
circle
of
radius
5
cm.
P
60
Chord
RS
is
8
cm.
Find
of
sector
in
a
the
minute,
S
angle
seconds
ORS
8
60
minutes
cm
in
5
cm
an
hour
angles
–
–
and
360°
=
R
5
cm
60×
6.
O
C
Relationships
in
●
a
2
How
Draw
a
Label
the
Draw
radii
center
are
3
By
circle
of
drawing
Explain
When
will
a
To
the
fraction
2
the
central
the
set
of
a
circle
related?
compasses
center
segment
show
the
the
the
angle
that
formula
pizza,
of
a
changes
describe
1
of
and
then
draw
a
chord
each
endpoint
that
is
not
the
diameter
of
the
circle.
c
line
to
your
a
parts
of
the
circle
to
of
the
chord.
Indicate
the
angle,
θ,
in
the
sector.
what
sharing
be.
with
from
the
that
the
2
chord
trigonometry
Verify
measurements
circle
Exploration
1
between
from
the
works
you
of
for
would
more
size
the
a
center
relationship
of
questions
have
to
equal
slices
pizza
slice
the
1,
make
you
you
circle
between
2
and
to
cut,
to
r
3
your
the
could
θ,
the
and
in
c
midpoint
can
be
Practice
formula
smaller
the
chord
as
θ
c,
=
use
2arcsin
(
).
1.
for
each
of
expressed
questions
4
and
5
in
Practice
1.
slice
measure:
pizza
θ
(angle
at
the
center
sector
of
the
sector),
measure
of
sometimes
the
called
the
sector
central
3
the
length
of
the
4
the
perimeter
crust,
or
arc
length,
angle
l
‘You
of
the
sector,
pizza
5
the
area
of
the
sector,
better
A
arc
length
in
because
enough
four
I’m
to
Lawrence
24 4
cut
the
P
7 Measurement
pieces
not
eat
Peter
hungry
six.’
“Yogi”
Berra
G E O M E T R Y
Objective:
B.
ii.
patter ns
In
describe
this
Investigating
Exploration
between
the
as
you
dierent
patter ns
general
will
r ules
search
for
measurements
in
consistent
patterns
a
and
with
ndings
generalize
the
relationships
circle.
ATL
Exploration
In
this
3
Exploration
you
will
generalize
the
relationship
between
the
measurements.
sector
For
a
pizza
of
radius
r
cut
into
four
equal
slices
or
sectors:
r
1
State
the
fraction
2
State
the
central
3
What
fraction
of
of
each
sector?
of
each
sector.
of
the
angle
the
pizza
for
in
each
each
sector.
circumference
Write
down
an
sector.
is
the
arc
expression
for
length
the
arc
length
When
answering
questions
4
Describe
of
a
how
sector.
you
Write
can
use
down
the
an
arc
length
expression
of
for
a
the
sector
to
nd
perimeter
of
the
the
perimeter
each
length
and
sector.
What
fraction
of
the
area
of
the
circle
is
the
area
of
each
an
expression
for
the
area
of
each
Copy
and
complete
Number
the
include
1
of
units.
Sector
Central
angle
sectors
(the
the
sector.
table.
Fraction
equal
should
sector?
appropriate
6
arcs
your
answer
always
Write
of
sectors,
nal
5
about
Arc
length
Sector
area
perimeter
whole
360°
circle)
2
3
4
60°
12°
10°
9°
n
360/θ
7.3 Going around and around
24 5
Arc
length
●
and
arc
sector
area
formulae
length
arc
length
r
θ
●
sector
area
sector
Example
Find
the
area
2
area
and
circumference
of
‘In
this
circle
in
terms
of
terms
means
your
9
2
Area
=
of
’
.
2
r
=
×
9
leave
in
answer.
cm
2
=
81
cm
Remember to include the correct units.
Circumference =
Example
Find
and
the
as
a
2
r
=
2
×
×
9
=
18
cm
3
length
of
number
arc
to
2
XY
in
terms
decimal
of
,
places.
X
6
cm
30°
Y
r
=
6
Arc
cm
length
XY
Use the arc length formula.
=
=
24 6
cm
3.14
cm
7 Measurement
G E O M E T R Y
Reect
Circle
to
a
and
and
given
sector
Which
●
What
gives
of
the
degree
Example
the
problems
number
●
Find
discuss
of
2
may
decimal
most
ask
accurate
accuracy
for
lengths
and
areas
in
terms
of
,
or
places.
do
value?
you
think
is
necessary?
4
area
of
sector
PQ.
P
Give
your
answer
to
3
s.f.
40°
Q
8
m
θ = 40°
r
=
8
m
Area
of
sector
PQ
Use the formula for the area of
=
a sector.
22.3402…
2
Area
of
sector
Practice
1
Find
the
terms
of
PQ
=
22.3
area
Give
s.f.)
Remember to include the units.
and
circumference
of
each
circle.
b
the
Give
your
answers
in
.
3
Find
(3
2
a
2
m
area
your
c
20
cm
and
perimeter
answers
to
a
2
of
the
half
0.4
circle
and
quarter
m
circle.
d.p.
b
7
mm
12
cm
mm
7.3 Going around and around
24 7
3
The
radius
of
this
circle
is
8
K
cm.
Tip
a
Find
the
arc
length
HI
Perimeter
b
Hence
nd
the
perimeter
of
sector
=
arc
HI
99°
135°
c
Find
the
perimeter
of
sector
length
KJ
(P
53°
H
=
l
+
+
2r
2r)
73°
d
Find
the
area
of
sector
HK
arc
r
J
length
I
r
A
4
In
a
b
this
question,
Find
the
Find
give
length
the
of
all
arc
perimeter
of
your
answers
in
terms
of
π
AB.
the
120°
shaded
sector.
B
7
c
5
Find
This
a
is
the
a
Find
area
sector
the
of
of
the
a
shaded
sector.
circle.
perimeter
of
the
sector.
14
b
Find
the
area
of
cm
the
m
sector.
240°
6
a
Find
the
length
of
the
major
arc.
36
b
c
Find
Find
the
the
perimeter
area
Problem
7
VX
and
VX
=
Find
UW
the
the
major
major
cm
45°
sector.
sector.
solving
UW
the
of
of
=
are
12
both
diameters
of
the
V
circle;
cm.
perimeter
of
sector
XOW
53°
U
O
W
X
8
A
frame
with
The
a
for
vertical
and
the
and
radius
of
this
a
curved
top
mirror
top
sides
is
20
an
as
of
arc
cm.
is
in
the
shown
the
shape
of
a
rectangle
in
the
diagram.
frame
are
50
with
central
Calculate
the
cm
angle
long,
97°
97°
20
cm
50
perimeter
frame.
30
diagram
24 8
7 Measurement
cm
not
to
scale
cm
G E O M E T R Y
9
Calculate
radius
4
the
volume
of
this
solid.
The
end
face
is
a
sector
of
a
circle,
cm.
4
cm
45°
10
10
A
light
hardwood
cross-section
of
dowel
radius
3
has
a
cm,
cm
semi-circular
and
is
2
m
2m
in
length.
in
the
Calculate
●
In
volume
of
wood
dowel.
3
Finding
D
the
Is
it
this
unknown
advantageous
section
you
to
will
values
generate
generate
your
more
in
own
of
cm
circles
formulae?
your
own
formulae.
ATL
Exploration
1
Calculate
the
4
circumference
of
each
whole
circle
in
terms
of
.
l
12
cm
3π
cm
θ
has
fascinated
r
people
The
for
Great
Pyramid
has
Use
the
arc
length
down
a
to
work
general
out
formula
the
for
fraction
nding
of
the
each
circle
angle
of
that
sector
is
θ
length
l
and
the
radius
relate
Calculate
the
area
of
each
very
the
to
it.
The
r
base
2
Giza
shaded.
given
closely
arc
at
dimensions
that
Write
centuries.
whole
circle
in
terms
of
.
the
is
square,
perimeter
which
the
a
is
of
equal
to
circumference
r
of
8
a
circle
with
a
mm
θ
A
radius
equal
pyramid’s
to
the
height.
2
6.4π
Use
the
Write
the
sector
down
area
A
3
Rearrange
4
Make
5
Rearrange
6
Solve
r
area
the
the
work
radius
of
the
for
the
for
fraction
nding
of
the
the
circle
angle
of
that
the
is
shaded.
sector
θ
given
r
you
formula
formula
out
formula
formula
subject
the
to
general
and
the
the
a
mm
found
in
step
1
to
make
arc
length
l
the
subject.
formula.
you
found
in
step
2
to
make
area
A
the
subject.
r
7.3 Going around and around
24 9
Example
Find
the
5
measure
of
arc
θ,
17.59
to
the
nearest
cm
degree.
θ
14
cm
Use the arc length formula.
Rearrange and solve to nd
⇒
The
=
measure
Example
Find
the
of
θ
.
71.988…
to
the
nearest
degree
is
72°
6
radius
of
the
circle
to
3
s.f.
r
31.42
mm
120°
arc
length
=
31.42
mm
Use the formula for the arc length.
Rearrange and solve for r
r
The
radius
25 0
=
of
15.0019...
the
circle
is
15.0
mm
7 Measurement
(3
s.f.)
G E O M E T R Y
Practice
1
Find
3
the
measure
of
the
central
angle
of
the
a
sector
in
each
b
9.599
2
mm
2.793
2
5
2
a
circle.
Find
the
m
m
mm
angle
in
the
sector.
B
A
2
b
Find
the
length
of
the
arc
AB
11.8
cm
7.5
3
a
Find
the
major
measure
of
the
central
angle
of
cm
the
sector.
2
22.34
b
Find
the
length
of
the
major
cm
arc.
4
c
4
Find
Find
the
the
perimeter
radius
of
of
each
the
major
circle.
cm
sector.
Give
your
answers
a
to
3
s.f.
b
24
cm
2
84
150°
m
200°
5
Find
the
perimeter
of
the
major
6
Find
the
area
of
the
minor
In
sector.
question
nd
16.6
6,
rst
sector.
the
angle
of
mm
the
3.14
44
sector.
cm
mm
3
cm
7.3 Going around and around
2 51
Problem
7
solving
Ellie plans to make a paper cone from a sector of a circle with angle 220°. What
size circle does she need to start with to make a cone with curved surface area
2
120 cm
8
Find
? Give your answer in terms of the circle's radius, accurate to 1 d.p.
the
length
of
chord
KM
9
Find
the
length
of
chord
PQ
2
K
12.57
cm
P
3.14
cm
Q
3
3
cm
cm
M
10
The
landing
radius
a
120
Draw
b
c
Find
Find
d
Find
a
zone
m.
for
The
diagram
the
the
a
to
perimeter
area
the
of
javelin
central
represent
of
the
shortest
competition
angle
the
of
the
this
landing
landing
landing
distance
is
sector
the
is
sector
of
a
circle
with
40°.
zone.
zone.
zone.
between
the
two
outer
corners
of
the
landing
zone.
Summary
●
A
chord
on
●
A
a
is
a
line
segment
with
its
endpoints
●
The
arc
circle.
diameter
is
a
chord
that
passes
through
larger
is
the
arc
is
minor
the
major
arc.
The
smaller
arc
the
A
center
of
a
circle.
major
●
A
secant
is
a
line
that
intersects
a
circle
at
two
minor
major
segment
segment
arc
points.
chord
●
A
tangent
one
is
a
line
that
touches
a
circle
at
only
minor
arc
point.
B
F
secant
E
●
diameter
A
sector
radii.
C
is
The
a
‘slice’
center
of
of
a
a
circle
circle
is
between
usually
D
O,
so
the
means
sector
the
is
minor
sector
BOC.
‘Sector’
sector.
A
chord
B
tangent
B
T
radius
minor
●
A
chord
divides
a
circle
into
two
minor
segments.
The
larger
The
smaller
segment
is
the
major
sector
segment.
radius
‘Segment’
●
An
arc
is
segment
usually
a
part
of
is
the
means
the
minor
the
minor
segment.
circumference
circle.
25 2
segment.
7 Measurement
of
a
major
O
arc
●
two
labelled
C
sector
usually
G E O M E T R Y
●
Two
radii
triangle.
to
the
and
A
a
line
from
midpoint
right-angled
chord
of
form
the
the
an
center
chord
isosceles
of
the
forms
●
Arc
●
Sector
length
circle
two
triangles.
area
arc
length
θ
sector
●
The
formula
sector
θ
=
for
radius
r
2arcsin(
the
with
length
angle
of
a
is
c
chord,
=
2r
c,
sin
in
area
a
or
).
r
c
Mixed
1
O
θ
practice
a
Find
the
arc
length
b
Find
the
perimeter
c
Find
the
area
of
of
of
sector
sector
sector
DC
3
BD.
EB
Find
the
perimeter
and
a
the
minor
sector
b
the
major
sector.
area
of:
B
2
mm
77°
C
315°
148°
9
ft
60°
75°
D
E
2
a
Find
the
length
b
Find
the
area
of
of
the
the
major
major
arc
QR
sector.
4
a
Find
the
length
of
the
b
Find
the
perimeter
c
Find
the
area
of
minor
the
arc.
minor
sector.
R
14
of
the
major
sector.
km
105°
Q
2
9.817
5
m
m
7.3 Going around and around
253
5
a
Find
the
measure
of
b
Find
the
length
of
c
Find
the
perimeter
d
Find
the
length
of
the
the
of
central
minor
the
the
angle.
Toni
has
lawn
sprinkler
a
rectangular
sprays
lawn
water
8
m
up
by
to
10
5
m.
Her
m.
arc.
shaded
chord
7
a
Draw
b
By
and
label
a
scale
diagram
of
the
lawn.
sector.
using
compasses
to
draw
the
region
MN
watered
M
the
best
by
the
sprinkler
position
for
the
to
scale,
determine
sprinkler
so
that
it
2
67.02 cm
waters
c
N
8
as
Divide
the
boundary
lawn
d
a
Find
the
measure
of
b
Find
the
perimeter
c
Find
the
length
the
of
central
the
the
angle.
shaded
watered
chord
A
regular
radius
cultural
dowsing’
as
a
means
of
recorded
in
or
even
answers
throughout
as
Egypt.
far
In
the
side
or
a
back
one
cloth
degree
or
to
the
to
history
as
the
version,
with
‘yes’
which
other
unlikely
the
by
the
the
area
of
the
sprinkler.
percentage
hexagon
5
cm.
of
the
lawn
that
is
constructed
your
the
in
a
circle
answer
perimeter
to
a
of
suitable
and
sector
lines
25
m
long,
starting
a
questions
and
time
has
of
the
center
is
the
is
the
an
event
or
‘no’
throwing
question
of
to
the
The
the
distance
circumference
imprint
made
in
the
pendulum
by
the
shot
in
the
landing
area.
a
Find
the
area
enclosed
by
the
entire
sector.
b
Find
the
area
enclosed
by
the
sector
inside
on
how
it
to
circle.
and
one
c
Hence,
nd
the
area
of
the
landing
area.
likely
going
to
has
a
total
length
of
30
cm,
Sector
calculate
total
of
the
45
distance
travelled
if
it
moved
a
line
25
m
degrees.
stop
b
If
the
and
pendulum
it
travels
an
starts
arc
of
facing
10
straight
cm,
down
calculate
10
board
cm
2
The
high
the
foul
50
turned
diagram
put.
2.135
m,
The
shows
the
and
throwing
landing
the
circle
area
landing
has
area
is
a
for
the
a
diameter
dividing
sector
of
with
2.135
25 4
line
mm
angle.
75
shot
7 Measurement
of
the
held
sways
is
circle
circle.
from
pharaohs
is
written
indication
in
the
has
been
the
pendulum
pendulum
of
measured
34.92°
If
the
degree
happen.
a
is
sprinkler.
Calculate
Give
throwing
above
calculate
nding
soil
used
garden
isosceles
solving
34.92°
thrown
expression
the
been
the
two
context
and
gold
and
km
from
water,
inside
sectors
accuracy.
angle
‘Pendulum
possible.
km
10
1
lies
as
UV
of
Personal
by
the
watered
hexagon.
in
that
lawn
sector.
of
Review
the
two
Hence,
Calculate
not
8
19.20
circle
into
Problem
of
of
cm
triangles.
6
much
m
cm
line
wide
the
G E O M E T R Y
3
‘Medicine
wheels’
Americans
Medicine
diameter
for
a
Wheel
of
the
a
28
days
of
Calculate
each
is
about
approximately
were
variety
a
24
equal
the
the
used
of
by
circle
of
meters.
The
stones
It
sections,
lunar
native
rituals.
is
c
Big
with
divided
likely
to
Horn
found
a
events
into
28
spots
represent
20
calendar.
approximate
Dierent
to
(winter
related
meters
distance
central
angle
of
d
sector.
spots
The
measures
b
Calculate
the
area
of
each
and
How
you
have
of
80
rising
rising
of
Calculate
along
the
circle
the
star
points
been
astronomical
of
the
their
stars).
star
Two
Sirius
between
approximate
Aldebaran
happens
central
Calculate
angle
the
shortest
them.
discuss
explored
the
statement
of
inquiry?
Give
specic
examples.
Statement of Inquiry:
Generalizing
and
analysis
relationships
of
activities
between
for
ritual
measurements
and
enables
the
are
circle.
whose
degrees.
have
specic
sector.
distance
Reect
the
apart.
two
the
to
solstice,
to
apart
rising
between
along
correspond
construction
play.
7.3 Going around and around
25 5
Which
triangle
is
just
right
7.4
for
you?
Global
context:
Objectives
●
Solving
problems
trigonometric
●
Knowing
the
Scientic
Inquiry
in
right-angled
triangles
●
using
F
ratios
properties
of
trigonometric
technical
innovation
questions
How
do
you
nd
immeasurable
●
ratios
and
What
are
the
measurements
of
objects?
relationships
in
the
special
triangles?
●
Solving
problems
SPIHSNOITA LER
elevation
and
that
angles
include
of
angles
of
depression
●
C
How
do
angles
nd
●
D
relationships
in
right-angled
real-life
How
does
ratios
Critical-thinking
Propose
and
evaluate
a
sides
help
understanding
help
and
you
measurements?
you
mathematical
ATL
between
triangles
create
the
and
trigonometric
understand
models?
skills
variety
of
solutions
7.4
14.2
Statement
of
Generalizing
Inquiry:
relationships
measurements
models
25 6
and
can
lead
methods.
to
between
better
G E O M E T R Y
Y
ou
●
should
use
the
in
know
Pythagorean
theorem
side
already
a
to
nd
the
1
missing
how
Find
in
T R I G O N O M E T R Y
to:
the
missing
accurate
right-angled
A N D
to
3
s.f.
side
All
in
each
triangle,
measurements
are
cm.
triangle
a
b
5
x
17
x
c
10
12
●
relate
angles
right-angled
the
and
sides
triangle
trigonometric
(sine,
cosine
and
of
a
2
Find
using
x
sin A,
solve
the
simple
tan A
and
tan B
A
ratios
tangent)
a
C
●
cos B,
problems
trigonometric
using
3
ratios
Find
B
the
value
of
x
in
a
each
triangle.
b
x
10 m
10 m
9 m
60°
x
Right-angled
F
triangles
trigonometric
The
●
How
●
What
right
angle
do
is
are
the
at
right
angles
to
their
foundations.
geometry
and
to
you
our
nd
the
most
This
ratios
measurements
relationships
earth,
common
and
and
the
in
the
angle
homes
fundamental
in
of
special
our
we
angle
immeasurable
is
triangles?
world
live
at
objects?
in
the
–
are
after
built
heart
of
all,
at
we
right
the
stand
angles
study
of
trigonometry.
7.4 Which triangle is just right for you?
2 57
Activity
ATL
1
Make
very
the
a
list
tall
of
Write
3
Discuss
these
method
rst.
If
down
you
were
the
Draw
and
a
methods
standing
6
surveyors
survey
distances
and
would
angles
land
have
already
but
distances?
did
In
Practice
an
use
to
where
nd
you
the
are
height
in
of
a
relation
to
ratios
to
ratios
do
of
the
front
job
in
of
a
order
each
starting
they
building,
to
relating
method.
with
the
best
these
and
what
calculate
its
height?
measurements.
prepare
always
surveyors
line
their
sine,
that
a
them
boundaries
land
use
are
maps
measuring
of
use
vision
to
and
instruments,
cosine,
and
measure
an
object.
together
tangent,
far
to
with
determine
angles?
about
so,
with
in
take
their
using
order.
equation
tools
and
know
to
rank
of
angles.
surveyors
learned
you
In
between
heights
order
trigonometric
of
and
ground
need
property
and
kind(s)
trigonometric
triangle,
Use
could
state
disadvantages
others
the
you
write
distances
immeasurable
1
and
and
ranking
accordingly.
what
the
on
determine
plots
Research
How
You
you
method,
with
your
would
diagram
immeasurable
5
methods
each
advantages
Explain
measurements
Land
For
building.’
2
4
dierent
building.
the
trigonometric
they
you
can
will
help
need
to
you
ratios
in
a
determine
review
how
to
right-angled
immeasurable
nd
the
calculator.
1
your
calculator
to
evaluate
these,
either
exact
or
to
three
signicant
gures.
Tip
a
sin
e
tan
47°
b
cos
32°
c
tan
17°
d
sin
f
cos
60°
g
tan
45.8°
h
sin
2°
Make
89°
sure
your
calculator
degree
i
cos
0°
j
tan
Find
the
angle
in
each
expression.
sin
A
=
0.467
b
cos
B
cos
α
=
0.898
e
sin
g
cos
φ
=
0.123
h
tan
Reect
Take
of
a
and
closer
Practice
●
What
●
Can
c
tan
C
=
1
look
at
the
β
=
A
angles.
0.5
=
f
tan
θ
=
3
27.3
1
values
of
the
trigonometric
ratios
1.
do
you
25 8
discuss
you
notice
explain
the
about
is
in
(not
the
reason
range
for
7 Measurement
your
of
values
for
observation?
each?
in
the
questions
mode)
you
degrees
=
2
d
radian
when
2
a
mode
30°
in
2
that
0.789°
to
are
using
measure
G E O M E T R Y
Some
people
use
trigonometric
with
acute
SOH
—
CAH
—
TOA
—
the
ratios
memory
in
2
1
knowledge
your
each
triangle.
aid
SOH-CAH-TOA
right-angled
triangle.
for
Given
remembering
a
T R I G O N O M E T R Y
the
right-angled
triangle
α:
angle
Practice
Use
a
A N D
of
Round
the
your
trigonometric
answers
to
the
ratios
to
nearest
nd
b
a
the
side
hundredth
c
4 cm
marked
where
x
in
necessary
.
11 cm
x
60°
37°
11 cm
x
x
41°
d
e
f
45°
47°
3 cm
50°
x
7 cm
x
x
10 cm
2
Find
All
the
angle
lengths
are
θ
in
in
each
triangle.
Round
your
answers
to
the
nearest
tenth.
meters.
a
b
13
θ
4
θ
3
3
c
d
11
θ
θ
10
11.9
4.4
3
Find
a
all
missing
sides
B
and
angles
A
in
these
right-angled
b
triangles.
c
C
C
B
62°
22.6 cm
53°
C
A
3 cm
40°
B
5 cm
A
d
B
e
A
B
28°
14 cm
6 cm
24°
A
C
C
7.4 Which triangle is just right for you?
25 9
Objective:
i.
use
C.
appropriate
terminology)
In
Communicating
this
in
mathematical
both
Exploration,
sketches
to
be
able
you
to
a
method
building
to
nd
out
Method
1
you
to
1:
Sketch
2
the
the
tall
use
appropriate
that
symbols
and
your
and
answers
accurate
are
notation
in
your
justied.
discussed
height.
your
school
Y
ou
is
in
the
are
Activity
now
without
going
actually
is
to
to
use
the
explore
measuring
shadow
two
its
of
methods
height.
and
its
that
shadow
is
in
your
represented
by
notebook,
this
and
situation.
draw
Label
the
the
sides
triangle.
what
these
your
the
sketch
that
represents
height
2:
1
small
a
can
see
to
you
calculate
would
the
height
need
of
to
the
make
in
building.
order
If
you
to
use
can,
measurements.
nd
Put
measurements
ratios
Use
Method
2
to
others
have
its
triangle
trigonometric
3
may
building
Determine
make
need
(notation,
explanations
Shadow
right-angled
of
will
language
written
1
measure
how
and
convince
Exploration
One
oral
to
of
determine
the
the
which
building.
situation.
trigonometric
Write
(Solve
down
it
if
the
you
ratio
will
help
trigonometric
have
actual
you
equation
measurements.)
Mirror
the
Measure
building
mirror
top
your
to
of
on
the
the
distance
the
ground
building
to
the
in
and
the
stand
far
middle
mirror,
your
of
enough
the
height
away
that
you
mirror.
and
the
distance
of
the
mirror.
3
Sketch
the
situation
4
Justify
how
5
Explain how you could calculate the height of the building a using
the
two
and
indicate
triangles
in
the
measurements
your
diagram
are
in
your
diagram.
related.
trigonometric ratios and b without using trigonometric ratios.
Reect
and
●
method
Which
method?
a
Using
of
the
the
as
Identify
nd
you
your
the
2
think
Pythagorean
a
is
more
accurate,
the
shadow
or
the
of
theorem,
this
nd
triangle.
the
Leave
length
your
radical.
what
type
missing
of
1
triangle
this
is,
and
thus
angles.
1
Continued
2 6 0
mirror
answer.
2
hypotenuse
answer
b
do
Justify
Exploration
1
discuss
7 Measurement
on
next
page
G E O M E T R Y
c
Hence,
use
triangle
to
your
nd
knowledge
the
of
following
trigonometric
values
(leaving
ratios
your
in
a
right-angled
answers
as
T R I G O N O M E T R Y
The
very
45°
cos
°
45°
tan
°
consider
the
45°
a
Using
nd
b
your
memorize,
as
equilateral
angle
knowledge
that
bisector
triangle
shown
of
equilateral
of
the
the
sum
Using
of
the
the
in
a
dotted
base.
line
Using
triangle,
Pythagorean
dotted
Hence,
ratios
use
in
a
following
sin
your
is
the
your
nd
4
right-angled
values
of
nd
the
the
β
β
2
length
help
cos
results
your
step
2,
which
was
other?
What
triangles
solve
Practice
By
using
answers
to
nd
answers
the
in
surd
tan
60°
tan
c
60°
°
and
2
d
by
comparing
the
surd
form
with
the
calculator
was
the
greatest
about
are
1
30°
the
known
longest
angle?
shortest
as
side
of
Where
side
‘special
the
were
and
right-angled
these
the
triangles’.
in
triangle
relation
smallest
to
and
each
angle?
Recognizing
these
ratios
can
problems.
3
the
in
the
in
form):
30°
°
your
α
2
trigonometric
°
60°
from
you
your
cos
values
These
of
°
which
angles.
β
angle
triangle
(leave
30°
Check
In
these
perpendicular
knowledge
theorem,
knowledge
°
3
nd
line.
°
sin
used
triangles,
α
d
methods
here:
2
c
are
°
α
angle
Assume
this
are
to
to
2
of
useful
the
Now
results
exploration
radicals):
sin
A N D
special
radical
triangles,
form
nd
where
1
the
unknown
measurements.
Leave
your
appropriate.
2
3
e
You
can
check
60°
b
12 cm
youranswers
c
5 m
4 m
Pythagoras’
using
theorem.
45°
30°
a
d
f
4
5
h
g
6 cm
x
6 cm
3 cm
7.4 Which triangle is just right for you?
2 61
Real-life
C
●
How
do
relationships
triangles
Example
Y
ou
are
applications
help
you
between
nd
real-life
sides
and
angles
in
right-angled
measurements?
1
standing
the
ground.
the
building
9
Using
is
m
a
50°.
away
from
sur veyor,
Find
the
a
you
building,
read
height
of
the
the
and
your
angle
of
building
eyes
are
elevation
to
the
160
of
cm
the
nearest
from
top
of
meter.
x
Draw
50°
9
a
50°
=
⇒
x
=
9
×
tan
50°
=
relation
side
10.726
+
10.726
=
to
adjacent
opposite
1.6
and
label
the
side,
An
angle
is
of
about
12
m
below
depression
An
the
the
angle
between
above
horizon
is
and
the
an
angle
of
depression
object
of
elevation
horizon
is
the
angle
an
object
and
horizon.
angle
Reect
●
Why
●
The
we
angles
units.
●
and
do
are
and
to
the
of
angles
of
cosine
and
right-angled
units
of
the
tangent
triangles
trigonometric
elevation
as
are
measured
ratios?
2
The
and
m
when
it
To
a
the
angle
of
is
2 6 2
Tower
ground.
standing
elevation
on
ying
the
the
originally
person
Gina
high
Leaning
was
limitations
3
answer.
of
depression
equal?
above
angle
specic
your
of
elevation
solving
famous
55.86
in
Justify
ratios?
4
Problem
1
elevation
trigonometric
angle
Practice
of
3
sine,
sides
are
the
depression
discuss
refer
What
Why
your
kite
you
are
information.
use
are
given
looking
for
the
the
tangent.
the
a
is
built
130
of
kite
that
above
m
away
top
of
of
‘leans’
a
ground
7 Measurement
the
at
from
a
your
makes
the
Pisa
(standing
the
accuracy
of
Determine
at
an
angle
height
right
the
of
that
4°,
top
to
a
is
of
its
angles
base
building
of
the
and
above
m
top
the
is
ground
ground).
building,
58
its
determine
high.
Explain
the
answer.
50°
angle
when
she
with
has
the
let
ground.
out
150
m
Determine
of
string.
height
round
horizon.
the
the
you
hence
tall.
angle
between
given
12.326
and
building
angle,
and
Add
The
the
m
In
tan
sketch
how
to
to
the
the
answer
nearest
meter.
G E O M E T R Y
4
A
vertical
altitude
5
A
ladder
angle
the
6
A
of
that
60°
ship
is
on
of
depth
Reect
●
For
For
is
get
the
of
the
long
is
a
of
far
up
of
away
shadow
which
against
Determine
it
the
is
1.3
m
long.
Determine
T R I G O N O M E T R Y
the
elevation).
leaning
ground.
how
is
on
the
water,
at
side
far
of
the
a
building
ladder’s
making
base
is
an
from
building.
and
underwater,
the
how
its
an
radar
angle
detects
of
a
submarine
depression
of
23°.
at
a
Find
submarine.
discuss
3,
question
to
3,
to
the
4
2,
as
4,
the
specic
as
happens
and
as
describe
sun’s
understand,
between
what
building,
question
stick
represented
Example
casts
angle
surface
the
important
m
m
closer
relationship
Find
the
long
(its
the
and
Practice
normally
6
and
shadow
It
is
with
238
of
1m
sun
Practice
you
●
the
building,
distance
the
stick
of
A N D
at
sides
either
a
what
altitude
this
point,
of
a
to
you
the
get
angle
farther
happens
elevation
the
the
as
building?
length
of
the
changes.
that
a
trigonometric
right-angled
fraction
to
of
from
or
triangle.
ratio
That
is
ratio
a
is
decimal.
2
length
of
the
side
labelled
x
to
3
s.f.
x
40°
500
25°
Separate
the
unknown
triangle
side
in
the
into
two
upper
right-angled
triangle
triangles.
and
use
to
s.f.,
the
trig
Label
ratios
the
to
nd
a
it.
Since
thenal
the
nal
answer
answer
must
be
is
required
carried
out
3
to
at
all
calculations
least
one
more
before
s.f.
40°
500
tan
40°
=
⇒
a
=
500
×
tan
40°
=
419.5
500
25°
Label
the
unknown
side
in
the
lower
b
triangle
tan
x
=
25°
a
+
=
⇒
b
=
500
×
tan
25°
=
use
trig
ratios
to
nd
it.
233.2
b
Add
=
and
419.5
+
233.2
=
652.7
=
653
(3
the
two
lengths
that
make
up
the
unknown
side,
s.f.)
and
round
to
the
required
degree
of
accuracy.
7.4 Which triangle is just right for you?
2 63
Example
In
the
diagram,
depression
of
3
AC
from
depression
C
=
52
to
from
B
C
m.
is
The
28°,
to
D
and
D
is
angle
and
42°.
the
of
C
angle
Find
28°
the
42°
distance
between
B
52
m
42°
28°
A
In
ΔABC,
In
ΔABD,
⇒
BD
=
tan 28°
tan 42°
97.80
Practice
1
Find
=
the
D
⇒
=
Use
B
the
trig
ratios
to
nd
the
required
sides.
⇒
57.75
=
40.0
m
(3
s.f.)
5
missing
length
x
in
each
diagram.
61°
35°
x
39°
cm
60°
9
38
2
Find
the
side
length
x
in
each
cm
x
cm
triangle.
a
In
b
2,
you
will
need
101°
23
71°
x
12
mm
x
mm
to
mm
decide
draw
mm
order
the
44°
Find
the
angle
θ
4
Find
the
distance
m
in
the
diagram.
30°
12.4
θ
58°
20.6
m
5
Find
the
of
B
A,
heights
and
from
the
ground
C
6
Find
the
side
length
x
diagram.
B
10.2
m
900
m
75°
A
20°
C
11.4
479
m
8.5
72°
3.2
m
58°
m
2 6 4
dotted
7 Measurement
m
x
m
in
this
to
solve
problem.
to
line
50°
in
3
a
where
G E O M E T R Y
Problem
7
A
single
an
speed
for
private
speed
an
determine
T R I G O N O M E T R Y
solving
engine
average
A N D
a
of
hour
how
plane
110
and
far
a
ies
km/h.
half
the
It
on
plane
due
North
for
then
tur ns
and
a
bearing
has
of
travelled,
an
130°.
and
hour
ies
b
at
and
the
18
minutes
same
At
the
its
bearing
end
of
at
average
this
from
time,
its
initial
position.
8
Two
The
ships
rst
travels
are
9
A
leave
one
on
from
a
and
60°.
of
travels
from
Reect
top
the
How
does
●
How
do
you
More
How
and
Using
What
3
Summarize
●
●
What
a
●
the
think
on
does
was
bigger
What
Why
0°
your
we
60°
at
km/h.
travels
23
on
km/h
Determine
a
straight
and
the
how
far
line
second
the
course.
one
two
ships
noon.
hill
that
angle
of
post,
lookout
is
100 m
above
depression
the
angle
of
of
sea
a
level.
boat
out
depression
From
at
of
sea
the
the
is
30°,
boat
is
post.
5
help
solve
the
determine
problems
immeasurable
when
the
distances?
triangle
understanding
the
isn’t
a
ratios
trigonometric
mathematical
and
ratios
help
you
create
models?
of
nd
values
cannot
greatest
could
your
no
of
in
sines,
cosines
and
tangents
of
several
a
each
of
the
three
ratios?
table.
6
right-angled
the
the
90°
discuss
angle,
there
trigonometric
ndings
angles
does
is
the
of
one
3
range
and
of
of
a
the
each
triangle?
Considering
cosines
of
28
lookout
calculator,
2
Reect
top
the
at
12:00
post,
of
between
is
at
and
bearing
understand
your
angles
a
115°
discuss
Exploration
1
10:00
trigonometry
right-angled
●
on
height
and
●
D
is
lookout
the
Find
of
another
post
the
at
on
bearing
one
lookout
base
port
triangles,
be
greater
value
you
you
have
calculator
maximum
say
give
found
found
when
value
reasons
than
for
for
an
a
tangent
even
you
the
why
the
sines
and
1.
greater
enter
tan
tangent
ratio?
By
choosing
value?
90°?
Why
is
this?
ratio?
7.4 Which triangle is just right for you?
2 6 5
So
to
far,
you
should
right-angled
●
The
the
lengths
angle.
●
the
side
of
the
The
sine
the
ratio
or
one
to
is
greater
of
the
Draw
legs
a
is
to
make
the
the
zero),
be
the
in
a
the
are
following
generalizations
related
is
shortest
side
length
the
is
of
proportional
side
will
be
triangle
one
of
hypotenuse
always
is
larger
will
opposite
is
to
therefore
90°,
the
the
be
smallest
longest
the
legs
always
than
the
divided
the
by
longest
numerator;
the
side,
hence
1.
ratio
of
the
leg
legs
that
make
other.
The
tangent
greater
are
longest
right-angled
the
the
The
triangle
The
hypotenuse.
Since
than
the
than
1,
or
up
equal
opposite
the
can
to
right
the
angle
angle
therefore
are
be
to
the
either
less
equal,
than
1
(but
1.
short
triangle
(very
close
where
to
one
length
of
0)
the
and
B
the
leg
4
right-angled
very
angles.
fraction
angle
than
right-angled
and
ratios
angle.
Exploration
1
to
less
an
larger
than
has
of
always
a
angle
hypotenuse.
the
tangent
of
angle,
cosine
denominator
adjacent
able
opposite
largest
the
The
sides
triangle
is
been
their
the
and
of
the
of
largest
Since
length
●
of
measures
opposite
have
triangles:
hypotenuse
is
5 cm.
Label
it
like
a
this:
A
2
Measure
3
Using
sin A,
4
lengths
the
and
denition
cos A,
Explain
a
sin B
what
b
of
and
b
sine
and
cosine,
nd
cos B
happens
to
sin
A
as
side
a
gets
You
might
look
at
the
eect
on
sin A
and
cos A
as
angle
A
gets
closer
to
In
Exploration
mathematical
the
5
eect
you
will
concept
5
1
(5,
the
points
0)
sin B
see
you
Exploration
Plot
on
and
how
have
and
the
cos B
as
tangent
angle
ratio
B
is
gets
closer
related
to
to
tan A
90°.
angle
closer
the
line
segment
(10,
joining
Measure
the
angle
between
to
zero.
4)
on
these
graph
two
paper.
Calculate
the
gradient
the
line
gradient
nd
the
tangent
of
the
and
the
x-axis
and
use
your
calculator
straight
between
line
(x
angle.
(x
,
2
3
Write
and
4
Plot
down
the
the
points
joining
segment
and
the
line
segment
two
parallel
Justify
the
to
the
the
2)
and
4),
the
Then,
line
of
(3,
Find
values,
tangent
the
and
x-axis.
the
you
notice
between
the
gradient
of
y
=
and
the
and
nd
−2.
the
nd
tangent
the
Write
of
the
tangent
down
relationship
angle
gradient
the
formed
by
angle
of
the
what
line
the
angle
you
between
the
of
line
line
between
the
notice
the
line
between
segment
the
between
gradient
and
of
a
the
line
x-axis.
relationships
you
noticed
in
steps
3
and
4
Continued
2 6 6
the
angle.
them.
the
tangent
and
relationship
of
(0,
segment
line
5
the
tangent
7 Measurement
of
points.
,
1
to
tan B
gets
another
a
2
A
studied.
The
of
and
zero.
as
Generalize
to
eects
smaller.
on
Generalize
want
the
on
next
page
y
)
2
is
m
y
)
1
and
G E O M E T R Y
6
Consider
the
y-axis.
and
the
What
x-axis
Plot
is
of
as
the
two
formed
of
of
the
and
line
Find
the
in
the
when
4)
be
line
why
a
1.
Write
move
tangent
point
you
and
to
move
the
down
the
of
closer
when
the
the
this
point
nd
what
point
(10,
angle
y-axis.
point
is
on
happens
4)
closer
formed
by
the
Determine
on
the
to
the
of
the
line
the
y-axis?
y-axis?
y
Note
segment
negative
that
you
nd
(0, 4)
the
should
T R I G O N O M E T R Y
to
to
between
tangent
the
you
the
angle
(0,
step
as
segment
Explain
angle
of
the
segment
between
x-axis.
tangent
move
0)
the
segment
segment
change
you
(5,
of
points.
the
the
line
points
gradient
and
on
line
tangent
the
angle
line
the
the
the
the
the
of
Reect
gradient
7
again
gradient
A N D
the
the
angle
the
line
positive
value.
is
between
and
the
x-axis.
obtuse
So
θ
(> 90°).
θ
x
(5, 0)
The
idea
of
trigonometric
right-angled
triangle
will
ratios
be
for
angles
explored
that
further
cannot
in
the
be
MYP
represented
4/5
in
Extended
a
book.
Summary
●
The
trigonometric
triangle
help
couldn’t
you
easily
ratios
make
make
in
a
right-angled
measurements
●
that
Some
results
worth
knowing:
you
otherwise.
2
√3
●
60°
√2
The
largest
angle
in
a
right-angled
triangle
30°
1
is
the
right
angle.
The
two
other
angles
2
30°
√3
are
1
45°
60°
always
acute
angles.
1
1
●
The
longest
side
in
a
right-angled
the
hypotenuse
(opposite
the
other
called
the
triangle
right
angle)
is
and
sin
●
In
a
cos
right-angled
are
value
●
The
and
sides,
always
of
tan
triangle,
between
can
following
legs,
be
any
results
0
are
the
and
value
hold
values
1.
of
sin
for
than
sin,
cos
the
30°
=
sin
60°
=
45°
=
cos30°
cos
=
60°
=
cos
tan
45°
=
tan
1
30°
=
tan
60°
=
the
tangent
tan:
0°
=
0
cos
0°
=
0
tan
0°
=
The
90°
=
1
cos
90°
=
0
tan
90°
gradient
of
a
line
is
equal
to
of
0
the
sin
sin
0.
●
sin
=
and
However,
greater
true
45°
shorter.
=
angle
that
the
line
makes
with
the
positive
undened
x-axis.
●
Angles
of
depression
angle
of
and
elevation:
depression
angle
of
elevation
7.4 Which triangle is just right for you?
2 67
Mixed
1
A
kite
the
practice
string
angle
from
27
48
m
between
degrees
increase
the
is
in
the
long.
the
to
54
As
kite
the
and
the
degrees.
vertical
height
wind
blew,
ground
Determine
of
the
kite
4
In a house that is being renovated, two corridors
meet at right angles, as shown in the diagram.
went
Workmen attempting to take a ladder around the
the
corner get the ladder stuck when it is at an angle of
above
45° to each wall.
ground.
48
a
Determine
the
b
Determine
the
stuck
it
length
length
of
of
the
a
ladder.
2nd
ladder
that
got
48
54°
27°
wall
when
of
the
was
1 m
at
an
angle
of
60°
with
the
corridor.
45°
2
Find
length
x
in
each
diagram.
2
m
a
1
6
7
m
4
0
25°
3
8
1
m
x
Problem
5
A
5
m
solving
ladder
is
resting
against
a
vertical
wall.
b
The
base
of
the
ladder
is
2
m
from
the
wall.
57°
Keeping
mc
that
4.8
1
it
m
which
2
2
is
its
now
base
rests
away.
the
xed,
the
against
the
Determine
ladder
has
ladder
rotated
opposite
the
been
is
angle
wall
so
which
through
rotated.
x
3
ABCD
and
is
PB
a
=
square,
12
where
AP
=
5
cm,
QC
=
7
cm
cm.
5
m
Calculate:
a
the
size
of
b
the
length
the
of
angle
marked
x
AB
x
2
c
the
length
of
DQ
d
the
length
of
PD
6
e
the
size
of
coast
guard
two
depression
∠BQD
C
D
Q
2 6 8
A
observes
P
7 Measurement
of
Determine
m
attendant
boats
45
how
in
m
atop
line
degrees
far
1
and
apart
a
with
the
cli
him
69
120
at
m
degrees.
boats
high
angles
are.
of
G E O M E T R Y
Review
in
A N D
T R I G O N O M E T R Y
context
Scientic and technical innovation
1
In
1852,
an
Indian
mathematician,
Radhanath
Legend
Sikdar,
used
measurements
and
has
calculated
to
calculate
the
height
of
Peak
XV
in
(later
to
be
named
Mount
required
the
use
of
a
device
that
from
the
ground
to
the
top
of
an
measuring
from
two
distance
come
the
between
up
angle
dierent
with
a
to
spots,
these
the
and
drawing
like
top
of
Peak
knowing
spots,
value
Sikdar
Sikdar
the
that
would
believe
him.
he
thought
Supposedly,
added
2
feet
to
make
it
more
believable!
books
record
his
calculated
height
object.
to
By
the
perfect
measured
History
angles
so
Everest).
he
This
that
was
the
nobody
Himalayas
it
trigonometry
be
29
002
feet.
XV
the
may
2
have
You
to
following:
are
the
the
pilot
aireld
altitude
is
of
after
3050
m
a
a
light
aircraft,
parachuting
and
you
are
returning
trip.
Your
heading
towards
P
the
landing
What
that
angle
you
strip
of
touch
which
is
depression
down
on
currently
should
the
14
you
landing
km
use
strip
away.
so
at
h
closest
45°
O
The
x
angle
(from
point
the
was
height
How
you
of
Generalizing
and
17 409.7
to
the
measured
closer
feet
top
to
(point
of
B
the
be
32°.
A),
the
peak
From
angle
a
of
45°
and
Statement
was
feet
Reect
have
32°
A
elevation
B)
17 409.7
elevation
Find
of
point
point?
Sikdar
calculated
for
Peak
XV.
discuss
explored
the
statement
of
inquiry?
Give
specic
examples.
Inquiry:
relationships
between
measurements
can
lead
to
better
models
methods.
7.4 Which triangle is just right for you?
2 6 9
the
8
Patterns
Sets of numbers or objects that follow a specic order or rule
Patterns
Patterns
play
in
an
understanding
follow
from
language
other
is
Subject,
Many
a
elmayı
for
languages,
the
of
is
use
apple
the
apple’.
structure
The
known
in
Steve
an
pattern
titled
and
musician
Search
online
performance
clap
1
clap
2
27 0
the
ate’,
as
and
in
that
and
in
using
a
order.
In
‘Iskender
word
the
vary
structure
built
order
as
English
would
of
often
structure.
in
and
be
patterns
‘Iskender
in
language
linguistics.
music
American
in
over
starting
for
but
study
SVO
sentence
translate
Music
repeated
SOV
learning
sentences
English,
are
Object,
the
structure
‘Clapping
rhythm
one
is
In
the
sentence
Patterns
Reich
an
the
patterns
sentences
and
in
Simple
language.
would
the
translation
those
example,
yedi’
‘Iskender
and
simple
Verb
role
language.
to
languages
Turkish,
ate
–
Romance
common:
important
of
patterns
language
for
half
if
music.
two
150
‘Steve
see
his
composer
beat
Reich
you
One
people’,
times,
a
who
but
his
and
with
later
follow
famous
consists
8
music’
any
of
variation
every
clapping
can
of
experiments
with
works
a
is
single
created
by
bars.
to
listen
patterns.
to
a
live
Patterns
in
physics
Johannes
Kepler
discovered
explain
planets
third
the
the
tells
it
laws
motion
around
time
orbit
physical
the
law
(1571–1630)
the
us
takes
sun,
of
sun.
that
a
and
that
the
if
His
T
is
planet
r
is
to
the
3
radius
What
of
its
does
orbit,
it
then
mean
to
T
∝
r
2
have
3
an
exponent
of
?
By
2
looking
When
bank
you
interest
What
the
invest
usually
savings;
in
with
can
fractional
pattern
you
money
banks
is
at
a
interest
bank,
on
compound
year
be
can
a
the
at
the
help
fair
needed
through
generated
account
Understanding
would
sensible
exponents,
we
exponents
denition.
the
your
the
annually.
bank
each
give
when
nance
awards
many
patterns
working
a
Patterns
for
to
the
by
end
pattern
you
value
to
for
withdraw
the
of
at
each
the
work
an
amount
year?
end
out
of
what
investment
money
in
if
partway
year.
2 71
What
comes
next?
8.1
Global context: Scientic and technological innovation
Objectives
●
Understanding
Inquiry
and
using
recursive
and
explicit
questions
●
What
●
How
is
a
sequence?
F
formulae
for
●
Recognizing
●
Finding
a
MROF
quadratic
sequences
linear
general
and
quadratic
formula
for
a
sequences
linear
a
●
or
sequence
●
Recognizing
●
Solving
problems
real-life
contexts
can
you
describe
the
terms
of
sequence?
What
is
a
general
formula
for
a
sequence?
patterns
in
real-life
contexts
●
What
●
How
can
●
How
do
types
of
sequence
are
there?
C
involving
sequences
in
a
●
you
you
identify
nd
a
sequences?
general
formula
for
sequence?
Can
you
always
predict
the
next
terms
D
of
ATL
a
sequence?
Critical-thinking
Identify
trends
and
forecast
possibilities
8.1
Statement of Inquiry:
Using
dierent
improve
forms
products,
to
generalize
processes
and
and
justify
patterns
can
help
solutions.
12.1
12.2
E8.1
272
A LG E B R A
Y
ou
●
should
identify
the
already
next
term
know
in
how
a
1
sequence
to:
Write
each
a
2,
b
c
●
substitute
into
formulae
2
4,
4,
0,
2,
8,
14,
6,
the
6n
b
6,
9,
Find
a
down
the
next
two
terms
in
sequence.
+
2n
…
19,
12,
…
…
value
3,
+
of :
when
7,
n
when
=
n
4
=
10
2
c
n
1,
when
n
=
2
2
d
●
solve
simple
linear
and
quadratic
3
equations
4n
2n
Solve
a
3n
b
5n
for
+
+
11,
when
n
=
1
n
7
=
19
2
=
14
2
c
●
write
a
simple
general
formula
number
for
4
patterns
5n
Find
b
4,
Introduction
●
What
●
How
●
What
is
a
can
is
a
to
1
=
181
formula
for
the
nth
term
of
sequence.
8,
12,
c
F
a
each
a
+
12,
24,
3,
16,
36,
6,
20,
48,
9,
…
…
12,
15,
…
sequences
sequence?
you
describe
general
the
formula
terms
for
a
of
a
sequence?
sequence?
ATL
Exploration
1
Describe
list
of
numbers.
a
1,
3,
5,
7,
9
b
d
1,
2,
1,
2,
1
e
Predict
with
2
each
1
Here
3,
1,
the
next
three
1,
6,
4,
Compare
9,
12,
numbers
16,
24,
in
your
descriptions
25
48,
each
c
96
f
list.
Compare
7,
0,
with
10,
1,
your
13,
8,
6,
others.
16,
5
results
others.
are
4,
Tip
two
1,
5,
19
dierent
1,
6,
1,
7,
ways
1,
of
continuing
the
list
3,
1,
4,
1,
5,
…
Search
online
for
‘3
5
5
1,
4,
1,
5,
9,
2,
6,
5,
3,
4
1
9
you
need
the
pattern
in
6
3
help
5
identifying
Explain
2
8
if
3,
1
each
one.
the
pattern.
8.1 What comes next?
273
5’
Reect
Here
is
0,
Is
a
1,
Try
way
8,
there
a
●
Could
●
If
In
5,
searching
Does
a
6,
of
discuss
1
continuing
5,
5,
6,
7,
6,
the
list
from
1f
in
Exploration
1:
7
pattern?
●
A
and
this
one
online
list
of
have
digit
was
is
an
the
numbers
numbers
you
sequence
for
count
identied
missing,
ordered
the
as
of
a
like
this:
01865
556
767.
pattern?
sequence
would
list
grouped
you
be
numbers.
without
able
to
Each
an
nd
internet
out
number
search?
what
in
the
it
list
was?
is
called
term
some
sequences
sequences
discuss
1,
in
the
Exploration
however,
Describing
terms
terms
is
in
1
not
a
follow
were
a
a
pattern,
generated
mathematical
specic
by
a
rule
pattern.
or
The
order.
list
in
All
the
Reect
and
pattern.
sequence
The
You
can
use
to
u
represent
the
rst
term
of
the
sequence,
u
1
thesecond
Forthe
to
subscript
sometimes
represent
and
sequence
term
1,
so
3,
on.
5,
7,
The
9,
subscripts
1,
2,
3,
match
the
term
number.
the
index
term.
...
2nd
term
3rd
term
4th
term
5th
As
=
1
u
1
=
3
u
2
=
5
u
3
=
7
u
4
could
also
use
u
for
the
rst
term,
so
the
the
is
word
also
=
9
to
mean
‘exponent’,
you
5
need
You
of
term
used
u
called
2
term,
‘index’
1st
is
to
work
out
that:
0
its
1st
term
u
=
2nd
1
u
0
term
=
3rd
3
u
1
Sequences
starting
term
=
4th
5
u
2
from
u
are
often
term
=
5th
7
u
3
the
sequence,
the
used
in
computer
=
do
not
match
the
programs.
term
When
u
Tip
number.
Be
Reect
and
discuss
context.
9
0
subscripts
the
2
careful
you
talk
rst
term’
mean
●
Why
is
the
notation
u
,
u
1
●
terms
a,
b,
What
does
c,
d,
u
e,
,
u
2
,
3
u
,
4
u
,
…
more
useful
than
labelling
the
5
…?
mean?
n
Describing
An
its
a
sequence
explicit
formula
using
uses
the
value.
2 74
an
8 Patterns
explicit
term’s
formula
position
(position-to-term
number,
from
4
0
starts
term
meaning
n,
to
rule)
calculate
u
0
when
about
–
or
do
u
?
1
‘the
you
A LG E B R A
The
formula
u
=
2n
1
for
n
≥
1
n
tells
us
that
the
value
of
the
nth
term
(u
)
is
given
by
2n
1
for
any
value
of
n
n
greater
an
than
or
equal
to
1.
When
you
are
working
with
sequences,
n
is
always
integer.
Using
the
above
u
=
2
=
1
formula,
×
1
letting
n
=
1
gives
the
following:
1
1
Similarly,
letting
=
2
=
3
u
×
n
=
2
2
gives:
1
2
and
so
on.
Example
1
2
A
sequence
is
given
by
the
explicit
formula
u
=
3n
2
for
n
≥
1.
n
a
Find:
i
the
rst,
ii
the
term
b
Determine
a
i
Since
n
second
of
the
and
sequence
whether
≥
1,
tenth
the
or
not
terms
terms
with
524
is
of
the
value
a
required
673.
term
are
sequence
u
of
,
the
u
1
sequence.
and
u
2
10
2
u
=
3
×
1
=
3
×
2
2
=
1
2
=
10
2
=
Substitute n = 1 for the 1st term.
1
2
u
Substitute n = 2 for the 2nd term.
2
2
u
=
3
×
10
298
10
Substitute
n
=
10
for
the
10th
term.
2
ii
3n
2
=
673
=
675
=
225
n
=
±15
≠
−15,
Solve
the
explicit
formula
for
the
specic
value
of
673.
2
3n
2
n
As
n
≥
1,
n
so
n
=
15.
2
When
n
=
15,
u
=
3
×
15
2
=
3
×
225
2
=
673
✓
Check
your
solution.
n
b
If
524
is
a
term
in
the
sequence,
then
2
2
3n
=
524
=
526
=
175.333…
2
3n
2
n
n
is
not
an
integer,
so
524
is
not
a
term
in
the
sequence.
8.1 What comes next?
2 75
Practice
1
Find
the
1
rst
ve
terms
of
each
sequence.
2
a
u
=
4n
1
for
n
≥
1
b
u
n
=
6n
+
2
for
n
≥
1
n
1
2
c
u
=
10
n
n
for
n
≥
1
d
u
n
=
2
2
4n
5
for
n
≥
1
n
4
e
u
=
(n
+
4)(n
3)
for
n
≥
1
f
u
n
25
155
for
n
≥
0
u
=
1051
1307
4
g
=
n
3
n
2
n
+
n
n
+
45
for
n
≥
1
n
24
2
Find
the
12
tenth
12
24
term
of
the
sequence
given
by
u
=
5n
3
1
for
for
n
≥
n
1.
1.
n
3
Determine
if
54
is
a
term
in
the
sequence
u
=
4n
≥
n
4
Determine
which
term
of
the
sequence
u
=
3n
5
has
value
61.
n
10
5
Find
the
fteenth
term
of
the
sequence
given
by
=
u
10
for
n
≥
1.
n
n
6
Find
u
=
the
2n
value
+
12
of
for
the
n
≥
fourth
term
of
the
sequence
given
by
0.
n
2
7
a
Find
the
term
of
the
sequence
u
=
2n
+
4
that
has
value
246.
n
b
Show
that
396
is
a
term
in
this
sequence.
2
n
8
Find
the
eleventh
term
of
the
sequence
given
by
u
=
100
for
n
≥
0.
n
5
Tip
Problem
solving
The
9
Identify
which
explicit
formula,
a
to
f,
corresponds
to
each
sequence,
i
to
vi
command
term
identify
requires
a
=
u
4n
+
1
for
n
≥
0
i
3,
6,
9,
12,
15,
b
u
state
=
2n
+
3
for
n
≥
1
ii
1,
3,
5,
7,
9,
…
you
n
c
u
=
5n
=
3n
=
3n
=
2n
4
for
n
≥
1
iii
2,
1,
4,
you
to
…
n
7,
10,
…
briey
have
your
how
made
decisions.
n
d
u
for
n
≥
1
2
for
n
≥
1
for
n
≥
iv
1,
5,
9,
13,
17,
…
0
v
5,
7,
9,
11,
13,
…
0
vi
1,
6,
11,
n
e
u
n
f
u
+
16,
21,
…
n
Describing
A
a
sequence
recursive
When
With
you
this
formula
know
recursive
using
gives
one
formula
the
term,
formula
a
relationship
you
you
(term-to-term
can
are
work
given
between
out
the
the
value
rule)
consecutive
terms.
next.
of
u
:
1
Tip
u
=
n
+
u
1
+
2,
u
n
=
1
for
n
≥
1
1
u
u
=
1
is
the
nth
term
of
n
1
a
sequence,
so
u
n
Substituting
n
=
1
gives
the
2nd
is
u
=
1
+
u
1
u
+
2
(n
1
=
1
+
2
=
+
1
term:
the
+
u
3
next
1)th
term
or
term.
is
the
term
before
the
nth term.
n
−
1
2
Substituting
n
u
=
=
2
+
u
u
1
=
2
gives
+
the
3rd
2
2
3
+
2
=
5
3
and
so
on.
27 6
8 Patterns
term:
A LG E B R A
Example
A
2
sequence
has
term-to-term
rule
u
=
n
Find
u
=
the
2nd
and
3rd
terms
of
the
+
u
1
+
8
for
n
≥
1,
and
u
n
=
12.
1
sequence.
12
The
rst
term
is
given
with
the
formula.
1
u
=
u
2
+
The
8
2nd
=
12
×
=
6
+
=
14
+
is
found
when
n
=
1.
8
8
Use
u
term
1
=
u
3
+
the
value
you
just
got
for
u
2
8
2
to
=
=
7
=
15
You
14
×
+
can
graphs
+
nd
the
3rd
term.
8
8
use
for
a
the
GDC
to
plot
formulae
in
explicit
and
Examples
1
recursive
and
1.1
formulae.
Here
are
the
2.
1.1
Tip
u
u
n
n
16
80
Some
GDCs
use
(4,15.5)
70
15
u
(3,15)
=
3
u
n
2
u
60
and
n
n
rather
1
2
n
than
u
14
and
n
50
u
+ 1
.
n
1
(2,14)
u
u
n
40
13
n
+
Inthis
8
1
example,
the
2
30
12
20
u
(1,12)
=
u
n
+
n
8.
1
11
10
0
10
n
0
1
2
Practice
1
Find
a
b
the
u
+
1
n
+
1
u
4
n
0
5
1
2
3
4
5
6
7
8
9
10
2
rst
=
n
3
ve
4u
1,
terms
of
u
n
=
2u
each
sequence.
=
1
for
n
≥
1
=
2
for
n
≥
1
Tip
1
+
1,
u
n
Look
when
c
u
=
n
+
u
1
for
patterns
1
+
7,
u
n
=
−4
for
n
≥
you
work
1
1
with
sequences.
1
d
=
u
n
+
u
1
+
1,
u
n
=
3
for
n
≥
Do
1
the
terms
of
the
1
2
sequence
e
u
=
n
f
+
u
n
(1
u
n
=
+
2u
1
4u
1
),
u
n
(1
u
n
=
0.8
for
n
≥
),
u
increase,
=
0.8
for
n
≥
to
1
1
n
seem
1
or
decrease,
bounce
around
1
chaotically?
g
u
n
=
u
n
+
−
1
3,
u
=
4
for
n
≥
2
1
8.1 What comes next?
277
2
A
sequence
is
given
by
u
=
3
u
n + 1
Find
the
rst
four
terms
of
and
u
n
the
=
−3.
0
sequence.
u
Tip
1
n
3
A
sequence
is
given
by
u
=
,
u
n + 1
=
2
for
n
≥
1.
It
can
be
useful
1
u
n
a
Find
the
rst
b
Describe
c
Predict
any
the
Problem
4
A
six
terms
patterns
next
few
of
the
you
of
the
is
given
by
u
=
2u
n + 1
5
A
the
term
sequence
the
of
is
the
given
value
The
C
fewterms
to
explore
an
unfamiliar sequence.
sequence.
sequence
by
of
the
u
=
What
●
How
can
●
How
do
types
largest
of
you
you
that
has
3u
2,
identify
a
2
for
n
≥
1.
value
u
=
2
≥
1.
257.
for
n
1
term
of
=
1
in
the
sequence
that
is
less
than
10
000.
sequences
sequence
nd
u
n
structure
●
1,
n
n + 1
Find
a
solving
sequence
Find
list
sequence.
notice.
terms
to
are
there?
sequences?
general
formula
for
a
sequence?
ATL
Exploration
1
For
each
a
3n
=
+
2
sequence,
write
7
c
n
b
a
=
5n
=
4n
the
rst
ve
terms.
2
n
=
3
2n
d
n
2
down
+
5
n
Copy
and
complete
the
diagram
for
the
sequence
given
by
a
=
3n
+
7,
n
lling
in
missing
Sequence
terms
in
13
10
the
sequence
16
and
the
rst
dierence
row.
19
The
is
F irst
difference
3
rst
the
3
between
b
3
Make
Look
at
a
similar
your
a
Describe
b
Compare
formula.
between
c
What
Sequences
by
a
like
constant
278
do
diagram
anything
rst
the
other
the
the
you
sequences
in
step
1
terms.
for
anything
you
explicit
explicit
ones
number,
notice.
dierence
Describe
the
for
diagrams.
the
dierence
dierence
in
are
formula
each
and
formulae
sequence
notice.
of
the
value
these
Exploration
2,
sequences.
8 Patterns
of
a
the
sequences
called
linear
with
Suggest
where
the
its
explicit
relationship
rst
dierence.
have
terms
in
common?
increase
or
decrease
consecutive
A LG E B R A
For
a
linear
constant
sequence,
and
the
the
explicit
dierence
formula
is
between
of
the
consecutive
form
u
=
a
+
terms
is
bn
n
Reect
●
If
you
Plot
discuss
plotted
graph
●
and
would
the
the
terms
you
graph
3
of
a
sequence
7
to
from
Exploration
2,
what
type
of
get?
of
a
=
3n
+
check
your
answer.
Plot
coordinate
pairs
n
as
(n,
a
).
n
ATL
Exploration
3
2
1
a
Copy
and
complete
this
diagram
for
the
sequence
given
by
e
=
n
+
n
+
5,
n
lling
in
missing
Sequence
terms
11
7
in
the
sequence
17
and
the
dierence
rows.
25
The
F irst
difference
4
second
dierence
is
dierence
between
consecutive
Second
b
difference
Make
2
a
similar
+
5n
the
6
rst
dierences.
diagram
for
each
of
these
sequences:
2
=
f
n
+
6
n
2
g
=
4
+
5n
+
n
n
h
=
n(n
+
4)
+
1
n
2
Look
a
Describe
For
a
the
and
second
dierences
dierences
between
in
these
your
diagrams.
sequences
and
the
sequences
2.
anything
you
notice
that
the
about
the
rst
and
second
dierences
of
sequence.
Describe
in
rst
Exploration
each
c
the
Describe
in
b
at
anything
explicit
formulae
for
these
sequences
have
common.
quadratic
sequence,
the
second
dierence
is
constant
and
the
2
explicit
formula
is
of
the
form
u
=
a
+
bn
+
cn
n
Reect
●
The
and
discuss
sequences
highest
power
in
4
Exploration
of
n
in
of
the
a
3
are
quadratic
quadratic
sequences.
What
is
the
sequence?
2
●
Plot
the
graph
sequence
e
=
n
+
n
+
5.
What
type
of
graph
do
n
quadratic
●
Other
sequences
than
sequence
linear
can
you
produce?
and
quadratic
sequences,
what
other
types
of
describe?
8.1 What comes next?
279
Practice
1
By
is
2
3
constructing
a
quadratic
Complete
in
the
a
these
F irst
c
11
4,
c
91,
3
e
5,
A
11,
17,
32,
64,
71, ...
any
missing
terms
in
the
sequence
and
27
3
23,
29,
5
each
sequence
…
a
the
is
b
…
for
Usually
6
…
51,
formula
term.
a
in
18
3
4
25,
75,
Example
Find
52,
8
whether
18,
general
each
36,
15
difference
84,
23,
4
difference
11,
13,
11
Determine
a
6,
solving
difference
Second
sequence
2
difference
Sequence
F irst
the
3
difference
Second
that
lling
Sequence
b
3
by
6
3
Problem
show
rows.
difference
Second
diagram,
diagrams
dierence
Sequence
F irst
a
sequence.
sequence
general
linear,
104,
97,
d
2,
4,
f
4,
12,
is
a
rule
formula
is
quadratic
7,
90,
11,
36,
that
an
83,
16,
108,
can
76,
neither.
…
…
324,
be
explicit
or
…
used
to
generate
formula.
3
general
formula
for
u
,
the
nth
term
of
the
sequence
11,
17,
23,
29, …
n
Sequence
17
11
23
29
Find
F irst
The
6
difference
rst
dierence
6
is
6,
the
rst
dierences.
6
so
compare
to
the
sequence
u
=
6n
The
rst
dierence
is
constant
n
sothis
n
u
1
2
3
4
11
17
23
29
6
12
18
24
is
a
linear
sequence.
n
+
6n
5
Look
for
a
pattern
connecting
5
to
6n
gives
u
n
The
general
formula
is
u
=
6n
n
2 8 0
8 Patterns
+
5.
u
and
n
Adding
6n
A LG E B R A
Notice
It
that
doesn’t
need
the
more
formula
mean
information
Practice
1
general
necessarily
you
that
about
the
the
nd
describes
pattern
will
sequence,
for
the
terms
continue.
example,
you
To
if
it
were
know
is
given.
that,
you
linear.
4
Find
a
general
a
=
8,
formula
to
describe
the
terms
in
each
sequence.
Tip
ATL
a
a
1
=
16,
a
2
=
24,
a
3
=
32
4
You
b
b
=
14,
b
1
c
=
=
−14,
e
e
c
=
49,
d
11,
−28,
38,
=
17,
f
=
29,
g
=
7
i
15,
i
1
2
set
=
of
a
=
9,
a
=
c
17,
b
=
13.5,
d
=
7,
d
string
=
of
g
=
=
18.5
6
7.8,
h
=
12.4
4
−21,
i
is
=
−39
part
of
a
describing
13,
a
b
=
=
11,
linear
the
sequence.
terms
of
each
sequence.
15
b
=
d
c
=
=
8
14
14.5,
c
=
15
12
23,
d
9
=
31
11
solving
decorative
then
d
=
11
connect
Let
f
8
14,
cable
connect
b
h
=
why
to
few
4
a
15,
rst
the
16
4
=
Explain
and
7,
below
7
Problem
a
=
10
5
A
g
14,
=
out
−56
65
13
c
is
working
terms.
=
18,
formula
=
by
4
12
9
d
=
7
11
c
f
i
11,
correct
3
6
=
=
e
3
general
b
=
3
terms
your
4
3.2,
−3,
a
5
3
=
Find
a
d
3
h
check
formula
1
,
2
Each
b
47,
2
=
c
3
−1.4,
can
general
4
=
2
=
27,
3
1
i
e
17.5,
3
h
−42,
3
=
1
h
23
2
,
8
=
4
=
2
g
=
3
=
1
g
c
d
2
1
b
3
=
e
1
f
=
2
=
20,
4
2
1
f
=
3
1
d
b
2
c
d
17,
it
takes
the
another
the
be
next
the
lights
500
rst
20
cm
are
wired
cm
of
as
shown.
bulb,
to
bulb.
total
length
of
10
cable
cm
n
needed
Find
a
to
connect
general
the
nth
formula
bulb.
for
d
250
cm
n
4
A
a
railway
depot
train
which
station.
It
is
is
then
stored
6
km
spends
overnight
from
all
its
day
in
starting
10
cm
10
cm
moving
etc.
from
its
along
in
a
starting
section
each
Explain
b
Find
c
a
and
The
of
complete
a
up
station,
why
down
the
can
14
journey
after
general
train
Find
track
back
its
rst
forth
long,
travels
so
28
complete
for
the
km.
journey
distance
it
of
the
has
day
it
has
travelled
travelled
after
n
34
km.
journeys
track.
travel
number
km
it
formula
the
and
of
400
km
before
complete
refuelling.
journeys
it
can
make
before
refuelling.
8.1 What comes next?
2 81
5
The
6
rst
ve
a
Find
b
Show
A
the
terms
100th
that
sequence
a
term
102
has
of
is
a
linear
of
sequence
the
u
of
=
a
formula
Problem
7
A
linear
for
the
nth
the
u
n + 1
Find
+
sequence
5
and
u
n
b
Write
term
of
the
A
the
linear
Find
has
terms
dierence
down
the
9
A
it
of
until
For
its
rst
type
rst
linear
Find
The
sequence
the
u
=
between
value
of
rst
the
in
The
i
25
and
is
less
and
the
ii
its
term
number.
−1.
2,
0,
4,
0
n
.
terms.
…
65.
and
second
is
formula
you
18,
4
for
can
28,
40
10
4
term
103.
a
sequence
conjecture
so
rst
a
10
is
to
identify
formula
and
the
modify
is
a
second
dierences
are:
40
12
2
this
and
28
8
2
constant
the
18
6
2
2
quadratic
sequence
and
is
related
2
Add
2
2
70.
exactly.
10,
2
for
=
u
155,
108
that,
2,
dierence
than
general
done
2
1
u
nd
term.
sequence
2
sequence
n
and
1
173,
term
the
0
difference
the
=
u
consecutive
2
to
25.
15
u
191,
rst
Having
2
2
second
has
nding
sequence
difference
Second
that
negative
describes
Sequence
F irst
term
sequence.
it
begins
sequence
step
18,
sequence.
9
8
11,
1
10
Find
4,
solving
sequence
a
3,
sequence.
member
formula
are
2
a
row
for
n
to
the
table:
3
4
5
6
7
8
0
4
10
18
28
40
9
16
25
36
49
64
n
2
n
1
4
2
Add
another
row
for
the
dierences
between
n
and
u
n
n
1
u
2
2
2
3
4
5
6
7
8
0
4
10
18
28
40
n
2
n
1
4
9
16
25
36
49
64
3
6
9
12
15
18
21
24
2
n
u
n
2
u
n
gives
multiples
of
3.
This
is
a
linear
sequence
with
general
n
2
So
2
n
u
=
3n
and
rearranging
gives
n
u
=
n
3n.
n
2
Check:
the
rst
ve
terms
of
u
n
2 8 2
8 Patterns
=
n
3n
are
2,
2,
0,
4,
10
✓
formula
3n
A LG E B R A
Practice
1
2
3
Find
a
a
4,
7,
c
3.5,
general
12,
1,
0,
g
0.5,
i
3,
9,
k
0,
1.5,
5,
b
Use
a
39,
24,
10,
53,
…
…
…
begins
formula
to
sequence
formula
38.5,
27,
27.5,
formula
quadratic
b
…
sequence
your
sequence:
…
18,
general
each
27.5,
17.5,
39,
10.5,
for
…
18.5,
15,
27,
quadratic
Find
A
8,
4.5,
17,
a
28,
11.5,
3,
1,
formula
19,
6.5,
e
A
5
18,
for
the
predict
begins
a
Find
a
for
the
b
Find
the
value
of
the
c
Find
the
value
of
u
4,
d
2,
6,
f
6,
14,
h
1,
2,
j
13,
l
4,
20,
20,
1,
12,
8,
20,
24,
17,
30,
36,
28,
42,
50,
…
…
66,
…
5,
10,
17,
26,
…
22,
33,
46,
61,
78,
3,
0,
18,
5,
14,
12,
8,
21,
0,
…
…
…
sequence.
the
3,
nth
7,
15th
4,
term
3,
0,
of
5,
the
12,
sequence.
…
term.
20th
term
of
the
sequence.
50
4
A
quadratic
sequence
25,
a
Show
that
the
16th
term
of
b
Show
that
the
17th
term
is
c
Explain
why
Problem
5
begins
A
its
the
49,
sequence
58,
is
65,
70,
….
negative.
positive.
subsequent
terms
will
be
positive.
solving
quadratic
Find
all
the
38,
sequence
rst
negative
begins
98,
100,
100,
98,
94,
…
term.
ATL
Exploration
1
Construct
4
dierence
diagrams
for
these
quadratic
sequences.
2
a
u
=
3n
=
2n
,
n
≥
1
4,
n
n
2
b
u
+
≥
1
n
2
c
u
=
5n
=
2n
=
n
=
2
n,
n
≥
1
n
2
d
u
+
3n,
n
≥
1
n
2
e
u
+
7,
n
≥
1
n
2
f
u
4n
,
n
≥
1
n
2
Look
at
the
dierence
diagrams.
The
a
Describe
the
relationship
between
the
explicit
formula
and
the
second
coecient
of
2
n
is
the
in
front
number
dierence.
b
Suggest
a
rule
linking
the
coecient
of
n
of
coecient
2
to
the
value
of
the
it.
The
of
second
2
3n
is3.
dierence.
Continued
on
next
page
8.1 What comes next?
2 83
3
Use
your
rule
to
predict
the
second
dierence
of
the
sequence
with
formula:
2
a
u
=
7n
–
2n
=
3
8n
+
1
+
2
n
2
b
u
n
2
c
u
=
n
–
4n
n
Check
six
your
terms
Example
Find
a
begins
=
each
4,
formula
u
1
=
12,
for
u
2
=
constructing
difference
the
26,
=
term
46,
a
dierence
20
the
=
diagram
quadratic
72,
u
5
46
14
of
u
4
26
8
nth
u
3
12
4
F irst
by
for
the
rst
sequence.
4
general
u
predictions
of
=
sequence
which
104.
6
72
104
26
32
The
Second
difference
6
6
6
constant
dierence
6
is
second
6,
so
the
2
coecient
n
u
1
2
3
4
5
4
12
26
46
72
104
3
12
27
48
75
108
of
n
is
6
÷
2
=
3.
6
n
2
Write
out
the
values
of
3n
and
2
3n
comparethem
to
the
values
of
u
.
n
2
u
3n
1
0
1
2
3
4
n
Find the general formula for the linear
The
sequence
1,
0,
1,
2,
3,
4,
...
has
nth
term
2
n.
sequence 1, 0,
1,
2,
3,
4 , …
2
Therefore,
u
3n
=
2
n
=
3n
n
2
Therefore
u
n
+
2
Rearrange.
n
2
In
a
quadratic
always
half
Practice
1
Find
a
a
2,
8,
c
3,
12,
e
general
18,
of
the
second
of
32,
formula
50,
for
the
nth
term
…
of
the
3,
d
8,
23,
44,
71,
…
f
51,
88,
135,
i
2,
6.5,
13.5,
7,
17,
quadratic
a
Find
the
b
Find
a
…
23,
35,
31,
…
sequence
next
three
formula
for
…
begins
10,
16,
terms.
the
nth
8 Patterns
term.
26,
each
0.5,
…
2 8 4
in
b
75,
1,
n
general
formula
dierence.
48,
24,
A
value
coecient
27,
7,
1,
the
the
6
g
k
2
1,
sequence,
9,
sequence.
2,
19,
4.5,
h
2,
7,
j
6,
17,
l
3,
8,
…
4.5,
0,
15,
8,
33,
7.5,
26,
34,
51,
18,
40,
57,
15.5,
12.5,
…
31.5,
…
…
86,
25.5,
…
…
38,
…
is
A LG E B R A
Problem
solving
Draw
3
The
nth
triangular
number,
T
,
is
the
number
of
dots
in
a
triangular
grid
of
n
dots
n
n
=
dots
a
dierence
diagram.
wide.
1
n
a
Write
b
Find
the
c
Find
a
=
2
down
n
the
value
=
3
of
n
=
4
T
4
value
of
T
5
general
formula
for
T
n
d
Use
your
formula
to
nd
the
value
of
T
and
T
7
your
answers.
e
Find
the
f
Verify
value
of
.
Draw
diagrams
to
check
8
T
15
that
T
is
a
square
number.
49
4
A
quadratic
sequence
has
rst
term
4
and
second
term
10.
2
The
coecient
Find
5
A
the
quadratic
Find
the
Objective:
ii.
next
select
of
n
three
in
D.
of
general
formula
is
2.
terms.
sequence
value
its
begins
3,
x,
29,
51, …
x
Applying
appropriate
mathematics
mathematical
in
real-life
strategies
contexts
when
solving
authentic
real-life
Tip
situations.
Think
‘Select
how
appropriate
you
tackle
mathematical
strategies’
means,
for
example,
that
it
is
up
to
you
Q6.
dene
draw
6
In
the
group
stage
of
a
volleyball
competition,
all
the
teams
in
the
to
play
each
other.
When
there
are
four
teams
in
a
group,
you
any
need
six
any
lists
and
or
games
tables.
are
needed
Find
the
teams
for
each
number
exactly
of
once
team
games
in
a
●
Can
you
play
always
each
needed
group
Continuing
D
to
of
n
for
of
the
each
other
team
teams
to
play
exactly
each
of
once.
the
other
teams.
sequences
predict
the
next
terms
of
a
sequence?
ATL
Exploration
In
these
circles,
segments,
n
=
and
1
the
n
The
rst
The
second
5
this
=
diagram
dots
on
divides
2
n
has
diagram
the
the
1
dot,
has
2
1
circumference
circle
=
into
3
are
connected
by
line
regions.
n
=
4
n
=
5
region.
dots,
2
regions.
Continued
on
next
page
8.1 What comes next?
to
variables,
diagrams,
group
make
have
about
whether
2 8 5
The
third
diagram
1
Write
down
2
Predict
the
reasons
3
Here
the
your
two
a
What
is
b
Explain
Reect
●
Do
the
●
regions
ways
you
the
for
n
=
would
4
and
n
expect
=
5.
when
n
=
6.
Give
could
you
the
draw
the
number
think
is
the
of
circle
for
regions
correct
way
n
in
to
=
6.
each
diagram?
continue
the
why.
5
diagrams
does
you
between
discuss
that
regions.
regions
diagram
Explain
of
of
dierence
and
either
of
4
answer.
which
prediction
dots,
sensible
the
sequence.
3
number
number
for
are
has
hold
t
your
true?
prediction?
Does
it
help
If
to
not,
look
can
at
you
the
form
next
a
term
in
sequence?
What
are
sequence
the
in
diculties
in
Exploration
trying
to
nd
a
general
formula
for
the
5?
Summary
●
A
sequence
Each
is
an
number
in
ordered
the
list
list
is
of
●
numbers.
called
a
term
For
a
quadratic
dierence
is
sequence,
constant
and
the
the
second
explicit
formula
2
is
●
An
explicit
number,
●
A
n,
formula
to
recursive
between
one
uses
calculate
formula
consecutive
term,
you
can
the
its
gives
the
you
know
A
general
can
be
used
general
●
decrease
●
For
a
to
is
where
by
a
for
a
generate
formula
Sequences
linear
formula
an
the
sequence
each
terms
constant
is
term.
explicit
a
rule
that
Usually
the
formula.
increase
number,
=
a
+
bn
+
cn
In
a
quadratic
the
general
the
second
sequence,
formula
is
the
coecient
always
half
the
of
linear
sequence,
are
The
of
triangular
dots
They
needed
form
a
formula
value
numbers
to
make
quadratic
describe
simple
the
formula
is
of
the
is
the
sequence
with
form
u
=
n
8 Patterns
a
and
+
of
bn
the
.
between
explicit
=
1
n
=
2
n
=
3
n
grids.
general
or
dierence
constant
in
number
triangular
called
n
terms
n
dierence.
sequences.
consecutive
2 8 6
u
next.
●
●
form
2
relationship
the
the
n
●
When
out
of
position
value.
terms.
work
term’s
=
4
A LG E B R A
Mixed
1
Find
a
practice
formula
for
the
nth
term
of
9
each
A
quadratic
sequence
begins
3,
x,
15,
25.5,
…
sequence.
Find
a
6,
10,
b
42,
14,
18,
22,
26,
10
35,
28,
21,
the
c
2,
5,
10,
17,
26,
37,
d
3,
8,
15,
24,
35,
…
e
2.5,
11.5,
16,
20.5,
Consider
2
0.5,
4,
g
9,
13,
h
6,
7,
A
13,
7,
linear
13.5,
6,
its
A
28,
9,
1,
4,
…
47.5,
Write
2
14,
17,
20,
Find
a
below,
hexagons
which
respectively.
3
…
u
=
u
the
rst
5
n
ve
and
u
1
=
the
side
formula
length
of
the
rst
pattern
that
for
the
1
11
In
a
610
terms.
nth
term
of
over
1000
hexagons.
12.
football
seats.
third
b
patterns
3
term.
formula
down
and
…
n
a
2
…
begins
hundredth
has
1,
…
sequence
sequence
honeycomb
length
contains
3
x
…
Find
Find
the
side
1
f
of
…
have
7,
value
…
has
stadium,
The
630
the
second
seats,
rst
row
and
so
row
of
has
620
on.
The
seats
seats,
has
the
stadium
the
sells
seats
from
the
middle
of
a
row
outwards.
sequence.
Each
sell
Problem
A
sequence
has
formula
u
=
u
n + 1
a
tickets
Write down
the
rst
ve
+
7
and
u
n
=
Find
Show
A
linear
that
85
is
sequence
the
the
completely
next
15.
a
terms
member
of
begins
18,
33,
rst
term
greater
than
number
the
of
the
sequence.
b
Find
front
the
the
48,
the
sold
before
they
row.
of
two
seats
rows
that
are
can
be
sold
if
used.
number
front
of
seats
that
three
rows
are
can
be
sold
if
used.
sequence.
Create
a
formula
for
the
total
number
of
...
seats
Find
be
0
c
5
for
the
only
only
b
must
solving
a
4
row
that
can
be
sold
if
the
front
n
rows
1000.
areused.
6
A
quadratic
sequence
begins
4,
10,
18,
28,
…
d
a
Find
a
formula
b
Write
down
c
Show
that
for
the
the
value
nth
of
sold
term.
the
Hence
seventh
e
term.
and
nd
its
is
term
a
member
of
the
A
quadratic
number.
sequence
begins
12
2,
4,
A
the
000
front
number
30
number
of
rows
of
seats
are
rows
that
can
be
used.
needed
for
the
rst
tickets.
12-storey
stops
,
the
the
sequence
20
7
if
Find
10
460
nd
10,
14,
at
tall
every
building
has
an
elevator
that
oor.
…
3
The
a
Find
a
formula
for
u
,
the
nth
term
of
elevator
takes
8
seconds
to
travel
one
oor,
the
n
14
seconds
to
travel
two
oors,
20
seconds
sequence.
totravel
b
Find
the
value
of
three
oors
and
26
seconds
to
travel
u
four
20
Let
oors.
be
T
the
time
taken
to
travel
n
oors.
n
Problem
solving
a
Show
that
T
forms
a
linear
sequence.
n
8
A
linear
sequence
has
terms
u
3
=
6
and
u
=
27.
6
b
Find
a
formula
for
T
n
a
Find
the
b
Write
dierence
between
successive
terms.
c
down
the
value
of
u
7
Find
the
elevator
greatest
could
number
travel
in
of
oors
under
8.1 What comes next?
a
that
the
minute.
2 87
Review
in
context
Scientic and technical innovation
1
Engineers
railway
are
laying
track.
separated
by
They
signalling
place
junction
cable
xed
alongside
lengths
of
If
a
you
add
possible
cable
in
a
third
station,
C,
there
are
six
journeys.
boxes.
A
The
total
length
of
cable
for
1,
2,
3
or
4
B
junction
C
boxes
Let
u
is
12
be
m,
the
18
m,
cable
24
m
length
or
for
30
n
m.
junction
boxes.
a
n
List
the
three
a
Find
a
formula
for
six
journeys
stations,
A,
B
that
and
are
possible
with
C.
u
n
b
b
Use
your
formula
to
predict
the
length
List
the
four
cable
needed
for
30
junction
The
number
of
passengers,
P
,
that
would
be
possible
with
stations,
A,
B,
C
and
D.
boxes.
Hence
2
journeys
of
who
write
down
the
total
number
of
will
n
possible
comfortably
by
the
t
into
following
an
table
n-carriage
for
small
train
values
is
of
n:
c
List
ve
n
1
2
3
4
40
110
180
250
the
journeys
stations,
Hence
P
journeys
with
four
stations.
given
write
A,
that
B,
down
C,
would
D
the
be
and
total
possible
with
E.
number
of
n
possible
a
Show
that
P
forms
a
linear
n
b
Find
a
d
formula
for
journeys
with
ve
stations.
sequence.
Create
a
stations
P
formula
on
the
linking
network
the
to
number
the
total
of
number
of
n
possible
c
Predict
the
number
eight-carriage
train
of
passengers
could
that
comfortably
an
The
hold.
Metropolitano
railway,
d
Suggest
might
a
reason
hold
more
passengers
3
In
a
subway
consists
If
A
you
and
AB
of
a
of
start
and
then
of
point
just
the
a
two-carriage
twice
one-carriage
network
consider
B,
a
why
than
of
a
55
de
of
e
Predict
the
total
f
Predict
the
number
every
stations,
possible
journeys
journey
end
stations
point.
call
that
were
built.
them
journeys
are
A
B
and
How
you
have
discuss
explored
the
statement
of
inquiry?
Give
specic
examples.
Statement of Inquiry:
Using
dierent
improve
2 8 8
forms
products,
to
generalize
processes
8 Patterns
Lisbon’s
number
there
BA.
Reect
Lisboa,
underground
stations.
train
number
dierent
the
two
has
of
possible
journeys.
train.
stations,
and
two
only
the
journeys.
and
and
justify
solutions.
patterns
can
help
of
additional
would
be
if
possible
two
extra
Back
to
the
beginning
8.2
Objectives
●
Factorizing
Inquiry
quadratic
expressions,
where
the
of
two
of
x
●
What
●
What
does
is
1,
including
the
dierence
brackets’
mean?
does
‘factorize
a
quadratic
squares.
expression’
●
‘expanding
F
2
coecient
questions
Factorizing
quadratic
expressions
where
mean?
the
coecient
of
x
is
not
1.
How
do
the
patterns
in
expanding
C
brackets
help
you
factorize
quadratic
expressions?
●
How
can
quadratic
easier
●
Can
to
patterns
help
expressions
you
in
a
write
form
that
everything
be
written
in
a
dierent
D
form?
ATL
Draw
Reection
reasonable
conclusions
and
generalizations
8. 2
Conceptual understanding:
Patterns
can
be
represented
in
equivalent
is
factorize?
forms.
9. 2
11. 2
2 8 9
MROF
●
2
Y
ou
●
should
expand
already
know
brackets
how
1
Expand:
a
(x
c
●
factorize
expressions
by
taking
to:
2
+
(x
3)(x
+
3)(x
4)
2)
b
(x
d
(x
5)(x
b
x
d
4x(3x
+
2)(x
+
5x
+
1)
2)
Factorize:
2
outa
common
factor
a
3x
+
12
2
c
Expanding
F
To
and
does
‘expanding
brackets’
●
What
does
‘factorize
quadratic
an
then
For
example:
algebraic
rewrite
the
a
expression
expression
means
without
2
7)(x
+
2)
symbol
take
the
case,
≡
≡
same
5) + 7(3x
5)
factorizing
What
‘expand’
The
+ 3x + 12
●
and
(x
6x
mean?
to
expression’
multiply
mean?
out
each
of
the
terms
brackets.
2
x
7x
means
value
+
‘is
for
2x
14
≡
identically
any
x
5x
equal
possible
14
to’.
value
of
It
shows
the
that
two
unknown
3x
expressions
variable
(in
this
For
example,
any
value
3(x
2)
≡
3x
6,
because
the
statement
is
always
true
=
17
of
Quadratic
an
an
because
only
it
when
=
4.
x
Exploration
the
true
is
not
for
x
at
5
identity,
x).
is
Look
+
equation,
1
following
expressions:
expressions
NOT
quadratic
2
expressions
3
5x
x
14
x
5x
14
2
3x
+
2x
8t
2
7w
+
21
x
2
m
+
+
3y
4
+
m
+
3
z
25
2
+
3z
2z
2
2y
12
2
+
1
2
5y
+
7
17
1
Based on your observations, suggest what is meant by ‘quadratic expression’.
2
Explain
satisfy
3
why
your
Explain
each
whether
quadratic
of
the
expressions
in
the
right-hand
column
do
not
description.
or
not
you
would
consider
(x
+
3)(x
5)
to
be
a
expression.
2
x
5x
whose
14
is
highest
a
quadratic
exponent
expression
is
because
it
contains
a
single
variable
(x)
An
2.
expression
highest
2
●
x
2
5x
14,
x
and
3x
are
all
quadratic
expressions.
3
like
x
●
5x
x
cubic
5x
14
2 9 0
14
is
is
not
3,
2
4
3
●
with
exponent
not
quadratic
quadratic
because
because
8 Patterns
its
its
highest
highest
exponent
power
is
less
is
greater
than
2.
than
2.
5x
14,
is
expression.
a
A LG E B R A
Quadratic
means
as
a
to
expressions
write
product
as
of
a
two
can
be
product
written
of
in
factors.
expressions
in
a
variety
‘Factorize
of
a
forms.
To
quadratic’
‘factorize’
means
‘write
it
brackets’.
ATL
Exploration
1
Copy
the
the
2
table.
For
each
row,
expand
the
brackets
to
help
you
complete
table.
Factorized
Expanded
expression
expression
2
(x
+
p)(x
+
q)
p
q
ax
+
bx
+
c
a
b
c
15
1
8
15
2
(x
+
3)(x
+
5)
(x
+
2)(x
+
9)
(x
(x
+
3)(x
6)
3)(x
4)
3
5
3
(x
2
+
2)(x
8)(x
+
4)
(x
)(x
+
6)
carefully
values
3
Copy
the
and
the
8x
+
4
1
at
signs
table
values
+
5)
(x
Look
x
of
a,
your
of
a,
below.
b
and
table.
b,
c,
p
Use
Describe
and
the
any
patterns
you
notice
in
the
q.
patterns
you
observed
in
step
2
to
predict
In
c
step
3,
you
shouldn’t
Factorized
Expanded
expression
expression
to
need
expand
brackets.
the
If
you
2
(x
+
p)(x
+
q)
(x
+
1)(x
+
7)
p
q
ax
+
bx
+
c
a
b
c
can’t
yet,
2
(x
3)(x
+
(x
1)(x
2)
5)(x
5)
see
go
and
a
pattern
back
to
discuss
step
your
8)
results.
(x
+
4
Verify
5
Copy
your
and
predictions
complete
by
this
expanding
each
factorized
expression.
table.
Factorized
Expanded
expression
expression
2
6
(x
+
p)(x
+
q)
(x
+
2)(x
+
2)
(x
+
4)(x
+
4)
(x
1)(x
1)
(x
6)(x
6)
Describe
any
p
patterns
q
you
notice
ax
in
+
bx
the
+
c
values
a
and
signs
b
c
of
a,
b,
c,
p
and
q
8.2 Back to the beginning
2 91
Practice
1
Copy
1
and
complete
this
table
based
on
the
patterns
you
observed
assuming
2
that
(x
+
p)(x
+
p
q)
=
ax
+
bx
q
2
+
c.
a
3
b
1
4
1
8
1
2
1
7
1
9
8
0
5
21
1
0
1
2
Find
values
such
3
N’nyree
has
whether
you
that
a
=
concluded
agree
or
●
How
do
the
quadratic
●
How
that
can
is
2
1,
p
that
=
if
c
disagree
Factorizing
C
c
q
and
=
0,
with
b
24
=
then
this
c
either
p
or
q
is
zero.
Explain
statement.
expressions
patterns
in
expanding
brackets
help
you
factorize
expressions?
patterns
easier
to
help
you
write
quadratic
expressions
in
a
form
factorize?
2
In
a
Exploration
=
1,
c
=
pq
2
and
you
b
=
p
found
+
that
in
all
the
expansions
of
(x
+
p)(x
+
q)
=
ax
+
bx
+
c,
q.
2
You
can
use
these
facts
to
factorize
expressions
such
as
x
+
x
12.
2
Suppose
Using
x
the
12
+
fact
and
Trying
x
12
that
1
c
12
each
pair
=
(x
+
=
pq
=
and
in
p)(x
+
−12
where
suggests
1
turn
q),
6
and
p
and
these
2
q
are
possible
6
and
whole
pairs
2
numbers.
of
4
values
and
for
3
p
4
and
and
q:
3
gives:
2
(x
12)(x
+
1)
≡
x
11x
12
11x
12
2
(x
+
12)(x
1)
≡
x
+
2
(x
6)(x
+
2)
≡
x
2)
≡
x
3)
≡
x
4x
12
4x
12
2
(x
+
6)(x
+
2
(x
4)(x
+
x
12
x
12
2
(x
(x
+
+
4)(x
4)(x
Also
3)
3)
using
found
which
has
x
gives
the
pairs
≡
of
+
the
fact
correct
that
b
numbers
sum
✓
=
p
+
which
x
q
makes
have
expression.
this
product
process
c,
you
more
just
ecient.
need
to
Having
nd
a
pair
Tip
b
Always
2
In
quadratic
+
x
12,
b
=
p
+
q
=
check
factorization
12
+
1
=
expanding
−11
brackets.
12
+
(
1)
=
2 9 2
your
1
11
8 Patterns
by
the
A LG E B R A
6
6
+
+
4
4
2
(
+
+
=
2)
3
(
−4
=
=
3)
4
−1
=
✓
1
2
So
x
+
x
12
≡
(x
+
4)(x
3).
2
To
q
In
factorize
which
a
have
Exploration
quadratic
product
2
steps
=
x
c
5
x
and
2
(x
You
+
p)(x
can
+
p)
also
2
(x
+
=
+
6
c
≡
(x
+
p)(x
+
q),
nd
two
numbers
p
and
b.
you
expanded
brackets
with
p
=
q:
2px
this
+
p
as:
2
x
+2px
Recognizing
bx
sum
2
+
2
p)
perfect
write
+
and
this
+
p
pattern
can
help
you
to
factorize
special
quadratics
called
squares.
Example
1
2
Factorize
x
+
10x
+
25.
2
2
x
+
10x
+
25
25
≡
(x
+
5)(x
≡
(x
+
5)
+
=
5
and
10
=
2
×
5
5)
2
If
all
This
three
makes
the
terms
have
a
factorization
Example
common
factor,
taking
this
common
factor
out
is
a
perfect square
rst
simpler.
2
2
Factorize
2a
+
6a
+
4.
2
2a
+
6a
+
4
2
≡
2(a
≡
2(a
+
+
3a
+
1)(a
Practice
1
Factorize
2)
+
Take
+
7x
each
x
+
12
b
g
x
7x
18
e
x
h
x
2
2
+
36
each
15
c
x
+
14x
+
49
each
3a
18
f
x
+
+
24
i
x
6a
+
3
b
5x
14
2
x
+
22x
+
121
c
x
12x
+
36
quadratic.
b
4b
2
+
2b
+
20e
c
5c
2
3d
18
3x
2
+
3x
2
11x
2
6d
+
2
7x
2
d
+
2
x
Factorize
a
8x
quadratic.
2
3
+
2
13x
Factorize
a
2
x
2
+
factor
quadratic.
2
d
common
2
2
x
the
2)
2
a
out
e
4e
10c
75
2
144
f
3f
24f
+
45
8.2 Back to the beginning
2 9 3
2.
4
Find
an
expression
for
the
unknown
side
in
these
rectangles:
a
b
Area
x
+
Area
2
?
2
2
x
+
7x
+
10
x
+
2x
3
?
x
Problem
5
Copy
and
1
solving
complete,
using
integer
values:
2
a
2
x
x
+
12
≡
(x
3)(x
)
b
x
d
x
2
c
x
+
16
≡
(x
8)(x
)
2
x
5x
≡
(x
+
)(x
7)
2
9x
+
20
≡
(x
+
5)(x
)
2
10x
16
e
x
A
rectangle
≡
(x
+
)(x
+
)
f
x
5x
24
≡
(x
)(x
)
2
6
has
area
x
10x
+
21.
Area
2
x
Its
perimeter
is
4x
Exploration
1
Copy
and
20.
Find,
in
10x
+
terms
21
of
x,
the
lengths
of
each
of
its
sides.
3
complete
this
table.
Factorized
Expanded
expression
expression
2
(x
+
p)(x
(x
+
3)(x
(x
2
In
9)(x
+
q
ax
+
bx
in
the
+
c
a
b
c
9)
6)(x
6)
(x
+
4)(x
4)
any
special
p
3)
+
Exploration
a
q)
(x
Describe
gave
+
3
patterns
you
you
found
pattern.
notice
that
When
expanding
they
are
values
and
brackets
expanded,
signs
in
the
the
of
a,
form
coecient
b,
(x
of
c,
+
p
p)(x
the
2
is
0
so
there
is
just
an
x
term
and
a
constant,
which
is
2
In
general,
expanding
(a
+
b )(a
b )
2
Any
expression
because
2
a
it
is
of
one
the
form
squared
≡
2 9 4
(a
+
b )(a
b )
8 Patterns
a
negative.
2
ab
+
ab
b
2
≡
a
2
b
2
b
quantity
2
b
a
≡
is
the
dierence
subtracted
from
of
two
squares,
another.
and
x
q
p)
term
A LG E B R A
Reect
and
discuss
1
2
Explain
how
this
diagram
illustrates
the
factorization
of
2
a
b
b
b
a
(a
b)
a
b
a
(a
If
you
recognize
factorize
it
an
expression
as
the
+
b)
dierence
of
two
squares,
you
can
easily.
Example
3
2
Factorize
x
64.
2
x
≡
2
64
(x
Dierence
8)(x
‘Fully
+
two
squares,
2
x
8
8)
factorize’
factorized
of
any
Example
means
write
as
a
product
of
expressions
which
cannot
be
further.
4
4
Fully
factorize
16x
81.
2
4
16x
(4x
81
2
2
)
9
2
2
≡
This
2
(4x
−
9)(4x
+
is
not
fully
factorized,
as
4x
2
dierence
2
≡
((2x)
≡
(2x
2
3
9
is
the
9)
of
two
squares:
(2x)
2
3
2
)(4x
+
9)
2
3)(2x
Practice
1
Fully
+
3)(4x
+
9)
3
factorize
each
expression.
2
a
x
d
x
2
25
2
b
2
x
2
121
e
j
16
1
f
x
2
169
h
81x
k
16
2
49x
9
4
9x
2
16x
4x
2
y
g
c
1
2
9
i
25u
l
1
4
x
2
16v
4
81y
8.2 Back to the beginning
2 9 5
Problem
2
Copy
and
solving
complete
each
identity,
using
integer
values.
2
a
2
x
100
2
c
≡
(3x
)(3x
+
)
b
25y
2
16a
b
≡
(
a
b)(
a
+
7b)
d
u
2
e
≡
(
y
4)(
y
+
4)
2
≡
(3u
+
2)(3u
+
)
2
t
≡
Reect
(3t
+
and
5)(
t
20)
discuss
f
x
≡
(3x
+
4)(6x
)
Quadratics
2
where
2
the
coecient
of
x
2
Aishah
tried
learned.
to
Here
factorize
is
a
3x
sample
+
of
13x
her
+
12
using
the
methods
you
have
2
In
the
quadratic
3x
In
the
expansion
(x
The
sum
p
+
q
=
+
+
b,
13x
p)
⇒
+
(x
b
12,
+
=
is
1
If
the
are
called
monic.
work:
a
q),
13,
=
3,
the
and
b
=
13
product
so
p
=
1
and
pq
and
c
=
=
c,
q
=
not
12.
⇒
c
=
is
12.
1,
coecient
the
is
quadratic
non-monic.
12.
2
Therefore,
●
Is
●
Why
doesn’t
●
How
is
Aishah’s
3x
+
13x
+
factorization
this
this
12
(x
+
correct?
method
expression
≡
work
1)(x
Is
it
for
dierent
+
12)
equivalent
this
from
to
the
original
expression?
expression?
the
other
expressions
you
have
factorized?
2
To
have
3x
as
the
rst
term
of
a
quadratic
expression,
the
factors
need
to
start
2
with
The
terms
that
factors
second
Look
of
term
at
the
multiply
3
in
are
1
each
and
make
3,
bracket
examples
Example
to
below
so
3x
the
can
be
and
factors
found
then
need
by
to
start
listing
attempt
(3x
)
and
possibilities
Reect
and
(x
).
The
systematically.
Discuss
3.
5
2
Factorize
3x
+
13x
+
12.
2
3x
+
13x
Pairs
1
of
and
12
+
12
≡
values
(3x
)
with
)
product
12
and
(x
1
12
are:
2
and
6
3
and
4
6
and
2
4
and
3
2
(3x
+
1)(x
+
(3x
+
12)(x
12)
≡
3x
≡
3x
+
37x
+
+
15x
+
12
2
+
1)
12
2
(3x
+
2)(x
+
6)
≡
3x
+
20x
+
+
12x
+
+
15x
+
+
13x
+
12
2
(3x
+
6)(x
+
2)
≡
3x
(3x
+
3)(x
+
4)
≡
3x
12
2
12
2
(3x
+
4)(x
2 9 6
+
3)
≡
3x
The
8 Patterns
12
✓
numbers
in
the
brackets
have
product
12.
A LG E B R A
Example
6
2
Fully
factorize
6x
7x
10.
2
6x
2
=
3x
×
2x
=
6x
×
x,
so
6x
7x
10
=
(3x
)
(2x
)
2
Find
terms
with
product
6x
2
or
6x
10
7x
=
−10
10
×
1
=
=
(6x
−5
×
)
2
=
(x
−2
)
×
5
=
−1
×
10
Find
numbers
with
product
10.
2
(3x
+
1)(2x
10)
≡
6x
28x
10
11x
10
2
(3x
+
2)(2x
5)
≡
6x
(3x
+
5)(2x
2)
≡
6x
10)(2x
1)
≡
6x
2
+
4x
10
+
17x
2
(3x
+
10
Try
the
terms
and
numbers
systematically.
2
(3x
10)(2x
+
1)
≡
17x
6x
None
10
of
these
is
correct.
2
(3x
5)(2x
+
2)
≡
6x
4x
(3x
2)(2x
+
5)
≡
6x
10)
≡
6x
10
2
+
11x
10
+
28x
10
2
(3x
1)(2x
+
Since
3x
and
2x
did
not
work,
try
terms
6x
and
x
2
(6x
10)(x
+
1)
≡
6x
4x
10
2
(6x
5)(x
+
2)
≡
+
6x
7x
10
Nearly
correct;
try
+5
and
2
2
(6x
You
+
5)(x
may
not
2)
≡
need
6x
to
7x
write
✓
10
down
all
the
possible
factorizations
to
test
them.
2
Each
one
gives
expanding
the
gives
correct
the
correct
Reect
and
●
factorization
Which
What
made
x
and
middle
discuss
them
constant
so
you
only
need
to
check
if
term.
3
was
easy
terms,
easier:
or
the
one
in
Example
5
or
in
Example
6?
dicult?
2
●
A
student
(3x
●
+
Why
5)(x
do
tries
+
you
consuming
Practice
Fully
2).
to
factorize
Show
think
way
to
that
3x
the
listing
the
tackle
this
+
17x
+
middle
10
term
options
and
is
gives
the
answer
incorrect.
systematically
might
be
a
time-
problem?
4
factorize
each
quadratic
expression.
Tip
2
1
2x
2
+
5x
+
3
2
3x
2
+
7x
+
2
3
3x
+
7x
6
Write
down
working
to
enough
be
sure
2
In
Example
constant
explore
identify
6
there
term,
which
another
the
are
way
factors
two
gives
to
possible
a
lot
of
factorize
straight
pairs
of
possible
quadratic
away.
factors
factors
for
to
the
test.
expressions
x
term
You
where
and
will
it
is
the
you
now
not
easy
have
answer.
to
the
correct
Look
opportunities
for
to
be
ecient:
it’s
do
working
some
your
8.2 Back to the beginning
OK
head.
2 97
to
in
Exploration
To
expand
a
pair
multiplication.
every
(2x
+
term
in
3)(x
4
brackets,
You
the
4)
of
≡
you
multiply
rst
use
every
the
term
distributive
in
the
property
second
of
bracket
by
bracket.
2x(x
4)
+
3(x
4)
If
you
use
a
2
≡
2x
8x
≡
2x
+
3x
12
dierent
method
for
2
5x
12
expanding
check
2
1
Use
2
the
The
distributive
table
shows
Working
from
brackets.
The
property
ve
left
top
to
dierent
to
right,
row
has
show
that
quadratic
each
been
cell
(2x
1)(x
+
expressions,
shows
one
completed.
step
Copy
5)
≡
one
in
and
2x
on
+
each
complete
+
3)(x
(2x
3)(x
+
4)
=
7)
=
2x(x
4)
+
3(x
4)
=
8x
+
3x
12
=
=
=
3x(2x
1)
+
4(2x
the
table.
2x
5x
12
5x
4
=
=
1)
the
2
2x
=
2
=
=
=
=
4x
+
12x
2x
=
6
2
3
Explain
to
4
which
you
think
is
easier:
=
completing
from
left
to
6x
+
right
or
right
left.
Comment
why
it
is
on
which
harder
of
than
the
the
steps
is
most
dicult
to
complete.
Explain
others.
2
5
Two
students
try
to
factorize
the
expression
Carmen
2
+
25x
Explain
Exploration
splitting
the
happens
help
In
4
B.
or
rule.
more
6x
25x
you
2
+
21x
+
student’s
shows
that
term
kind
5
you
Then
you
of
6x
helps
make
correct
some
+
14.
2
+
to
a
25x
+
14
factorize
quadratic
way.
factorized
≡
6x
+
the
+
5x
+
14
expression.
easier
Exploration
expressions
20x
to
5
to
factorize
by
investigates
nd
a
what
pattern
to
quadratic.
patterns
general
gather
should
14
can
justify,
should
+
work
the
Investigating
and
4x
you
in
expand
this
verify
Exploration
few
≡
which
factorize
prove,
general
14
middle
Objective:
iii.
+
when
you
+
Miranda
2
6x
6x
rules
enough
verify
it,
by
information
showing
that
so
you
your
can
conjecture
conjecture
holds
a
for
a
examples.
Exploration
5
ATL
Investigate
the
relationship
between
the
coecients
of
the
split
middle
term
2
and
To
the
coecients
work
out
examples,
how
you
in
to
could
ax
split
+
bx
the
make
a
+
c
middle
table
term
like
of
a
quadratic
by
studying
2 9 8
8 Patterns
some
this:
Continued
that
it
gives
5.
the
same
(2x
+
result
row
.
expanding
2
(2x
9x
brackets,
on
next
page
3)(x
4).
for
A LG E B R A
You
Coecients
Factorized
Coecients
Expanded
of
the
split
middle
more
+
ax
expression
expression
can
bx
+
information
c
term
by
a
gather
in
2
b
making
up
your
c
own
examples
2
(2x
+
(2x
+
3)(x
+
4)
1)(x
2x
+
3x
+
8x
+12
3
8
2
11
12
for
the
left-hand
column.
2)
make
You
new
can
examples
...
by
Add
nd
more
the
rows
until
product
and
you
the
discover
sum
of
the
the
pattern.
You
coecients
of
may
the
nd
split
it
useful
middle
to
changing
numbers
term.
examples
already
This
skill
Reect
and
discuss
in
is
you
been
an
what
Example
you
found
have
given.
important
when
4
investigating
Compare
the
the
in
Exploration
5
with
others
before
moving
on.
in
mathematics.
7
2
Factorize
15x
2x
24.
2
15x
2x
24
Find two numbers with product 15 × −24 = −360 and sum
360
=
18
×
−20
2
=
18
+
−20
24
≡
15x
Split
2
middle
term
into
18x
20x
2
2x
15x
+
18x
20x
24
Take
Does
the
the
order
factorizing
the
≡
3x(5x
≡
(3x
of
+
6)
4(5x
4)(5x
the
+
middle
quadratic
in
+
the
common
factors
6)
(5x
from
+
6)
each
is
a
pair
of
terms.
common
factor.
6)
terms
matter?
Example
Look
at
these
two
ways
of
7:
2
15x
2x
24
2
2
+
≡
15x
≡
3x(5x
≡
(3x
They
2.
18x
+
6)
4)(5x
both
20x
4(5x
+
give
24
≡
15x
6)
≡
5x(3x
≡
(5x
+
6)
the
same
20x
+
4)
+
+
6)(3x
18x
6(3x
24
4)
4)
factorization.
2
Quadratics
where
the
coecient
of
x
is
not
1
may
be
factorized
by
2
splitting
whose
the
sum
middle
is
b
and
term.
To
whose
factorize
product
is
ax
+
bx
+
c,
look
for
two
numbers
ac
8.2 Back to the beginning
2 9 9
Practice
5
Problem
1
2
Find
pairs
solving
of
numbers
a
product
240
c
product
96
Find
values
and
and
to
ll
which
sum
sum
these
have:
32
b
product
180
35
d
product
1000
boxes
and
complete
the
2
a
and
sum
and
3
sum
117
identities.
2
x
b
x
4x
+
2
27x
+
2
=
x
+
=
5x(2x
=
(
x
+
4x
7)
x
14
(2x
)(
x
+
+
x
=
7)
)
+
=
x(4x
=
(x
+
+
3x
+
x
3)
+
(4x
)(
x
+
+
+
3)
Use
)
any
methods
3
Factorize
each
learned.
quadratic.
2
2
9x
a
5x
d
6x
g
6x
+
4
b
6x
e
8x
h
6x
2
+
4
j
15x
m
2x
5x
+
1
7x
2
k
8x
+
1
n
2x
+
7x
15
q
4x
2x
3
t
7x
14x
3
Problem
These
could
w
expressions
ll
3x
x
2
i
6x
the
x
can
empty
+
Can
17x
+
6
5x
1
+
10x
3
l
21x
o
15x
23x
+
6
2
9x
+
10
40x
15
2
+
8x
21
r
12x
u
8x
+
4x
5
2
16x
+
4
+
24x
+
16
2
9x
+
11x
+
2
x
10x
x
21
be
factorized.
Determine
the
dierent
b
everything
the
values
that
boxes.
2
Exploration
Factorize
3
2
2
+
●
1
12x
+
2
+
x
2
+
9x
Non-factorizable
D
f
8x
solving
2
a
15
2
5x
+
2
14x
2
2
4
4x
2
2
v
c
2
3x
2
5x
2
2
2
s
+
2
+
2
4x
7x
2
11x
2
p
2
+
be
+
2
c
6x
+
10x
+
expressions
written
in
a
dierent
form?
6
following
expressions
if
possible.
Verify
your
solution
by
expanding.
factorized
form
factorized
form
expanded
2
x
16
2
49x
121
2
x
+
25
2
25x
+
81
2
4x
+
9
Continued
3 0 0
8 Patterns
on
next
page
of
the
you
have
A LG E B R A
2
Describe
anything
you
notice
about
the
expressions
you
could
not
Tip
factorize.
Explain
Factorize
the
why
they
are
not
factorizable.
If
3
following
expressions
if
possible.
Verify
your
you
yet
by
have
not
solution
learned
how
expanding.
to
factorized
form
factorized
form
multiply
brackets
expanded
in
step
out
like
5,
those
either
use
2
x
+
6x
+
4
a
computer
algebra
2
2x
3x
system
1
it
as
a
or
complete
group
with
2
x
–
10x
24
your
teacher’s
2
x
+
10x
+
24
assistance.
2
4x
4
+
x
10
For
those
why
5
the
methods
Expand
these
Describe
Reect
●
When
When
other
using
are
if
that
that
power
you
as
take
HL
Fundamental
+ 3 +
cannot
learnt
5 )( x
factorize,
did
+ 3 −
not
explain
clearly
work.
5 ).
expression
be
you
factorized
you
of
can
not
of
be
factorized
work
with
algebra
be
in
‘not
factorizable’,
integers.
we
What
other
as
set
You
part
algebra
says
is
of
that
of
any
there
quadratic
numbers.
learn
the
integers,
althoughitmight
real
will
using
are
quadratics.
factorized,
the
numbers.
Mathematics
is
using
use?
cannot
help
are
theorem
not
5
theorem
which
could
already
quadratic
could
complex
you
notice.
polynomial)
numbers
known
a
it
will
Fundamental
higher
(x
expressions
techniques
The
you
that
numbers
quadratic
have
discuss
say
mean
of
you
anything
we
which
brackets:
and
usually
types
expressions
about
Diploma.
university-level
These
numbers
complex
Proving
(or
involve
numbers
the
mathematics.
Summary
2
●
The
It
symbol
shows
≡
that
means
two
‘is
identically
expressions
take
equal
the
to’.
●
Where
one
same
the
coecient
method
of
of
x
factorizing
does
not
involves
equal
listing
1,
all
2
value
for
any
possible
value
of
the
the
unknown.
pairs
pairs
●
To
factorize
a
quadratic
written
in
the
of
of
terms
numbers
with
with
product
product
ax
c.
and
Then
all
try
the
the
form
dierent
combinations
until
you
nd
pairs
2
x
+
bx
+
c
≡
(x
+
p)(x
+
q),
nd
two
numbers
which
p
and
q
which
have
product
c
and
sum
●
2
●
Any
expression
of
the
form
a
Alternatively,
is
of
two
2
a
quantity
squares,
because
subtracted
is
often
quicker
to
look
for
the
it
is
numbers
from
whose
sum
is
b
and
whose
one
product
squared
it
2
b
two
dierence
work.
b.
is
ac
another.
2
b
≡
(a
+
b)(a
b)
8.2 Back to the beginning
3 01
Mixed
1
practice
6
Factorize:
Factorize:
2
2
a
x
c
x
e
x
g
x
i
x
k
x
2
+ 4 x
+ 4
b
x
− 13 x
2
a
a
c
c
e
4e
g
3g
5a
36
b
16b
d
4d
f
f
h
16h
2
2
2
+ 5x
− 14
d
x
− 7x
− 8
f
x
h
x
+ 18 x
+ 81
− 11x
+ 24
− 15 x
− 100
+ 11x
− 42
2
+
11c
+
24
2
2
9
+ 36
17d
2
3e
+
2f
2
2
23g
+ 100
2
Problem
2
+ 4 x
+ 3
j
x
+ 4 x
− 96
l
x
2
7
2
+ 7x
2
Ornella
Factorize
+ 56
n
x
of
a
whole
result,
number,
and
adds
subtracts
four
to
9.
− 11x
− 60
By
letting
the
original
anexpression
number
for
this
be
n,
write
process.
fully:
b
2
Hence
show
the
number
that
Ornella
obtains
2
x
49
b
x
169
can
2
64
always
be
written
as
the
product
of
two
2
x
d
25 x
integers
4
2
e
the
2
− 15 x
down
c
1
solving
thinks
squares
a
a
14
+ 6
it,
2
+
2
− 20 x
x
48
2
2
m
15
8
2
144 x
81
f
256 x
Copy
and
with
a
dierence
of
6.
complete:
169
2
2
g
2
4 x
2
y
h
16 x
a
x
b
x
x
+
15
≡
(x
3)(x
)
2
9 y
2
2
i
j
289 y
x
+
20
≡
(x
+
4)(x
)
4
2
25 x
x
16
2
4
k
4
16 x
1
l
c
x
d
x
8x
≡
(x
+
)(x
11)
4
9x
729 y
2
3
Factorize
9
2
a
2x
11x
2
+ 8x
+ 6
b
2x
+ 14 x
Dagmar
it,
+ 20
and
2
2x
e
2x
≡
thinks
(x
of
subtracts
number
c
+
5)(x
)
completely:
in
the
an
1.
even
By
form
number,
expressing
2n,
or
squares
his
original
otherwise,
show
that
2
− 12 x
+ 10
d
3x
f
3x
− 9x
+ 6
the
result
product
2
of
of
this
two
process
can
consecutive
be
odd
written
as
the
integers.
2
+ 8x
− 24
+ 21x
− 24
3
10
2
g
4 x
You
have
two
− 8x
− 32
h
2x
− 2x
− 40
one
measuring
measuring
4
rectangular
grids
of
1 cm
cubes,
2
n
+
n
3
+
1
by
by
n
+
n
+
6 cm,
3 cm,
and
where
n
the
is
other
a
Factorize:
positive
2
a
2x
c
2x
e
5x
integer.
2
+ 5x
+ 2
b
2x
3x
+ 13 x
+ 20
n
2
+
6
n
+ 12
d
− 12 x
+ 4
f
− 8x
7x
+ 38 x
− 24
n
n
+
+
3
1
2
2
6x
3
− 3
2
2
g
+
2
− 11x
+ 31x
+ 5
h
6x
j
8x
− 17 x
+ 5
− 41x
+ 5
2
2
i
8x
+ 18 x
+ 9
2
k
9x
2
+ 12 x
− 32
l
10 x
− 13 x
− 30
a
5
4a
3
3a
and
number
2
a
Find
simplify
an
expression
for
the
total
3
Factorize:
b
7b
2
35b
grids
168b
b
of
1 cm
cubes
in
the
two
rectangular
combined.
Show
that
if
you
take
apart
the
rectangles
3
and
recombine
always
cubes
left
simply
3 0 2
8 Patterns
be
be
able
over
a
the
to
1 cm
form
(where
straight
a
cubes
rectangle
the
line
you
rectangle
of
cubes).
will
with
will
no
not
A LG E B R A
11
A
rectangular
nmeters
on
sports
each
pitch
side.
has
The
a
border
total
area
They
of
occupied
repeat
number
of
the
process,
rows
as
adding
before
and
the
the
same
same
number
2
by
the
pitch
and
its
border
is
4n
+
28n
+
45.
of
columns
as
before.
Now
the
grid
contains
112cubes.
n
They
repeat
resulting
repetition,
n
Pitch
the
grid
there
Construct
sequence
a
8,
dierences
n
follow
the
dimensions
are
a
third
209
336
time,
cubes.
cubes
in
and
After
the
a
the
nal
grid.
n
a
Find
process
contains
of
the
a
dierence
45,
and
112,
209,
show
quadratic
diagram
336.
that
showing
Analyze
these
the
the
numbers
sequence.
pitch.
b
Letting
u
be
the
number
of
cubes
in
the
nth
n
12
Start
with
a
rectangular
grid
of
8
grid
cubes:
(so
u
=
8),
nd
a
formula
for
1
u
in
the
n
2
form
2
4
an
+
bn
+
c
rows
columns
c
Factorize
your
formula
for
u
.
n
d
Hence
determine
columns
Somebody
columns
many
grid
else
to
the
they’ve
now
then
grid,
adds
but
added.
contains
Reect
and
How
have
you
Give
specic
45
some
doesn’t
They
do
extra
tell
tell
rows
you
you
being
the
added
number
each
of
rows
and
time.
and
how
that
the
cubes.
discuss
explored
the
statement
of
conceptual
understanding?
examples.
Conceptual understanding:
Patterns
can
be
represented
in
equivalent
forms.
8.2 Back to the beginning
3 0 3
9
Space
The frame of geometrical dimensions describing an entity
Dimensions
We
live
located
in
a
using
dimension,
We
a
can
three
people
imagine
mirror,
familiar
space
three-dimensional
is
or
order
with
Cartesian
There
a
is
a
the
could
of
a
a
can
longitude
be
2D
on
the
coordinate
plane,
that
and
at
altitude.
any
universe
plane
named
everything
These
to
after
the
be
images
create
our
Letting
particular
would
in
and
if
possess
graphs.
French
time
place
like,
world
This
we
no
be
can
the
be
4th
moment
simply
in
look
thickness.
scientist
into
You
representation
philosopher,
time.
of
are
2D
and
Descartes.
point
book
in
3D
written
Dimensions,
mysterious
and
located
ground.
three-dimensional
A
meaning
two-dimensional
Cartesian
Many
exist
objects
the
wonderful
encounters
3 0 4
locate
latitude,
shadows
Rene
plane.
Romance
b eing
what
using
called
to
and
observe
mathematician
In
aspects:
world,
visitor
what
it
objects,
would
over
about
from
might
a
the
look
therefore
hundred
a
we
need
have
years
third
like.
dimension
add
three
ago
two-dimensional
to
by
third
axis
coordinates
Edwin
world
and
a
Abbott
where
struggles
one
to
of
to
the
instead
of
two.
called Flatland,
its
inhabitants
comprehend
how
such
a
Animation
Have
you
ever
understanding
achieve
the
use
this
of
barycentric
the
the
effect.
how
Therefore,
in
coordinates,
is
and
that
effort
it
animated
three-dimensional
or
large
are
the
using
simplies
complete
the
characters
space
part
of
that
this
different
using
the
a
can
we
look
live
modern
so
in
process
of
set
and
of
and
move
in
gures.
coordinates,
triple
uid
success
animated
harmonic
coordinates
greatly
to
a
creating
mathematicians
method
time
of
mathematics
animations,
this
wondered
To
is
create
are
numbers.
a
in
An
essential
animations
which
animating
real?
a
The
is
your
form
due
to
favorite
of
advantage
character,
to
and
of
reduces
animation.
More
In
the
on
than
1990s,
the
string
discovered
at
least
But
theories
any
dimensions
does
this
number
including
spaces.
If
the
then
than
ten
of
in
there
being
existence.
mean?
always
worked
dimensional
innite
mathematics
explaining
more
of
working
physics
really
have
spaces,
universe,
of
likelihood
Mathematicians
with
dimensions
mathematicians
the
ten
what
3
is
inherent
perhaps
dimensional
capable
of
structure
there
dimensions
are
of
our
even
waiting
to
discovered.
3 0 5
be
Spacious
interiors
9.1
Global
context:
Objectives
●
Finding
the
(including
●
Finding
Inquiry
surface
area
pyramids,
the
(including
Personal
volume
of
pyramids,
of
cones
any
cones
any
and
3D
3D
●
shape
F
spheres)
What
is
SPIHSNOITA LER
●
expression
What
the
How
dierence
between
area
and
area?
are
some
cylinders,
spheres)
cultural
questions
surface
●
shape
and
and
are
properties
pyramids
the
surface
and
of
prisms,
cones?
areas
of
pyramids,
C
cones
●
D
ATL
Is
and
there
a
spheres
best
related?
method
for
nding
volume?
Creative -thinking
Apply
existing
knowledge
to
generate
relationships
measurements
construction
activities
for
enables
and
and
products
or
processes
between
the
analysis
ritual
ideas,
7.3
Statement of Inquiry:
Generalizing
new
of
9.1
play.
E9.1
3 0 6
G E O M E T R Y
Y
ou
●
should
use
already
know
Pythagoras’ theorem
how
In
A N D
T R I G O N O M E T R Y
to:
each
triangle,
missing
answer
side.
as
a
If
nd
the
length
necessar y,
give
of
the
your
radical.
1
a
5
m
5
m
2
41
9
cm
cm
b
●
nd
area
the
of
volume
cuboids,
and
sur face
prisms
and
cylinders
Find
each
the
volume
solid.
answer
to
If
3
and
sur face
necessar y,
signicant
4
cm
cm
5
3
of
your
gures.
6
3
area
round
cm
cm
4
12
8
5
4
m
m
mm
Ara
6
3.5
mm
4
2
●
nd
the
length
area
of
an
of
arc
a
sector
and
6
Find
and
the
the
area
of
length
8
of
a
trapzium:
mm
mm
mm
the
of
shaded
its
sector
arc.
cm
110°
9.1 Spacious interiors
3 07
Proper ties
F
●
What
is
●
What
ar
Exploration
1
Mak
a
list
of
2
Mak
anothr
th
of
3D
dirn
som
solids
twn
proprtis
of
ara
prisms,
and
surfa
ylindrs,
ara?
pyramids
and
ons?
1
shaps
list
of
that
hav
shaps
an
that
ara
hav
that
a
you
surfa
an
ara
alulat.
that
you
an
alulat.
3
Nxt
to
ah
4
Compar
and
dirns
A
only
in
is
plan
stp
ontrast
twn
polyhedron
has
shap
a
3D
(at)
2,
th
th
shaps
ways
solid
stat
of
th
in
numr
ah
list.
alulating
A
that
pyramid
A
cone
irular
A
on
is
a
3D
as
is
solid
and
not
a
an
with
apx
A
a
or
(th
A
You
alrady
ylindrs
volum
know
and
of
pyramids,
Th
surface
Th
volume
3 0 8
how
prisms.
area
of
a
to
nd
th
surfa
Now
you
will
ons
and
sphrs.
is
th
total
ara
3-dimnsional
9 Space
look
of
all
solid
is
th
Th
mt
othr
at
is
a
3D
ar
distan)
and
not
a
of
a
amount
a
fas
ar
alld
polyhdron.
with
points
on
on
th
quidistant
from
of
surfa
th
ntr.
uoids,
ara
and
3-dimnsional
of
a
polyhdron.
volum
th
with
point
is
solid
surfa
is
a
and
ara.
solid
th
nding
fas
3D
All
sphr
th
a
similaritis
surfa
pyramid
fa.
sam
ara
at
A
sphere
sphr’s
polyhdron.
is
that
apex.
urvd
vrtx.
fas.
and
as.
triangls
th
its
Idntify
ara
polygon
fas.
of
spa
it
solid.
oupis.
G E O M E T R Y
Exploration
Copy
3D
and
A N D
T R I G O N O M E T R Y
2
omplt
this
tal
with
your
osrvations
aout
th
dirnt
solids.
Prism
●
any
Cylinder
Pyramid
Cone
shape,
often
a
regular
polygon
●
rectangular
based
prisms
Base
are
called
cuboids
●
two
opposite
faces
each
could
be
the
base
●
always
●
the
a
circle
Crosssame
section
radius
parallel
the
the
as
to
base
all
base
along
the
cylinder
●
triangles
●
if
is
the
a
base
regular
polygon,
the
triangles
Sides
are
all
congruent
●
the
triangles
meet
at
all
the
apex
Reect
●
●
and
How
is
a
How
is
it
How
do
givn
its
discuss
sphr
similar
1
to
th
3D
solids
in
th
tal
in
Exploration
2?
dirnt?
you
think
th
dirns
volum
ompard
and
to
surfa
othr
3D
ara
of
a
sphr
is
found,
solids?
9.1 Spacious interiors
3 0 9
Finding
C
●
Sur face
How
th
surfa
area
aras
of
pyramids,
ons
and
sphrs
rlatd?
area
Pyramids,
surfa
ar
sur face
ons
and
sphrs
ah
rquir
a
dirnt
mthod
for
nding
ara.
Apex
Th
slant
distan
as
to
height
from
th
of
th
apx
a
pyramid
ntr
of
th
of
is
on
th
dg
of
th
Slant
height
pyramid.
Vertical
height
Th
vertical
distan
height
from
th
is
th
apx
prpndiular
to
th
as.
ATL
Exploration
3
Tip
1
a
Dtrmin
whih
2D
(at)
shaps
mak
up
th
7
dirnt
fas
of
this
squar-asd
cm
Th
pyramid.
ar
b
Idntify
th
information
you
nd
to
nd
th
7
four
triangls
isosls
cm
ongrunt.
surfa
ara
of
squar-asd
pyramids.
10
2
a
Us
th
information
nd
th
ara
of
th
givn
as
in
of
th
diagram
pyramid
cm
to
A.
10
b
Justify
whthr
th
information
in
cm
th
Pyramid
diagram
th
3
is
nough
triangls
c
Calulat
a
In
on
th
pyramid
to
th
sid
surfa
B,
th
nd
th
of
ara
ara
th
of
of
on
A
of
pyramid.
pyramid
sid
of
th
squar
hight
of
th
pyramid
A.
as
Apex
and
th
givn.
vrtial
Suggst
how
you
ould
ar
Slant
height
alulat
Vertical
height
th
slant
hight
of
th
pyramid
using
this
information.
b
Find
c
Us
th
slant
hight
of
th
pyramid.
6
ara
th
of
slant
on
hight
of
th
to
alulat
triangular
cm
6
th
fas
of
th
Pyramid
pyramid.
d
Hn
310
nd
th
total
9 Space
surfa
ara
of
pyramid
B.
B
cm
and
G E O M E T R Y
Example
Find
th
A N D
T R I G O N O M E T R Y
1
surfa
ara
of
this
rtangular-asd
6
cm
pyramid.
10
of
as
=
8
×
10
=
80
cm
There are three dierent side areas
2
Ara
8
cm
m
to nd; the rst one is the rectangular base.
6
cm
Use Pythagoras to calculate the slant height
s
1
6
cm
s
for the front and back triangular faces.
1
8
2
s
2
=
4
cm
cm
2
6
+
4
=
Us
s
and
h
for
slant
hight
52
1
⇒
=
s
m
of
front
vrtial
hight.
1
Ara
for
triangl
The area of the back triangle is the same as the
front triangle. Note: keep answers exact (using
radicals) at each stage. Use your calculator
only at the end to give your nal answer
Similarly,
alulat
th
slant
hight
s
for
th
2
lft
and
right
2
s
=
6
triangular
correct to 3 s.f.
fas:
2
+
5
=
61
2
⇒
The area of the right side triangle
Ara
of
lft
triangl
=
is the same as the left side triangle.
Total
ara
=
=
ara
of
as
+
2
tims
th
ara
of
front
+
2
tims
th
ara
of
sid
80
+
2
×
10
+
2
×
triangl
Find the total area of all 5 faces.
triangl
4
2
=
Reect
●
215
m
and
Compar
●
How
do
pyramid
you
●
How
as
would
hxagon
as
ontrast
pyramid
think
with
hxagon)
s.f.)
discuss
and
squar-asd
(3
its
prosss
a
for
nding
rtangular-asd
ompars
rgular
to
nding
polygon
(for
th
th
surfa
ara
of
a
pyramid.
surfa
xampl,
a
ara
of
a
pntagon
or
as?
you
its
th
and
this
anothr
2
nd
as?
th
ara
How
of
would
th
as
you
nd
of
a
th
pyramid
slant
with
hight
of
a
rgular
this
pyramid?
9.1 Spacious interiors
31 1
Th
S
=
surfa
A
+
ara
You
Find
1
th
4.5
4
with
an
n-sidd
rgular
polygon
as
is
1
qustion,
may
pyramid
triangl
Practice
ah
a
nA
as
In
of
nd
opy
to
surfa
th
draw
ara
diagram
and
omplt
mor
diagrams
for
ah
pyramid.
If
of
m
it
dirnt
as
stps
nssary,
5
2
you
solv
in
round
th
th
prolm.
solution.
your
answr
to
3
s.f.
cm
m
4
m
5
5
cm
cm
3
6.5
cm
3
6.2
cm
5
4
This
dg
cm
cm
squar-asd
masurs
8
pyramid
has
four
idntial
8
5
th
b
Hn
Problem
5
A
th
Gorg
Egypt.
Gorg
7
wants
b
Hn,
surfa
a
a
m
m
ara.
of
48
paint
whih
pyramid
solid
whit
is
to
nd
3D
ara
hight
Explain
This
is
ought
a
m
solving
surfa
Its
has
th
ara
a
all
four
ttrahdron
plastr
mm
fas
th
a
with
rplia
and
th
with
of
sid
pyramid
in
th
pyramid
of
th
surfa
as
with
th
quilatral
sid
lngth
Grat
lngth
of
h
of
nds
th
7
triangls.
m.
Pyramid
its
ston-olor
of
squar
fas
squar
to
to
mak
on
his
as
it
m
Find
and
its
total
surfa
ara
2400
pyramid
that
h
.
cm
30
312
9 Space
to
mm.
mor
nds
hight.
30
trip
75
ral.
paint.
sid
m
is
look
2
30
sloping
hight.
its
ttrahdron
Find
6
slant
nd
Eah
m
5
Find
fas.
m.
8
a
triangular
cm
to
paint.
G E O M E T R Y
8
Ths
two
squar-asd
pyramids
ar
mathmatially
A N D
T R I G O N O M E T R Y
similar.
Whn
sal
ara
4
th
linar
fator
sal
is
r,
th
fator
2
cm
6
4
is
cm
cm
6
Pyramid A
Pyramid
r
cm
B
2
Pyramid
A
Calulat
Th
surfa
unravlld
has
th
ara
into
surfa
surfa
of
a
a
ara
on
irl
ara
of
with
and
a
64
m
Pyramid
as
B.
radius
r
and
slant
hight
s
an

stor:
2πr
r
s
Th
ar
lngth
2πr.
Th
Th
formula
of
radius
th
of
major
th
that
stor
stor
show
th
is
is
qual
qual
to
to
th
rlationships
th
slant
in
a
irumfrn
hight
irl
s
of
th
of
th
irl,
on.
ar:
2
Ara
of
a
irl
=
Ara
of
a
stor
πr
θ
2
π s
=
360
θ
Ar
lngth
of
a
stor
2π s
=
360
Howvr,
you
Thrfor
2π r
know
that
ar
lngth
l
=
2πr
θ
2π s
=
360
360 r
⇒
θ
=
s
And
th
ara
of
a
stor
 360 r



s
2
π s
=
360
r
2
π s
=
=
π rs
s
2
Thrfor
th
surfa
ara
of
a
on
=
πr
πr s
+
2
Th
surfa
th
radius
th
on.
ara
of
th
of
a
as
on
and
=
s
πr
is
+
th
πr s
whr
slant
r
hight
is
of
s
Th
urvd
surfa
ara
of
th
on
is
πr s
r
9.1 Spacious interiors
3 13
Practice
Find
th
2
surfa
8.1
ara
of
ah
solid
on.
If
nssary,
m
5
13
1
3
Th
diagram
Calulat
shows
th
ara
th
of
round
your
4
Blgium
from
pag
is
famous
his
loal
a
Draw
b
Dtrmin
nd
c
a
th
Find
3
s.f.
cm
m
dimnsions
papr
usd
to
of
a
papr
mak
th
up.
6
cm
up.
cm
solving
nwspapr.
of
to
cm
9
Problem
answr
for
its
Rn
Frnh
maks
nwspapr.
diagram
th
to
show
largst
a
fris,
papr
Th
th
pag
whih
on
masurs
smiirl
possil
ar
on
diamtr
oftn
from
of
a
410
th
th
srvd
smiirl
mm
pag
y
of
in
ons
ut
315
from
mad
a
mm.
nwspapr.
smiirl.
Hn,
radius.
th
ara
of
th
largst
on
Rn
an
mak
from
a
pag
4
m.
of
this
nwspapr.
d
Find
th
e
Hn,
irumfrn
nd
th
radius
of
of
th
th
irular
as
of
as.
th
papr
on.
2
5
A
solid
on
has
surfa
a
Find
th
urvd
b
Find
th
slant
c
Hn,
As
th
irls
ara
(whih
hav
as
no
mor
Reect
Suppos
sam
into
●
small
How
pl
●
straight
dgs
for
at
th
pis,
many
ara
th
is
of
th
and
as
radius
on.
of
you
not
th
on.
annot
Finding
possil
us
th
with
th
sam
surfa
mthods
mthods
ara
that
of
a
you
for
nding
sphr
know
and
3
an
orang
Suppos
ompltly
irls
svral
you
lld
would
thn
as
you
tims,
pld
many
xpt
of
to
rating
th
th
ll
irls
orang,
irls
with
as
th
with
raking
th
it
possil.
orang
pis?
How
dos
this
dmonstrat
th
formula
2
Th
,
mathmatis.
around
th
m
on.
dgs,
orang.
and
of
hight
discuss
trad
as
125.7
polygons.
all)
advand
you
of
vrtial
would
and
radius
surfa
hight
th
no
you
has
rquirs
nd
ara
surfa
31 4
ara
of
a
sphr
9 Space
=
4πr
for
th
surfa
ara
of
a
sphr?
G E O M E T R Y
Practice
Find
th
T R I G O N O M E T R Y
3
surfa
ara
of
ah
sphr.
If
nssary,
round
your
answr
to
3
s.f.
2
1
radius
=
2
cm
diameter
3
A N D
Calulat
th
surfa
ara
of
ths
=
24
cm
solids.
b
5
cm
4
a
cm
9
Problem
cm
solving
2
4
Th
surfa
5
How
dos
ylindr
ara
th
in
of
●
Is
sphr
surfa
whih
it
Finding
D
a
thr
ara
ts
is
of
400
a
m
sphr
prftly?
.
Calulat
ompar
(Diagram
to
its
radius.
th
shown
at
surfa
ara
of
a
right.)
volume
a
st
mthod
for
nding
volum?
ATL
Exploration
Arhimds
a
similar
You
Part
1
1
Put
so
2
–
disovrd
mthod
will
ylindr
nd
and
a
that
golf
th
Rord
watr
th
volum
of
a
sphr
in
225
bce,
in
watr),
and
you
will
us
hr.
golf
all
(or
any
all
that
sinks
a
masuring
watr.
nding
th
4
the
all
in
watr
th
volume
th
a
golf
masuring
ovrs
watr
of
lvl
th
and
ball
ylindr.
Fill
th
ylindr
with
watr
all.
alulat
th
volum
of
th
all
and
th
togthr.
3
3
Rmov
volum
4
Us
th
of
your
golf
th
all.
Rord
th
nw
watr
lvl,
and
alulat
th
1
ml
=
1
m
watr.
rsults
from
stps
2
and
3
to
nd
th
volum
of
th
golf
Continued
all.
on
next
page
9.1 Spacious interiors
31 5
Part
5
2
–
comparing
Th
diagram
Th
radius
a
Stat
shows
of
th
spheres
a
golf
radius
a
golf
all
of
and
all
is
th
cylinders
in
a
ylindr.
approximatly
4.2
4.2
m.
cm
Whn
ylindr
ts
b
Stat
th
hight
of
th
Masur
nd
golf
th
th
diamtr
volum
all
with
of
th
its
of
th
golf
all
irumsrid
volum
of
its
that
you
ylindr.
irumsrid
usd
in
stp
Compar
ylindr.
1.
th
Us
this
volum
Us
your
of
to
th
ndings
th
possil
th
suggst
Arhimds
and
formula
of
invntor.
tomston,
In
a
Exploration
Syraus
It
to
for
is
said
4,
th
(287
h
rprsnt
th
volum
bce
had
his
volum
a
–
212
sphr
favourit
of
of
th
a
sphr.
bce)
was
a
Grk
irumsrid
mathmatial
sphr
is
y
mathmatiian
a
ylindr
on
his
proofs.
two-thirds
of
th
volum
of
its
3
irumsrid
2
ylindr,
4
(2π r
=
and
2πr
.
Two-thirds
of
this
volum
3
Th
You
volum
π r
) =
3
has
3
3
is
whih
volum
an
us
volum
a
of
a
sphr
similar
of
a
mthod
to
vrify
th
formula
for
volum
of
a
pyramid
on.
Volum
of
Volum
of
uoid:
l
pyramid
1
×
w
×
with
sam
as
×
h
ara
=
ara
of
of
as
×
as
×
hight
hight
3
and
h
hight:
l
w
This
formula
an

gnralizd
to
a
pyramid
with
any
as.
1
Similarly,
th
volum
of
a
on
is
th
volum
of
a
ylindr
with
th
sam
3
as
radius
and
hight.
1
Volum
of
on
=
×
volum
3
h
1
2
V
π r
=
3
r
Volum
of
pyramid
Volum
of
on
316
=
×
9 Space
ara
of
as
×
hight
h
of
ylindr
smallst
ylindr,
ylindr
circumscribes
sphr.
to
sphr
ylindr
insid
6
a
prftly
th
G E O M E T R Y
Practice
Find
th
volum
of
ah
10
3D
solid.
2
cm
6
7
cm
8
cm
cm
3
24
4
7
cm
cm
6
cm
cm
24
10
12
5
cm
cm
cm
6
7.4
10
8
10
In
cm
cm
7
10
cm
cm
8
8
T R I G O N O M E T R Y
4
1
8
A N D
Prati
2
6
9
cm
18
cm
cm
Q4,
you
found
th
ara
of
a
nwspapr
on
mad
from
a
smiirl.
a
Find
b
Explain

th
lss
volum
why
than
Problem
11
Ths
of
th
this
nwspapr
volum
of
potato
answr
to
part
your
on.
in
this
on
of
Frnh
Fris
would
a
solving
two
ons
ar
mathmatially
similar.
Whn
3
Con
A
has
volum
2000
sal
10
Calulat
th
volum
of
th
linar
m
Con
cm
12
fator
volum
Cone A
Cone
is
r,
th
cm
B.
sal
fator
B
3
is
12
a
Explain
b
Th
why
any
two
sphrs
ar
mathmatially
r
.
similar.
3
volum
Find
i
th
of
a
sphr
volum
radius
8
of
a
of
radius
sphr
4
m
is
268
(to
m
3
s.f.)
of
m
ii
radius
2
m
3
13
Find
th
hight
of
a
on
that
14
Find
th
radius
of
a
on
15
Find
th
radius
of
a
sphr
16
Suppos
has
volum
270π
mm
and
as
radius
9
mm.
3
that
has
volum
8.38
m
and
hight
2
m.
3
mpty
Find
you
ll
ylindr
th
answr
hight
as
a
an
mpty
that
up
has
th
fration
of
with
sphr
th
sam
ylindr
th
volum
total
with
523.6
watr
hight
that
th
hight
and
and
th
thn
diamtr
watr
of
m
would
pour
as
it
th
rah.
into
an
sphr.
Lav
your
ylindr.
9.1 Spacious interiors
317
Summary
●
A
polyhedron
(at)
is
a
3D
solid
that
has
only
plan
●
fas.
Th
surface
fas
●
Th
of
a
Th
of
of
a
spa
surfa
rgular
is
th
total
3-dimnsional
volume
amount
●
area
ara
polygon
ara
●
A
pyramid
as.
Th
is
a
3D
othr
solid
fas
ar
with
a
Volum
of
th
solid
is
th
oupis.
of
a
pyramid
as
is
S
=
with
A
+
pyramid
=
×
ara
of
an
n-sidd
nA
as
●
all
solid.
3-dimnsional
it
of
triangl
as
×
hight
polygon
triangls
that
mt
h
at
a
point
alld
th
apex.
A
pyramid
is
a
polyhdron.
2
●
Surfa
radius
ara
of
th
of
a
on
as
and
=
s
πr
is
+
th
πrs
s
h
●
A
cone
an
is
apx
a
or
3D
solid
vrtx.
A
with
on
a
irular
is
not
a
as
and
polyhdron.
r
●
Volum
●
Surfa
of
a
on
2
●
A
sphr
is
a
3D
solid
with
on
urvd
ara
of
a
sphr
=
fa.
r
All
th
points
equidistant
A
sphr
on
(th
is
th
sam
not
a
sphr’s
surfa
distan)
from
ar
th
ntr.
polyhdron.
●
●
Th
slant
from
th
apx
of
height
ntr
th
of
of
a
pyramid
on
dg
of
pyramid.
Slant
Height
height
318
9 Space
is
th
th
distan
as
to
th
Volum
of
a
sphr
4πr
whr
slant
r
is
hight.
th
G E O M E T R Y
Mixed
A N D
T R I G O N O M E T R Y
practice
2
1
Find
3D
th
surfa
ara
and
volum
of
3
ah
A
on
radius
solid.
a
12
b
cm
6
has
5
surfa
ara
204.2
m
cm
4
12
c
9
3
5
cm
as
cm
cm
cm
cm
cm
and
m.
5
12
,
cm
a
Find
th
slant
hight
b
Find
th
volum
of
of
th
th
on.
on.
d
3
cm
3
4
Th
volum
Aording
of
th
all
to
5
e
A
sor
th
sphrial
all
is
rgulations,

twn
whthr
rgulations.
cm
a
must
Determine
1
of
Justify
soop
this
i
th
68
dm
.
irumfrn
m
sor
your
of
5.6
and
all
70
m.
satiss
th
answr.
ram
is
ut
in
half,
and
f
ovrd
in
hoolat
(inluding
th
at
fa).
2
Th
a
6
16.4
cm
total
Find
of
cm
b
ara
th
i
6
solving
Hn,
Two
Th
2
All
ight
dgs
of
a
squar-asd
pyramid
m
radius
nd
papr
in
of
th
is
85
m
sphrial
soop
th
volum
of
th
i
ram
hoolat.
ons
apaity
of
ar
th
mathmatially
largr
on
is
8
similar.
tims
th
ar
apaity
12
hoolat
ram.
ovrd
Problem
of
of
th
smallr
on.
long.
Th
surfa
ara
2
th
smallr
Find
th
on
surfa
is
50
ara
m
of
.
th
largr
on.
12 cm
12 cm
12 cm
a
Find
its
surfa
b
Find
th
surfa
mathmatially
as
a
ara
squar
of
and
ara
and
similar
sid
its
9
volum.
volum
pyramid,
of
a
with
m.
9.1 Spacious interiors
319
of
Objective:
ii.
slt
In
these
surface
D.
appropriat
real-life
area
of
Review
Th
as
th
pharaohs.
select
to
the
answer
in
ral-lif
stratgis
strategies
the
ontxts
whn
you
have
solving
learned
authnti
for
cultural
uilt
squar-asd
urial
Th
ar
ritual
thr
for
largst
loatd
Many
liv
inund
y
a
th
pyramids
quns
and
a
Find
and
st
i
in
th
town
of
th
that
th
Goldn
of
Giza,
sids
of
th
as
masurs
230.4
th
mtrs
Find
3
masurs
145.5
mtrs.
th
ratio
of
sid
lngth
to
whil
How
hight
th
is
Th
ara
c
a
pyramid
layr
of
of
th
was
whit
limston
Compare
th
rlat
d
ara
to
of
th
Calculate
Calculate
to
gam
pyramid
originally
limston.
ovrd
Calculate
was
Nar
in
th
as
total
th
th
surfa
squar
ara
as.
of
primtr
as
th
radius
Describe
of
th
is
th
of
How
dos
,
hoop
Th
Srpnt,
what
on
usd
with
m
th
to
as
th
you
of
gan
dorat
12.5
a
yllow
and
How
you
a
A
allgam,
vry
larg
arhaologial
Playrs
havy
on
of
ity
a
of
hundrds
of
sit
ould
all
through
nd
a
ring
th
wall
of
th
Gnralizing
analysis
m.
and
Th
matrial
ativitis
surfa
why
ara
in
putting
nding
tradition
of
in
Grmany
in
onial
thm
(No
of
ourt
of
th
th
th
all
th
all
(no
ring
was
through
gam.
dorating
th
and
th
vrgrn
9 Space
njoy
it
16th
was
ntury
.
thir
wr
possil
With
auty
put
from
to
all
undrnath
th
fa.)
a
of
th
a
aout
Determine
ara
for
m
whih
tr
dorations:
and
a
or
tr
a
would
on
maximum
with
a
hav
with
a
mor
hight
of
irumfrn
hight
of
180
of
m
and
a
th
of
irumfrn
of
210
m?
Justify
A
in
Rokfllr
your
answr.
pyramids
found
ranging
sphrs
alld
tr
loatd
ritual
Cntr
in
Nw
hundrds
City
is
lit
vr y
yar
in
arly
Dmr.
from
largst
tr
masurd
100
ft
tall
and
wr
a
volum
quivalnt
to
65 500
ui
ft.
jarosit.
statmnt
twn
for
shap,
dorations
th
ara
was
discuss
rlationships
of
irumfrn
Totihuaan.
root
xplord
th
th
of
inquiry?
Giv
spi
xampls.
Statement of Inquiry:
320
sport,
th
of
hight
noti
Tmpl
irumfrns
to
Reect
and
th
Mxio.
If
rsultd
roughly
Find
hav
in
justify
thir
had
in
at
gtting
and
trs
Th
ovrd
Itza
Christmas
Y
ork
3.5
ah
this
pyramid.
irumfrn
qual
Th
Msoamrian
sphrs
for
sids
b
aout
tam
Msoamria.
visil
vrtially
m
m
215
City
Arhaologists
of
ndd
Ratio?
volum
wll
a
Mxio
th
lay
arlist
allowd).
95
2830
rsults.
Fathrd
in
y
200m
2
of
2
th
usd.
Goldn
th
with
pyramid.
ths
all.
th
irular
irl
th
still
Chihn
sids.
e
paint:
los
4
to
to
largst
Ratio?
ntir
with
ndd
th
th
hands
b
volum
playd
mountd
Goldn
ii
Grat
th
is
jarosit
all.
Potntially
of
hight
of
all
wr
(1.62).
of
amount
smallst
nar
ourt
Pyramid
and
prsrvd
uildrs
Ratio
th
th
was
Eah
volume
expression
siz
Cairo.
the
situations
questions.
b
pyramids
nding
ral-lif
context
and
of
mathmatial
shapes
Egyptians
part
mathmatis
situations,
3D
in
Personal
1
Applying
masurmnts
and
play.
nals
th
onstrution
availal
for
dorations.
A
parable
about
parabolas
9.2
Global
context:
Objectives
●
Finding
the
quadratic
●
forms:
axis
a
a
symmetry
and
vertex
of
●
a
F
quadratic
standard,
Finding
of
function
factorized
quadratic
and
function
in
three
vertex
given
on
its
What
shape
How
do
Finding
a
function
to
model
a
real-life
Understanding
an
object
in
a
how
given
many
unique
dimension
of
points
dene
the
parameters
aect
How
can
you
in
the
of
of
shape
express
three
What
are
the
a
a
of
a
quadratic
quadratic
its
graph?
quadratic
dierent
advantages
disadvantages
space
ofa
●
D
ATL
graph
ways?
parabola
●
●
the
C
graph
function
●
is
distinct
●
points
innovation
questions
function
three
technical
function?
●
dierent
and
MROF
●
Inquiry
function
Expressing
Scientic
quadratic
What
than
makes
of
the
and
dierent
forms
function?
one
quadratic
form
better
another?
Critical-thinking
Apply
existing
knowledge
to
generate
new
ideas
or
processes
8.2
9.2
11.2
Statement of Inquiry:
Representing
systems,
patterns
models
and
with
equivalent
forms
can
lead
to
better
methods.
3 21
Y
ou
●
should
interpret
already
graphs
of
know
linear
1
functions
In
how
the
do
m
to:
linear
and
c
function
y
=
mx
+
represent?
c,
what
y
5
2
From
the
graph,
nd:
4
a
the
x-intercept
3
b
the
y-intercept
2
3
a
Determine
if
the
gradient
1
is
b
positive
Find
the
or
negative.
equation
0
of
x
1
1
this
●
factorize
a
quadratic
4
line.
Factorize:
2
expression
a
x
b
2x
+
c
x
d
3
5x
+
6
2
3x
2
2
49
2
Quadratic
F
You
are
●
What
●
How
do
of
graph?
its
shape
surrounded
the
by
2x
functions:
is
the
graph
parameters
many
of
of
dierent
a
a
x
standard
quadratic
quadratic
shapes
form
function?
function
which
can
aect
be
the
shape
classied
A
according
to
their
mathematical
properties.
One
of
the
most
common
shapes
trajectory
the
an
arch.
Arches
can
be
seen
everywhere,
from
the
shape
of
a
banana
to
ight
as
the
from
path
Reect
The
you
knew
or
How
the
many
ball
a
just
32 2
discuss
shows
one
the
or
the
of
ball
bridges
and
buildings,
and
in
trajectories,
such
1
trajectory
of
a
basketball.
The
trajectory
is
a
the
ball’s
would
go
positions
in
the
in
the
basket?
air,
could
What
if
you
you
tell
knew
two
of
three?
positions
would
question
to
parabola.
not
positions,
fountains
basketball.
and
called
whether
its
a
photograph
curve,
If
water
of
go
in
do
the
you
think
basket?
later.
9 Space
you
Make
would
a
good
need
to
guess.
know
You
to
will
be
sure
return
path
of
mouth
moving
guards,
is
is
to
that
this
object.
a
A LG E B R A
Exploration
1
Tip
2
Graph
the
quadratic
function
y
=
x
.
The
graph
is
a
parabola.
If
your
graphing
program
1.1
has
y
a
or
slider
function,
insert
a
change
in
a
the
GDC
bar
you
slider
the
can
to
values
functions
2
0
f1(x)
=
easily.
ax
1
10
10
x
0
1
2
1
Graph
y
=
ax
non-integer
for
a
few
values
of
a
between
10
and
+10.
Include
some
values.
2
2
The
parabola
basketball’s
in
the
graph
trajectory
is
of
y
=
concave
x
is
concave
down
up
.
The
parabola
of
the
.
Tip
a
Find
the
values
of
a
for
which
the
parabola
is
concave
up.
A
b
Find
the
values
of
a
for
which
the
parabola
is
concave
parabola
resemble
c
Find
the
values
of
a
that
make
the
parabola
narrower
can
down.
than
the
a
smile
or
graph
a
frown;
could
this
2
of
y
=
x
help
you
remember
2
d
Find
the
values
of
a
that
make
the
parabola
wider
than
the
graph
of
y
=
x
2
3
Graph
some
y
=
x
+
c
for
non-integer
dierent
values
of
c
between
10
and
+10.
the
for
Include
values
which
of
the
parabolawill
values.
a
be
concave
down
concave
up?
or
1.1
y
c
0
5
5
1
2
f 2(x)
=
x
+
c
x
0
1
Describe
the
eect
of
changing
the
value
of
c
2
4
Graph
some
y
=
ax
+
bx
non-integer
(Leave
a
=
1
for
for
dierent
values.
the
values
Describe
the
of
b
between
eect
of
10
and
changing
+10.
the
Include
value
of
b.
moment.)
1.1
y
b
0
5
a
5
1
5
5
1
2
f 3(x)
=
a
x
+
b
x
x
0
1
Continued
on
next
page
9.2 A parable about parabolas
323
5
Graph
the
the
graph
Go
back
linear
function
changes
for
y
=
bx
dierent
for
dierent
values
of
values
of
b.
Describe
how
b
2
6
xed
to
and
dierent
7
Use
the
parabola
change
values
your
b.
of
y
=
ax
Describe
+
bx.
how
Give
the
a
any
shape
of
value
the
except
parabola
0.
Keep
changes
a
for
b
ndings
to
describe
the
similarities
and
dierences
in
the
shape
Try
of
the
graphs
of
each
pair
of
quadratic
to
these
2
a
y
=
2
2x
+
x
+
1;
y
=
2
8x
+
x
+
answer
functions.
1
b
y
=
questions
2
3x
1;
y
=
−3x
without
1
graphs.
2
c
y
=
10x
2
;
d
y
=
x
drawing
You
e
y
=
the
use
2
+
3;
y
=
x
1
technology
2
can
to
check
2
+
x
x
1;
y
=
x
of
a
x
+
1
your
answers.
2
The
and
A
standard
c
are
real
parameter
●
form
numbers,
of
a
of
quadratic
and
function
a
linear
a
≠
graph
The
parameter
m
function
The
graph
quadratic
The
parameters
denes
its
is
y
=
ax
+
bx
+
c,
where
a,
b
0.
denes
The
function
the
y
=
form
mx
+
gradient;
of
c
its
is
the
graph.
always
a
parameter
straight
c
line.
denes
its
y-intercept.
2
●
concave
The
of
up
a
or
parameters
a,
b
and
concave
a,
b
and
c
function
c
dene
down,
are
y
=
ax
+
whether
and
called
its
the
bx
the
+
c
is
always
parabola
is
a
parabola.
slim
or
wide,
y-intercept.
coecients
of
the
quadratic
function.
2
Parameter
is
called
a,
the
Quadratic
A
coecient
of
x
,
is
called
the
leading
coecient.
Parameter
linear
functions
functions.
integer
These
belong
are
to
the
functions
family
where
of
the
functions
variable
x
called
has
only
exponents.
polynomial
is
a
mathematical
expression
involving
a
sumofpowers
4
one
The
A
or
more
degree
linear
function
c
constant.
and
polynomial
positive
the
variables
of
a
polynomial
function
is
a
multiplied
is
a
by
function
polynomial
polynomial
coecients,
is
the
value
function
function
of
of
degree
forexample,
of
its
degree
largest
1.
A
of
A
constant
2
x
3x
+
exponent
1.
of
x.
quadratic
polynomial
degree
0.
example,
2.
constant
y
=
2
y
=
2x
can
has
For
the
function
be
written
0
ATL
Reect
●
Why
is
What
●
Why
●
One
for
and
the
type
do
parameter
of
you
ordered
example:
line,
for
How
324
such
a
function
think
pair
(2,
of
points
as
the
2
in
the
would
quadratic
result
parameter
7).
example:
many
space,
discuss
c
is
coordinates
Two
(1,
do
3)
points
and
you
graph
9 Space
(5,
think
of
a
if
a
called
denes
dene
function
=
a
the
a
not
allowed
to
equal
constant?
0-dimensional
1-dimensional
space,
space,
or
or
point,
straight
2).
are
required
quadratic
0?
0?
to
dene
function?
a
2-dimensional
,
0
since
x
=
1.
A LG E B R A
A
straight
on
the
from
the
line
has
parabola
positive
to
a
has
constant
its
own
negative,
gradient.
gradient.
or
vice
A
The
versa,
is
parabola
point
does
where
called
the
not.
its
In
fact,
gradient
vertex
or
each
point
changes
turning
point
of
parabola.
vertex
negative
positive
negative
positive
gradient
gradient
gradient
gradient
vertex
A
concave
maximum
down
parabola
turning
has
a
A
concave
minimum
point.
=
ax
parabola
has
a
point.
2
2
y
up
turning
+
bx
+
c,
a
<
y
0
=
ax
+
bx
+
c,
a
>
0
maximum
turning
point
minimum
turning
point
ATL
Exploration
2
2
Here
is
the
graph
of
y
=
x
2x
3.
y
(
(3, 0)
1, 0)
0
x
2
y
=
x
−
2x
−
3
(1,
1
Describe
the
2
State
the
equation
3
State
the
coordinates
4
Describe
and
the
the
symmetry
of
of
the
of
of
axis
the
relationship
x-coordinate
the
parabola.
of
symmetry.
vertex.
between
the
4)
the
equation
of
the
axis
of
symmetry
vertex.
Continued
on
next
page
9.2 A parable about parabolas
325
5
State
the
x-intercepts.
x-intercepts
x
and
x
1
vertex
x
.
Determine
of
the
the
function
relationship
and
the
Graph
step
5
this
relationship
y
=
The
the
as
a
formula:
x
=
a
______.
the
following
works
for
quadratic
these
functions.
Test
that
your
4x
function
formula
the
points
the
graph
All
the
Now
5
b
quadratic
test
whose
your
y
=
functions
formula
leading
8
y
=
+
6x
7
c
y
=
−x
+
2x
+
10
in
from
coecient
a
step
step
≠
6
5
have
on
leading
the
Copy
coecient
following
a
=
quadratic
1
or
1.
functions
±1.
2
2x
+
and
4x
1
b
complete
x-coordinate
Quadratic
of
the
this
c
table
vertex
function
for
with
all
the
the
parameters
quadratic
Parameters
a
2x
and
a
+
5x
3
and
b
functions
b
and
in
the
steps
6
x-coordinate
the
vertex,
and
7
of
x
v
2
y
=
x
−
2x
−
3
a
=
1;
b
=
−2
x
=
1
v
Find
a
pattern
relating
x
to
a
and
b.
Write
this
relationship
as
a
formula.
v
9
10
Create your own quadratic functions, and test your formula in your examples.
Explain
how
to
nd
the
y-coordinate
of
the
vertex,
when
you
know
its
x-coordinate.
11
Factorize
between
Explain
12
the
the
Determine
quadratic
expressions
x-intercepts
why
this
how
and
the
relationship
to
nd
the
in
step
factors
of
6.
Describe
the
the
quadratic
relationship
expression.
holds.
y-intercept
of
any
quadratic
equation
of
the
2
form
y
=
Reect
●
What
Is
it
Do
do
you
main
+
c.
discuss
think
for
think
would
it
3
happens
to
it’s
it
still
if
have
possible
look
characteristics
●
x-intercepts
●
axis
●
vertex
of
you
bx
a
quadratic
function
x-intercepts?
If
so,
isn’t
how
factorizable?
would
you
them?
What
The
+
and
possible
nd
●
ax
and
like?
of
32 6
9 Space
for
a
quadratic
When
the
y-intercepts
symmetry
where
crosses
the
x-axis.
are
also
They
2
x
2
a
the
of
from
zeros
7
are
of
graphs.
2
x
x-intercepts
x-coordinates
v
2
a
the
of
2
Write
v
6
between
x-coordinate
do
graph
you
of
a
function
think
this
quadratic
to
have
would
no
x-intercepts?
happen?
function
are:
of
called
the
the
function
A LG E B R A
Properties
of
quadratic
functions
2
For
a
quadratic
function
●
the
x-coordinate
●
the
equation
●
the
coordinates
●
the
y-intercept
For
a
quadratic
of
of
its
of
is
the
f (x)
(0,
ax
vertex
axis
its
=
of
+
bx
c,
a
≠
0:
is
symmetry
vertex
+
is
are
c)
function
f (x)
with
x-intercepts
x
and
x
1
●
the
x-coordinate
of
the
vertex
is
●
the
y-coordinate
of
the
vertex
is
f (x
:
2
)
v
Practice
For
each
1
quadratic
function
in
questions
1
to
6:
Use
i
nd
the
coordinates
of
the
your
check
ii
nd
the
equation
of
its
axis
of
only
symmetry
your
after
worked
iii
determine
iv
nd
v
if
vi
draw
the
the
whether
function
is
concave
up
or
is
sketch
factorizable,
of
the
nd
quadratic
the
y
=
function
using
your
y
=
x
5
y
=
−2x
6
2
y
=
−x
4x
2
+
4
y
=
−8x
6
y
=
3x
2
20x
51
Write
a
concave
8
Write
a
quadratic
can
the
from
i
to
v
2x
4
+
16x
11
use
up
graphs
modelled
ball
at
6x
+
1
solving
7
hits
results
2
+
Problem
be
down
2
3
can
have
out.
2
x
x
2
You
concave
you
them
x-intercepts
2
1
to
results
y-intercept
function
a
the
GDC
vertex
a
as
quadratic
function
of
with
quadratic
parabolas.
height
of
1
function
axis
with
of
functions
For
x-intercepts
symmetry
to
example,
solve
during
x
=
2
3.
4.
real-world
a
and
baseball
problems
match
a
that
player
m.
9.2 A parable about parabolas
327
The
height
h
(meters)
of
the
ball
at
time
t
(seconds)
can
be
modelled
by
the
2
quadratic
function
maximum
ball
●
hits
height
the
=
the
−5t
ball
+
14t
+
reaches,
1.
You
and
can
how
graph
many
this
function
seconds
it
to
takes
nd
the
before
the
ground.
Represent
Time
h
time
cannot
on
be
a
the
x-axis,
negative
since
value,
so
time
t
≥
is
0.
the
At
t
independent
=
0,
the
variable.
height
of
the
ball
is
1
m.
The
time
depend
●
height,
≥
y
height
(1.4, 10.8)
(m)
2
=
−
5x
+
14x
+
0
x
time
hits
How
the
the
the
graph
ground
could
you
function
give
the
y
two
=
you
can
again
nd
8.
see
after
the
times:
0.65
the
2.87
times
Using
a
1
(2.87, 0)
(0, 1)
maximum
height
of
(s)
the
ball
is
10.8
m
and
it
seconds.
when
GDC,
seconds
the
the
and
ball’s
points
2.2
height
of
is
8
m?
intersection
seconds,
both
Add
of
to
the
rounded
your
two
to
1
graph
graphs
d.p.
y
height
y
=
(1.4, 10.8)
(m)
8
(2.2, 8)
(0.65, 8)
2
y
=
−
5x
+
14x
0
1
x
time
Practice
+
(s)
2
ATL
Graph
the
functions
reasonable
Problem
The
the
US
to
viewing
these
questions.
When
using
a
GDC,
rst
set
window.
solving
Food
number
answer
of
and
Drug
bacteria
Administration
B
in
food
uses
refrigerated
this
at
mathematical
temperature
T
model
for
(°C):
2
B
The
temperature
2°C
will
and
be
at
32 8
14°C.
a
in
a
=
20T
particular
Determine
minimum.
9 Space
the
−
20T
+
refrigerator
temperature
120
can
be
where
set
the
time
is
independent
variable.
y
From
so
0.
the
1
not
the
Represent height on the y-axis, since height is the dependent variable.
The ball will always be above the ground, or on the ground, so h
a
does
on
anywhere
number
between
of
bacteria
up
A LG E B R A
2
A
boy
time
throws
sees
it
a
fall
stone
into
into
the
the
air
while
water.The
standing
height
h (m)
on
of
a
the
cli
and
stone
at
after
any
some
time
t (s)
2
can
be
modelled
a
the
height
b
the
time
above
it
takes
maximum
c
3
the
A
time
tennis
by
=
the
takes
−4.8t
sea
height
it
ball
h
it
+
16t
+
45.
level
at
which
stone
to
reach
Determine:
the
its
stone
was
maximum
thrown
height,
and
the
reaches
the
stone
manufacturer
to
hit
the
estimates
water.
its
daily
costs
using
the
function
2
C (t)
of
200
tennis
the
4
=
10t
balls
where
0.114t
produced.
minimum
Sarah’s
+
C
is
Determine
the
the
cost
in
dollars,
number
of
and
tennis
t
is
balls
the
number
that
produce
cost.
banker
tells
her
that
the
value
of
her
investment
can
be
modelled
by
2
the
function
thousands
5
a
the
b
how
A
v (t)
of
initial
=
45
euros
+
after
amount
many
company’s
75t
it
where
months.
that
months
weekly
t
4t
Sarah
takes
prot
P
v
is
the
value
of
her
investment
in
Determine:
invested
for
Sarah’s
from
investment
selling
x
items
to
can
reach
be
maximum
modelled
value.
using
2
the
function
company
P (x)
needs
=
to
−0.48x
sell
Algebraic
C
●
How
●
What
of
a
can
for
38x
forms
you
are
+
the
295.
maximum
express
a
of
a
and
the
number
of
items
the
prot.
quadratic
quadratic
advantages
quadratic
Determine
weekly
function
in
disadvantages
function
three
of
the
dierent
dierent
ways?
forms
function?
2
The
quadratic
either
of
these
function
y
functions
=
x
x
gives
the
2
factorizes
same
to
y
=
(x
+
1)(x
2).
Graphing
parabola:
y
6
5
4
3
2
2
y
y
=
(x
+
1)(x
−
=
x
−
x
2
2)
1
0
4
3
2
1
x
1
2
3
4
1
2
3
The
factorized
form
of
a
quadratic
function
is
y
=
a(x
p)(x
q),
a
≠
0.
9.2 A parable about parabolas
32 9
Reect
●
The
and
x-intercepts
appropriate
●
How
do
of
the
of
a
4
function
are
also
called
its
zeros.
Why
is
this
an
name?
you
function
Most
discuss
determine
when
it
is
quadratics
in
the
coordinates
factorized
you
have
of
the
vertex
of
a
quadratic
form?
considered
so
far
have
had
two
distinct
2
zeros.
Now
consider
the
quadratic
function
y
=
x
+
2x
+
1,
which
factorizes
2
toy
=
(x
+
1)(x
+
1)
=
(x
+
1)
and
has
graph:
y
2
y
=
x
+
2x
+
1
6
5
4
3
2
1
0
5
4
3
2
x
1
1
2
2
The
one
function
unique
quadratic
Using
y
x
zero,
also
the
=
+
x
=
work
formula
2x
−1.
for
+
x
has
Does
a
only
the
one
the
( −1) +
unique
formula
quadratic
nding
x
1
function,
1
when
x
this
+
you
function
x-coordinate
factor,
(x
developed
has
of
only
the
+
1),
for
one
vertex
and
the
only
vertex
unique
using
of
a
factor?
the
zeros
of
( −1)
2
=
=
= −1
v
2
2
b
Using
the
formula
x
=
2
−
=
−
=
−1
v
2a
Both
formulae
work
Exploration
1
Graph
these
coordinates
equal
to
for
this
the
quadratic
the
functions
vertex.
Verify
2
y
Write
the
3
=
(x
Based
down
on
of
each
a
y
b
the
=
of
the
ndings
function
(x
+
(x
+
For
each,
state
x-coordinate
of
the
the
vertex
y
between
quadratic
in
below.
c
steps
Verify
1
=
−(x
the
2
1)
d
x-coordinate
y
of
b
y
=
and
your
(x
−(x
the
+
9 Space
1)
vertex
and
function.
2,
state
answers
the
by
coordinates
graphing
the
2
2)
=
4)
of
the
vertex
functions.
2
c
y
=
−(x
+
4)
Continued
3 3 0
is
2
3)
relationship
factor
your
y
2
=
the
2
2)
unique
separately.
that
x-intercept.
2
a
function.
3
quadratic
of
2 (1)
on
next
page
A LG E B R A
4
In
all
1.
the
quadratics
Explore
the
the
x-intercept
coecients.
in
steps
1
relationship
in
the
Write
to
quadratic
down
3
the
between
your
5
y
=
The
2(x
below
with
of
was
the
b
form
of
a
y
=
either
vertex
dierent
1
or
and
leading
ndings.
2
3)
general
coecient
x-coordinate
functions
2
a
leading
the
−3(x
quadratic
+
2
2)
c
function
with
y
=
only
−2(x
one
2)
unique
2
x-intercept
and
When
the
a
the
y
at
and,
=
a(x
x-intercept
quadratic
x-axis
factor
is
one
h)
in
therefore,
the
a
Write
terms
function
point,
.
has
of
repeated
to
the
y
=
(x
function
coordinates
of
the
vertex
the
zero
unique
graph.
at
this
solution,
The
its
function
graph
has
a
intercepts
repeated
point.
2
and
2)
one
of
2
Graphing
the
h
only
vertex
down
y
translates
=
(x
the
2)
+
graph
4
4
on
units
the
in
same
the
axes
shows
positive
y
that
adding
direction.
4
The
2
graph
of
y
=
(x
2)
+
4
has
vertex
(2,
4)
and
no
x-intercepts.
y
2
y
=
(x
−
2)
+
4
(2, 4)
2
y
=
(x
−
2)
0
x
(2, 0)
2
Graphing
y
subtracting
=
3
(x
2)
from
2
and
the
y
=
(x
2)
function
3
on
translates
the
the
same
graph
3
axes
shows
units
in
the
that
negative
2
y
direction.
The
graph
of
y
=
(x
2)
3
has
vertex
(2,
3)
and
two
x-intercepts.
y
2
y
=
(x
−
2)
2
y
=
(x
−
2)
−
3
(2, 0)
0
x
(2,
3)
2
The
quadratic
vertex
functions
y
=
(x
2)
2
+
4
and
y
=
(x
2)
3
are
written
in
form.
2
The
(h,
vertex
k)
is
the
form
of
a
quadratic
function
is
y
=
a
(x
h)
+
k,
a
≠
0,
where
vertex.
9.2 A parable about parabolas
3 31
Practice
For
1
to
6,
determine
3
state
if
the
the
coordinates
quadratic
is
of
the
concave
vertex
up
or
of
each
concave
quadratic
y
=
(x
3
y
=
2(x
5
y
=
−x
2
3)
+
1
2
y
=
−(x
4)
4
y
=
−3(x
6
y
=
3x
3
2
+
1)
+
2
2
1
2
A
the
Conver ting
To
2)
+
1
convert
1
solving
quadratic
Write
+
2
Problem
7
and
down.
2
1
function
function
function
quadratic
from
with
in
leading
vertex
coecient
2
has
a
repeated
zero
at
x
=
form.
forms
factorized
form
to
standard
form,
expand
the
brackets:
2
y
If
=
you
form
(x
can
to
+
3)(x
2)
factorize
factorized
→
a
y
=
x
+
quadratic
x
6
function,
you
can
convert
it
from
standard
form:
2
y
To
=
x
x
convert
2
from
→
y
vertex
=
(x
+
form
1)(x
to
2)
standard
form,
expand
and
simplify:
2
y
+
=
(x
1)
=
(x
1)(x
=
x
=
x
2
1)
+
2
2
2x
+
1
+
2
2
⇒
y
How
do
2x
you
Exploration
+
3
convert
will
Exploration
1
Expand
from
help
these
you
standard
answer
form
this
(x
+
(x
b
(x
at
e
your
relationships
3
For
these
into
two
linear
results
in
between
quadratic
identical
factor.
(x
step
p
+
c
x
c
x
f
Write
In
b,
functions,
factors
(x
down
+
1)
+
the
(x
and
+
+
x
+
p
values
p),
2)
2
=
between
choose
p)(x
p)
(x
in
bx
of
other
factorization
+
and
for
c
c,
nd
the
c
so
that
words:
each
they
the
factorize
square
of
a
one.
2
4x
+
c
b
x
d
x
2
4
(x
2
3)
1.
and
2
a
next
2
3)
2
1)
Look
(The
2
2)
2
2
form?
expressions:
2
d
vertex
4
2
a
to
question.)
+
4x
+
c
2
+
Write
x
a
+
c
general
rule
for
what
x
+
c
you
did
in
step
3.
For
a
quadratic
function
2
x
+
bx
linear
3 32
+
c,
express
c
in
factor.
9 Space
terms
of
b
for
the
quadratic
to
be
the
square
of
a
3.
A LG E B R A
In
steps
rst
3
two
and
4
terms
of
of
Exploration
a
quadratic
4
you
were
expression,
completing
you
found
c
the
so
square.
that
the
Given
the
quadratic
2
factorizes
into
coecient
You
can
of
use
standard
the
the
square
x
Example
to
a
linear
factor.
For
this,
c

b 


=
,
2
where
b
is
the
term.
completing
form
of
vertex
the
square
to
convert
a
quadratic
function
from
form.
1
2
Write
y
=
x
+
2x
2
in
vertex
form.
2
y
=
x
+
bx
+
c
2
=
=
2
1
=
1
2
x
Use
to complete the square for the x
and x terms.
2
+
2x
+
1
=
(x
+
1)
Write
the
completed
square
in
the
right
hand
2
y
=
x
+
2x
2
side.
As
this
adds
1,
subtract
1
at
the
end.
2
=
(x
=
(x
+
2x
+
1)
2
1
2
y
+
1)
You
could
check
and
simplifying.
by
expanding
3
Example
2
2
Sketch
the
Vertex
is
(
graph
1,
of
y
=
(x
+
1)
3.
3)
2
For
Axis
of
symmetry
2
y
=
(x
+
1)
is
x
=
y
=
h)
+
k
the
vertex
is
(h,
k).
−1
2
–
3
=
=
−2
x
+
2x
2
In
y-intercept
a(x
standard
form,
y-intercept
=
c
2
Graph
is
concave
up.
Positive
Find
2
When
x
=
1,
y
=
3
2
=
some
other
points
coecient
on
the
of
curve
x
.
by
1
substituting
a
few
values
for
x
2
When
x
=
−3,
y
=
(
2)
3
=
1
y
Connect
0
x
a
the
smooth
points
parabolic
with
curve.
2
(
1,
3)
9.2 A parable about parabolas
3 3 3
Reect
Consider
and
the
discuss
two
equivalent
5
quadratic
functions:
2
y
●
=
2
x
+
2x
Which
2
form
information
●
In
is
●
How
the
use
the
What
many
Practice
The
2,
standard
is
form
quadratic
in
y
=
function
sketching
y-intercept
another
and
its
of
+
gives
1)
3
you
in
vertex
more
form.
immediate
graph?
was
way
(x
found
by
nding
expanding
the
the
y-intercept?
original
Which
way
easier?
function
1
of
to
Example
function.
in
points
or
draw
are
its
necessary
graph?
to
determine
a
unique
quadratic
Explain.
4
quadratic
functions
below
are
given
in
standard
form.
Tip
i
Convert
ii
Sketch
each
function
into
vertex
form.
Find
the
graph
of
the
quadratic
by
using
the
vertex
and
two
and
points
on
the
y
=
2
+
x
2x
+
1
b
y
=
y
=
−x
g
y
=
x
+
1
e
y
=
x
h
y
=
x
2
2
For
the
4x
x
+
2
quadratics
nd
the
ii
nd
two
iii
sketch
=
vertex
other
the
1
Match
using
the
points
of
on
the
formula
y
=
the
y
=
f
y
=
i
y
=
+
6x
+
x
+
−x
quadratic
3
1
2x
+
x
value
nd
third
1
3x
a
2
1
x
the
x-coordinate
of
the
vertex
using
the
points
you
have
found.
2
x
b
quadratic
1)
y
=
2
2x
4x
+
1
c
y
=
1
c
y
=
(x
6x
3x
function
with
its
graph.
2
3
b
y
=
x
2
+
2
2)
2
+
‘easy’
quadratic
2
d
3
solving
each
(x
+
x
for
2
a
=
below:
graph
4x
Problem
3
y
2
2x
2
y
c
2
+
i
a
2
2
6x
2
e
y
=
x
+
1
2
3x
1
f
y
=
2
x
x
y
y
y
3
2
2
1
1
1
x
0
2
1
1
2
3
4
0
5
4
1
3
2
1
3
2
1
x
1
2
3
4
2
1
2
3
2
3
4
3
4
9 Space
x
1
1
0
3 34
an
of
x
to
2
x
2
d
y-intercept,
substitute
quadratic.
2
a
the
other
2
point.
A LG E B R A
y
y
3
1
y
5
5
4
4
3
3
2
0
4
3
2
x
1
1
1
1
2
0
4
3
2
Reect
and
discuss
a
quadratic
function
that
●
Find
a
quadratic
function
whose
●
Find
a
quadratic
function
●
1,
Compare
2
0
3
2
x
1
1
2
3
4
5
1
6
Find
(
1
1
●
zeros
x
1
3
with
goes
through
zeros
vertex
are
(1,
x
2)
the
point
=
and
0
and
one
(0,
x
=
of
1).
4.
its
3).
your
answers
with
others.
Did
you
all
nd
the
same
functions?
Explain.
Solving
To
solve
from
to
real-life
real-life
the
use
given
to
a
b
a
have
Find
=
2(l
+
the
and
may
then
have
to
decide
derive
on
the
the
quadratic
best
form
of
function
the
quadratic
question.
meters
plot’s
Determine
P
you
3
100
the
problems,
information,
answer
Example
You
problems
w)
the
=
of
fencing
maximum
enclose
possible
dimensions
100
to
a
rectangular
plot.
area.
dimensions
give
the
maximum
area.
[1]
Write
A
=
lw
From
2(l
+
and
[2]
equations
area
of
a
for
w)
+
=
100
w
=
50
l
=
50
=
(50
l
in
terms
of
w)w
=
50w
Substitute
by
the
into
[2],
to
get
w
equation
linking
A
and
w
2
Area
a
is
given
maximum
w
w
2
A
perimeter
[1]:
Write
l
the
rectangle.
value
quadratic
at
its
function
50w
w
,
which
has
vertex.
Continued
on
next
page
9.2 A parable about parabolas
3 3 5
an
Method
1–nding
the
vertex
using
the
factorized
form
2
50w
w
=
w
0,
=
w
=
w(50
–
w)
=
0
Find
the
zeros
of
the
function.
50
A
maximum
Sketch
and
the
graph
identify
its
of
the
function
maximum
point.
The maximum area of the
0
w
25
rectangular plot occurs when w = 25.
When
A
=
w
=
25:
25(50
25)
=
625
Calculate
the
maximum
area.
2
Maximum
Method
area
=
625
2–nding
m
the
vertex
using
the
standard
form
2
A
=
w (50
w)
=
50w
w
Find
When
A
=
w
=
the
x-coordinate
of
the
vertex.
25:
25(50
25)
=
625
area
=
625
2
Maximum
Method
3–nding
2
m
the
vertex
using
vertex
form
2
w
50w
=
(w
25)
2
625
Complete
2
Hence
A
Vertex
=
=
+
−(w
(25,
2
50w)
=
−(w
25)
+
625
625)
2
Maximum
b
When
w
=
maximum
Reect
When
and
area
25,
l
area
and
sketching
=
625
=
50
are
standard
●
factorized
●
vertex
3 3 6
w
=
width
graphs
of
of:
form
form
form?
9 Space
25.
=
discuss
disadvantages
●
m
The
length
dimensions
=
25
m
(a
that
produce
the
square).
7
quadratic
functions,
what
are
the
advantages
the
square
for
w
50w.
A LG E B R A
Practice
In
these
use
the
5
questions,
most
Problem
1
You
side
have
of
a
derive
ecient
the
quadratic
method
for
function
answering
the
that
best
ts
the
situation
and
question.
solving
300
m
river,
of
as
fencing
to
enclose
a
rectangular
plot
along
the
shown.
River
Plot
Determine
of
2
the
You
the
have
a
rectangular
3
total
of
plots.
enclosed
is
Xixi
wants
fence
as
possible
The
that
use
to
one
exercise
and
a
would
these
square
as
side
perimeter
sides
area
of
200
m
of
that
you
fencing
Determine
area
house
4
maximum
can
enclose
and
the
dimensions
plot.
big
as
o
of
a
the
the
to
make
dimensions
two
of
equal,
each
adjacent,
plot,
such
that
the
total
possible.
rectangular
area.
She
exercise
has
10
m
area
of
for
her
fencing.
dog,
Find
using
the
the
maximum
area.
an
athletics
semicircle
at
maximize
dimensions
the
to
track
each
end.
area
nd
of
the
is
0.4
km.
The
Determine
the
rectangular
area
of
the
track
the
exact
part
entire
of
track
has
two
values
the
parallel
of
track
eld
to
x
and
eld,
the
r
and
nearest
meter.
X
r
Form
D
●
Given
a
What
quadratic
y-intercepts
quadratic
and
and
makes
function
one
equation,
vertex.
If
quadratic
you
you
can
start
form
sketch
with
a
better
its
than
graph
parabola,
by
another?
nding
how
can
its
you
x-
and
nd
its
function?
9.2 A parable about parabolas
3 37
This
photograph
Consider
The
the
x-intercepts
enough
of
water
parabolic
and
information
fountain
fountains
shape
of
the
y-intercept,
to
nd
the
is
superimposed
2nd
which
fountain
is
quadratic
also
upon
stream
the
vertex,
function
coordinate
from
are
modelling
axes.
left.
labelled.
the
Is
this
parabolic
stream?
Tip
Some
and
GDCs
graphing
software
have
functionality
the
of
superimposing
picture
on
coordinate
2
The
general
form
Substituting
the
of
a
quadratic
coordinates
2
y
=
a(x
How
the
(
+
you
parabola
4.4,
0)
or
the
vertex
with
(0,
vertex
7)
for
h
is
(h,
k)
is
y
=
a(x
and
k
gives
have
a
negative
h)
7
or
nd
is
y
=
the
ax
×
4.4
+
7
0).
coecient
down.
You
Substituting
can
the
x-
a?
use
and
It
must
one
of
the
y-values
at
two
other
point
(4.4,
value
points
0)
since
given:
gives
=
0.
Solving
for
a
gives:
7
a
=
=
−0.361
(3
d.p.)
2
4.4
2
The
You
quadratic
can
function
graph
3 3 8
k
7.
leading
concave
(4.4,
+
2
a
+
2
0)
can
of
function
this
that
models
quadratic
9 Space
on
the
the
fountain
same
stream
coordinate
is
axes
y
=
to
−0.361x
check:
+
7.
a
a
pair
axes.
of
A LG E B R A
ATL
Reect
●
and
Again,
nd
stream,
but
coecient
●
Does
is
●
the
this
a.
your
an
What
a
●
it
discuss
Is
minimum
When
nding
standard
ii
vertex
iii
factorized
and
basket?
A
point
in
space
A
line
one
A
curve
Can
has
has
you
object
in
two
a
to
models
nd
the
the
fountain
leading
function
exactly
model
the
parabolic
fountain,
or
Explain.
do
you
need
to
determine
the
equation
of
equation
you
of
a
quadratic
function,
under
what
use:
form
to
form?
1
asked:
know
to
answer
has
no
How
be
is
how
many
sure
and
many
space?
that
positions
the
ball
do
you
would
think
go
into
the
‘three’.
dimension,
dimensions
three-dimensional
an
that
function?
dimension
guess
quadratic
x-intercept
function?
model?
would
correct
it.
same
the
other
information
the
discuss
determine
of
the
form
need
The
the
quadratic
i
would
it
form
use
quadratic
circumstances
you
time
approximate
unique
Reect
general
8
and
needs
and
four-dimensional
two
needs
points
How
needs
points
three
determine
many
one
points
a
to
point
to
determine
points
to
unique
would
it.
determine
object
you
it.
in
need
to
uniquely
dene
universe?
You
3D
a
Exploration
5
2D
as
can
project
objects
onto
space,
with
such
television
transmission.
The
Golden
spanning
The
Gate
the
diagram
Bridge
channel
shows
in
San
between
the
Francisco,
San
central
USA
Francisco
section
of
the
is
Bay
a
suspension
and
bridge.
the
The
Pacic
the
same
a
way,
4D
objects
Ocean.
suspension
can
be
cable
projected
is
In
bridge
onto
a
parabola.
3D
space.
y
x
Continued
on
next
page
9.2 A parable about parabolas
3 3 9
1
Research
2
Trace
the
3
for
4
the
axis
Use
the
the
Using
the
heights
parabola.
of
and
your
y
the
Draw
symmetry
dimensions
x
of
of
towers
a
the
you
pair
and
of
the
length
coordinate
of
axes,
the
central
with
the
y
section.
axis
as
parabola.
found
from
step
1
to
determine
a
suitable
scale
axes.
scale,
nd
the
coordinates
of
the
vertex,
and
the
x-
and
y-intercepts.
5
Using
your
parameter
6
Compare
Practice
1
Trace
Find
your
function
and
another
point
on
the
curve,
nd
the
model
with
others.
Discuss
any
similarities
and
dierences.
6
each
a
quadratic
a
parabola.
quadratic
Draw
function
coordinate
to
model
the
axes
and
decide
on
a
suitable
scale.
parabola.
a
b
c
d
Exploration
Take
photos
technology
tracing
paper
Reect
Why
of
to
34 0
parabolic
it
objects
nd
the
quadratic
over
the
photo.
and
might
6
be
discuss
that
so
encounter
that
in
your
denes
everyday
your
life.
object,
or
Use
use
9
many
9 Space
you
function
natural
and
built
structures
are
parabolic?
A LG E B R A
Objective:
v.
Justify
real-life
In
this
will
Applying
whether
a
mathematics
solution
makes
in
real-life
sense
in
the
contexts
context
of
the
authentic
situation.
activity
then
your
D.
you
decide
will
which
write
one
is
quadratic
better
in
models
that
describing
the
describe
given
the
given
scenarios,
situation.
and
You
explain
choice.
Massachusetts
Activity
physical
For
a
free
throw
shot
in
basketball
(in
US
standard
teacher
●
●
●
education
units):
James
Naismith
the
the
the
free
throw
basket
is
player
line
10
ft
is
15
ft
above
releases
the
away
the
ball
from
the
foot
of
the
basketball
in
the
late
to
keep
ground
from
an
approximate
height
of
8
his
ft
1800s
class
indoors
occupied
in
weather.
peach
ball
1
Find
two
Make
2
Decide
for
3
life
4
5
the
your
the
ball,
of
these
intercept
the
the
not
might
can
ball
that
could
satises
quadratic
reasons
select
the
model
the
your
one
shot
into
the
basket
like
a
used
and
soccer
that
that
choice
you
is
most
of
basket.
appropriate
form.
think
best
models
a
take
the
into
account
shot
goes
real-
the
factors
basket.
that
aect
Suggest
the
what
path
some
be.
opposing
that
jump
in
some
into
an
to
your
reaching
team
could
opposing
reach
a
model?
the
intercept
player
maximum
If
so,
is
of
modify
5
the
ft
10
ball
away
ft.
your
on
its
from
Could
the
this
function
way
so
to
free
player
that
it
basket.
9.2 A parable about parabolas
a
a
ball.
specications.
function
for
the
bad
He
why.
the
ball
the
give
models,
Assume
and
of
whether
from
basket.
thrower
function
and
does
factors
player
models
form
two
or
functions
each
Explain
model
of
the
that
models,
shot.
Your
A
on
your
From
dierent
sure
invented
basket
341
Summary
●
The
standard
form
of
a
quadratic
function
is
●
A
concave
up
parabola
has
a
2
y
=
ax
+
bx
+
c
where
a,
b
and
c
are
real
minimum
turning
point.
2
numbers,
and
a
≠
0.
y
=
ax
+
bx
+
c,
a
>
0
minimum
turning
●
The
y
●
=
factorized
a(x
The
form
p)(x
vertex
q),
form
a
of
of
≠
a
a
quadratic
function
point
is
2
0.
●
quadratic
function
For
is
a
quadratic
function
the
x-coordinate
the
equation
of
f
(x)
the
=
ax
vertex
+
bx
+
c,
a
≠
0:
is
°
2
y
=
a(x
h)
+
k,
a
≠
0,
where
(h,
k)
is
the
vertex.
of
its
axis
of
symmetry
°
●
The
degree
of
a
polynomial
function
is
the
is
value
of
its
function
A
of
●
A
is
largest
a
quadratic
degree
turning
polynomial
function
2.
concave
exponent
A
is
down
x.
A
function
a
constant
of
linear
of
degree
polynomial
function
parabola
has
a
has
1.
the
coordinates
the
y-intercept
of
its
vertex
are
°
function
degree
0.
is
c
°
maximum
●
point.
For
a
quadratic
function
f (x)
with
x-intercepts
2
y
=
ax
+
bx
+
c,
a
<
x
0
and
x
1
:
2
the
x-coordinate
of
the
vertex
is
x
°
v
maximum
turning
point
the
y-coordinate
of
the
vertex
is
°
Mixed
1
For
practice
each
quadratic
function:
a
Express
terms
i
nd
the
ii
state
vertex,
x-intercepts
and
y-intercept
and
whether
b
the
axis
of
symmetry,
it
up
or
concave
sketch
a
y
the
total
Find
x
12
b
y
=
x
+
7x
+
=
x
3
d
y
=
2
x
These
−6x
Soraya
5x
1
f
y
=
2x
9x
each
the
maximum
has
40
functions
are
in
standard
one
to
vertex
form,
and
of
the
c
y
=
x
+
6
b
y
=
x
The
x
2x
Kanye
fencing
Find
to
make
the
a
dimensions
+
has
state
by
P
of
the
the
an
animal
species
function
2
=
−0.38t
time
in
+
134t
months
+
1100,
since
where
the
t
is
population
9
d
y
=
x
+
6x
+
8
2x
+
7
observed.
was
x
5
f
y
=
x
x
+
the
7
at
number
its
of
months
until
the
population
maximum
b
the
maximum
population
c
the
number
months
solving
600
m
of
fencing
to
make
of
before
the
species
ve
pens
like
this:
6
A
5
ball
m
the
x
It
is
lands
9 Space
and
the
on
the
this
vertex
up
into
ground.
reaches
thrown.
models
y
thrown
above
ball
was
34 2
rst
Determine:
disappears.
adjacent
for
area.
2
Problem
3
play
population
modelled
a
=
of
area.
2
+
2
y
area.
2
4x
2
e
maximize
vertex.
2
x
will
form.
the
=
m
play
maximum
P (t)
coordinates
y
that
5
is
a
x
2
+
quadratic
Change
of
3x
5
2
value
2
2x
2
y
the
12
the
e
in
2
+
x
2
=
pens
area.
rectangular
y
the
graph.
4
c
of
down
2
=
area
x
Determine
c
iii
of
total
is
the
concave
the
a
a
5
2
from
seconds,
height
seconds
quadratic
situation.
form.
air,
After
maximum
ground
Find
the
Write
it
of
9
after
it
function
in
m.
that
standard
is
A LG E B R A
Review
in
Scientic
context
and
technical
innovation
d
The
area
of
is
of
,
the
a
region
where
parabola,
distance
from
b
formed
is
and
its
the
h
is
vertex
by
a
length
its
to
parabola
of
the
height,
the
a
base
i.e.,
Because
truck
enough
the
fully.
base.
and
of
1
This
rollercoaster
is
in
Münich,
Germany.
e
and
that
of
the
outline
of
of
foreground
the
parabolic
truck
section
of
Draw
axes
and
use
the
c
Find
a
d
Use
this
2
At
a
If
the
is
with
center
0.2
m
can
width
in
the
line.
0.8
open
m
to
of
open
2.6
right
The
below
opens
door
a
stuck
need
door
the
to
top
top
the
fully,
of
map
the
function
to
of
of
to
is
the
Hill
the
same
in
to
determine
model
estimate
tunnel
maximum
as
entrance
Budapest,
side.
steps.
how
close
be
in
passenger
to
d
the
be
side
and
of
still
the
be
tunnel
able
its
determine
maximum
a
of
the
that
the
width,
quadratic
Describe
some
architects
issues/concerns
take
into
that
account
you
when
a
a
tunnel
before
its
construction.
the
shape
of
area
under
runs
through
tunnel
9.8
model
to
is
the
meters,
represent
thetunnel.
b
State
a
geometric
represent
c
If
a
a
typical
determine
be
put
travel
on
truck
truck
the
show
through
all
of
and
How
you
have
a
that
the
width
if
of
they
tunnel
you
steps
could
through
limitation
trucks
that
the
Reect
has
height
these
assumptions
model
passing
in
your
be
the
2.62
that
are
safely.
made
and
to
m,
should
going
State
be
used
tunnel.
to
any
sure
to
solution.
discuss
explored
the
statement
of
inquiry?
Give
specic
examples.
Statement of Inquiry:
Representing
models
and
patterns
with
to
door.
Hungary.
height
the
showing
track.
function
m
lane
section.
right
Castle
is
meters,
would
passenger
truck
m
350
tunnel
scale.
suitable
section
the
door
the
could
modelling
suitable
the
almost
the
track.
think
b
the
by
is
section
f
the
your
a
3.4
door
Determine
open
Trace
tunnel
driver’s
Show
the
a
of
in
for
Suppose
truck,
all
length
room
height
the
the
its
stopped
equivalent
forms
can
lead
to
better
systems,
methods.
9.2 A parable about parabolas
34 3
10
Change
A variation in size, amount or behavior
Changing
One
strategy
problems
can’t
you
is
solve
can
the
for
to
solving
change
into
problem
Changing
mathematics
them
equivalent
from
ones
problems
you
a
that
fractions,
changes
the
decimals
way
you
solve.
example,
fractions
you
with
cannot
different
3
directly
can
you
calculations
add
denominators,
say:
out
change
to
make
the
numbers
them
easier
in
to
mentally?
2
7
you
can
equivalent
change
form
5
this
which
15
Changing
A
formula
calculate
speed
●
=
the
a
b
c
distance
else
=
=
can
calculate
the
speed
34 4
of
a
car
or
50km,
426
+
99
65%
of
20
solve:
50km,
s
by
=
substituting
d/t
time
time
time
when
and/or
between
when:
80km,
=
=
=
2
2
4
hours
hours
hours.
you
change
time?
change
distance,
and
24
subject
speed
you
to
×
35
relationship
changes
distance
How
to
=
distance
What
●
the
distance
the
the
distance/time
Calculate
possible
0.75
an
29
35
speed
into
=
the
shows
is
14
+
35
addition
time?
the
formula
given
quantities.
values
into
You
the
can
formula:
14
÷
these
work
+
But
and
express
number.
How
For
between
percentages
0.07
Evolutionary
The
theory
over
time.
presented
voyage
In
adapted
On
the
He
to
now
the
suited
reproduce.
to
Individuals
that
a
that
had
are
collected
these
pass
genes
a
in
the
most
their
are
less
likely
have
of
individuals
to
to
but
selection.
genes
likely
genes
and
origins
environment,
are
Darwin
ve-year
natural
in
change
plants
common
through
any
on
Charles
during
differences
In
species
variation
have
environment
then
living
(1859),
studying
species.
will
that
Species
species
there
without
in
he
how
environments
that
They
of
world,
claimed
within
describes
Origin
evidence
know
reproduce
the
different
individuals
most
evolution
around
animals.
We
of
change
survive
their
survive
and
offspring.
and
environment.
A
girae
can
with
reach
giraes
a
food
longer
that
neck
other
cannot.
A
The
girae’s
girae
than
inherit
the
long
Climate
ice
changed
periods,
in
the
debate
–
rate
than
is
Scientists
help
us
of
know
years,
exact
all
causes
that
from
global
that
between
10 000
agree
atmosphere
food
other
to
giraes
sur vive
is
and
more
reproduce.
the
ice
years
of
ages
and
amount
burning
fossil
temperatures
at
warmer
them.
interglacial
of
carbon
fuels
a
climatehas
with
between
glacial
the
Earth’s
–
present
oil,
coal
faster
before.
analyze
How
to
we
about
increasing
ever
change.
of
the
almost
Earth’s
gas
data
millions
periods
but
and
●
sample
over
Scientists
more
change
core
interglacial
gets
neck.
likely
From
that
ospring
can
make
data
to
observe
predicting
and
change
predict
also
changes?
34 5
A
frog
into
a
prince
10.1
Global
context:
Objectives
●
Understanding
Orientation
Inquiry
how
various
parameters
●
aect
in
space
and
time
questions
What
transformations
can
be
applied
to
F
the
●
shape
Applying
and
position
translations,
of
a
graph
reections
lines
and
MROF
●
What
are
linear
and
Describing
the
transformation
of
a
and
Describing
combinations
of
transformations
How
algebraically
and
graphically
the
can
If
equation
of
a
graph
following
the
graph
use
of
transformations
a
graph?
to
nd
of
a
transformed
function
the
same
as
the
graph
of
the
original
one
function,
more
you
equation
D
is
or
for
graphs?
of
●
function
Writing
transformations
graphically
the
●
three
quadratic
function
●
a
the
C
graphs
algebraically
●
curves?
dilations
●
to
and
has
the
function
been
transformations
transformed?
●
If
two
graph
are
ATL
dierent
of
the
the
transformations
same
transformed
transformations
really
give
the
function,
dierent?
Creative -thinking
Generating
novel
ideas
and
considering
new
perspectives
10.1
14.1
Statement of Inquiry:
Relationships
patterns
of
that
can
help
and
predict
frequency
34 6
model
change
clarify
duration,
and
variability.
A LG E B R A
Y
ou
●
should
sketch
already
graphs
of
linear
know
and
how
1
to:
Sketch
a
graph
of :
2
quadratic
●
write
functions
quadratic
functions
in
a
f(x)
a
Write
2
=
2x
+
the
3
b
quadratic
f(x)
=
x
function
2
ver tex
form
y
b
=
x
+
Write
6x
+
8
down
in
the
ver tex
form.
coordinates
of
the
2
ver tex
Changing
F
●
What
Changing
the
the
the
graph
of
y
=
x
+
6x
+
8.
graphs
transformations
parameters
coordinate
of
of
a
can
line
or
be
applied
quadratic
to
lines
curve
and
changes
curves?
its
position
on
plane.
For
a
straight
line
For
a
quadratic
y
=
mx
+
c,
the
y
=
(x
parameters
are
gradient
m
and
y-intercept
c.
2
function,
Exploration
1
Use
your
given
must
be
In
y
=
y
=
and
x-values
the
to
to
draw
the
parameters
match
y
of
=
3
x
to
13x
following
briey
the
are
h
and
k
x-values
54x
of
2
2x
2x
pairs
how
of
curves
the
on
the
rst
curve
have
used
curve.
1.
3
and
y-values
72
and
y
and
y-values
to
words
3
+
of
3
+
in
second
2
x
Use
k,
describe
4
2x
3
b
case,
3
x
Use
software
each
transformed
4
a
+
1
graphing
axes.
h)
8
=
3
to
5.
2
x
4x
+
3x
of
4
of
20
of
1
+
to
2.
4.
2
c
y
=
x
+
x
6
Use
x-values
Use
x-values
and
of
y
=
3(x +
3)(x
4
to
3
and
4
to
4
and
and
d
2).
y-values
y
=
y-values
and
x-values
of
x
0
to
4
f
2
When
x-values
words
terminology
Research
three
of
describing
everyday
4.
and
.
y-values
of
6
to
and
Use
to
2
x
Use
6.
.
of
2
e
to
the
have
to
10
the
like
to
.
10
and
y-values
transformations
‘shift’
make
sure
meanings
of
6.
and
your
the
non-mathematical
in
step
‘stretch’.
It
meaning
words
of
is
is
6
1,
6.
you
might
important
fully
reection,
meanings
to
to
use
the
correct
understood.
dilation
as
well
as
from
step
1
and
specic
translation.
All
mathematical
meanings.
3
Describe
each
reection,
of
the
dilation
transformations
or
using
one
of
the
terms
translation.
10.1 A frog into a prince
34 7
In
this
the
section,
reections
will
A
be
measured
translation
amount
A
in
factor.
to
by
Practice
curve
but
or
when
it
or
all
is
same
these
either
three
the
types
x-axis
or
of
the
transformation.
y-axis.
The
All
dilations
y-axis.
every
point
on
a
graph
moves
by
the
same
a
x-values
dilated
scale
points
on
compression.
the
parallel
a
graph
increased
to
the
undergoes
by
y-axis,
the
all
dilation
same
the
scale
y-values
are
factor.
on
the
If
are
one
other
side
side
of
of
a
mirror
the
line
mirror
to
points
the
same
line.
1
diagrams
describe
if
x-
only
in
direction.
stretch
maps
away
the
x-axis,
the
reection
distance
These
same
a
encounter
reections
from
Similarly,
increased
1
is
the
will
be
occurs
the
dilation
parallel
A
you
will
the
into
show
single
the
examples
of
transformation
blue
graph
that
transformations.
has
taken
place
to
In
each
case,
transform
the
red
curve.
y
a
4
3
2
1
0
3
2
1
x
1
2
3
4
5
6
1
2
3
4
y
b
4
3
2
x
0
9
8
7
6
5
4
2
1
1
2
3
4
5
6
7
8
9
1
2
3
4
c
d
y
y
7
4
6
3
5
2
4
1
3
0
3
2
x
1
2
1
1
2
3
0
5
4
3
2
1
2
34 8
10 Change
x
1
1
4
2
3
4
5
A LG E B R A
e
y
8
6
4
2
0
4
4.5
3.5
3
2.5
2
1.5
1
0.5
x
0.5
1
1.5
2
2.5
3
3.5
4
4.5
2
4
6
8
y
f
6
5
4
3
2
1
0
4
3
2
x
1
1
2
3
4
5
1
2
3
2
These
case,
the
diagrams
identify
blue
show
the
two
further
examples
transformations
of
graph
which
transformations.
transform
the
red
In
curve
each
into
curve.
y
y
2
4
3
2
0
4
3
2
1
x
1
2
3
4
1
1
2
0
4
3
2
1
x
1
3
1
4
2
5
3
2
3
4
5
4
y
y
5
6
4
5
3
4
2
3
1
2
1
0
6
5
4
3
2
1
x
1
2
3
4
5
6
7
1
0
4
2
3
2
x
1
1
2
10.1 A frog into a prince
34 9
Problem
solving
2
3
Use
your
Ronnie
Reggie
GDC
says
says
Decide
what
Reect
the
Using
y
=
+
Predict
Check
Do
y
will
also
f (x
dierent
of
How
more
exploration,
aects
as
sense.
If
the
y
k
x
=
are
where
f (x)
of
of
2
2)
and
the
the
y
=
(x +
2)
other.
other.
correct.
f (x)
of
y
=
=
1,
f (x
4x
predict
+
calculator.
+
4
1)
what
will
What
represent
the
+
quadratic
transforming
as
the
graph
of
look
do
like.
you
same
notice?
transformation
or
are
as
it
does
not,
Once
it
equation
aect
its
graph?
is
a
to
make
you
good
observed
you
idea
you
with
observations
have
Then
perhaps
consistent
should
to
need
ask
to
try
about
the
how
pattern,
check
a
ndings
your
friend
if
rule
they
rephrasing
varying
you
a
should
by
try
testing
think
your
a
to
few
rule
it.
2
linear
on
r ules
asked
possible.
Sometimes
an
transformations
patter ns
general
graph.
precisely
Graph
=
(x
translation
reection
Exploration
graph
y
you
the
Exploration
1
f (x)
Investigating
examples.
makes
=
like.
does
patter ns
it
=
and
B.
describe
describe
a
a
y
1
graphical
ii.
this
claims
to
the
Objective:
parameter
is
is
of
transformations?
●
In
y
and
Linear
C
graphs
notice.
look
your
1)
graph
discuss
what
+
the
graph
their
answers
using
=
one
you
graph
4
draw
one
and
your
f (x)
that
that
whether
Explain
Draw
to
functions
your
graphing
y
=
x
and
1.1
software,
y
and
insert
of
between
k
Try
a
slider
dierent
10
to
change
and
values
of
10.
k
the
Start
to
value
with
explore
k
y
=
=
x
0.
how
y
=
x
+
k
(1, 1)
changing
y
=
x
to
y
=
x
+
k
values
graph.
k
2
Repeat
step
functions:
and
y
=
3x
Describe
y
=
f (x)
y
+
+
1
for
=
2x
k,
y
more
=
−x
linear
+
0
10
changing
changes
the
y
=
f (x)
to
graph.
Continued
3 5 0
10
k
k
how
k
three
+
10 Change
dierent
x
0
the
Try
changes
on
next
page
of
k
A LG E B R A
3
Graph
to
Try
y
4
the
change
=
dierent
4(x
h)
Repeat
y
=
y
=
h
values
of
h
changes
3
for
+
3,
3(x
Describe
y
of
h)
=
=
to
the
three
y
y
=
4x
and
between
10
explore
y
=
4(x
and
how
h).
+10.
Insert
Start
changing
y
a
with
=
4x
slider
h
=
0.
to
graph.
more
(x
linear
h)
functions:
1.1
2
y
h)
how
f (x
functions
value
step
2(x
and
to
linear
the
h)
changing
changes
y
=
the
f (x)
graph.
2
y
2
5
Graph
the
quadratic
functions
y
=
x
=
x
and
2
y
=
(x −
h)
+
k
2
y
=
(x
h)
+
k
on
your
graphing
software,
0
and
insert
two
sliders
of
and
between
to
change
the
values
h
h
k
10
and
10.
x
Start
0
k
0
with
10
10
10
10
2
h
6
=
k
=
Find
0,
the
so
y
=
x
values
.
of
h
and
k
and
write
the
equation
the
left
of
the
parabola
2
y
=
(x
+
h)
k
that
is:
2
a
4
units
higher
b
4
units
to
c
both
a
Explain
than
y
=
x
2
the
left
of
y
=
x
2
7
its
b
4
units
Vertical
how
and
4
units
to
of
y
=
x
changing
the
function
y
=
f (x)
to
y
=
f (x
changing
the
function
y
=
f (x)
to
y
=
f (x)
h)
changes
graph.
Explain
its
higher
how
+
k
changes
graph.
and
horizontal
shifts
in
the
graph
of
a
function
are
called
translations
●
●
●
y
=
f (x
translates
y
=
f (x)
by
h
units
in
the
When
h
>
0,
the
graph
moves
in
the
positive
When
h
<
0,
the
graph
moves
in
the
negative
y
=
f (x)
+
k
translates
y
=
f (x)
by
k
units
in
>
0,
the
graph
moves
in
the
positive
When
k
<
0,
the
graph
moves
in
the
negative
=
f (x
Reect
When
the
h
>
h)
in
+
the
and
0,
right
k
y
=
f (x)
by
h
units
to
right.
the
left.
y-direction.
y-direction,
y-direction,
in
the
the
up.
down.
x-direction
and
y-direction.
discuss
why
and
translates
to
x-direction,
the
k
y
x-direction.
x-direction,
When
kunits
to
h)
does
not
to
y
=
the
2
f (x
h)
translate
the
function
y
=
f (x)
by
h
units
left?
10.1 A frog into a prince
3 51
Example
The
red,
The
red
1
green
and
blue
lines
are
y
parallel.
y
Find
in
line
the
terms
has
equation
equation
of
of
the
y
=
f (x).
other
=
f(x)
7
lines
6
f (x).
5
3
2
1
0
4
3
2
1
x
1
2
3
4
1
The
in
blue
the
line
is
a
translation
of
the
red
line
by
4
units
y-direction.
y
7
6
Find
that
points
on
the
correspond
two
to
graphs
each
other.
3
2
1
0
4
3
2
1
x
1
2
3
4
1
Hence
The
the
green
right
by
3
blue
line
line
is
a
has
equation
translation
of
y
=
the
f (x)
blue
+
4.
line
to
the
units.
y
7
6
3
2
1
It
is
easiest
between
0
4
3
2
nd
blue
corresponding
and
green
points
lines.
x
1
1
to
the
2
3
4
1
Horizontal
Hence
the
green
line
has
equation
y
=
f (x
−
3)
+
are
3 5 2
10 Change
translations
4.
of
the
form
f (x
h).
A LG E B R A
Example
The
red
Find
2
curve
the
is
a
equation
translation
of
the
red
of
the
curve
blue
in
curve,
terms
of
which
has
equation
y
=
f (x).
f (x).
y
y
=
f(x)
4
3
2
1
0
4
3
x
1
1
2
3
4
1
2
3
4
The
vertex
This
3
is
units
The
a
In
moved
translation
of
from
2
(
units
1,
1)
to
the
to
(1,
right
4).
Find corresponding points.
and
Identify the translation.
down.
blue
curve
Practice
1
has
each
graphs
has
equation
y
=
f (x
2)
3.
2
graph
are
you
are
given
translations
of
the
that
equation
function.
of
one
Find
of
the
the
functions.
equation
of
y
y
=
The
each
other
function.
y
−3x
3
A
C
B
4
2
y
=
0.5
x
+
2
3
1
2
x
0
7
6
5
4
3
2
1
1
3
5
1
D
2
0
3
2
x
1
1
3
E
2
F
4
3
4
10.1 A frog into a prince
3 53
y
y
7
2
2
G
y
=
H
x
6
1
2
y
=
−0.5 x
5
J
0
10
9
8
7
6
5
4
3
2
1
4
x
1
2
3
1
3
2
2
3
1
4
L
5
0
4
3
2
x
1
1
2
3
4
5
6
7
8
9
10
1
6
K
M
2
7
3
4
Problem
2
Without
solving
using
coordinates
Label
any
of
graphing
points,
intercepts
software
sketch
with
the
the
or
a
calculator
graphs
of
or
these
nding
any
specic
functions.
axes.
Start
with
y
=
x
or
2
a
y
=
x
+
5
b
y
=
2x
3
c
y
=
−2x
y
3
=
the
2
d
y
=
2
x
+
5
e
Exploration
1
Use
graphing
y
=
(x
+
2
2)
f
y
=
(x
1)
1
3
software
to
graph
the
linear
function
f
(x)
=
mx
+
c
and
1
insert
with
a
f
slider
(x)
=
x,
to
change
with
m
=
the
1
values
and
c
=
of
m
and
c
between
10
and
10.
Start
0.
1
2
Add
the
graphs
of
y
=
(x)
−f
and
y
=
1
3
Change
using
the
the
values
slider.
relationship
of
m
and
the
(
x).
1
Describe
between
f
c
1.1
y
the
graphs
f
(x)
=
m
x
+
c
1
y
=
f
(x)
and
y
=
−f
1
y
=
Identify
the
1
transformation
of
(x).
f
(x)
to
y
that
=
−f
1
takes
the
graph
(x).
1
f
4
Continue
of
m
and
to
c
vary
using
the
the
values
slider.
(x)
3
your
graphs,
of
y
=
f
(x)
to
that
the
takes
graph
of
1
(
x
)
1
0
describe
the
y
=
0
c
10
−10
10
graph
f
(
x).
1
f
(x)
2
=
−f
(x
10 Change
)
1
Continued
3 5 4
x
0
the
10
transformation
f
By
m
examining
=
on
next
page
x
and
graph.
translate
4
A LG E B R A
2
5
On
a
clear
screen,
graph
the
quadratic
function
y
=
f
(x)
=
(x
h)
+
k.
1
Insert
6
Add
sliders
the
to
graphs
change
y
=
−f
the
(x)
values
and
y
=
f
1
in
steps
3
and
4,
nd
of
(
h
and
x).
By
k
between
changing
10
the
and
sliders
10.
as
you
did
1
the
relationship
between
the
graph
of
y
=
f
(x)
and
1
the
graphs
of
y
=
− f
(x)
and
y
=
f
1
7
Explain
for
whether
quadratic
(
x).
1
the
generalizations
you
made
in
steps
3
and
4
hold
true
functions.
The
graph
of
y
=
−f (x)
is
a
reection
of
the
graph
of
y
=
f (x)
in
the
x-axis.
The
graph
of
y
=
f (
is
a
reection
of
the
graph
of
y
=
f (x)
in
the
y-axis.
Example
3
Here
is
the
graph
x)
of
y
2
y
=
(x
2)
+
3.
8
A
Find
of
the
6
equations
graphs
A
and
B
4
2
y
=
(x
−
2)
+
3
2
0
6
5
4
3
2
1
x
1
2
3
4
5
6
2
4
6
B
8
2
Graph
A
is
a
reection
of
y
=
(x
2)
+
3
in
the
y-axis.
2
Equation
of
A
=
f (
x)
=
(
x
2)
+
=
(
x
2)(
=
(x
=
(x
Reection of f(x) in the y-axis is f(
3
x
2)
+
3
Substitute
x).
x into f(x).
2
+
4x
+
4)
+
3
2
+
+
2)
3
Expand, then write in vertex form.
2
Graph
B
is
a
reection
of
y
=
(x
2)
+
3
in
the
x-axis.
Reection of f(x) in the x-axis is
2
Equation
of
Practice
B
=
−f (x)
=
−(x
2)
3
3
y
1
Copy
this
Sketch
sketch
and
label
graph
the
of
f (x)
graph
of
=
f (
3x
x)
+
2.
on
the
3
same
2
axes.
f (x)
=
3x
+
2
1
Sketch
and
label
the
graph
of
f (x)
on
the
same
axes.
x
0
Label
any
intercepts
with
the
y-axis.
—2
—1
1
3
10.1 A frog into a prince
3 5 5
f(x).
2
2
Copy
this
sketch
graph
of
f (x)
=
x
6x
+
10.
y
2
f (x)
Sketch
the
and
same
label
the
graph
of
f (
x)
=
x
−
6x
+
10
10
on
8
axes.
6
Sketch
and
label
the
graph
of
f (x)
4
on
the
same
axes.
2
0
x
1
3
In
each
other
graph
graphs
below
are
you
are
reections
given
of
that
the
equation
function.
y
a
of
Find
one
the
of
2
the
4
5
functions.
equation
b
3
of
each
6
The
function.
y
4
4
L
3
L
3
1
3
L
2
2
1
1
0
x
3
2
1
2
1
3
4
4
2
x
1
5
1
1
L
y
=
− x
−
4
1
2
2
y
=
−
4
x
−
4
3
3
y
3
3
2
2
L
7
L
5
1
1
2
y
=
(x
3)
+
1
0
5
4
3
2
x
1
1
2
3
4
x
0
5
6
5
4
3
2
1
1
1
2
2
2
L
L
6
y
=
−(x
3)
+
2
8
3
4
The
the
graph
of
y
y-direction
y-axis
Find
to
the
give
3
=
x
+
and
line
1
is
then
translated
reected
y
in
in
the
4
y
L
equation
of
line
=
x
+
1
L
1
0
x
L
2
5
The
graph
x-axis
to
of
give
y
=
x
curve
is
translated
C.
Find
the
in
the
x-direction
equation
of
curve
and
then
reected
in
the
C
y
2
y
=
x
0
x
2
C
3 5 6
10 Change
A LG E B R A
Exploration
1
Draw
Plot
(1,
a
coordinate
these
1),
This
A
points:
(4,
set
smooth
2
4
1)
of
(
and
eight
plane
4,
(5,
with
5),
axes
(
3,
from
2),
(
10
2,
to
0),
+10.
(
1,
1),
(0,
0),
3).
points
denes
the
function
f (x).
Join
the
points
with
a
curve.
change
is
applied
to
f (x)
to
give
a
new
function
g
(x).
This
change
is
1
dened
as
g
(x)
=
2f (x).
1
a
Write
down
the
coordinates
of
the
eight
points
of
g
(x).
1
b
Draw
and
label
the
graph
of
g
(x)
on
the
same
coordinate
plane
as
f
(x).
1
c
Describe
the
change
between
f (x)
and
g
(x).
1
3
Repeat
step
2
for
a
new
change
g
(x)
=
f (x).
2
4
Make
a
generalization
changes
a
5
when
Draw
is
to
g(x)
0
the
<
a
=
af (x):
<
1
graph
applied
to
of
f (x)
for
what
happens
b
f (x)
to
again
give
a
on
new
a
when
when
new
a
>
the
h
(x).
f (x)
1.
coordinate
function
function
plane.
This
A
change
change
is
1
dened
as
h
(x)
=
f (2x).
1
a
Write
down
the
coordinates
of
the
eight
points
of
h
(x),
for
example
1
the
b
point
Draw
(
and
4,
5)
label
will
the
become
graph
of
h
(
2,
(x)
5).
on
the
same
coordinate
plane
as
f (x).
1
c
Describe
the
change
between
f (x)
and
h
(x).
1
6
Repeat
step
5
for
a
new
change
h
(x)
=
f
2
7
Make
a
changes
a
8
generalization
to
when
Use
0
<
graphing
quadratic
a
h(x)
Graph
a
=
<
for
what
happens
1
b
software
to
see
if
when
a
=
the
insert
1
the
function
f (x)
your
a
>
1.
generalizations
apply
to
linear
and
functions.
linear
functions
f
(x)
=
1
and
when
f (ax):
and
a
slider
then
to
change
explore
what
x,
f
(x)
=
a
×
2
the
happens
f
(x)
and
1
value
of
a
when
0
<
f
(x)
=
3
between
a
<
=
x
0
1
and
f
(a
×
x)
1
and
3.
Start
when
a
with
>
1.
2
b
Repeat
step
8a
for
the
quadratic
function
f
(x)
.
1
10.1 A frog into a prince
3 57
The
mathematical
y
=
a f
y
=
f
term
for
(x)
is
a
vertical
(ax)
is
a
horizontal
stretch
dilation
of
or
f
dilation
compression
(x),
of
f
scale
(x),
is
factor
scale
dilation.
a,
parallel
factor
,
to
the
parallel
to
y-axis.
the
x-axis.
af (x)
f (ax)
y
y
C
C
C
2
C
2
1
1
2
C
f (x)
=
x
f (x)
=
(4x)
1
2
a
>
1
C
f (x)
=
x
f (x)
=
4x
a
1
>
2
1
C
2
2
C
2
0
0
x
x
1
vertical
dilation
scale
factor
4
horizontal
dilation
scale
factor
4
y
y
C
C
1
1
2
C
f (x)
=
x
1
2
C
f (x)
=
x
2
1
1
C
C
f (x)
2
2
4
0
<
a
<
1
C
2
1
C
0
<
a
<
f (x)
=
2
x
2
1
4
0
0
x
x
1
vertical
dilation
scale
factor
horizontal
4
3 5 8
10 Change
dilation
scale
factor
4
A LG E B R A
Example
4
2
Here
is
the
Describe
graph
the
eect
transformation,
of
how
of
of
=
and
y
x
the
giving
individual
transformed,
y
examples
points
sketch
are
the
graphs
of
2
a
b
2f
f
y
(x)
=
x
(2x).
0
a
The
(0,
graph
0)
→
of
(0,
2f
(x)
is
a
dilation,
scale
factor
2,
x
parallel
to
the
y-axis.
0)
y-coordinates of points on the
(
2,
4)
→
(
2,
8)
graph increase by scale factor 2.
(1,
1)
→
(1,
2)
y
y
=
y
2f(x)
(
=
f(x)
2, 8)
Sketch the graph. Label some points.
(
2, 4)
(1, 2)
(1, 1)
x
0
1
b
Graph
of
f
(2x)
is
a
dilation,
scale
factor
,
parallel
to
the
x-axis.
2
(0,
0)
→
(0,
0)
(2,
4)
→
(1,
4)
x-coordinates
of
points
on
the
1
graph
increase
by
scale
factor
.
2
(
2.5,
6.25)
→
(
1.25,
6.25)
y
(
1.25, 6.25)
y
(
=
f(2x)
y
=
f(x)
2.5, 6.25)
(1, 4)
(2, 4)
0
x
10.1 A frog into a prince
3 5 9
y
Practice
4
2
1
Copy
this
Sketch
same
sketch
the
of
graph
f
of
(x)
=
g (x)
(x
=
1)
2f
(x)
.
on
the
axes.
1
2
f (x)
=
0
(x
1)
x
1
y
2
2
Copy
this
sketch
of
f
(x)
=
−x
+
4.
2
f (x)
Sketch
same
the
graph
of
h (x)
=
f
(2x)
on
the
=
−x
+
axes.
0
x
2
2
y
2
3
4
4
Copy
this
sketch
of
f
(x)
=
x
+
2x
3.
2
Sketch
Sketch
the
the
graph
graph
of
of
g (x)
=
h (x)
=
f
(3x)
−f
on
(3x)
the
on
same
the
axes.
same
f (x)
=
x
+
2x
3
axes.
0
3
1
3
2
4
Draw
the
graph
of
Draw
the
graphs
f
(x)
=
x
 1
of
g (x)
=
f


x
 2
on
the
same

and
h (x)
=
f
(2x)

axes.
2
5
Describe
the
transformation
that
maps
the
graph
of
y
=
x
to:
2
1
2
a
y
=
2
3x
b
y
=

2
x
c
y
=
(4x)
d
y
=

2
1

x

3


2
6
Write
is
down
the
transformed
equation
by
a
of
the
graph
which
results
when
y
=
x
dilation:
a
scale
factor
5
parallel
to
the
y-axis
b
scale
factor
4
parallel
to
the
x-axis
parallel
to
the
y-axis.
1
c
scale
factor
3
7
Describe
the
transformation
that
takes
the
blue
graph
y
a
2
y
=
red
x
y
3x
the
2
b
y
to
7
+
2
2
x
2
y
=
x
x
2
y
3 6 0
10 Change
=
−(x
7)
+
2
graph:
x
A LG E B R A
You
need
to
quadratic
be
familiar
functions
Combinations
●
Any
linear
applied
●
Any
–
of
with
types
of
transformation
reection
and
for
linear
and
dilation.
transformations:
function
one
three
translation,
after
quadratic
can
the
be
other
function
dened
to
can
the
be
by
one
function
dened
or
y
by
more
=
transformations
x.
one
or
more
transformations
2
applied
Example
one
after
the
other
to
the
function
y
=
x
5
2
List
the
transformations
applied
to
the
function
2
a
y
=
x
6x
the
+
4
b
transformed
2
a
x
=
x
to
give:
2
+
Sketch
y
y
=
−3x
12x
11
graphs.
2
+
6x
+
4
=
(x
+
6x)
+
4
Write the function in vertex form.
2
=
[(x
=
(x
+
3)
9]
+
4
2
+
3)
5
2
Start with the original function y = x
2
Start
with
f
(x)
=
and
x
apply one transformation at a time.
y
f
(x
+
3)
is
a
8
horizontal
2
y
translation
of
=
(x
+
2
3)
y
=
x
3.
4
f
(x
+
3)
5
translation
is
a
vertical
of
5
on
f
(x
+
2
3).
0
6
5
4
3
x
1
2
4
2
y
=
(x
+
3)
5
6
2
b
2
3x
12x
11
=
−3(x
+
4x)
11
2
=
−3[(x
=
−3(x
+
2)
4]
11
y
2
+
2)
+
1
3
2
y
2
Start
with
f
(x)
=
=
3x
2
2
y
=
x
3x
=
parallel
3f
to
(x)
the
is
a
dilation
y-axis,
2
scale
2
y
=
−3(x
+
2)
+
y
1
=
x
1
factor
3.
2
y
=
−3x
=
−3f
(x)
is
the
x
reection
in
the
2
x-axis.
1
2
y
is
=
a
−3(x
+
2)
=
horizontal
−3f
(x
+
2)
translation
of
2.
2
y
=
−3x
2
2
y
=
−3(x
+
2)
+
1
=
−
3f
(x
+
2)
+
1
2
is
a
vertical
translation
of
1.
y
=
−3(x
+
2)
3
10.1 A frog into a prince
3 61
Reect
●
Why
and
is
the
Example
●
In
5
discuss
quadratic
in
Example
order
5b,
3
changed
to
does
show
it
from
the
matter
standard
form
transformations
in
which
order
to
ver tex
that
you
have
apply
form
been
in
applied?
the
transformations?
●
Why
is
Practice
1
List
a
y
the
=
it
best
to
do
the
translations
last?
5
transformations
−4x
b
y
=
applied
2x
to
the
9
c
linear
y
=
function
0.5x
+
y
=
4
x
d
to
y
give:
=
−1.2x
2.5
2
2
List
the
transformations
function.
Sketch
the
applied
graphs
of
to
the
the
y
=
3
y
=
x
Express
y
=
+
8x
g (x)
a
f (x)
=
x
c
f (x)
=
−2x
+
16
in
5,
e
f (x)
=
2x
g
f (x)
=
x
3(x
e
terms
g (x)
5,
=
of
y
f (x)
4x
g (x)
=
h
f (x)
=
4x
=
and
+
+
3x,
g (x)
=
a
to
f (x)
=
+
6,
y
=
−x
+
4
3x
state
+
the
5
9
f
y
=
−0.5x
transformation
b
f (x)
=
2x
+
d
f (x)
=
−2x
5,
4x
applied
g (x)
5,
+
3x
12
f
f (x)
=
−3x
=
g (x)
6x
=
+
to
+
1
f (x)
to
get
g (x):
5
2x
5
2
+
=
(x
2)
5(x
2)
+
6x
7,
g (x)
=
−3x
6x
7
6
2
8x
in
+
12,
terms
g (x)
of
=
f (x)
x
2x
and
+
state
3
the
combination
8,
g (x)
=
−3x
+
3
b
f (x)
=
2x
2
f (x)
=
x
e
f (x)
=
4x
of
transformations
+
g
f (x)
=
x
4x,
g (x)
=
x
4x
+
+
7,
g (x)
=
−4x
2
Problem
4
d
f (x)
=
2x
f
f (x)
=
x
2
5x
3,
g (x)
6,
g (x)
=
x
applied
to
3
=
x
2
2x
3,
g (x)
=
2
+
5x
+
2
2
2x
+
2
2
each
12x
g (x):
c
For
each
2
2x
g (x)
2
5
give
2
5x
g (x)
get
to
2
2
2
Express
c
20
2x
x
2
9)
2x
2
f (x)
=
2
2
4
y
2
b
25x
2
d
function
transformations.
2
a
quadratic
−x
6x
+
4
h
f (x)
=
(x)
is
x
x
+
1.5
2
+
4x
+
4,
g (x)
=
x
2
+
+
6x
+
9
2
+
x
+
1,
g (x)
=
9x
3x
+
1
solving
pair
of
graphs,
the
orange
function
f
obtained
by
applying
one
2
transformation
to
the
blue
function
f
(x).
Express
1
f
(x)
in
terms
2
of
f
(x),
and
state
the
1
transformation.
a
b
y
f
c
y
y
(x)
1
f
(x)
2
f
(x)
1
0
x
0
x
f
(x)
2
f
(x)
1
0
f
(x)
2
3 6 2
10 Change
x
A LG E B R A
d
e
y
y
f
y
f
(x)
1
(x)
f
1
0
x
f
(x)
2
f
f
(x)
(x)
f
2
2
(x)
1
0
0
6
For
each
pair
of
graphs,
the
orange
function
f
(x)
is
x
x
obtained
by
applying
2
two
or
more
transformations
to
the
blue
function
f
(x).
Express
f
1
of
f
(x),
and
state
the
(x)
in
terms
2
transformations.
1
y
a
f
y
b
f
(x)
(x)
1
1
x
0
f
y
c
(x)
2
x
0
f
0
(x)
x
1
f
(x)
2
f
(x)
2
Translation,
of
any
reection
and
dilation
transformations
can
be
used
on
the
graph
function.
Example
Here
is
the
6
graph
of
function
f
(x)
and
y
the
1
transformed
function
f
3
(x).
2
f
f
(x)
(x)
1
2
2
Identify
take
f
the
(x)
to
transformations
f
1
that
1
(x).
2
Hence
write
the
function
f
(x)
0
as
8
2
7
6
5
4
3
2
1
x
1
1
a
transformation
of
f
(x).
1
2
3
4
There
has
been
a
translation
There
has
been
a
dilation
3
units
to
the
left.
The minimum points at y =
parallel
to
y-axis
scale
factor
1 have been
3.
transformed to minimum points at y =
f
(x)
→
f (x
+
→
3f (x
3)
translation
3
units
left
1
Therefore,
f
+
3)
(x)
dilation
=
3f (x
+
scale
factor
3
parallel
to
y-axis
3)
2
10.1 A frog into a prince
3 63
3.
Example
Given
the
7
graph
of
f
(x),
draw
the
graph
y
1
of
f
(x)
=
5f
2
(
x).
1
f
(x)
1
x
f
(
x):
reection
in
the
y-axis.
Identify the transformations.
1
5f
(
x):
vertical
dilation
of
scale
factor
5
Multiply the y–value of all points by 5.
1
y
y
f
(
x)
f
1
(x)
1
5f
(
x)
1
x
Draw the graph of each transformation.
x
f
(
x)
1
Practice
1
For
each
6
pair
of
graphs,
nd
the
transformations
applied
to
f
(x)
to
get
1
Hence
write
the
function
f
(x)
as
a
transformation
of
2
f
f
(x).
2
(x).
1
y
a
y
b
6
6
f
f
(x)
(x)
1
1
5
5
4
4
3
3
2
2
1
1
(x)
f
2
0
9
8
7
6
5
4
3
2
1
0
x
1
2
3
4
5
7
6
5
4
3
2
1
1
2
2
3
3
4
4
5
5
f
(x)
2
6
3 6 4
10 Change
x
1
1
6
2
A LG E B R A
c
y
y
d
5
5
f
f
(x)
f
2
(x)
1
(x)
1
4
4
3
3
2
2
f
(x)
2
1
1
0
4
3
2
0
x
1
1
2
3
4
5
4
6
3
2
6
graph
of
f
(x),
draw
the
graph
of
f
1
(x)
=
2f
2
(x
5
5
6
f
4
4
5
a
3
3
4
the
2
2
3
Given
1
1
2
2
x
1
1
(x).
2
3)
b
1
f
(x)
=
2
f
(
x
+
5)
c
1
f
(x)
=
0.5f
2
(
x)
1
5
5
5
f
(x)
1
f
4
4
(x)
4
1
f
(x)
1
3
3
3
2
2
2
1
1
1
0
x
d
f
(x)
0
x
2
2
3
4
2
3
5
4
3
1
1
2
2
2
3
3
3
4
4
4
5
5
=
2
3f
(x
5)
e
1
f
(x)
=
−2f
2
(x)
f
1
y
f
(x)
=
f
2
y
(
x
2
1
x)
+
3
1
y
7
5
4
4
3
3
2
2
1
f
6
(x)
1
f
5
(x)
1
4
3
f
(x)
1
2
1
1
0
1
x
1
2
3
4
5
0
1
0
4
3
2
1
x
x
1
1
2
3
4
5
6
7
8
1
1
1
2
2
3
2
3
4
3
5
4
10.1 A frog into a prince
3 6 5
Transformations
D
●
If
the
the
●
If
graph
of
original
two
dierent
Exploration
Use
f
2
(x)
Find
is
mx
c
value
these
to
and
functions
the
In
+
of
has
the
function
is
function
been
transformations
function,
are
the
give
the
the
same
as
the
graph
of
transformed?
graph
transformations
of
the
really
same
dierent?
5
technology
=
not?
transformed
function,
transformed
1
a
or
graph
their
where
c
in
graphs,
the
these
are
dierent
reections
the
graphs
linear
in
of
the
f
(x)
functions
y-axis
and
and
f
(
of
the
the
x)
form
x-axis,
are
the
f
(
same.
x).
What
graphs?
original
function
and
the
transformed
function
identical?
3
For
one
of
your
graphs
transformation
the
y-axis.
original
4
What
f
Does
point?
single
(
x)
the
How
in
to
step
that
resulting
does
pick
point
this
transformation
2,
point
point
on
reecting
have
aect
is
a
by
the
your
same
answer
equivalent
to
the
it
the
line.
in
the
Apply
x-axis
coordinates
to
step
two
the
and
as
then
the
2?
combined
reections?
Example
8
2
Consider
and
h (x)
the
=
f
function
(2x)
both
f
(x)
=
2x
generate
.
Show
the
that
same
the
transformations
g (x)
=
4f
(x)
graph.
Find the function generated by
g (x)
=
4 f
(x)
=
4(2x
=
8x
=
f
=
2(2x)
=
8x
=
h (x)
the rst transformation.
2
)
2
Find the function generated by
h (x)
(2x)
the second transformation.
2
2
Both the functions generated by the
g (x)
two dierent transformations are equal.
The
two
transformations
Reect
●
In
and
Example
generate
discuss
8
the
the
same
graph.
4
function
g (x)
is
a
vertical
dilation
by
scale
factor
4
of
the
1
function
f
(x).
The
function
h (x)
is
a
horizontal
dilation
by
scale
factor
.
2
Do
g (x)
and
h (x)
transformations
represent
the
same
of
the
function
on
the
parabola
f
transformation
or
dierent
(x)?
2
●
Choose
to
that
other?
3 6 6
a
point
point.
How
Do
does
the
two
this
new
aect
10 Change
f
(x)
=
points
your
2x
.
have
answer
Apply
the
each
same
above?
of
the
transformations
coordinates
as
each
A LG E B R A
Summary
Translations
●
y
=
f
(x
Reections
h)
translates
y
=
f
(x)
by
h
units
in
the
●
x-direction.
●
y
=
f
(x)
+
y
=
the
f
(x
graph
graph
k
translates
y
=
f
(x)
by
k
units
in
the
●
y-direction.
●
The
The
of
graph
graph
h)
+
k
x-direction
translates
and
k
y
=
units
f
in
(x)
by
the
h
units
y
of
y
of
=
f
of
=
f
y
=
(x)
y
−f
=
(x)
(x)
in
f
is
the
(
x)
in
a
reection
of
the
of
the
x-axis.
is
the
a
reection
y-axis.
in
y-direction.
Dilations
●
y
=
a f
factor
●
y
=
f
(x)
a,
(ax)
factor
,
is
a
vertical
parallel
is
a
to
dilation
the
horizontal
parallel
to
the
of
f
(x),
scale
y-axis.
dilation
of
f
(x),
scale
x-axis.
af (x)
f (ax)
y
y
C
C
C
2
C
2
1
1
2
C
f (x)
=
x
f (x)
=
(4x)
1
2
a
>
1
C
f (x)
=
x
f (x)
=
4x
a
1
>
2
1
C
2
2
C
2
0
0
x
x
1
vertical
dilation
scale
factor
4
horizontal
dilation
scale
factor
4
y
y
C
C
1
1
2
C
f (x)
=
f (x)
=
x
1
2
C
x
1
2
1
C
2
C
f (x)
2
4
0
<
a
<
1
C
2
1
C
0
<
a
<
f (x)
2
=
x
2
1
4
0
vertical
Combinations
●
Any
linear
more
of
0
x
dilation
scale
factor
horizontal
dilation
x
scale
factor
4
transformations
function
can
transformations
be
dened
applied
one
by
one
after
or
the
●
Any
or
quadratic
more
function
can
transformations
be
dened
applied
one
by
after
one
the
2
other
to
the
function
y
=
x.
other
to
the
function
y
=
x
10.1 A frog into a prince
3 67
Mixed
practice
2
1
Here
is
the
graph
of
f
(x)
=
x
2x
1.
b
6
5
7
f
(x)
2
4
6
5
2
4
2
f(x)
=
x
2
x
1
1
3
2
0
4
3
x
2
1
1
f
(x)
1
2
x
4
3
2
1
c
2
6
Use
this
graph
to
help
you
sketch
the
graph
f
of
(x)
2
5
each
function
and
write
down
the
coordinates
4
of
the
new
vertex.
3
a
f
(x
+
2)
b
f
(x)
4
2
f
(x)
1
c
f
(x)
d
f
(
x)
e
2f
(x)
f
f
(
3x)
g
f
1
0
7
(x
+
1)
3
h
3
f
j
5f
6
5
4
3
x
2
1
(x)
2
i
2f
(x)
1
(
0.5x)
+
4
3
2
State
which
combination
of
transformations
has
4
been
applied
to
f (x)
to
get
g (x):
d
a
f
(x)
=
x,
g(x)
b
f
(x)
=
−3x
c
f
(x)
=
12x
=
2x
8
8
6,
g(x)
=
x
+
5,
g(x)
=
6x
2
6
f
(x)
1
+
+
6
4
2
d
f
(x)
=
2
x
+
4x
+
4,
g(x)
=
x
5
2
2
e
f
(x)
=
(x
2)
2
+
1,
g(x)
=
(x
+
1)
2
0
x
2
2
f
3
f
For
(x)
=
each
2x
+
pair
8x
of
transformation
+
4,
g(x)
graphs,
applied
=
x
nd
to
f
+
4x
the
down
the
function
f
(x)
to
+
4
2
4
single
get
1
Write
2
2
f
f
(x)
2
6
(x).
2
(x)
as
a
8
2
transformation
of
f
(x).
1
e
a
4
f
4
(x)
1
3
3
2
f
(x)
f
2
(x)
f
1
(x)
2
2
1
1
0
4
0
4
3
2
x
2
3
1
4
1
2
3
3
5
5
3 6 8
10 Change
x
A LG E B R A
c
f
6
f
4
(x)
f
2
(x)
1
5
3
4
2
3
1
2
0
f
(x)
5
1
4
x
2
3
2
1
1
2
0
4
x
3
1
3
f
(x)
2
4
d
6
5
4
4
For
each
pair
of
graphs,
nd
the
transformations
f
(x)
2
applied
to
f
(x)
to
get
1
function
f
(x)
f
(x).
Write
down
3
the
2
as
a
transformation
of
f
2
(x).
2
1
1
a
0
8
7
6
5
3
x
2
2
1
5
f
(x)
1
2
4
3
3
f
(x)
1
4
2
5
1
0
3
x
2
2
e
1
6
2
f
(x)
2
5
f
(x)
2
4
b
3
6
2
f
5
(x)
1
1
4
f
(x)
1
3
x
4
3
1
5
1
2
1
3
0
4
3
x
2
2
4
1
5
2
3
f
(x)
2
4
5
10.1 A frog into a prince
3 6 9
5
f
Given
the
graph
of
f
(x),
draw
the
graph
of
f
1
(x).
2
6
5
4
f
(x)
f
2
(x)
1
4
3
2
1
1
0
x
2
0
6
5
4
1
3
1
x
1
1
2
2
3
f
(x)
1
3
4
a
f
(x)
=
2f
2
(x
+
4)
b
f
1
(x)
=
f
2
(2x)
+
4
1
5
c
f
(x)
=
−f
2
Review
in
ies
an
freely
d
f
(x)
=
2f
2
(
x)
1
context
Orientation
Whenever
(2x)
1
in
object
through
space
such
the
as
air,
and
a
ball,
its
time
arrow,
motion
is
or
dart
aected
2
by
A
student
throws
a
ball
from
a
window
in
a
tall
building.
2
gravity.
Any
object
which
isn’t
self-propelled
x
(like
Its
a
bird
by
a
or
its
rocket
parabola.
directly
The
a
This
downward
path
that
trajectory.
gives
would
the
an
and
takes
in
of
terms
a
path
gravity
exerts
equation
(y)
follows
because
object
The
height
is
be)
a
in
a
ight
is
the
acts
(0,
known
u
as
travelled
h)
is
a
usually
A
frog
leaps
its
is
on
unit
a
lily
pad.
Viewed
from
position
horizontal
represents
trajectory
1
of
is
and
given
by
(0,
0).
on
eect
graph
the
y-axis
is
the
where
,
of
which
velocity
of
the
it
is
in
thrown
and
m/s.
increasing
the
value
of
the
value
of
trajectory.
the
eect
of
increasing
A
of
the
trajectory.
baseball
player
is
practicing
her
batting
on
an
range.
Let
(0,
0)
be
the
point
at
which
vertical;
ight
is
given
by
y
=
f
hits
the
ball.
The
indoor
range
is
about
than
the
point
at
which
she
hits
the
5 m
ball.
(x)
a
( x ) =
graph
The
Her
rst
hit
follows
a
trajectory
given
by
1
2
f
h − 5
meter.
its
3
where
=
the
higher
The
the
from
horizontal
the
Describe
she
1
position
initial
indoor
x-axis
y
(x).
from
initial
the
Describe
h
3
side,
is
its
u
1
by
horizontal
b
distance
given
u
force.
trajectory
of
is
given
always
constant
trajectory
x
−
y
4
2
1
x
= −
(
4
x
− 10
+ 5.
)
20
a
Draw
a
graph
of
its
trajectory.
i
Describe
the
transformations
that
map
a
2
b
Find
the
distance
it
travels
before
returning
curve
to
with
equation
y
=
x
onto
starting
curve
2
1
its
a
height.
with
equation
y
= −
x
(
− 10
)
+ 5.
20
c
The
frog
can
change
its
jump
by
modifying
ii
the
angle
and
speed
with
which
it
Show
point
o.
Describe
clearly
how
its
jump
would
its
new
trajectory
were
given
(0,
y
=
f
(2x)
ii
y
=
1.2
f
Use
(0.7x)
iii
y
=
f
(
the
graph
curve
hitting
Explain
model
37 0
a
clearly
why
trajectory
y
for
=
−0.8 f
the
10 Change
curve
passes
through
the
described.
transformations
(x)
frog’s
does
jump.
not
whether
the
to
maximum
height
of
x)
this
d
as
by
determine
i
this
0)
vary
iii
if
that
launches
is
point
more
and
than
hence
4
meters
would
above
hit
the
her
roof.
A LG E B R A
b
Her
second
shot
follows

y
a
trajectory
given
by
b
Draw
c
Find
a
graph
of
this
relationship,
b(t).
2
1
= −

(
x
− 10
− 100
)


40
.


b(50).
State
what
this
amount
represents.
By
considering
the
graph
coordinates
of
transformations,
the
maximum
nd
point
of
the
d
curve.
Hence
determine
whether
or
not
Find
the
ball
will
hit
the
Car
loans
month,
(the
use
the
loan
simple
amount
balance)
interest
you
owe
decreases
so
that,
on
by
the
the
every
(House
loans
or
Suggest
this
way.)
car
Suppose
loansand
$150per
do
Write
you
have
$12
you
are
making
payments
to
represent
as
months
that
a
function
of
the
Reect
and
How
you
have
gone
by
change(s)
produce
the
what
a
same
State
o
a
parallel
to
the
graph
scenario
with
a
higher
change(s)
graph
with
a
to
the
lower
scenario
could
y-intercept
how
the
increase
graph
your
will
be
original
transformed
payment
to
the
number
month.
Write
down
an
equation
for
of
scenario.
(t).
statement
of
inquiry?
Give
specic
examples.
Statement of Inquiry:
Relationships
predict
model
duration,
patterns
frequency
of
and
if
$250
discuss
explored
but
x-intercept.
loan
new
have
what
Suggest
per
(b)
pay
of
g
equation
balance
to
000
month.
an
you
not
you
a
take
same
mortgages
the
in
will
y-intercept.
produce
work
it
loan.
could
loan
f
amount.
long
ceiling.
e
4
how
the
change
that
can
help
clarify
and
variability.
10.1 A frog into a prince
3 71
this
A
10.2
thin
Global
line
context:
Objectives
●
Changing
the
divides
Scientic
Inquiry
subject
of
a
formula
and
us
technical
innovation
questions
●
What
●
How
is
the
subject
of
a
formula?
F
●
●
Simplifying
rational
algebraic
expressions
can
you
change
the
you
simplify
subject
of
a
formula?
Performing
algebraic
mathematical
operations
on
rational
●
expressions
How
can
a
rational
MROF
expression?
●
How
do
operations
on
rational
algebraic
C
expressions
rational
●
Does
compare
numerical
technology
to
operations
on
expressions?
help
or
hinder
D
understanding?
ATL
Use
Organization
appropriate
skills
strategies
for
organizing
complex
information
10.2
Statement of Inquiry:
Representing
helped
change
humans
apply
and
equivalence
their
in
a
understanding
variety
of
of
forms
scientic
has
principles.
11.3
E10.2
372
A LG E B R A
Y
ou
●
should
simplify
already
know
how
fractions
1
to:
Simplify
each
fraction.
4
14
a
b
16
35
2
2
20 x
x
c
d
2
3
( 20 x )
x
●
factorize
quadratic
expressions
2
Factorize.
2
a
x
+
5x
+
6
2
b
x
3x
10
2
c
●
solve a linear equation algebraically
3
x
Solve.
a
2x
1
=
b
5x
7
=
c
Rearranging
F
You
What
●
How
can
you
change
●
How
can
you
simplify
already
solution
x
equation
=
to
equations
2
c
2
=
know
3.
solve
+
b
=
8
for
show
when
a
=
=
a
2
7
A
or
2
2
By
+
an
the
is
b
is
innite
In
by
an
You
3
and
b
a
=
8
4x
+
17
formula?
the
a
subject
of
rational
linear
a
like
‘isolating
know
how
between
formula?
expression?
equations
called
also
=
to
the
use
5x
x’
7
or
=
8,
for
which
has
‘rearranging
substitution
variables,
to
the
solve
example,
solving
4.
2
+
b
2
are
equation
sets
of
both
that
values.
on
for
values
words,
the
equations,
describes
Each
values
an
variable
of
the
but
c
2
=
+
algebraic
in
a
other
2
a
b
is
also
a
relationship
formula
can
take
formula.
between
dierent
variables.
=
3,
of
for
b
=
other
a
and
values
you
can
4,
c
=
formula:
b
you
for
solve
b
can
and
for
is
also
of
nd
c
a
you
any
of
valid
for
values
a,
a
=
b,
5,
and
corresponding
can
the
nd
a
b
=
12,
c
=
c
value
for
c.
corresponding
variables,
13
given
the
value
values
of
1
between
F
=
C
+
degrees
Celsius
the
temperature
in
°F
2
Find
the
temperature
in
Celsius
°C,
you
used
need
to
to
solve
solve
the
and
degrees
Fahrenheit
is
given
32.
Find
have
It
combinations
1
in
5.
two.
relationship
the
a
number
Exploration
by
x’.
substituting
other
other
The
solve
sometimes
2
valid
substituting
a.
of
relationships
depending
Similarly
,
for
to
1
2
a
and
c
more
values,
=
and
formula
two
c
is
+
2
a
5x
subject
how
This
that
the
8x
5
formulas
●
is
9
an
when
it
is
35°C.
when
equation.
it
is
77°F.
Carefully
To
nd
write
a
temperature
down
the
steps
you
equation.
Continued
on
next
page
10.2 A thin line divides us
373
3
Using
the
nding
same
F
and
steps
nish
as
you
with
a
used
in
formula
step
for
2,
C.
begin
C
is
with
now
the
the
formula
‘subject’
of
for
The
the
formula.
a
Use
your
new
formula
from
step
3
to
nd
the
temperature
in
it
is
77°F.
Compare
your
answer
to
that
in
step
Write
down
formula’,
what
and
the
you
think
necessary
it
means
steps
to
to
do
‘change
the
0°F.
subject
of
the
formula
is
a
subject
of
the
and
this.
way
of
In
boils
The
variable
in
Reect
●
In
When
the
and
are
formula
●
discuss
Exploration
What
When
1,
does
between
the
rearranging
the
solving
it
be
theorem
the
side
other
and
and
an
based
2
2
c
2
c
=
a
=
a
2
2
you
solve
a
=
a
=
c
and
100°C.
dierent
the
subject
of
the
formula
change
the
C?
dierences
between
changing
the
subject
of
a
to
change
the
subject
of
a
formula?
states
that
a
is
c
in
a
right-angled
2
=
a
triangle,
the
relationship
2
+
b
,
where
c
is
the
hypotenuse
and
a
and
b
2
+
b
to
make
a
the
subject
of
the
formula.
+
b
+
b
2
2
2
b
Subtract
Rearrange
2
2
b
from
both
sides.
Take
the
soyou
square
can
root
ignore
the
variable
factorize
3 74
you
so
are
that
trying
the
to
isolate
variable
10 Change
appears
appears
only
more
once.
than
once
in
the
of
the
Now
formula,
the
terms
so
that
the
2
b
variable
When
at
2
b
a
0°C
algebraic
2
2
b
at
scale
water
sides.
2
=
c
2
c
formula,
on
equation?
useful
lengths
two
2
Rearrange
a
212°F.
1
2
are
of
at
32°F
1
Pythagoras’
between
subject
changing
F
similarities
and
could
Example
change
scale,
at
formula.
relationship
●
you
this
Celsius
boiling
relationships.
salt
freezing
freezes
freezing
the
on
2
is
Changing
of
water
water
5
based
Celsius
at
when
was
mixture
and
4
Fahrenheit
scale
a
is
a
is
on
both
the
sides.
negative
the
left
a
is
square
subject
of
hand
a
side.
length,
root.
the
formula.
A LG E B R A
Example
2
ATL
Rearrange
to
make
m
the
subject.
Multiply
Ft
=
mv
Ft
=
m
Practice
1
both
sides
by
the
denominator
mu
(v
of
the
Factor
u)
Divide
right
out
both
m
hand
to
side.
isolate
sides
by
v
it.
u.
1
Rearrange
each
formula
to
make
the
variable
in
brackets
the
subject.
1
2
a
y
=
5x
4
[x]
b
y
=
x
+ 3
[x]
3
2
c
p
e
V
=
m + 2
2
[m]
d
V
[V ]
f
V
=
(2w
5)
[w]
You
may
seen
some
have
2
=
V
f
+
2ad
i
i
=
V
f
+
at
[a]
of
these
i
in
science.
2
V
g
I
mV
=
[R]
R
h
F
=
[r]
+ r
r
2
1
mV
2
i
F
=
[V]
j
F
=
mV
+
mgh
[m]
2
r
3 RT
2
k
μ
=
[M]
l
c
=
(3a
+
2b)
–
1
[a]
M
2
Write
down
subject
3
The
of
the
the
formula
formula
for
the
area
A
of
a
circle
with
radius
r.
Make
r
the
formula.
for
the
volume
V
of
a
cone
with
height
h
and
base
1
2
of
radius
r
is V
πr
=
h.
Make
r
the
subject
of
the
formula.
3
4
When
driving
a
car,
if
you
accelerate
from
rest
at
a
constant
rate
a
for
an
1
2
amount
of
time
t,
the
distance
d
travelled
is
given
by
the
formula
d
=
at
.
L
2
Rearrange
5
The
time
the
formula
taken
(T )
for
to
a
make
t
the
pendulum
subject.
to
swing
once
each
way
and
back
L
to
its
starting
position
is
given
by
the
formula T
,
= 2π
where
L
is
the
g
length
of
the
Rearrange
the
Problem
6
Between
pendulum.
formula
to
make
L
the
subject.
solving
any
two
charged
particles
with
charge
q
and
q
1
,
there
kq
force.
You
can
calculate
this
force
using
the
formula
is
an
electric
2
q
1
F
=
2
,
where
r
is
the
2
r
r
q
1
distance
between
the
particles.
Rearrange
the
formula
to
make
r
the
q
2
subject.
10.2 A thin line divides us
3 75
A
rational
algebraic
expression
is
a
fraction
that
contains
The
variables.
Ancient
Egyptians
images
Factorizing
the
can
numerator
simplify
the
change
and
the
fraction,
rational
as
you
would
12
For
algebraic
denominator
fractions
have
a
a
to
common
numerical
a
simpler
factor,
form.
you
can
When
divide
to
fraction.
fractions
same
to
20
in
way
the
they
represent
did
words.
Horus
was
a
sky
5
god
(x
used
represent
3
=
example
x
to
2)
whose
shattered
x
eye
into
was
six
=
(x
−
2) ( x
+ 5)
x
pieces.
+ 5
of
his
eye
Each
shattered
represented
dierent
Exploration
fraction.
Interestingly
Simplify
these
fractions
completely.
Justify
each
add
step.
the
six
together
a
b
Some
d
e
you
3
h
a
For
the
one
fraction.
each
simplify
of
it
a
4
For
each
5
Based
x
●
The
on
before
●
To
Why
and/or
the
or
a
of
in
step
to
discuss
a
variable
you
need
rational
=
3,
state
steps
to
have
in
the
denominator
explain
the
steps
needed
to
that
on
nd
the
10 Change
3
any
and
c
value
3,
that
write
x
=
cannot
down
the
x
take.
steps
for
expression.
are
not
allowed
variable.
Justify
you
Do
your
restrictions
expression,
rst.
1
x
2
simplify?
to
allowed
expressions,
algebraic
restrictions
you
not
is.
expression.
answers
after
are
that
b
denominator
37 6
rational
given
rational
and
do
simplify
a
you
why
3
your
values
called
+
the
i
that
expression
simplifying
Reect
these
to
=
value
Explain
1.
consider
demonstrate
possible.
of
get
than
perfection
State
you
this
f
to
g
if
fractions
c
rather
2
a
2
ATL
1
part
on
may
in
you
the
nd
denominator
these
are
restrictions
answer.
the
need
variable?
to
What
factorize
the
are
they
for?
numerator
is
that
not
A LG E B R A
Example
Simplify
3
completely
and
state
any
restrictions
on
x
Factorize
The
the
x
ATL
∉{0,
2,
are
denominator
0,
numerator
any
values
including
and
that
any
denominator
make
the
cancelled
completely.
expressions
when
in
simplifying.
7}
Practice
Write
restrictions
the
these
2
expressions
in
their
simplest
form.
State
any
restriction(s)
on
the
variable.
In
3
5
10 a
4
5x
1
6
+ 3x
42 x
2
the
4
2a
simplest
3
2x
35 x
numerator
7
y
and
(x
− 7 )( x
+ 8)
(a
− b )( a
+
2b )
x
5
+
6x
+ 7 )( x
+ 8)
(a
2
x
x
− 12
−
a)
+ 7x
− 3x
x
− 10
2
− 12 x
x
−
+ 32 x
9
2
2
−
+ 10
3
x
common
factors.
x
8
2
x
2b )( b
2
+
7
−
no
+ 8
6
2
(x
denominator
have
2
4
form
5
y
3
6
x
+
2x
x
− 35
− 7x
+ 12
2
x
+
4 x
−
21
10
2
x
− 3x
− 18
Problem
11
Write
an
solving
unsimplied
expression
in
the
rational
numerator
expression
and
the
in
x
that
restriction
on
x
12
Write
a
rational
expression
in
x
which
simplies
Operations
on
rational
x
a
that
quadratic
x
∉{
2,
5}.
+ 3
to
,
x
C
has
and
where
x
∉{6,
10}.
10
algebraic
expressions
●
How
do
operations
operations
You
can
multiply
and
on
on
rational
divide
rational
algebraic
numerical
algebraic
expressions
compare
to
expressions?
fractions
in
the
same
way
as
numeric
fractions.
Exploration
3
1
fractions
Multiply
these
i
simplifying
ii
multiplying
rst,
rst,
by:
and
and
then
then
multiplying,
simplifying.
Continued
on
next
page
10.2 A thin line divides us
377
Explain
which
method
a
2
Use
you
found
b
your
easier.
c
preferred
method
from
step
1
to
multiply
these
algebraic
fractions.
a
,
x
≠
−3,
x
≠
b
−5
,
x
c
3
,
Divide
your
these
fractions,
Use
x
3,
1,
∉{2,
leaving
your
±2}
3,
±4}
answers
in
simplest
form.
Explain
method.
a
4
∉{
b
your
your
method
in
c
step
3
to
divide
these
algebraic
fractions.
Explain
method.
a
,
x
≠
−3,
x
≠
−5
b
,
c
,
x
x
∉{±3,
∉{2,
0,
3,
5}
±4}
The
Reect
and
discuss
reciprocal
3
of
●
Write
down
two
equivalent
fractions.
Take
the
reciprocal
of
.
It
is
the
each.
multiplicative inverse.
Are
●
the
What
new
are
fractions
Example
Simplify
fractions
the
and
also
dierences
equivalent?
between
multiplying
and
multiplying
dividing
and
numeric
dividing
algebraic
fractions?
4
completely
and
state
the
restrictions
on
the
variable.
Factorize
To
divide
by
a
the
fraction,
quadratic
multiply
by
expressions.
its
reciprocal.
Simplify.
Include
,
x
∉{± 3,
values
of
x
that
original expressions
378
make
the
4}
10 Change
equal
to
0.
denominators
of
the
A LG E B R A
Practice
Simplify
3
each
rational
expression.
State
any
restrictions
3
25 p
66 p
3 p
×
1
4
5x
5k
+ 10
24 p
6x
10 c
6x
18
12 y
4 x
x
+ 8
2
+ 3 y
3
x
−
x
−
2
x
1
a
×
5
2
+
2x
x
+ 1
− 10
2a
+
2x
− 3
−
6
2b )
2
x
+ 5x
−
24
x
2
x
+
x
−
+
2x
−
48
÷
2
x
(a
8
2
+
2
b
2
x
×
x
+ b
2
a
2
− 3x
2
a
÷
+ 1
2
x
7
+ 9y
2
ab
3
−
2x
6
2
x
3
÷
4
9
c
3
5c
×
3x
variables.
÷
5
3
the
5
c
2
55 k
on
2
x
2
− 9x
+ 18
x
2
+ 3x
− 18
2
2
2
x
9
2x
4 x
+ 3
Reect
●
before
●
In
a
+ 5x
and
Should
you
or
nd
nding
are
series
two
or
resistance
in
(R
6
the
+
− 10
2x
2
x
6x
4
+ 12
2x
1
Find
a
R
the
=
on
the
Explain
when
would
variable
your
the
the
ways
For
you
Explain
of
nd
your
connecting
components
circuit
R
ohms,
R
in
a
multiplication
the
restrictions
=
R
2
ohms,
R
=
3
ohms,
or
electrical
connected
is:
components
in
parallel,
(where
together:
the
R
,
total
R
,
...
are
2
in
a
circuit
=
8
ohms
=
6
ohms
=
5
ohms
with:
R
2
When
two
combined
3
before
2
1
2
–
reasoning.
2
1
c
4
components).
resistance
1
b
+
reasoning.
1
of
+ 5
6x
4
of
total
4
+
4
reciprocal?
possible
resistances
6x
2
x
tot
the
−
÷
2
4 x
restrictions
question,
the
parallel.
)
2x
multiplying?
Exploration
There
+
discuss
after
division
after
10
2
x
− 3x
÷
2
−
x
6x
÷
2
x
in
2
×
9
Write
resistances
resistance
down
a
are
is
connected
greater
necessary
step
or
less
before
in
parallel,
than
the
adding
determine
individual
fractions
if
the
resistances.
together,
and
explain.
4
Find
twice
5
Find
an
expression
the
an
resistance
expression
resistance
2
ohms
for
of
for
the
the
the
more
total
resistance
when
one
component
has
when
one
component
has
other.
total
than
resistance
the
a
other.
10.2 A thin line divides us
379
Example
Find
each
5
rational
a
expression
in
simplest
form.
b
If
a
×
×
there
is
between
the
LCM
no
the
is
common
factor
denominators
simply
the
then
product
of
the
denominators.
Simplify
x
∉{
5,
your
nal
answer
completely.
3}
There
is
a
common
factor
between
b
the
two
denominators:
(x
2).
×
With
now
x
Practice
In
C.
organize
Practice
any
3
4,
show
each
on
the
the
using
steps
in
a
3}
logical
your
str ucture.
working
expression.
4
x
the
in
a
logical
order.
Remember
to
include
variables.
rational
+
1
add
Communicating
information
restrictions
Simplify
common
4
Objective:
v.
∉{2,
a
State
restrictions
1
3
1
2x
4 x
2
y
any
on
variables.
1
2
+
3
the
3
+
4
2
3x
ab
6x
5
−
ac
bc
2
1
1
x
+
2
x
−
1
2
x
+ 5
x
3 8 0
+
4
− 10
4
8
10 Change
x
+
4
− 5x
−
24
x
+ 3
− 1
2x
+ 1
x
x
2
6
+
x
−
8
x
7
11
2
x
x
+
x
− 3
+
10
+ 3x
x
5x
2x
2
2
7
6
9
x
x
5
x
+
5
3
5
x
−
−
x
3
5
+
12
x
+ 1
x
+ 3
4
x
− 5
denominator,
terms.
you
can
A LG E B R A
You
have
fractions.
●
seen
Simplify
dividing
●
To
that
algebraic
Simplifying,
rational
by
multiply
multiplying
●
To
divide
●
To
add
rational
It
is
behave
and
by
in
rst
the
expressions,
ecient
in
much
dividing
factorizing
numerator
multiply
to
the
involve
the
simplify
the
and
same
very
as
numerical
processes:
expression,
and
then
denominator.
numerator
the
way
similar
rational
and
multiply
expressions
the
before
them.
expressions,
subtract
rational
multiply
by
expressions,
the
rst
reciprocal
rewrite
the
of
the
divisor.
fractions
with
a
denominator.
Finding
D
factors
more
rational
and
common
expressions
common
denominator.
fractions
multiplying,
restrictions
on
variables
using
technology
●
Does
technology
Exploration
1
Exploration
or
graphing
Describe
3
State
4
Explain
5
Repeat
6
Explain
you
saw
to
hinder
or
not
that
draw
similarities
whether
expressions
2
software
any
or
understanding?
5
In
2
help
the
or
simplies
the
graphs
dierences
GDC
or
of
to
both
between
graphing
x
3.
Use
your
GDC
expressions.
the
graphs.
software
has
graphed
both
correctly.
whether
steps
2
or
to
whether
4
not
you
think
for
you
both
which
should
expressions
simplies
always
trust
to
results
x
are
+
you
equal.
3.
obtain
using
technology.
Summary
●
A
formula
algebraic
is
an
equation
relationship
that
describes
between
two
or
an
Rational
●
sets
of
values.
Each
variable
in
a
expressions:
more
formula
must
exclude
denominator
take
of
●
dierent
the
other
Changing
way
of
the
A
It
or
on
in
the
algebraic
contains
of
the
the
formula
rearranging
usually
variable
rational
that
subject
changing,
dierent
depending
involves
is
the
values
of
x
that
make
a
in
the
expression
equal
to
0
values
variables.
relationships.
●
values,
all
can
a
algebraic
isolating
a
●
can
be
simplied
●
can
be
multiplied,
subtracted
using
by
dividing
divided,
similar
common
added,
rules
as
factors
or
for
numerical
fractions.
formula.
expression
is
a
fraction
variables.
10.2 A thin line divides us
3 81
Mixed
practice
3
2
Rearrange
the
formulae
to
make
the
variable
in
y
16 x
+
11
2
y
15 z
×
12
3
x
brackets
the
500T
3
3
V
πr
=
[r]
2
P
=
x
[T]
3
2
v
2
y
+
m
3
y
3
x
−
3
y
4
y
x
4
y
1
m
1
F
x
÷
13
x
3
8 xy
5z
subject.
4
1
5
2
2
[r]
= G
W
4
=
CU
[C]
2
2
r
Simplify
2
5
E
=
mc
[c]
v
6
c
v
c
+ v
the
= v
these
expressions.
State
the
restrictions
on
variable.
[c]
0
x
−
2 y
1 +
2 xy
7
14
4
Simplify
each
restrictions
rational
on
the
expression.
State
x
x
− 7
x
x
+ 1
x
x
1
x
+ 5
any
variables.
3
2
+
16
7
4
12 x
5
+
15
5
5x
y
7
x
4
17
x
+ 9
8
+
4
6x
8
3
36 xy
4 x
9
9x
18
2
x
−
4 x
−
21
x
9
+
4
3
1
x
1
a
+ b
−
a
b
4
4
10
2
x
2
− 8x
− 33
x
16
x
+ 1
2
+
20
6
Review
in
Scientic
x
−
x
x
2
context
and
technical
innovation
h
Find
the
time
described
1
5
+
19
2
x
An
important
relationship
in
the
study
of
i
,
=
where
V
is
the
velocity
of
an
is
Find
or
distance
and
t
is
the
time
taken.
has
exercise,
speed
a
cruising
consider
for
the
speed
this
entire
of
speed
870
to
An
the
In
On
a
ight
ying
with
from
j
average
ight.
a
headwind
to
of
Montreal,
40
wind
speed
km/h.
if
the
and
ight
speed
of
the
it
the
wind
tailwind)
minutes
if
speed
the
and
30
and
ight
direction
took
Find
the
2
When
resistors
are
connected
in
the
A333.
return
ight,
it
is
ying
with
a
km/h.
Find
the
actual
speed
of
the
Write down
of
the
A333
an
expression
with
a
for
headwind
the
of
x
is
R
given
+
R
Write down
of
the
A333
an
expression
with
a
for
headwind
the
of
y
actual
speed
distance
resistors
actual
Write
5560
between
Brussels
and
e
Make
Find
connected
an
of
one
a
of
smallest
.
R
3
the
sum
in
series,
of
the
the
total
expression
for
individual
circuit
them
is
resistor
with
3
the
total
resistors,
4ohms
and
the
larger
largest
than
resistor
Montreal
2
ohms
t
the
the
subject
time
it
of
the
larger
than
twice
the
smallest
takes
to
travel
Simplify
Find
the
without
b
any
Write
down
resistance
time
it
took
on
the
ight
the
described
an
expression
when
one
in
a
a
is
in
a.
Then
a
a
10 Change
into
a
single
resistor
for
the
connected
the
total
same
in
as
series
the
to
the
in
circuit
3 8 2
expression
fraction.
formula.
largest
part
=
R
2
km.
wind.
g
are
simply
down
where
one.
f
is
resistance
speed
km/h.
is
is
+
R
total
1
resistances.
the
The
the
1
+
by
parallel,
km/h.
a
d
(headwind
hours
A333.
resistance
c
and
tailwind
When
30
6
1
1
of
(headwind
hours
seconds.
tot
On
6
is
resistance
b
ight
direction
took
1
actual
return
minutes.
Find
or
Brussels
the
this
2
a
on
b
Airbus
km/h.
be
the
tailwind)
57
A333
took
object,
t
d
it
part
motion
d
is V
in
single
fraction.
simplify
the
expression
into
A LG E B R A
3
Ohm’s
an
Law
states
electrical
that
circuit
the
with
voltage
current
(V ),
(I )
in
and
Problem
4
resistors,
R
and
r
connected
in
series,
is
given
Scientists
=
IR
+
calculate
the
gravitational
force
by
between
V
solving
two
any
two
masses
(m
and
m
1
Ir.
Gm
formula
F
using
2
m
1
the
)
2
=
,
where
r
is
the
distance
2
a
Make
I
b
Circuit
the
subject
of
the
formula.
r
between
A
has
a
resistor
(R)
of
6
ohms
and
a
smaller
resistor
with
resistance
r,
series.
Circuit
B
has
a
resistor
Determine
(R)
of
and
smaller
a
smaller
resistor
in
resistor
Circuit
three
A.
times
Circuit
A
a
B
are
voltage
the
total
of
connected
12
volts.
current
in
parallel,
Find
owing
an
in
of
the
gravitational
constant.
many
times
larger
the
force
gravity
is
when
masses
is
the
distance
between
the
halved.
the
and
One
mass
is
doubled
and
the
distance
have
expression
both
how
circuits
between
the
masses
is
factor
Determine
increased
by
scale
for
4.
the
eect
on
the
in
gravitational
terms
is
and
b
circuit
G
5
two
ohms
and
connected
of
in
them
a
force
between
them.
r
c
One
the
mass
is
distance
scale
factor
doubled,
between
4.
the
other
them
Determine
is
is
tripled
reduced
the
eect
and
by
on
the
Tip
gravitational
The
total
current
is
divided
between
if
they
are
connected
in
One
mass
force
minimum
a
gravitational
equation
speed
rocket)
for
eld
the
required
to
break
is
called
escape
for
free
an
of
halved
to
the
larger.
and
change
Find
the
other
to
keep
how
the
the
is
made
distance
gravitational
same.
object
Earth’s
‘escape
velocity
is
times
needs
(usually
them.
parallel.
eight
The
between
the
d
circuits
force
v
velocity’.
of
an
The
object
is
e
2G m
E
given
by:
v
e
=
where
G
is
the
gravitational
r
constant,
m
is
Earth’s
mass,
and
r
is
the
distance
E
between
The
is
Earth’s
escape
center
velocity
approximately
Now,
fuel
rockets
to
This
break
fuel
more
lighter
objects
000
a
mass
miles
you
need
and
How
you
object.
the
Earth
hour.
amount
of
gravitational
weight
more
the
leaving
per
Earth’s
discovering
Reect
and
tremendous
from
considerable
vehicles,
have
for
require
away
adds
thrust,
25
of
fuel.
to
It’s
more
the
a
pull.
rocket,
vicious
ecient
and
circle
fuels
and
thus
that
new
it
takes
more
scientists
methods
thrust
hope
of
to
to
lift
it.
overcome
But
by
to
create
building
propulsion.
discuss
explored
the
statement
of
inquiry?
Give
specic
examples.
Statement of Inquiry:
Representing
helped
change
humans
apply
and
equivalence
their
in
a
understanding
variety
of
of
forms
scientic
has
principles.
10.2 A thin line divides us
3 83
r
Getting
more
done
10.3
in
less
Global
time
context:
Objectives
●
Finding
a
Globalization
Inquiry
constant
of
●
proportionality
and
sustainability
questions
How
do
you
nd
a
constant
of
F
●
proportionality?
Setting
up
equations
direct
to
and
model
indirect
a
proportion
●
situation
What
MROF
look
●
Graphing
●
Recognizing
direct
and
direct
indirect
and
does
a
proportional
relationship
like?
relationships
inverse
proportion
●
What
●
How
does
it
mean
to
be
proportional?
C
from
●
graphs
Identifying
tables
of
direct
and
inverse
proportion
does
changing
proportional
from
one
variable
relationship
aect
in
the
a
other?
values
●
Can
situations
seem
proportional
when
D
they
●
ATL
Is
are
not?
simpler
better?
Transfer
Combine
knowledge,
understanding
and
skills
to
create
products
or
solutions
5.1
Statement of Inquiry:
Changing
and
always
to
simplied
forms
can
help
analyze
the
eects
of
consumption
conservation.
10.3
E5.1
3 8 4
N U M B E R
Y
ou
●
should
express
a
already
ratio
as
a
know
how
fraction
1
Express
in
its
a
5
c
●
write
linear
real-life
functions
to
model
to:
2
:
a
:
earns
function
b
A
graphs
of
linear
functions
3
fraction
2
d
0.2
:
per
the
4
:
1.2
hour. Write
amount
he
a
earns
in
hours.
taxi
any
draw
a
b
£10
for
ride
costs
mile. Write
●
as
form.
13.3
John
h
ratio
3
4.5
situations
each
simplest
Draw
a
$2
plus
function
$4.50
for
the
per
cost
of
journey.
graphs
of
the
two
functions
above.
2
●
draw
graphs
of
quadratic
4
Draw
functions
If
you
the
●
How
●
What
sell
make
much
is
a
of
you
does
as
We
variables
constant
∝ x
do
means
say
the
two
to
sell.
If
of
in
of
you
you
directly
as
x’
the
sell
sell
direct
each
of
from
y
3
=
to
x
for
integer
+3.
proportionality?
mats,
If
x
relationship
quantities
be
values
varies
mouse
you
of
graph
variation
constant
money.
said
all
a
proportional
much
are
‘y
a
mats
say
for
inverse
nd
items,
mouse
twice
money.
Two
and
identical
number
will
y
values
Direct
F
the
money
twice
half
are
look
as
like?
you
as
many,
directly
proportion
make
many
varies
mouse
you
make
with
mats,
half
you
as
proportional.
if,
and
only
if,
their
ratio
variable.
or
‘y
is
directly
proportional
to
x’.
ATL
Exploration
A
store
1
sells
Write
regular
down
regular
Determine
3
Draw
to
4
10
From
a
To
by
meet
LED
2
Euros
represent
your
variable
the
variables
is
representing
per
bulb.
store’s
and
dependent,
the
by
4
growing
for
determine
number
b
the
bulbs
to
for
income
justify
your
from
selling
choice
of
letters
to
variable.
graph,
the
2
bulbs
amount
and
of
which
money
one
made
is
independent.
from
selling
up
bulbs.
your
multiply
Identify
which
graph
light
light
function
each
2
a
a
bulbs.
represent
1
7
of
c
light
by
demand
Euros
per
what
bulbs
5
for
happens
to
the
income
when
you
sold:
d
more
by
12
ecient
e
by
bulbs,
14
the
station
also
sells
bulb.
Continued
on
next
page
10.3 Getting more done in less time
3 8 5
5
Using
the
represent
your
6
same
the
graph
Determine
to
the
Justify
●
Describe
a
increases,
∝ x
k
is
In
x
means
called
The
and
y
a
●
In
°
●
a
kx
y
is
shop
is
a
1,
money
Justify
if
in
one
made
your
new
function
bulbs.
Draw
a
to
line
on
is
in
direct
proportion
answer.
Exploration
quantity
constant
called
the
a
LED
1
go
through
increases,
the
the
origin.
other
proportion.
a
linear
form
k,
where
constant,
proportional
of
of
sold.
drew
direct
for
down
y
to
=
or
then
≠
0.
constant
variation
x,
k
the
of
variation.
function
relationship
between
kx.
2
identify
the
proportionality
constant
for
the
bulbs
bulbs.
the
of
constant
the
form
ratio
of
x
y
=
kx,
and
y.
the
proportionality
Explain
why
you
constant
think
this
k
is.
1
down
number
kx
write
selling
1
you
relationships
charges
Write
amount
discuss
light
called
Example
A
if
light
linear
also
=
1,
from
relationship.
where,
in
step
for:
regular
For
is
=
Exploration
LED
this
proportionality
and
function
°
y
in
bulbs
graphs
not
function
Reect
the
light
situation
y
words,
is
the
that
the
of
as
income
discuss
but
function
other
whether
why
of
represent
and
●
y
to
number
Reect
variables
amount
of
€1.50
a
for
a
function
liters
of
liter
to
of
mineral
represent
mineral
water
water.
the
and
relationship
the
total
price
between
of
the
mineral
water
purchased.
b
Hence,
c
Show
a
determine
that
the
number
of
proportionality
liters
and
price
constant.
are
in
direct
proportion
by
drawing
graph.
Continued
3 8 6
10 Change
on
next
page
N U M B E R
a
P
l
=
=
price
number
P (l )
b
k
=
=
so
of
liters
1.50l
For every extra liter the price increases by
1.50
The
c
Dene the variables.
P
graph
and
l
is
a
are
straight
in
direct
line
that
passes
through
the
€1.50.
origin,
proportion.
P
6
5
soruE
4
ni
3
The total price depends on
the number of liters, so P is the
ecirP
dependent variable and N is the
2
independent variable.
1
0
l
1
2
Number
If
the
graph
graph
can
is
be
Reect
●
How
is
linear
●
a
4
liters
straight
written
and
the
as
line
y
=
passing
kx,
discuss
and
through
x
is
the
directly
origin
then
the
proportional
to
equation
of
the
y
3
proportionality
constant
represented
on
the
graph
of
a
direct
proportion?
Explain
why
represent
Example
The
3
of
a
a
linear
graph
proportional
that
does
not
go
through
the
origin
does
not
relationship.
2
variable
y
varies
as
x.
When
x
=
4,
y
=
10.
‘y
Find
the
value
of
y
when
x
=
varies
means
7.
as
‘y
is
x’
directly
proportional
y
∝ x
10
=
means
k
×
that
y
=
to
x’.
kx
4
⇒
When
x
=
7,
y
=
2.5
×
7
=
17.5
Use
the
proportionality
constant
to
nd
10.3 Getting more done in less time
y
when
x
=
3 87
7.
Practice
1
The
1
variable
represent
a
c
if
b
c
this
is
is
directly
proportional
relationship.
State
to
what
doubled
the
variable
happens
b
b
d
c
if
b.
Write
an
equation
to
to:
c
is
tripled
4
is
added
2
c
c
if
b
is
multiplied
by
if
to
b
3
2
3
4
The
variable
a
Write
b
Find
the
c
Find
v
P
is
the
v
is
function
value
when
directly
t
V
Two
variables
Find
the
when
P
x
the
represents
constant
to
V.
to
waste
a
Q
are
in
direct
from
4.7
reduce
Mass
of
Verify
When
of
t
=
5,
v
=
40.
proportionality.
V
=
3.2,
P
=
8.64.
this
proportion.
table:
6.1
13.72
trash
in
Composting
science
dumped
Number
relationship
t.
solving
waste.
In
variable
proportionality.
When
6.58
order
of
the
the
5.4.
values
0
farms.
a
that
to
25.
and
missing
Problem
food
=
=
Q
In
of
proportional
proportional
Find
x
5
directly
of
in
landlls,
produces
experiment,
a
composter
containers
soil
that
(kg)
the
number
many
rich
Kent
at
one
people
soil
that
records
time
the
and
are
can
beginning
then
be
number
the
mass
of
2
3
7
9
1.6
2.4
5.6
7.2
of
containers
and
mass
of
of
soil
to
used
compost
in
gardens
containers
soil
are
or
offood
produced.
in
direct
proportion.
b
Find
the
constant
c
Calculate
the
of
proportionality.
amount
of
soil
produced
from
y
5
containers
of
food
waste.
6
6
Determine
which
of
these
functions
represent
5
d
direct
proportion,
and
nd
the
constant
of
4
proportionality
for
those
that
a
do.
3
2
e
c
1
x
0
6
5
4
3
2
1
1
2
1
2
3
4
b
f
5
6
3 8 8
10 Change
3
4
5
6
N U M B E R
Objective:
B.
ii.
patter ns
describe
In
this
Investigating
Exploration,
models
the
as
you
relationship
patter ns
general
will
r ules
nd
between
a
consistent
general
the
time
rule
in
needed
with
terms
to
plant
ndings
of
a
specic
trees
and
function
hourly
that
income.
ATL
Exploration
In
an
eort
products,
they
1
the
One
total
Hence,
Copy
hour
and
Time
a
can
from
it
needed
been
summer
take
table
to
of
harvested
that
trees
2400
income
groups
job
300
planting
would
the
dierent
have
plant
student’s
complete
2400
that
take
long
this
the
(t)
plant
can
student
how
nd
for
trees
income
Determine
2
replace
students
plant.
Find
to
2
pays
one
8
consumer
cents
per
tree
that
hour.
trees.
one
per
in
for
student
to
plant
2400
trees.
hour.
show
the
time
taken
and
income
per
students.
1 student
2 students
8
4
4 students
8 students
to
trees
(h)
Income (I) per
24
hour ($/h)
3
Predict
the
time
calculations
your
4
down
a
a
the
to
of
students
trees
is
them
halves
by
a
same
If
and
y
∝
are
Write
taken
●
Does
in
x’
and
Reect
●
per
hour
Add
for
the
16
students.
values
for
16
Do
some
students
to
time
and
the
income
between
the
taken
y
Generalize
this
relationship
the
per
pattern,
between
and
the
write
number
of
hour.
number
of
students
and
the
time
to
are
is
directly
income
to
plant
the
results
proportional:
hour.
2400
number
inversely
number
per
But
trees.
of
The
students
proportional
in
the
other
doubling
doubling
time
taken
doing
if
the
the
to
plant
of
the
planting.
multiplying
variable
the
number
being
one
of
divided
by
number.
an
or
the
trees.
proportional
x
table.
relationship
non-zero
as
and
doubles
non-zero
y
inversely
this
trees
2400
one
the
variables
the
x
2,
students
inversely
Two
income
predictions.
represent
relationship
plant
Exploration
number
and
your
from
to
planting
Describe
needed
pattern
function
students
In
needed
check
table.
Identify
5
to
‘y
inverse
is
linear
inversely
proportion,
proportional
you
to
can
x’,
say
and
‘y
you
varies
can
write
.
and
the
to
relationship
plant
the
discuss
2400
function
4
between
trees
have
a
(t)
the
from
number
of
Exploration
constant
of
students
2
as
a
(s)
and
time
function.
proportionality?
Explain.
10.3 Getting more done in less time
3 8 9
Graphs
If
y
is
inversely
proportional
to
x,
then
y
is
directly
proportional
to
of
.
are
An
equation
or
represents
a
relationship
of
called
inverse
reciprocal
proportion
Example
It
takes
there
the
b
Determine
c
Write
Draw
a
2
×
time
if
down
the
the
16
=
2
hectares
Find
prepare
inverse
person
16
a
d
an
graphs.
relationship
3
one
are
or
for
this
a
to
be
one
prepare
relationship
and
of
to
the
this
one
hectare
of
eld
for
planting
and
prepared.
person
function
eld
graph
32
days
to
to
prepare
represents
represent
number
of
the
the
16
direct
hectares
or
inverse
relationship
for
planting.
proportion.
between
the
time
to
workers.
function.
days
One
person,
two
days’
prep
time
per
person,
total
time
=
32
days.
b
w:
t:
number
time
The
prep
This
c
t
to
of
workers
prepare
time
describes
16
halves
an
hectares
as
the
inverse
1
2
4
8
32
16
8
4
number
of
workers
doubles.
proportion.
⇒
∝
k
Substitute
=
values
for
t
and
w,
and
solve
for
k
32
t
d
35
30
Plot the points from the table of values.
emit
Even
25
noitaraperP
own
though
right
a
the
function
smooth
is
curve,
in
its
only
the
20
points
with
integer
values
on
the
n
15
axis
are
meaningful
since
there
can’t
10
be
5
0
w
0
1
2
3
4
Number
3 9 0
10 Change
5
of
6
workers
7
8
9
10
a
non-integer
number
of
workers.
N U M B E R
Practice
1
For
i
2
each
part
a
to
e:
Determine
whether
proportion
or
ii
If
there
is
a
a
The
b
The
number
that
3
there
neither.
proportional
perimeter
p
of
of
an
liters
is
a
Justify
relationship
your
relationship,
equilateral
n
of
paint
of
direct
proportion,
inverse
answer.
nd
triangle
needed
k
with
to
side
paint
an
length
area
l.
A,
given
2
liters
will
paint
a
15
m
area.
3
c
y
=
4x
x
y =
d
e
y =
x
2
3
The
variable
a
Write
b
Find
the
c
Find
f
y
is
in
Find
4
5
T
is
the
when
inverse
y
when
x
when
Two
variables
to
their
d
Write
b
Mark
a
of
t
to
are
values
each
of
When
v
=
6,
f
=
4.
proportion.
proportionality.
x.
When
x
=
5.2,
y
=
2.
to
m.
When
m
=
1.8,
T
=
4.
in
in
inverse
this
proportion.
s
4
t
0.8
table:
4.8
2
Archimedes’
person
from
the
Law
of
pivot
the
must
Lever
be
says
that
inversely
the
proportional
w
a
function
20
when
kg
to
and
Emilie
represent
is
1.5
sits
m
1.25
the
from
m
Law
the
from
of
the
pivot.
the
Lever.
The
pivot.
seesaw
Use
this
is
information
k
nd
Emilie’s
has
Sumatran
Tiger’
endangered
of
decided
animals.
people
of
Cost
person
per
weight.
to
which
Number
Verify
relationship
v.
3.1.
seesaw,
Milijenko
a
variable
2.
and
weighs
Hence,
number
of
the
the
1.6.
=
s
down
nd
constant
to
solving
balanced
c
m
weight
a
to
=
missing
balance
distance
=
the
represents
proportional
T
To
v
of
proportional
that
proportion
Find
the
inversely
value
Problem
7
is
function
inversely
Find
6
f
5
encourage
funds
The
cost
that
the
of
participating
people
number
her
research
class
and
adoption
as
shown
to
participate
conservation
per
in
person
the
table
in
of
‘Adopt
these
depends
a
critically
on
the
below:
1
4
8
10
20
$300
$75
$37.50
$30
$15
of
people
and
the
cost
per
person
are
inversely
proportional.
b
Find
the
constant
c
Calculate
the
cost
of
proportionality.
per
person
when
25
people
participate.
10.3 Getting more done in less time
3 91
8
Conservation
number
related
table
of
tigers
tothe
Increase
of
in
US
on
the
wild.
in
the
adoption
The
program,
amount
number
of
spent
wild
are
on
tigers
trying
to
double
conservation
as
shown
in
the
is
the
conservation
population
Determine
whether
the
proportion.
type
of
Write
c
Determine
goal
down
of
their
an
how
adding
2.4
4.9
8.3
390
624
1274
2158
or
not
equation
much
3 200
this
Justify
relation
your
relating
will
tigers
need
to
the
to
the
is
in
a
proportion,
and
if
so,
state
answer.
two
be
wild
variables.
spent
to
be
on
conservation
met.
(This
will
for
the
double
population.)
Determine
whether
relationships.
i
1.5
$)
tiger
b
a
the
increase
spent
(millions
9
in
like
below:
Amount
a
eorts,
The
the
Justify
relationship
following
your
relationships
are
proportional
answer.
between
the
distance
and
the
time
to
see
the
between
the
distance
and
the
time
to
hear
lightning
ii
The
relationship
the
thunder
iii
The
relationship
hear
b
the
Lightning
i
ii
iii
strikes
time
km
long
it
takes
to
see
Find
how
long
it
takes
to
hear
verify
●
What
●
How
You
have
●
=
seen
is
a
to
see
the
lightning
and
the
time
to
away.
how
Hence,
kx
the
Find
your
does
does
aect
y
5
Dierent
C
between
thunder
the
answer
types
it
mean
to
changing
to
of
be
one
the
a
lightning.
the
thunder.
iii
variation
proportional?
variable
in
a
proportional
relationship
other?
examples
of
relationship
these
of
proportional
direct
proportion
relationships:
between
x
and
y
k
●
=
is
a
relationship
of
inverse
proportion
between
x
and
y
x
In
the
higher
next
Exploration,
powers
3 9 2
of
x
as
you
will
well.
10 Change
see
that
y
can
vary,
directly
or
inversely,
with
N U M B E R
ATL
Exploration
This
circle
has
3
radius
down
1
Write
2
a
Find
the
new
area
if
you
double
b
Find
the
new
area
if
you
triple
c
Find
the
new
area
if
you
halve
d
Multiply
the
e
Suggest
of
3
new
the
Justify
Write
Find
5
This
a
radius
general
is
or
down
not
of
of
side
the
rule
and
constant
what
by
a
the
few
for
area
A
of
the
the
the
circle,
in
terms
of
r
radius.
radius.
the
more
what
multiplied
area
has
for
radius.
r
factors,
and
nd
eachtime.
statement
cube
Find
a
circle
the
the
Write
the
formula
area
whether
between
4
the
r.
the
the
a
happens
constant
formula
radius
to
c,
represents
of
the
proportionality
the
area
where
a
c
when
≠
the
radius
0.
proportional
relationship
circle.
linking
area
A
and
radius
r.
proportionality.
length
formula
happens
by
to
x
for
the
the
volume
volume
if
V
you
the
cube.
double,
of
halve
x
or
triple
what
the
side
happens
multiplied
by
length.
to
a
the
Suggest
volume
constant
c,
a
general
when
where
c
the
≠
rule
side
for
length
is
0.
x
x
6
Justify
whether
between
7
Write
Find
Two
to
a
a
the
or
not
volume
statement
the
variables
power
of
constant
of
x
x,
of
and
y
the
and
formula
the
side
represents
length
proportionality
of
a
the
linking
proportional
relationship
cube.
volume
V
and
side
length
x.
proportionality.
are
in
direct
or
non-linear
proportion
if
y
is
proportional
.
n
The
variation
function
is
y
=
kx
,
k
≠
0
and
n
>
0.
n
y
varies
directly
as
Exploration
Diego
1
plans
Write
to
down
x
n
or
y
is
in
direct
proportion
to
x
4
tile
a
the
wall
with
formula
for
square
the
tiles
area
A
of
side
of
a
length
square
l.
tile.
t
2
Let
A
represent
the
area
of
the
wall.
w
Let
a
the
number
Write
down
of
an
tiles
=
n.
expression
for
n
in
terms
of
A
and
A
and
l
w
b
Write
down
an
expression
for
n
in
terms
of
A
t
w
Continued
on
next
page
10.3 Getting more done in less time
3 9 3
3
Determine
whether
proportional
4
State
side
the
5
Write
the
the
of
number
area
proportional
length
length
to
each
of
the
of
tiles
relationship
tile.
Find
is
directly
or
inversely
tile.
the
between
constant
the
of
number
of
tiles
and
the
proportionality.
l
down
the
formula
for
the
area
A
of
this
smaller
tile.
s
6
Find
the
number
m
of
new
tiles
he
needs
to
cover
Write
A
down
an
w
expression
a
in
for
terms
of
m:
A
and
A
w
b
in
terms
of
s
A
and
l
w
7
Compare
n.
number
b
the
proportionality
you
length
Does
Two
and
this
The
that
height
aect
variables
proportional
tiles
x
to
variation
does
the
and
a
are
function
with
the
to
be
a
whole
number
of
tiles
in
of
of
an
,
tiles
you
inverse
need
cover
non-linear
or
.
is
to
and
n
You
>
the
In
varies
inversely
Exploration
4
as
you
the
smaller
●
the
number
●
the
proportionality
the
problem
of
tile,
tiles
Explain
what
variation
=
the
wall
also
entirely?
if
y
write
is
this
as
0.
y
is
inversely
have
more
tiles
inversely
constant
found
k
proportional
to
x
that:
you
need
for
proportional
changes
only
to
the
the
when
given
side
the
area
length
initial
of
each
tile
conditions
of
5
happens
to
y
when
x
is
multiplied
by
a
constant
c
in
these
functions:
2
y
along
change.
Exploration
●
the
is
aect:
n
or
should
●
the
x
length
proportion
can
n
y
side
wall?
number
y
tiles
5
has
the
power
to
constant?
there
of
changing
needed
discuss
think
and
of
How
the
Do
●
and
a
Reect
●
m
kx
y
=
3
kx
y
=
kx
n
●
Generalize
3 9 4
your
results
rst
10 Change
for
y
=
kx
,
n
>
0,
and
then
for
,
n
>
0.
N U M B E R
n
●
y
=
kx
is
a
relationship
of
direct
variation.
When
x
is
multiplied
by
a
n
constant
●
is
c,
a
then
y
is
multiplied
relationship
of
by
inverse
c
variation.
When
x
is
multiplied
by
a
n
constant
Example
The
light
distance
a
b
c
the
The
light
Draw
a
is
divided
a
movie
screen
is
3
m
intensity
what
distance
the
on
the
screen
projector
Find
d
intensity
the
y
by
c
4
Describe
the
then
between
When
Find
c,
the
on
to
to
the
the
the
the
when
varies
screen
light
is
represent
is
the
6
light
m
intensity
and
intensity
the
with
the
square
of
on
the
the
variation
a
intensity
is
24
units.
away.
when
the
distance
between
screen
is
increased
by
25%.
screen.
function.
Choose
sensible
variable
names
Substitute
k
=
24
×
9
=
the
halved.
projector
light
inversely
projector.
projector,
the
the
screen
between
eect
graph
from
happens
and
screen
and
and
the
write
the
known
variation
values
and
function.
solve
for
k
216
Find I when d = 6.
⇒
The
light
intensity
at
6
m
is
6
units
of
illumination.
b
Replace
this
When
the
distance
is
halved,
the
light
intensity
is
4
times
as
d
=
1.25d
with
the
d and
variation
substitute
function.
great.
Increasing
5
c
into
d
d
by
25%
gives
d
=
n
o
1.25d
4
=
.
0.64
When
64%
or
the
of
distance
the
is
increased
by
25%
the
light
intensity
is
reduced
to
original.
Continued
on
next
page
10.3 Getting more done in less time
3 9 5
d
d
is
the
independent
illumination
variable,
I depends
as
the
on
the
distance.
on
the
curve
I
35
ytisnetni
30
25
All
20
thgiL
15
points
have
meaning,
have
any
as
you
can
10
distance
d
on
5
the
0
horizontal
axis.
d
0
1
2
3
4
5
6
Distance
Example
The
table
each
a
shows
of
a
of
Determine
cable
the
is
9
10 11
(m)
1
of
four
(mm)
m
if
one
resistances
1
m
cables
of
the
same
material,
but
radius.
cable
Resistance
8
5
dierent
Radius
7
of
this
of
cable
(ohms)
relationship
direct
1
2
5
10
1.5
0.375
0.06
0.015
between
variation,
the
inverse
radius
variation
and
or
the
resistance
neither.
Justify
of
the
your
answer.
b
Hence,
of
a
the
Let
R
r
=
As
if
the
relationship
=
radius
variation
function,
write
down
the
equation
increases,
R
decreases.
variation,
If
,
=
then
Dene
k
=
or
r
×
no
pair
2nd
k
is
of
pair
not
values
of
variables.
There
is
either
Compare
the
constant,
k
gives
so
in
the
variables.
R
gives
values
changes
variation.
For
1st
the
(ohms)
inverse
R
a
(mm)
resistance
r
is
function.
R
=
k
1
=
×
2
1.5
×
=
R
=
,
k
must
be
constant.
1.5
0.375
=
0.75
Test
with
pairs
of
values
from
the
table.
≠
2
If
R
=
then
k
=
r
×
R
Try
2
1st
pair
of
values
gives
k
=
1
×
1.5
=
1.5
2
2nd
pair
of
values
gives
k
=
2
×
0.375
=
1.5
Test
with
pairs
of
values
from
the
table.
2
3rd
pair
of
values
gives
k
=
5
4th
pair
of
values
gives
k
=
10
×
0.06
=
1.5
2
Therefore,
R
=
,
k
is
which
×
0.015
=
1.5
constant.
is
an
inverse
b
variation
relationship,
because
k
is
constant.
Use
3 9 6
10 Change
the
value
of
k
from
part
a
N U M B E R
Reect
discuss
●
What
conditions
●
What
do
Practice
1
and
i
For
you
If
each
it
is
think
it
always
means
be
to
met
be
for
direct
or
variation?
‘proportional’?
table
a
an
of
values,
inverse
variation
determine
variation
function,
whether
function
write
or
down
the
functions
neither.
the
Justify
equation
a
are
a
your
of
the
direct
variation
answer.
function.
b
x
1
2
3
4
x
1
2
3
4
y
1
8
27
64
y
24
12
8
6
d
c
x
2
0
3
4
y
20
0
45
80
e
2
inverse
3
function,
ii
must
6
x
2
4
6
12
y
30
15
10
5
f
When
x
2
0
2
5
x
1
2
4
9
y
0.5
undened
0.5
0.032
y
40
22
13
8
x
=
2,
y
=
12.
Find
the
value
of
y
when
x
=
4
if:
2
a
y
varies
directly
as
c
y
varies
inversely
3
x
b
y
varies
directly
as
d
y
varies
inversely
x
2
3
y
=
24
when
x
=
5.
as
x
Find
the
value
of
x
when
y
=
2
as
x
if:
3
a
y
varies
directly
as
c
y
varies
inversely
x
b
y
varies
directly
as
d
y
varies
inversely
x
2
4
as
3
x
The
surface
area
of
a
sphere
varies
The
surface
area
of
a
sphere
with
directly
as
the
as
x
square
of
its
radius.
2
a
Find
b
From
the
c
the
surface
the
what
enlarged
d
weight
distance
the
of
in
Earth.
of
a
factor
a
sphere
above,
or
5
with
cm
is
radius
otherwise,
100π
2
cm
cm.
write
down
the
formula
for
sphere.
to
the
of
eect
surface
area
of
a
sphere
when
the
radius
is
5.
on
the
radius
of
halving
the
surface
area
of
a
sphere.
solving
an
km
object
from
3670
Find
International
surface
a
the
approximately
of
of
happens
Determine
The
its
area
by
Problem
5
of
information
surface
State
area
radius
the
the
km.
the
Space
in
A
Newtons
center
of
certain
weight
Station,
of
varies
the
Earth.
astronaut
the
which
(N)
same
orbits
inversely
The
weighs
astronaut
at
an
with
radius
850
when
average
of
N
of
at
the
square
of
the
Earth
the
surface
she’s
400
on
km
is
the
from
the
Earth.
10.3 Getting more done in less time
3 97
6
The
rings
sizes.
of
of
the
particle.
abundance
7
Under
shape
with
the
of
its
The
of
If
a
a
a
use
2-meter
perfect
of
a
low-ow
consumption.
while
low-ow
water
shower
should
is
be
the
If
a
If
your
the
graph
Direct
an
in
ounce
of
8
a
low-ow
this
an
30
of
per
shower
form
per
of
the
size
the
to
per
per
ounce.
volume
pipe
of
5
our
minute
This
the
radius
the
directly
reduce
water
of
in
ounce,
$40
minute.
a
of
determine
varies
costs
radius
with
will
$24
of
variety
percent.
way
liters
a
cube
10%,
a
pyrite
that
the
pipe
of
of
the
gold)
easy
liters
to
of
costs
pyrite
square
has
tenth
cm
use
only
the
particles
fool’s
of
5
become
head
a
of
heads
use
to
called
piece
of
abundance
nearest
length
a
up
proportional
an
the
has
may
pipe
in
cm
the
what
head?
6
calculator/software
change
the
a
of
shower
made
has
to
shower
proportional
of
of
heads
heads
be
(often
side
side
Regular
radius
Exploration
cost
with
the
regular
to
inversely
particle
pyrite
shower
shower
directly
head.
of
is
particle,
The
cube
length
found
size
size
cube.
If
water
of
been
abundance
conditions,
volume.
the
have
the
3-meter
right
determine
8
Saturn
Amazingly,
parameters
of
has
the
a
slider
variation
bar
or
similar
function
and
function
see
how
you
the
can
shape
of
changes.
variation
functions
n
Graph
the
change
1
and
graph
direct
the
3.
If
and
variation
value
your
table
of
k
between
calculator
of
function
values
10
has
a
y
=
and
kx
.
10,
Insert
and
split-screen,
a
the
you
slider
value
will
be
so
of
that
n
able
you
can
between
to
see
the
simultaneously.
1.1
f1(x):
y
k*x^n
10
n
1
0.
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
7.
7.
8.
8.
8
1
k
0.
10
6
1
4
n
10
10
y
=
kx
2
0
10
8
6
4
2
x
2
4
6
8
10
2
4
9.
9.
10.
10.
6
8
n
10
f1(x):
1
Start
with
slider,
in
the
the
and
graph
2
State
what
3
Repeat
values
describe
k
steps
as
you
k
=
and
and
the
change
represents
1
1
how
2
on
for
n
n
the
the
=
=
graph
2
1.
k
x
Change
changes.
value
of
the
value
State
what
of
k
using
does
not
k
graph.
and
n
=
3.
Continued
3 9 8
10 Change
the
change
on
next
page
N U M B E R
Inverse
variation
functions
This
reciprocal
includes
Graph
the
inverse
variation
function
graph
negative
.
values
of
x
1.1
f1(x):
k /x^n
10
0.
n
1
#U…
8
1.
1
3
k
1.
6
2.
0.5
3.
0.333333
4.
0.25
1
4
k
y
=
10
10
n
2
x
0
10
8
6
5.
0.2
6.
0.166667
7.
0.142857
x
4
2
4
6
8
10
4
6
8.
0.125
9.
0.111111
8
k
f1(x):
10
n
x
4
Start
with
slider,
5
the
and
values
describe
k
=
how
1
and
the
n
=
1.
graph
Change
State
what
happens
as
x
gets
closer
to
+10
State
what
happens
as
x
gets
closer
to
0
the
negative
6
Explain
7
Repeat
value
of
k
using
the
and
(from
10.
the
positive
side
and
from
side).
how
steps
Reect
the
changes.
the
4
and
to
proportionality
6
for
n
=
discuss
2
and
constant
n
=
aects
the
graph.
3.
7
n
●
Consider
°
°
°
°
the
variation
function
y
=
kx
Describe
how
n
aects
the
shape
of
the
graph.
Describe
how
k
aects
the
shape
of
the
graph.
Justify
what
happens
when
Justify
what
happens
as
positive
●
Repeat
●
Describe
from
●
direct
a
these
a
the
four
how
negative
steps
for
=
0.
becomes
extremely
large
in
both
the
directions.
the
inverse
you
can
recognize
a
you
can
recognize
an
variation
direct
function
variation
.
function
graph.
Describe
from
and
x
x
how
inverse
variation
function
graph.
10.3 Getting more done in less time
3 9 9
Practice
1
Write
4
down
the
variation
a
y
varies
directly
as
b
y
varies
inversely
c
y
varies
directly
d
y
varies
inversely
e
y
varies
directly
x,
as
as
as
and
x,
the
as
function
and
y
when
square
the
the
when
and
of
square
cube
of
=
y
x,
of
x,
sketch
30,
=
x
2,
x
and
x,
graph.
5.
=
6.
when
and
and
=
its
x
when
when
x
=
x
=
4,
=
2,
y
4,
y
=
y
=
120.
=
5.
62.5.
1
f
y
varies
inversely
as
the
cube
of
x,
and
when
x
=
,
y
=
4.
2
Problem
2
Determine
an
inverse
solving
whether
variation
a
the
following
function,
or
graphs
neither.
show
y
3
2
1
variation
3
2
2
1
1
x
0
3
2
1
1
2
2
3
3
3
2
2
1
1
x
1
1
2
3
3
2
0
1
1
1
2
2
3
3
3
3
2
2
1
1
x
1
1
2
the
2
1
1
2
2
3
3
graph
of
the
variation
function
y
.
=
2
x
a
Determine
Justify
whether
your
or
not
y
can
have
a
negative
value.
answer.
b
Explain
what
happens
to
y
as
x
gets
extremely
c
Explain
what
happens
to
y
as
x
gets
closer
d
Hence,
4 0 0
3
1
2
3
1
2
3
explain
why
x
x
0
3
3
1
2
Sketch
2
y
f
0
3
1
d
3
e
2
x
0
1
1
0
3
function,
answer.
b
y
2
direct
your
3
c
3
a
Justify
this
10 Change
is
an
inverse
large.
and
variation
closer
to
0.
function.
N U M B E R
Modelling
D
Until
●
Can
●
Is
now
variation
situations
simpler
you
or
have
inverse
one
change
in
inverse
variation?
you
favourite
each
1
rell
Write
go
will
3
a
Suggest
a
dealing
the
an
coee
you
a
variation
there
similar
shop,
can
you
they
are
not?
functions
situations
change
in
in
that
which,
another,
are
even
there
either
direct
though
is
no
a
direct
or
buy
equation
total
cost
for
as
option
for
you
a
cup
can
for
either
€5
pay
and
€4.50
rell
it
for
every
your
time,
though
€3.50.
variation.
rule
with
Are
causes
which
direct
when
7
or
the
Determine
not
been
proportional
relationships
better?
variation.
cost
down
represents
2
to
drink,
seem
always
variable
Exploration
When
propor tional
it
each
option
relates
represents
Justify
your
determining
to
a
and
the
sketch
number
direct
a
of
graph
drinks
variation,
and
that
bought.
which
one
is
answers.
whether
or
not
a
function
is
a
variation
function:
a
b
4
In
from
its
from
its
Describe
but
that
not
a
most
equation
graph.
a
is
situation
not
an
variation
of
the
denitions
of
technically,
it
could
Exploration
1
Sketch
a
y
=
the
Explain
3
For
and
be
b
why
each
k
practice
has
been
inverse
of
y
they
are
increases
function.
problems
positive.
proportion,
k
can
these
the
other
how
you
decreases,
know
it
is
have
be
seen
so
far,
according
any
number
to
the
the
except
0,
so
=
are
functions:
−3x
all
c
variation
function,
true
or
functions.
determine
whether
increases,
b
As
x
decreases,
c
As
x
increases,
d
As
x
decreases,
e
Explain
what
happens
as
x
gets
closer
f
Explain
what
happens
as
x
gets
farther
y
y
y
the
following
false:
x
positive
y
d
As
the
you
However,
a
in
as
Explain
negative.
variation
statements
quantity
8
graphs
3x
2
and
constant
direct
one
variation
function.
examples
proportionality
where
inverse
increases
decreases
decreases
increases
and
negative
and
closer
and
to
farther
0.
away
from
0,
directions.
10.3 Getting more done in less time
4 01
Reect
●
and
Summarize
discuss
your
proportionality
●
Describe
between
a
ndings.
and
real-life
two
8
What
inverse
is
the
dierence
between
negative
proportionality?
situation
in
which
there
is
a
negative
variation
variables.
Summary
Propor tional
●
Two
variables
are
said
to
relationships
be
in
General
direct
proportion
for
values
if,
and
k
>
shape
of
the
0
graph
k
y
only
if,
their
ratio
is
a
constant
all
of
<
0
y
each
variable.
0
k
y
=
x
●
y
∝ x
means
‘y
proportional
●
y
∝ x
k
is
means
called
varies
to
as
x’
or
‘y
is
0
x
directly
x
x’.
that
the
directly
y
=
kx
for
a
constant
proportionality
k,
where
constant,
or
k
≠
direct
0.
constant
linear
propor tion
of
variation.
●
The
●
Two
function
y
variables
=
x
kx
is
and
called
y
are
a
linear
inversely
variation
function
proportional
k
if
>
0
k
<
0
y
multiplying
one
in
variable
the
other
of
them
by
being
a
non-zero
divided
by
number
the
same
results
non-zero
0
k
y
number.
●
If
x
‘y
and
y
varies
are
in
an
inversely
inverse
as
x’
or
linear
‘y
is
proportion,
inversely
you
can
proportional
●
If
and
y
is
you
can
inversely
proportional
●
An
of
●
write
y
proportional
to
Two
to
say
or
proportion
variables
linear
propor tion
.
x,
then
y
is
directly
x
and
or
y
represents
a
are
reciprocal
in
direct
a
relationship
relationship
non-linear
proportion
k
>
0
k
y
if
y
is
x
0
.
equation
inverse
and
∝
x
to
inverse
x’,
=
x
proportional
to
a
power
of
x,
or
<
0
y
.
n
●
The
variation
function
is
y
=
kx
,
k
≠
0
and
n
>
0.
0
n
y
n
●
y
varies
●
y
=
directly
as
x
n
or
y
is
in
direct
proportion
to
=
kx
0
x
direct
non-linear
propor tion,
n
kx
is
a
relationship
of
direct
variation.
When
x
is
when
n
is
even
n
multiplied
by
a
constant
c,
then
y
is
multiplied
by
c
k
>
0
k
y
<
0
y
n
y
0
direct
=
10 Change
0
non-linear
when
4 0 2
kx
x
n
is
propor tion,
odd
x
N U M B E R
●
Two
variables
x
and
y
are
in
an
inverse
k
non-linear
>
0
k
proportion
if
y
or
is
.
proportional
Y
ou
can
also
to
a
write
power
this
of
as
.
variation
function
is
and
n
>
varies
inversely
as
x
=
n
x
n
or
y
is
inversely
proportional
to
x
indirect
non-linear
when
●
is
a
relationship
x
x
0.
n
y
0
k
y
The
0
,
0
●
<
y
y
of
inverse
variation.
When
x
n
is
propor tion,
even
is
k
>
0
k
<
0
n
multiplied
by
a
constant
c,
then
y
is
divided
by
c
y
y
k
y
=
n
x
0
indirect
non-linear
when
Mixed
1
is
propor tion,
odd
practice
Sketch
and
n
x
0
x
the
graph
determine
of
each
whether
variation
it
is
a
5
function
direct
or
p
is
inversely
Find
inverse
the
proportional
missing
values
to
in
the
this
square
of
q.
table:
relationship.
p
100
1
x
a
y
=
5x
b
y =
q
3
2
c
y
=
0.4
2
2
1.4x
d
y =
6
Write
down
the
function
connecting
x
and
y
for
2
x
each
f
y =
y =
values:
a
3x
x
2
x
1
3
5
8
2.25
6.25
3
3x
g
of
1
7
e
table
y =
h
y =
y
0.25
x
2
y
2.5
16
2
5x
5
2
The
variable
variable
a
Find
t.
x
is
directly
When
the
t
value
=
of
4.6,
the
proportional
x
=
to
the
b
constant
7
A
supermarket
£1.30
Write
c
Find
down
the
function
relating
t
and
when
t
=
a
P
y
is
=
inversely
0.2.
Find
when
y
=
M
varies
When
M
directly
=
12.6,
as
c
1.25
sells
300ml
bottles
of
ketchup
for
Explain
why
to
y.
When
P
=
15,
the
=
3.
to
total
the
cost
of
number
ketchup
of
is
bottles
in
of
direct
ketchup
bought.
1.5
b
4
10
8.2
proportional
P
5
bottle.
proportion
3
4
x
a
x
0.5
of
proportionality.
b
1
3.45
square
Find
c
of
c.
when
In
a
get
M
=
special
one
oer,
if
you
buy
two
bottles
you
free.
17.15
Is
the
the
total
cost
number
of
still
in
bottles
direct
proportion
bought?
Justify
to
your
answer.
10.3 Getting more done in less time
4 0 3
Problem
12
solving
The
with
8
The
force
needed
to
break
a
board
varies
the
length
of
the
board.
It
takes
a
force
Newtons
to
break
a
board
60
cm
long.
aperture
force
needed
to
break
a
board
20
cm
The
number
of
tennis
balls
you
can
125.
varies
inversely
Write down
the
the
relationship
can
t
in
a
box
with
the
pack
variation
between
and
the
volume
function
the
radius
of
of
in
the
A
girae’s
weight
The
volume
the
animal’s
varies
m
tall
and
weight
of
height.
directly
weighs
a
2m
1.1
tall
An
metric
baby
V
base
and
balls
that
a
the
If
the
the
girae
During
freefall,
the
an
is
spent
Find
proportional
falling.
how
Review
An
far
in
it
to
the
of
a
πr
=
h
is
Find
object
object
falls
in
falls
1
40.8
of
m
speed
cylinder
h,
the
where
height
cylinder
variation
b
the
If
the
is
r
When
speed
when
given
is
of
has
falls
the
in
a
the
the
xed
function
write
down
in
this
cylinder
down
square
shutter
the
4.
by
the
radius
of
the
cylinder.
height,
between
the
V
write down
and
r.
proportionality
case.
the
andh.
has
a
xed
variation
Hence,
radius,
function
write
write
between
down
the
is
proportionality
directly
the
5
girae.
distance
8,
cube
is
V
11
is
balls.
balls.
tonnes.
inversely
setting.
expresses
of
with
adult
varies
2
constant
of
camera
aperture
shutter
setting
Hence,
10
the
a
the
that
number
setting
Find
formula
box
a
the
long.
13
9
of
of
Find
aperture
the
speed
square
of
is
120
the
inversely
the
with
shutter
constant
in
this
case.
time
20
s.
minute.
context
Globalization and sustainability
Problem
c
solving
Draw
a
graph
represents
1
In
order
to
reduce
overcrowding
in
their
and
Alphatown
have
begun
real
estate
in
a
way
that
they
hope
developers
to
construct
the
city
Now
Boomcity,
the
with
use
price
of
a
building
plot
the
distance
of
the
plot
from
Use
the
the
square
of
price
the
of
a
plot
distance
varies
from
both
Alphatown
the
center
Find
the
each
city.
cost
and
costs
Boomcity,
$250
is
of
of
directly
diameter
a
plot
5
the
a
plot
10
km
from
the
center
of
be
25
to
estimate
served
by
a
the
number
water
pipe
cm.
of
the
of
water
that
serve
ows
proportional
the
50
pipe.
A
Make
a
variation
function
number
water
30
cm,
pipe
to
through
the
of
a
square
diameter
water
of
10
the
of
pipes
40
for
of
houses
cm
the
of
number
diameter
and
4 0 4
table
that
diameter
with
o
your
number
world,
so
is
yet
many
of
trade-os
diameter
conserve
houses
that
diameter
25
can
be
cm.
estimations
of
were
houses.
a
priority
of
humans
in
many
our
activities.
might
pipes
want
be
and
and
between
our
using
need
to
water.
average
of
10
houses
cm,
20
50
varies
water
consumption
25
to
can
between
the
per
northern
served
southern
hemispheres.
But
there
is
cm,
great
variation
between
European
cm.
estimate
be
widely
served
cm.
10 Change
the
by
a
number
water
of
pipe
80cubic
with
5
times
400
Luxembourg’s
meters
less
cubic
per
than
meters
average
person
per
Germany’s
per
person.
of
year
need
Discuss
cm
countries.
your
far
actual
the
the
larger
also
Use
pipe
houses.
table
by
water
conservation
of
water
and
b
graph
can
actual
a
how
the
Water
head
a
of
km
The
can
number
000.
what
amount
pipe
that
equation
by
Discuss
parts
The
the
center.
g
2
and
inversely
the
from
from
pipe
the
f
In
water
center.
Alphatown,
with
your
diameter
the
served
In
the
varies
tond
city’s
of
center.
e
inversely
that
the
served.
houses
with
In
function
between
buildings
of
outside
variation
will
d
encourage
the
relationship
pricing
houses
oce
of
cities,
diameter
Boomcity
the
is
average
of
N U M B E R
Problem
Suppose
solving
with
3
Generating
electricity,
whether
nuclearenergy,
on
to
the
fossil
can
have
environment.
decrease
something
the
that
fuels,
We
amount
will
wind
serious
should
of
also
the
construction
heating
engineer
of
tries
an
to
negative
where
it
will
be
with
a
at
X
is
oce
inversely
we
part
a
the
Find
of
the
distance
consume;
building,
a
heater
ecient.
The
If
a
A,
engineer
however,
position
plans
the
the
to
put
the
company’s
directly
trees
1.5
meters
planted.
produce
change
of
an
0.3°C.
tree
change
with
a
that
would
diameter
of
2
be
meters.
heat
heater
change
what
size
of
1°C
trees
was
should
desired,
be
used.
the
Companies
use
to
to
a
variety
buy
their
of
ways
to
products.
try
One
to
of
entice
those
is
the
setting
a
price
that
will
encourage
people
to
source.
at
manager
their
goods.
A
clothing
company
makes
position
jeans
in
jeans
sell
a
variety
of
price
ranges.
The
Bootcut
prefers
for
$25
while
the
Classic
jeans
sell
for
B.
Determine
how
reduce
answer
varies
the
heat
$160.
will
a
temperature
purchase
The
for
determine
a
in
proportional
from
of
temperature
expected
by
square
diameter
temperature
consumers
received
of
money.
place
most
change
diameter
eects
doour
5
spot
the
average
b
In
of
or
all
energy
save
temperature
cube
through
Trees
hydroelectricity,
the
the
as
the
a
much
heat
moving
received
the
at
X.
heater
Give
to
The
inversely
B
your
a
If
percentage.
50
demand
proportional
000
bought,
were
for
pairs
nd
of
this
to
the
how
company’s
the
square
Bootcut
many
of
jeans
of
jeans
the
the
is
price.
were
Classic
jeans
bought.
heater
b
position
If
the
company
wanted
to
double
the
number
B
of
pairs
what
of
Classic
price
should
be
they
to
jeans
should
the
bought,
be
nearest
sold.
determine
Your
at
answer
dollar.
heater
4
2.5
3
m
6
position A
Since
the
volume
of
a
sphere
is
by V
given
πr
=
,
3
you
would
should
2
the
m
of
3
The
‘urban
buildings
heat
and
island
parking
eect’
lots
occurs
grass
and
dirt
are
with
normal,
in
these
leading
areas
to
that
this,
require
shade
and
some
tend
health
the
and
How
you
now
planting
decrease
Reect
have
cities
of
these
the
Dave’s
and
=
5
the
cm)
the
of
of
a
cream
of
the
one
Cream
price
is
ice
cube
price
Ice
of
Shop
large
radius
small
(radius
scoop
of
to
be
issues.
have
Show
that
this
does
not
represent
a
direct
trees
to
variation.
higher
If
this
were
a
direct
cube
variation,
determine
To
price
of
one
large
scoop.
ordinances
c
provide
temperature
the
Suggest
to
changes.
vary
reasons
based
on
why
the
the
price
volume
of
does
the
not
ice
seem
cream.
statement
of
inquiry?
Give
specic
examples.
Statement of Inquiry:
Changing
and
to
simplied
=
ice
$5.00.
discuss
explored
of
scoop
constructed,
the
combat
with
price
concrete.
b
than
at
$2.50
(radius
cube
Temperatures
directly
the
when
a
covering
is
that
However,
cream
cm)
cream
4
vary
scoop.
ice
think
forms
can
help
analyze
the
eects
of
consumption
conservation.
10.3 Getting more done in less time
4 0 5
11
Equivalence
The state of being identically equal to or interchangeable, applied to
statements, quantities or expressions
0.999...
Did
you
know
that
0.999...
=
is
meaning
0.999…
is
You
prove
can
not
almost
this
1,
but
result
is
actually
using
the
identical
to
1.
Read
the
that
following
understand
each
you
and
will
make
study
sure
n
Then
=
in
this
that
=
=
forever
after
keeps
the
step.
the
show
same
that
decimal
rst
line
from
the
second
way,
any
is
you
can
innite
the
same
as
equivalent
.
procedure
as
also
repeating
its
Use
gives
same
9n
numb er
chapter.
9.999…
the
the
point.
fractional
Subtracting
decimal,
you
0.999…
10n
that
repeating
In
Let
recurring
equivalence
decimal
transformations
a
1?
the
above
to
9
prove
1
that
0.333…
is
equal
to
,
3
Dividing
both
sides
by
9
gives
or
that
0.2424….
is
equal
to
8
n
=
1
(Hint:
when
two
places
33
repeating,
the
In
1847,
Byrne
of
Englishman
published
Euclid’s
used
Elements
color
theorems,
an
in
Oliver
edition
where
diagrams
instead
of
to
text
he
prove
and
labels.
●
Do
of
you
a
think
theorem
equivalent
4 0 6
to
a
visual
can
a
‘proof’
be
written
one?
multiply
equation
by
both
100
are
sides
instead
of
of
10).
Logical
In
equivalence
mathematics,
(or
not
The
●
to
Ellie
then
If
Ellie
she
has
has
can’t
from
have
are
be
logically
Madagascar
crossed
not
can
proved
to
be
equivalent.
statements
travels
she
statements
logically
following
If
●
be)
two
the
crossed
to
Iceland
equator.
the
travelled
equivalent:
equator,
from
then
Madagascar
to
Iceland.
The
of
second
the
For
rst
each
statement
of
the
contrapositive
logically
●
If
a
●
Ellie
on
●
If
Ellie
to
●
If
passes
state
whether
with
its
or
write
not
it
the
is
partner.
then
she
speaks
will
hard
then
she
will
do
well
test.
her
algebra
take
her
test
brother
zoo.
Ellie’s
she
contrapositive
language.
algebra
she
the
the
statements,
Portuguese
studies
her
then
and
speaks
romance
If
called
following
equivalent
Ellie
is
statement.
will
bus
is
arrive
on
at
time
then
school
on
time.
4 07
A
11.1
model
Global
of
context:
Objectives
●
Solving
linear
equations
●
Using
linear
equality
Identities
Inquiry
equations
algebraically
equivalence
equations
and
and
systems
linear
systems
of
to
MROF
Creating
a
mathematical
questions
●
What
●
How
solve
using
is
an
equivalence
transformation?
can
you
solve
equivalence
linear
equations
transformations?
equations
●
●
relationships
F
graphically
transformations
and
of
and
model
to
solve
C
real-life
Are
all
solution
equations
methods
for
systems
of
equivalent?
problems
●
●
Determining
if
a
model
solution
is
equivalent
How
do
the
equations
the
real-life
Evaluating
and
light
real-life
interpreting
your
solutions
the
to
of
systems
the
types
of
of
may
solutions
have?
in
●
of
relate
solution
they
●
graphs
to
Can
good
decisions
be
calculated?
problems
D
ATL
Transfer
Apply
skills
and
knowledge
in
unfamiliar
situations
11.1
Statement of Inquiry:
Modelling
with
equivalent
forms
of
representation
can
improve
decision
making.
14.3
E14.1
4 0 8
A LG E B R A
Y
ou
●
should
expand
already
and
factorize
expressions
to
know
how
algebraic
obtain
1
to:
Expand:
equivalent
a
4(x
+
3)
b
5(2x
1)
d
4x(7
+
15
expressions
2
c
2
●
solve
linear
equations
3
3(8
6x)
x
x
)
Factorize:
a
3x
+
6
b
5x
c
14
+
35x
d
34
Solve
these
85x
equations.
x
a
=
3
b
4x
=
52
5
x
c
5x
7
=
8
d
+
4
=
32
2
●
nd the equation of a line by nding
4
Find the equations of these two lines.
its gradient and y-intercept
y
4
3
2
1
0
3
2
x
1
2
1
2
3
Linear
F
You
●
What
●
How an you solve linear equations using equivalene transformations?
know
priniples
how
that
is
to
The
examples
equations.
an
use
elow
Deide
priniple’
a
5
=
2
3x
=
7
d
6
2x
2x
=
11
=
5
linear
to
do
show
or
‘the
b
e
h
2
+
Lael
or
1
=
the
‘the
–
6x
steps
you
ut
6x
=
12
x
=
2
4n
+
16x
4x
stages
in
the
multipliation
are
the
do
you
know
solution
=
9n
2
=
5n
+
+
that
is
4
1
example
the
mathematial
used
4x
=
x
it’s
eing
as
to
solve
linear
demonstrates
‘the
priniple’.
f
=
elow
If
typially
c
2
priniple’.
property
that
think
multipliation
6
mathematial
2(3x
equations,
this?
whether
g
4x
transformation?
1
addition
3x
equivalene
solve
you
Exploration
1
equations
i
either
neither
=
3
x
=
12
7x
1
=
2x
5x
=
9
5x
=
2x
3x
=
9
‘the
of
x
+
then
8
9
addition
these,
+
priniple’
state
the
used.
4)
=
3x
+
7
8
=
3x
+
7
3x
=
15
_________________
x
=
5
_________________
_________________
11.1 A model of equality
4 0 9
Reect
●
These
an
●
priniples
don’t
solving
language
formal
to
terms
solve
we
need
linear
for
an
are
the
1
alled
‘equivalene
transformations’.
Why
is
this
name?
desrie
Equivalence
To
discuss
appropriate
Why
When
and
a
sutration
equations
your
in
the
working
mathematial
priniple
past,
steps.
you
In
a
may
this
priniples
or
division
have
setion,
used
in
priniple?
used
you
solving
very
will
informal
learn
linear
the
equations.
transformations
equation
you
an
use
these
Principle
mathematial
priniples:
Example
An
Addition
value
or
Principle:
variable
Add
to
the
both
same
sides
of
Add
an
get
equation.
2x
Multiplication
Principle:
Multiply
by
3
to
the
=
both
sides
equivalent
of
2x
3
=
5
to
equation
uses
8.
Multiply
both
sides
of
11
=
same
non-zero
value
or
variable
by
both
sides
of
an
priniples
to
transform
an
equation
to
get
the
into
an
equivalent
5
on
mathematial
5x
1
the
equivalence
transformation
equivalent equation.
equation.
11
equation
=
x.
5
Example
Solve
used
the
at
1
equation
eah
step.
.
Rememer
to
hek
Show
your
the
equivalene
transformation
solution.
Multiply
x
2
=
2(3x
+
x
2
=
6x
+
8
2
8
=
6x
+
8
x
10
=
6x
x
10
=
−x
+
6x
10
=
5x
x
−
x
+
both
sides
by
4.
4)
8
Add
Add
8 to both sides (or subtract 8 from both sides).
x
to
both
sides
(or
subtract
x
from
both
sides).
Multiply both sides by 5.
2
Chek:
=
x
LHS:
RHS:
LHS
4 10
=
RHS
✓
11 Equivalence
A LG E B R A
Practice
1
Solve
eah
a
1
these
step.
2x
+ 3
equations.
Rememer
=
x
Show
to
the
hek
equivalene
your
transformation
that
you
use
at
solutions.
− 7
b
x
2x
6
1
c
−(x
+ 2) − 3 x
=
2( x
+ 1)
d
1 − 3( x + 2) =
(2 x − 8) + 3
2
x
x
1
e
+ 2 =
x − 4
3
the
equals
x
=
equation
what
Reect
and
Can
you
Explain
oth
think
why
●
Are
●
How
system
Solving
a
of
an
all
of
and
transformations
the
variale
x
on
to
write
oth
an
sides
of
the
the
equations
every
seen
how
of
of
an
may
of
two
an
method
you
5( x
think
+ 4) − 8 +
this
x
=
6( x
+ 2)
means.
y
zero’
is
not
an
equivalene
in
are
do
not
not
equivalene
result
in
transformations?
equivalent
equations.
equations
for
systems
systems
of
of
equations
equations
relate
equivalent?
to
the
types
of
have?
or
equations
you
that
methods
equation
what
equation
linear
is
equation
why.
graphs
they
the
2
operations
of
to
explain
operations
solution
do
algerai
and
sides
these
system
satisfy
have
using
rakets
Explain
solutions
Y
ou
with
equivalene
transformations
Systems
that
use
discuss
transformation.
A
3,
happens,
‘Multiplying
C
(9 − 8 x )
solving
equivalene
Desrie
●
=
sign.
Apply
●
2x
5
equation
equivalent
3
1
h
(8 − 4 x )
4
Problem
4
2
1
(6 x − 3) =
3
For
2x
=
5
1
g
2
+ 2
f
2
more
means
the
solve
equations
nding
with
values
the
for
same
eah
unknowns.
unknown
system.
an
equation
(equivalene
like
3(x
+
2)
transformations),
6
ut
=
4(2x
did
you
3)
+
1
y
know
8
y
that
you
an
also
solve
it
using
a
graphial
method?
=
3(x
+
6
2)
7
(2.2,
6.6)
6
You
an
onsider
the
two
sides
of
the
equation
as
two
separate
linear
5
equations,
eah
equal
to
the
variale
y.
You
an
then
graph
eah
equation
4
onthe
same
set
of
axes.
3
Here
is
a
graph
of
y
=
3(x
+
2)
2
6
y
=
4(2x
3)
1
and
y
=
4(2x
3)
+
1
0
3
2
1
11.1 A model of equality
x
1
2
3
411
+
1
The
to
graph
the
shows
same
that
value,
simultaneouslyis
Reect
●
Why
y
x
and
is
the
=
=
value
6.6.
So
of
the
x
=
2.2
makes
solution
that
oth
equations
satises
oth
equal
equations
2.2.
discuss
drawing
a
graph
3
y
hand
not
the
best
method
to
solve
this
equation?
●
When
is
using
solution
using
You
an
of
an
tehnology
equation?
not
an
(Hint:
appropriate
Can
you
method
always
nd
to
nding
exat
the
solutions
tehnology?)
solve
the
two
equations
y
=
3(x
+
2)
6
and
y
=
4(2x
3)
+
1
algeraially.
Expanding
y
=
3(x
y
=
4(2x
So,
we
+
and
2)
have
=
3x
[2]
y
=
8x
a
an
⇒
1
y
⇒
=
y
3x
=
8x
equations
11
in
two
unknowns,
third
equivalene
Principle:
equation
by
an
transformation
Replace
par t
y
=
[4]
=
8x
11 =
5x
[5]
x
=
a
Chek
[1]
the
y
one
value
of
(x,
an
use
example,
value
to
x
into
oth
here:
for
the
y = 8x
of
give
equations
in
equations
two
y
3x
RHS:
3
=
×
pair
2.2
RHS
substitution
gives
unknowns
satises
oth
original
[2]
6.6
4 12
two
of
3x
LHS:
single
solve.
from
=
equations
11,
substitute
[1]
8x
into
equation
11.
y
=
the
6.6.
solution
is
an
y).
solution
LHS
The
this
pair
=
to
− 11
system
ordered
y,
2.2
Sustituting
For
and
3x and
the
[2]
3x
you
For
equivalent
expression.
[3]
x
− 11
Substitution
of
+
two
y
is
6
3)
[1]
There
simplifying:
=
✓
method
equation
6.6
in
equations:
x
=
x
− 11
LHS:
3
×
2.2
RHS:
8
×
2.2
LHS
redues
one
two
separate
unknown.
11 Equivalence
=
RHS
equations
in
=
6.6
11
=
17.6
11
=
6.6
✓
two
unknowns
into
A LG E B R A
Example
Use
7x
the
+
2y
x
2
method
=
y
19
=
of
and
4
⇒
sustitution
x
y
y
=
=
x
4,
and
to
solve
hek
the
your
system
=
x
The
4
⇒
y
solution
=
is
3
4
(3,
equations
4
Choose
Substitute
y
of
solution.
=
−
the
expression
for
y
one
into
the
2y
⇒
7(3)
+
2(
1)
=
21
2
=
19
y
⇒
3
(−
and
solve
for
y.
equation,
and
solve
for
x
the
1)
=
4
solution
as
algebraically
an
by
ordered
pair
substituting
(x,
the
y).
x
✓
and
x
other
equations
1).
Check
+
the
Substitute the value for x into one of the equations to nd the value of y
1
Write
7x
of
y
values
into
both
original
equations.
✓
y
19
y
7x
=
2
3
2
1
y
=
x
4
Check
0
graphically
equations
and
by
graphing
nding
their
the
two
intersection.
x
2
3
4
5
1
(3,
1)
2
3
Practice
Solve
1
3x
3
x
5
2x
these
− 2y
=
2
systems
=
0
25 + 9 y
+ 3y
=
Rupa
and
and
−6
Problem
6
and
of
y
=
6x
and
equations
using
the
sustitution
7 − 2x
− 5y
3x
2
=
+ 2y
3
=
4
y
=
3x
9 − 2x
3x
and
− 4 y
x
=
+ 4y
8
= 11
25
solving
George
[1]
5x
+
y
[2]
x
+
3y
x
+
3(8
and
and
= 12 − 5y
solve
this
system
of
equations:
5x
Rupa
Copy
method.
=
8
=
⇒
y
=
omplete
=
y
=
8
and
x
+
3y
=
10
George
8
5x
10
5x)
+
[1]
5x
+
y
[2]
x
+
3y
10
their
working
5(10
to
nd
their
=
8
=
10
3y)
⇒
+
y
x
=
=
10
3y
8
solutions.
11.1 A model of equality
4 13
Reect
●
●
Does
and
your
nal
variale
you
Pratie
2,
When
is
discuss
solution
hoose
to
question
it
easier
Exploration
to
4
for
the
solve
system
for
and
of
equations
sustitute?
depend
Hint:
look
on
whih
ak
at
6
sustitute
i
for
x
ii
for
y?
2
Tip
1
Find
a
solution
for
the
following
system
of
equations:
If
x
+
y
=
the
question
4
doesn’t
2x
y
=
5
whih
to
2
Write
the
equations
one
aove
the
other
(as
aove)
and
then
add
one
term
at
a
you
Explain
4
Find
5
Starting
x
+
what
the
value
with
y
=
4
y
=
5
2x
multiply
6
Rewrite
this
Add
the
8
as
the
time.
new
the
the
to
either
system
in
the
this
original
the
in
y’s
in
new
the
equation.
algerai
graphial
or
method.
equation.
system,
rst
equation
did
step
equations
value
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another
to
from
term
you
two
x
either
equation
and
the
y
seond
2.
equation
diretly
aove
one
2
together
term
y
term.
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what
happens
to
x’s.
Find
solve
of
the
eah
another,
7
happens
an
them
the
3
you
method
use,
hoose
together,
tell
for
x
or
that
equations
[1]
3y
x
=
−3
[2]
y
x
=
1
Sutrating
[2]
y
from
sustitution
method
of
of
from
y,
or
uses
3y
[1]
this
method
eause
the
x
new
=
ideas
−3
an
e
a
it
equations
from
and
eliminates
equation.
y
the
x
triky,
with
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=
eause
frations
2.
it
is
result.
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use
it
diult
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to
is
solve
the
1:
variale
x
Tip
3y
x
=
−3
x
=
1
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y
that
sutration
2y
=
−4
y
=
−2
Sustituting
system
of
y
=
addition
−2
into
equations
414
of
opposite.
is
(
either
3,
[1]
or
2).
11 Equivalence
[2]
gives
x
=
−3.
The
solution
to
this
is
the
the
A LG E B R A
Cheking
the
(1)
=
y
x
2
This
Can
see
solve
sutrat
touse
1
3
is
1
original
equations:
an
x
3(
alled
the
elimination
method
=
2)
for
−3
(
3)
solving
=
−6
+
3
systems
=
of
−3
✓
equations.
why?
of
equations
equations
equivalene
to
using
eliminate
the
one
transformation
elimination
of
on
the
one
method,
variales.
or
oth
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add
may
equations
or
need
rst.
3
system
your
3y
✓
systems
eah
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=
the
Practice
Solve
oth
(2)
3)
method
To
in
1:
(
you
solution
of
equations
solutions
[1]
5x
+
y
=
27
[2]
2x
+
y
=
12
[1]
2x
[2]
2x
3y
+
y
Reect
=
=
using
algeraially
or
the
elimination
2
−16
4
0
and
discuss
How
do
you
deide
whih
●
How
do
you
deide
what
a
Exploration
[1]
2y
[2]
4y
[1]
3x
[2]
5x
x
=
7
+
x
=
11
+
2y
=
11
2y
=
13
5
●
toeliminate
method.
graphially.
variale
to
to
multiply
eliminate?
one
of
the
equations
y
in
order
variale?
As
3
early
the
In
Pratie
2,
you
solved
the
following
system
of
equations
y
as
3x
+
+
3y
2y
=
=
−6
solving
of
two
the
if
it
is
equations
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system
possile
y
an
to
eliminate
integer
and
a
then
variale
adding.
y
multiplying
just
one
Explain.
unknowns.
method
in
the
if
it
is
possile
to
eliminate
x
y
multiplying
eah
equation
is
in
dierent
integer
and
then
adding.
Jiuzhang
(Nine
y
in
the
Explain.
Mathematial
3
Solve
the
system
y
eliminating
4
Solve
the
system
y
using
5
Chek
6
Explain
your
answer
how
you
y
similar
sustituting
would
deide
Art),
x
written
a
The
desried
ook
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a
a
equations
suanshu
2
method
25
Determine
of
a
of
two
1
bc
had
sustitution:
invented
2x
200
Chinese
method
it
into
whih
to
eliminate
oth
variale
of
to
the
y
instead.
original
eliminate
Han
during
the
Dynasty.
equations.
rst.
11.1 A model of equality
4 15
Example
Solve
the
3
system
elimination
of
equations
method.
[1]
5x
7y
=
27
[2]
3x
4y
=
16
Chek
5x
your
7y
=
27
result
and
3x
16
algeraially
=
or
4y,
using
the
graphially.
Rearrange
the
equations
Choose
Multiply
oth
sides
of
[1]
y
4,
and
oth
sides
of
[2]
y
7.
that
to
line
equivalence
give
the
same
up
like
terms.
transformations
coecient
for
one
variable.
[3]
20x
[4]
21x
5(4)
7y
+
=
28y
=
108
28y
=
−112
x
=
−4
⇒
x
=
4
Add
Substitute
into
to
value
one
[3]
of
and
the
[4]
to
eliminate
original
y
equations
27
nd
20
7y
=
27
7y
=
7
y
=
−1
is
(4,
Solution
nd
the
Write
1)
the
of
the
2
solution
variable.
as
an
ordered
pair.
Check algebraically by substituting the
5(4)
7(
1)
=
20
+
7
=
27
✓
solution into both original equations.
3(4)
4(
1)
=
12
+
4
=
16
✓
y
2
3x
y
1
16
=
4
0
Check
x
2
5
6
graphically
to
nd
7
1
(4,
point
1)
2
5x
3
y
27
=
7
4
Practice
Solve
eah
Chek
1
2x
3
x
+
system
your
+
5y
3y
4
of
answers
=
24
7
=
and
0
equations
either
4x
and
2y
x
5
3x
=
17
+
y
and
3y
x
the
=
=
20
3
elimination
or
method.
graphially.
2
2a
4
5r
6
4x
+
=
3b
=
23
−9
3s
and
and
4a
4s
+
=
b
=
12
13
2r
y
+
5
4 16
+
using
algeraially
= 0
2
11 Equivalence
0.5y
=
12.5
and
3x
=
8.2
0.8y
of
intersection.
the
A LG E B R A
Reect
●
and
Would
it
systems
●
When
a
Practice
Use
the
1
3x
3
3y
5
4x
+
+
system
y
9
and
2x
=
11
3y
=
−2
Draw
the
a
+
x
4x
2
of
eient
you
5x
and
and
of
linear
use
the
4?
linear
sustitution
If
so,
explain
equations,
Write
yourself
equations
algerai
+
y
4y
+
how
a
set
method
on
any
of
the
why.
do
of
you
selet
guidelines
the
on
most
how
to
eiently.
your
method
hosen
for
solving
method
eah
eah
system
time.
of
Chek
equations.
your
graphially.
=
2x
4x
22
=
y
=
9
6
2
y
=
2x
+
4
and
3x
+
y
4
3x
+
2y
=
16
and
7x
6
3x
+
2y
=
19
and
x
=
+
+
9
y
y
=
=
19
8
4
2y
=
16
8y
=
−64
the
or
systems
Generalize
down
to
Pratie
method?
seleted
Exploration
1
system
algeraially
=
in
5
why
solutions
a
6
eient
algerai
most
Explain
more
equations
solving
eient
solve
e
of
discuss
of
equations
on
separate
b
your
x
graphs:
+
4x
results
onditions
from
your
neessary
work
for
a
so
2y
=
16
8y
=
24
far,
system
of
and
two
from
step
linear
1.
Write
equations
to
have:
3
●
a
●
no
●
a
unique
non-unique
Desrie
Without
reasons,
a
2x
x
c
+
how
6
5
2y
=
0
+
in
or
5
2y
=
−10
it
is
not
unique
why
any
of
not
(an
of
innite
systems
Inlude
these
the
a
numer
of
sketh
has
of
a
for
for
a
relate
eah
2x
+
system
of
state,
and
3y
=
12
3y
=
6
two
the
numer
of
ase.
equations,
solution,
to
giving
explain
linear
why.
equations
to
solutions.
you
the
possile
solutions).
equations
systems
system
of
2x
=
two
Explain
not
pair)
b
y
why
have.
solving
whether
=
graphs
may
atually
y
Explain
have
ordered
solution
the
they
2x
4x
5
(an
solution
solutions
4
solution
should
test
equivalent
your
solution
equations
you
in
the
get
original
from
equations,
using
and
equivalene
transformations.
11.1 A model of equality
4 17
Applications
D
●
You
are
Can
going
80questions.
●
eah
good
to
take
The
orret
of
deisions
a
written
soring
answer
is
e
1
of
equations
alulated?
driver’s
suh
sores
systems
liense
exam
that
ontains
that:
point
1
●
eah
inorret
answer
is
a
point
dedution.
4
If
you
order
Step
answer
to
1:
earn
all
70
of
the
points
Identify
the
questions,
(the
how
minimum
x
represent
the
numer
of
orret
Let
y
represent
the
numer
of
inorret
2:
Identify
do
you
need
to
get
orret
in
pass)?
variables:
Let
Step
many
to
the
answers.
answers.
constraints:
Conditions
Both
x
and
negative
y
must
numer
e
of
positive
numers
or
zero,
sine
you
annot
answer
a
the
questions.
values
variales
on
of
are
the
alled
onstraints.
There
are
80
Expressed
Step
3:
Write
The
questions,
the
mathematially:
Create
the
equations
80
so
to
questions
numer
of
0
maximum
≤
x
≤
80;
0
for
≤
y
x
≤
or
for
y
is
80.
80
model:
represent
are
orret
orret
the
or
answers
situation.
inorret,
+
numer
so:
of
inorret
answers
=
80
⇒
x
+
y
=
80
1
Total
sore
=
(numer
of
orret
×
answers)
(numer
of
inorret
answers)
=
70
4
1
⇒
x
y
=
70
4
The
model
[1]
x
[2]
x
+
is
y
=
a
system
of
two
equations:
80
1
y
=
70
4
Solve
your
model:
Finish
Use
an
eient
method
to
solve
the
system
of
out
Check
your
working
equations.
and
solution:
the
prolem
then
answers
Chek
your
solution
algeraially,
or
y
using
a
GDC
or
graphing
in
software.
sentene.
Interpret
To
earn
answer
your
70
solution
points,
you
questions
4 18
in
the
need
to
context
answer
inorretly.
11 Equivalence
of
the
problem:
questions
orretly,
and
you
an
ll
in
the
the
last
A LG E B R A
Example
From
an
the
in
1996
e
4
to
2012,
modelled
total
exports
years.
The
or
year
y
=
1310x
Imports:
y
=
725x
analyzing
imports
0
and
x
=
x
(from
y
represents
the
the
imports
+
+
1996
in
and
system
millions
represented
exports
of
of
y
from
equations,
dollars,
x
=
and
a
partiular
where
x
y
ompany
represents
represents
the
time
0.
7430
of
equations,
etween
year
2012)
the
is
system
the
to
imports
5165
exports
represents
total
following
2000
Exports:
By
the
y
total
1996
2000,
is
4
≤
exports
so
x
≤
or
desrie
and
the
the
ompany’s
pattern
of
2012.
range
of
Identify
the
variables
and
constraints.
12.
imports,
thus
y
>
0.
16000
14000
12000
Graph
Imports:
the
system
of
equations.
10000
(3.87,
10 237)
y
=
725x
+
7430
The
8000
intersection
which
6000
is
around
is
at
the
x
=
year
3.87
≈
4,
2004.
Exports:
4000
y
=
1310x
+
5165
2000
0
3
4
The
it
ompany
spent
the
more
had
on
ompany’s
Practice
2
1
no
1
exports
imports
export
2
in
than
inome
it
3
4
1996.
made
5
6
Then,
on
exeeded
7
8
until
exports.
its
9
aout
After
import
10
11
Interpret
2004,
the
2004
in
expenses.
the
graph
the
in
information
your
context
of
own
the
from
words,
and
problem.
6
ATL
Problem
In
the
following
answer
of
1
the
You
your
prolems,
question.
equations,
open
running
and
a
for
your
Between
x
Coal
=
and
Petroleum
Calulate
the
y
oal
e
the
=
in
the
given
eient
how
the
and
year
y
whih
initial
$7800.
mathematial
method
=
investment
The
long
amount
y
the
weekly
will
you
petroleum
modeled
93.2x
prodution:
year
an
are
mathes
an
represents
use
Determine
2010,
prodution:
with
usiness
prot
or
most
for
model
solving
the
to
system
solution.
usiness
the
ountries
0
the
your
$8800.
total
2000
produing
where
is
reate
Choose
hek
small
osts
usiness
(when
2
solving
it
of
$90 000.
revenue
e
have
efore
weekly
you
reak
from
even
invested).
prodution
following
The
(inome)
from
system
the
of
fuel-
equations,
2005:
+
3100
−29.1x
+
prodution
2942
from
oth
forms
of
fuel
was
equal.
11.1 A model of equality
4 19
3
A
phone
ompany
operating
fee
of
$5.20,
whih
4
You
fee
plan
gave
fund
muh
of
Why
interest
5
to
est
an
15.5
per
for
interest
the
year
discuss
of
minute.
you,
hoose
explain
roker,
and
the
rest
how
in
a
into
Plan
B
you
A
harges
harges
would
an
an
operating
deide
on
reasoning.
put
$210
plaed
Plan
your
who
reeive
was
from.
minute.
part
fund
in
of
the
paying
interest.
eah
money
5%
in
a
annual
Determine
how
fund.
7
want
oers
to
per
Determine
and
you
investment
them
plans
ents
investment
annual
of
weekly
plus
someone
of
kind
two
ents
e
end
and
one
What
37
initial
would
when
●
2%
the
your
Reect
●
plus
$4500
At
has
$31.45,
would
paying
interest.
of
a
advantages
to
plae
lower
might
their
return
the
money
than
ank
the
oer
in
two
other
in
the
dierent
funds,
one?
fund
with
the
lower
rate?
Your
unle
week
(2 kg
Karl
wants
to
lose
some
weight
y
urning
extra
alories
eah
iyle,
whih
During
=
approx.
3000
alories).
He
will
do
it
y
riding
a
de
urns
350
alories
an
hour,
and
y
walking,
whih
urns
200
hour.
If
unle
Karl
has
10
hours
per
week
to
devote
to
exerise,
ylist
of
walking
and
yling
will
allow
him
to
ahieve
his
goal
exatly
3000
etween
of
4000
urning
will
what
urn
omination
Tour
rae,
alories
a
an
the
Frane
and
5000
alories?
alories
That
123
for
is
per
900
the
stage.
around
alories
entire
ATL
Reect
and
discuss
8
rae,
to
●
Is
it
possile
to
reate
more
than
one
orret
mathematial
model
equivalent
252
doule
to
heeseurgers.
solve
a
partiular
●
Are
●
Explain
all
methods
in
onepts
your
of
prolem?
for
solving
own
equality
the
words
and
model
the
equal
similarities
or
and
equivalent?
dierenes
etween
the
equivalene.
Summary
1
An
equivalence
appliation
priniples
of
in
transformation
one
or
solving
more
an
of
is
the
equation
the
2
following
so
that
Methods
of
solution
equations
are
Priniple:
values
oth
to
sides
Multipliation
same
an
non-zero
Adding
of
an
Priniple:
value
or
the
same
value
or
equation.
Multiplying
values
on
one
y
oth
the
sides
of
Elimination:
of
to
Priniple:
4 20
an
Replaing
equivalent
part
expression.
of
11 Equivalence
the
in
of
the
By
one
equation
this
and
other
one
two
linear
equations
in
redue
one
value
sustitute
equation,
this
resulting
unknown.
applying
transformation(s),
equivalene
the
two
unknown,
into
one
of
equations
and
the
solve.
original
an
equations
y
one
variales,
into
equation
Sustitute
equation
Solve
the
expression
one
equation.
Sustitution
of
equivalent:
for
Addition
systems
all
Sustitution:
resulting
for
equations:
to
solve
for
the
other
variale.
in
A LG E B R A
Graphial:
enter
the
software.
solution
Solve
oth
equations
The
of
point
the
equations
into
of
a
GDC
for
or
y
and
the
graphing
intersetion
is
the
lines
Solution
are
of
equations.The
are
oinident,
atually
the
graphs
whih
same
of
the
means
the
two
line.
system.
4
3
system
equations
senarios
for
a
system
of
two
linear
Creating
of
a
mathematial
model
using
systems
equations:
equations:
One
solution:
system
graphs
No
of
of
the
oth
equations
Innitely
numer
Mixed
1
Solve
There
an
is
no
parallel
to
pair.
interset
ordered
The
Identify
●
Translate
the
system
equations
pairs
these
at
one
3x
b
4 − 3x
of
that
equations
the
●
Solve
two
●
Chek
that
is
will
an
●
innite
system
the
of
prolem
into
a
equations
solution
the
real-world
satisfy
y
identifying
you
the
use
2
at
+ 4
Use
= 19
=
a
5x
either
eah
of
− 1)
=
in
the
solution
original
in
the
ontext
of
the
prolem
4 (2 x
− 3)
elimination
system
method.
y
=
x
b
− 3
x + 2 = 4 −
4 x
of
Chek
+ 3
− 7y
and
= 19
or
sustitution
equations.
your
y
and
=
solutions
− x
5x
Justify
to
your
solve
hoie
graphially.
− 2
− 8 y
=
−13
1
x
c
+
y
= 1 and
x
−
y
= 3
6
1
x
d
3
1
x
+
1
y
= −
1
and
5
x
−
11
y
=
2
3
e
Objective:
solve
the
ecient
A.
solution
in
for
that
a
variety
describes
of
each
systems
of
real-world
problem,
instead
of
A
taking
you
y
vitamin
An
and
of
will
get
drinking
your
milk
and
the
this
oune
of
milk
ontains
56 μg
(mirograms)
of
orange
juie
ontains
5
60
μg
of
vitamin
A.
μg
a
of
many
ounes
need
A
realisti
hemist
−33
to
of
drink
milk
daily
and
in
requirement
of
amount
Deide
for
whether
you
to
or
not
drink.
has
een
asked
to
make
a
solution
of
liters
mg
has
ontaining
only
two
20%
aidi
aid.
The
solutions,
prolem
one
is
that
ontaining
of
aid
and
the
other
ontaining
32%
aid.
Determine
orange
order
550
A.
vitaminA;
to
juie
meet
mg
of
how
many
liters
of
eah
he
should
you
in
order
to
reate
the
solution
he
has
een
your
asked
minimum
=
38mg
use
would
− 7.5 y
most
vitamin
Determine
how
− 1.5 x
and
12%
alium
select
1200
he
oune
and
16
equations.
8
ofalium
= 17
6
relationships
vitamin
juie.
+ 2.5 y
6
ontexts
2
orange
3.5 x
4
understanding
in
solving
supplements
and
2
context
that
mineral
alium
model
method
and
deide
and
orretly
mathematical
Identities
You
Knowing
prolems
Review
an
real-life
1
d
and
the
Interpret
4
1
of
equations
There
transformations
2
For
onstraints
will
1
iii.
and
point.
step:
3(3 x
variales
practice
a
c
the
The
pair
graphs
●
the
lines.
solutions:
ordered
solution
ordered
will
equations.
many
of
is
equations
are
equivalene
eah
algerai
equations
solution:
satisfy
The
alium
to
make.
and
11.1 A model of equality
4 21
3
A
oee
The
distriutor
premium
kilogram,
$8.25
these
and
20
kilograms
lends
Determine
should
e
is
in
the
ommodity,
Sine
Brazil
oee,
oee
In
the
1991,
pot,
they
a
a
the
it
is
Before
of
you
have
oee.
4
per
sells
most
y
for
wishes
to
in
has
lend
on
live
in
of
the
fator
1.5
printer,
an
is
(e.g.
the
with
world’s
in
University
on
world
sientists
the
we
or
not.
oee
so
that
This
weam.
US
in
the
Amerians
1700s,
was
the
eer.
discuss
explored
the
statement
equivalent
forms
making.
4 2 2
the
etter
of
inquiry?
Give
spei
examples.
Statement of Inquiry:
Modelling
ost
11 Equivalence
of
representation
an
improve
deision
a
printer
two
ut
whih
average
onditions
uy
to
ents.
traded
department’s
empty
the
of
valuale
$150
Brazil.
their
was
drink
just
inkjet
40 %
footage
rst
page
is
to
down
osts
ontaining
eah
it
printer
kilogram.
of
deided
narrowed
petroleum.
inuential
on
You’ve
printer
only
pot
aught
How
per
kilograms
around
live
reakfast
and
$9
Camridge
the
Reect
lend
mixture
at
weather
world’s
oee
preferred
of
$10.50
distriutor
seond
most
the
if
types
for
mixture.
amera
tell
a
many
the
group
the
of
sell
produes
streaming
made
The
surpassed
xed
ould
to
world’s
single
pries
aimed
how
two
sells
standard
kilogram.
two
Coee
the
per
reate
has
lend
laser
average
ost
of
The
osts
6
other
$30,
ents.
numer
uy.
have
The
the
of
and
hoies.
of
option
ut
is
eah
eah
an
page
Determine
pages)
the
where
eah
More
than
one
way
to
11.2
solve
Global
a
problem
context:
Objectives
●
Solving
Scientic
and
Inquiry
quadratic
equations
algebraically
technical
innovation
questions
●
What
●
How
is
the
null
factor
law?
F
and
●
graphically
Solving
using
problems
quadratic
by
creating
in
do
you
factorized
solve
a
quadratic
equation
form?
models
●
How
can
you
use
MROF
and
real-life
equivalence
C
transformations
to
solve
quadratic
equations?
●
How
are
the
quadratic
●
How
do
three
methods
equations
you
for
solving
equivalent?
determine
a
‘best
method’
D
among
●
Do
equivalent
systems,
problems
ATL
or
methods?
models
create
and
methods
solve
them?
Critical-thinking
Propose
and
evaluate
a
variety
of
solutions
8.2
9.2
Statement of Inquiry:
Representing
patterns
with
equivalent
forms
11.2
can
lead
to
better
systems,
models
and
methods.
4 23
Y
ou
●
should
conver t
into
already
quadratic
standard
know
how
1
expressions
to:
Rearrange
a
form
(x
+
into
2)(3x
standard
form:
1)
2
b
●
conver t
quadratic
expressions
into
2
(x
1)
16
Factorize:
2
factorized
form
a
x
+
x
6
2
b
2x
+
5x
3
2
c
4x
81
2
●
conver t
ver tex
quadratic
functions
into
3
Write
form
in
where
the
(h,
k)
form
is
y
the
=
a(x
h)
+
k,
ver tex.
2
a
y
=
x
6x
+
14
2
b
●
use
in
equivalence
solving
linear
transformations
4
y
=
3x
6x
+
Solve 3(x + 2)
equations
7
6 = 4(2x
3) + 1.
State the equivalence transformation
you use in each step.
Quadratic
F
Th
●
What
●
How
graph
is
do
shows
th
null
you
th
and
fator
solv
two
linear
a
linar
equations
law?
quadrati
funtions
quation
f
(x)
=
x
in
+
1
fatorizd
and
1
and
th
f
=
(x)
funtion
(x
+
1)(2
whih
is
th
produt
of
f
(x)
=
form?
2
x,
2
ths
two
linar
funtions,
x).
3
y
7
6
f
5
(x)
=
x
+
1
1
4
3
2
f
(x)
=
3
f
(x)·
f
1
(x)
2
0
7
6
5
4
3
2
1
x
3
1
2
f
(x)
=
2
x
2
3
4
5
Expanding
6
(x
7
+
1)(2
givs
Th
graph
baus
f
of
(x)
th
=
(x
produt
of
th
two
+
x)
is
quadrati
1)(2
a
3
4 24
11 Equivalence
linar
funtions
funtion.
is
a
parabola,
a
x)
quadrati
xprssion.
A LG E B R A
Exploration
1
Writ
ths
a
=
1
quadrati
funtions
as
th
produt
2
f (x)
f (x)
=
2
Graph
3
Writ
two
linar
funtions:
2
+
x
3x
+
2
b
f (x)
=
−x
x
d
f (x)
=
−3x
2
c
of
+
2
2
2x
5x
ah
3
quadrati
funtion
with
its
+
two
4x
1
rlatd
linar
funtions.
2
th
Graph
th
Reect
●
What
funtion
●
is
Whih
bing
Th
zros
rosss
th
a
th
nd
funtion
of
up
So,
th
Exploration
4x
or
th
btwn
of
linar
th
nd
1
as
its
a
produt
two
rlatd
of
two
linar
linar
fators.
funtions.
1
onav
ar
to
+
and
x-intrpts
funtion
x-axis.
to
2x
rlationship
and
paramtr
of
=
discuss
onav
th
yound
quadrati
and
funtions
f (x)
x-intrpts
of
th
original
linar
parabola?
funtions
is
rsponsibl
for
th
parabola
down?
x-oordinats
th
x-valu(s)
th
th
x-intrpts
suh
that
f (x)
of
th
of
any
=
points
whr
funtion
th
graph
algbraially,
0.
2
2
1
a
Plot
th
graph
b
Fatoriz
of
f (x)
=
x
+
2x
3,
and
nd
th
zros
of
this
funtion.
2
produt
c
Solv
d
Chk
x
is
g (x)
+
=
your
vrifying
2x
3
to
nd
two
linar
funtions
g
and
h
whos
f (x).
0
and
h (x)
solutions
that
thy
=
0.
algbraially
satisfy
f (x)
=
by
substituting
thm
into
f (x)
and
0.
2
2
Plot
3
Explain
th
You
For
th
two
=
of
th
at
of
th
f
(x)
or
=
3x
4
law
last
g (x)
h (x)
fator
=
a
×
0,
of
and
or
0
and
if
th
to
that
th
b,
h (x),
and
law
stats
on
=
btwn
funtion
null
numbrs
0
x
rlationship
factor
thn
Whn
g (x)
us
null
zro,
graph
zros
an
Th
th
th
th
whr
=
g
0
produt
must
thn
and
of
quadrati
numbrs
ab
parts
solutions
solutions
solv
if
rpat
h
a
=
ar
of
b
0
a
of
th
to
d
th
in
stp
linar
quadrati
1
quations,
quation.
quations.
two
or
mor
numbrs
is
zro.
or
b
=
0,
funtions,
or
if
both.
f (x)
=
0
thn
both.
11.2 More than one way to solve a problem
4 25
Example
1
2
Solv
2x
6x
=
0.
2
6x
2x
2x
=
x
0
3
=
x
=
⇒
=
0;
0
x
⇒
=
⇒
x
1
0
=
2x (x
3)
=
0
Factorize
0
x
Use
=
the
the
quadratic.
null
factor
law.
3
3
2
x
=
0:
2(0)
=
3:
2(3)
6(0)
=
0,
LHS
=
RHS
✓
Check
both
solutions
in
the
original
equation.
1
2
x
6(3)
=
0,
LHS
=
RHS
✓
2
Example
2
2
Solv
(2x
2x
+
+
2x
5x
1)(x
1
=
3)
0;
x
3
=
0
3
=
=
0.
Factorize the quadratic.
0
Set
x
=
both
Check
✓
linear
the
functions
solutions
in
equal
the
to
0
original
and
solve.
quadratic.
1
2
x
=
3:
2(3)
5(3)
3
=
0
✓
2
Practice
1
Solv
1
ah
quation,
laving
your
answrs
xat.
Tip
2
a
2
x
8x
=
0
b
2
x
5x
+
6
=
0
c
6x
+
4x
16
=
0
To
2
d
2
x
9x
=
0
e
4x
=
1
2
g
10
3x
=
f
3x
2
x
h
x
=
us
fator
2
48
=
2
i
9x
=
10x
2
4
=
Problem
2
Writ
a
23
k
13
2x
2
=
5
l
18
quation
in
standard
form
that
has
2
and
4
6
c
a
quadrati
quation
with
roots
and
4
a
whos
b
with
4 2 6
lading
only
oint
intgr
and
5
3
Find
+
2
and
b
2
3
9x
solutions:
1
a
=
solving
quadrati
is
1
oints.
11 Equivalence
rst
rarrang
th
quadrati
so
2
6x
2
3x
null
0
qual
j
th
law,
5:
3
2x
to
it
zro.
is
A LG E B R A
You
hav
now
Howmany
sn
quadrati
solutions
Reect
and
an
quations
quadrati
discuss
Graph
●
Explain
●
Can
a
f (x)
=
two
uniqu
solutions.
hav?
2
2x
why
hav
2
2
●
that
quations
6x
thy
quadrati
and
hav
a
g (x)
=
4x
dirnt
quation
hav
4x
+
1.
numbr
of
mor
than
uniqu
two
solutions.
uniqu
solutions?
Explain.
Practice
1
Solv
2
ah
quation,
if
possibl,
2
a
+
2x
+
1
=
0
answrs
xat,
x
6x
e
4x
+
9
=
0
c
x
2
9x
6x
+
Problem
Writ
a
1
=
0
10x
x
=
3
s.f.
=
−25
9
=
12x
f
9x
=
−4
12x
solving
quadrati
quation
−2
in
x
b
standard
form
whos
only
solution
is
3
=
c
x
=
5
Example
to
2
+
1
a
or
2
b
2
2
your
2
x
d
laving
4
3
2
Th
lngth
a
Writ
b
Solv
c
a
Writ
Lt
A
x
=
of
an
a
rtangl
quation
th
th
width.
x(x
+
that
4
m
mor
rprsnts
than
th
its
ara
width.
A
of
Its
th
ara
is
21
m
.
rtangl.
quation.
down
=
is
4)
=
dimnsions
Thn
lngth
of
=
th
x
+
rtangl.
4.
State
your
variables
and
write
the
equation.
21
2
b
+
4x
x
+
4x
(x
+
7)(x
x
=
Rearrange,
21
factorize,
and
solve.
2
x
+
7
=
21
=
3)
0
⇒
0
=
x
0
=
−7
=
3
1
x
3
=
0
⇒
x
2
x
c
width
=
3
m,
lngth
=
width
+
4
m
=
7
is
a
length
and
cannot
be
negative
m
hence
x
is
not
a
solution.
1
Practice
1
Th
3
width
of
a
rtangl
is
3
m
lss
than
its
lngth.
Th
ara
of
th
2
rtangl
2
Th
is
lngth
18
of
m
a
.
Find
rtangl
th
is
1
lngth
m
and
lss
width
than
2
of
th
tims
its
rtangl.
width.
Its
ara
is
2
45m
.
Find
its
dimnsions.
11.2 More than one way to solve a problem
4 27
3
A
rtangular
shapd
gardn’s
lngth
is
2
m
lss
than
twi
its
width.
2
If
th
ara
of
th
gardn
is
420
m
,
nd
th
dimnsions
of
th
gardn.
2
4
Th
th
bas
bas
of
a
and
hight
Problem
5
A
5
squar
m
th
xds
of
th
its
hight
by
17
m.
If
its
ara
is
55
m
,
nd
triangl.
solving
and
mor
rtangl
of
triangl
a
rtangl
than
is
6
twi
m
lss
hav
th
th
lngth
than
th
sam
of
sid
th
of
ara.
sid
th
Th
of
th
squar.
lngth
of
squar.
Find
th
th
Th
rtangl
width
lngth
of
of
th
is
th
sid
squar.
4x
x
6
Find
7
A
th
pol
hight
th
lngths
lans
of
wall.
15
of
th
against
m.
Find
Th
th
●
How
of
vrtial
lngth
of
distan
Solving
C
a
sids
this
wall.
th
from
triangl.
Th
pol
th
you
us
is
1
wall
quadratic
an
top
of
m
to
th
pol
mor
th
touhs
than
bottom
twi
of
th
th
its
wall
at
distan
a
from
pol.
3x
equations
quivaln
transformations
to
solv
quadrati
quations?
●
How
ar
th
thr
mthods
for
solving
quadrati
quations
quivalnt?
ATL
Reect
You
Do
hav
you
What
and
sn
think
would
discuss
that
it
its
is
quadrati
possibl
graph
Exploration
3
quations
for
look
a
an
quadrati
hav
on
quation
or
to
two
hav
solutions.
no
solutions?
lik?
3
2
1
Using
a
GDC
a
Find
th
b
Construt
or
graphing
points
lins
whr
softwar,
th
through
graph
parabola
th
th
funtion
intrsts
intrstion
points
th
f (x)
=
x
2x
3.
x-axis.
that
ar
paralll
to
th
y-axis.
c
Grab
what
its
d
th
parabola
happns
vrtx
Th
vrtx
happns
th
is
of
to
x-axis,
to
to
th
th
th
and
and
th
mov
two
lft
and
original
vrtial
whn
it
th
around
lins
right
parabola
lins
is
it
vrtial
as
you
xatly
of
is
th
as
th
th
mov
plan.
th
urv
Dsrib
so
that
y-axis.
blow
mov
on
oordinat
you
th
th
x-axis.
vrtx
so
Dsrib
that
it
is
what
abov
x-axis.
2
2
Graph
th
funtion
f (x)
=
3
+
2x
x
.
Rpat
parts
a
to
d
in
stp
Continued
4 2 8
11 Equivalence
1
on
next
page
+
3
3
A LG E B R A
3
Basd
a
on
your
whthr
linar
b
or
abov
th
not
a
to
stps
1
and
2,
dtrmin:
quadrati
funtion
is
always
quadrati
funtion
has
th
produt
of
two
funtions.
whthr
c
answrs
or
or
not
a
blow
numbr
of
th
zros
if
th
funtion
is
always
x-axis.
zros
of
th
quadrati,
if
its
vrtx
lis
on
th
x-axis.
ATL
Reect
and
discuss
4
2
f (x)
=
x
2
annot
b
fatorizd
using
only
intgrs.
2
●
Solv
th
quadrati
dirn
of
two
quation
x
2
=
0
by
fatorizing
2
=
0
by
rst
using
th
squars.
2
●
Solv
th
quadrati
quation
x
using
th
addition
prinipl.
●
Do
your
two
mthods
giv
th
sam
solution?
2
Whn
a
quadrati
is
of
th
form
ax
c
=
0,
c
>
0,
you
an
us
th
addition
2
prinipl
.
Taking
to
Th
th
rarrang
solution
squar
to
ax
=
c,
is
root
and
th
multipliation
sids
is
quation.
of
lavs
not
an
1,
and
Disuss
x
=
both
sids
=
th
2x
xampl
whih
quivaln
Reect
●
1
For
to
gt
.
of
an
quation
2
quivalnt
prinipl
is
(
1)
obviously
dos
not
always
giv
an
2
=
1
,
fals.
but
taking
Thrfor
th
squar
‘taking
root
th
of
squar
both
root’
transformation.
discuss
rasons
for
5
ah
stp
in
solving
this
quation.
+ 3
2
[1]
x
=
2x
+
3
2
[2]
x
2x
[3]
(x
3)(x
[4]
x
[5]
x
3
=
3
=
3,
0
x
1
●
Is
Whn
a
intgr
fators.
quadrati
th
+
0
1
=
0
−1
of
sids’
funtion
oints,
omplting
=
x
both
quation
Instad
1)
or
=
0
2
‘squaring
quadrati
+
=
has
it
an
an
quivaln
b
solutions.
may
b
writtn
Whn
diult
fatorizing,
to
anothr
as
transformation?
th
produt
quadratis
xprss
mthod
do
thm
for
of
not
as
linar
hav
th
solving
Explain.
fators,
fators
produt
quadrati
of
th
with
linar
quations
is
squar.
11.2 More than one way to solve a problem
4 2 9
Example
4
2
Solv
x
2x
4
=
0.
2
x
2x
4
is
2x
4
=
not
fatorizabl.
2
x
0
‘Complete
the
square’
to
write
2
x
2x
=
4
x
2x
+
1
(x
1)
the
quadratic
in
vertex
form.
2
=
4
+
1
Take
the
square
root
of
both
sides.
Don’t
2
=
5
forget
the
positive
and
negative
Check
the
both
original
square
roots.
solutions
in
equation.
✓
✓
Example
5
2
Solv
th
quadrati
quation
2x
4x
3.
Rationaliz
th
dnominator
in
your
answr.
2
4x
2x
Rearrange
3
the
equation
and
factorize
2
so
that
x
has
coecient
1.
2
2(x
2x)
3
Complete
the
square.
Solve
4 3 0
11 Equivalence
for
x.
A LG E B R A
Practice
Solv,
if
4
possibl,
by
omplting
2
1
x
4
x
7
x
=
−1
Lav
your
answrs
2x
+
2
x
+
5
x
8
2x
2x
=
1
4x
+
6
3
x
6
x
9
3x
2
=
0
2x
20x
+
40
=
0
2
0
Whn
a
20x
+
3
=
0
quadrati
‘omplting
‘quadrati
qual
th
11
quation
squar’
formula’.
to
zro
to
Just
bfor
4x
+
=
4x
annot
using
0
+
6x
+
1
=
0
1
+
6x
=
+
12x
2
2
4x
solv
lik
=
2
+
2
+
2x
2
2
=
2
+
xat.
2
2
2
st
squar.
2
4x
2
10
th
it.
with
th
=
b
3
12
fatorizd,
You
an
also
fatorizd
you
us
form,
2x
an
a
th
us
=
th
mthod
−9
mthod
that
quation
of
involvs
should
th
rst
b
formula.
2
Th
quadrati
Example
formula
to
solv
ax
+
bx
+
c
=
0,
a
≠
0,
is
.
6
2
Solv
x
+
3x
1
=
0,
giving
your
solutions
to
1
d.p.
Write
a
=
1,
b
=
3,
c
=
the
quadratic
formula
and
the
values
the
values
into
the
formula
Separate
1
Solv
for
answrs
4p
th
variabl,
xat.
if
possibl,
Rmmbr
to
st
8p
1
=
0
b
q
g
9x
2q
=
− 5
e
+
if
Chk
11
=
6x
h
possibl,
your
th
solutions
x
d
5a
3a
=
4b
qual
formula,
to
0,
using
a
x
3
=
0
b
2
+
1
=
0
c
− 6c
+
3
f
4b
8
=
1
quations
GDC
or
i
giving
graphing
two
solutions.
3a
+
1
=
0
e
your
nssary.
r
+
3r
6
=
0
2m
+
23
+
8
=
14m
your
4t
=
answrs
−t
to
1
d.p.
softwar.
2
x
2x
4
=
0
c
2
+
laving
whn
2
+
c
simplify.
2
+
quadrati
2
a
quadrati
2
Solv,
the
and
2
3c
2
2
th
quation
2
+
and
2
5a
2
d
using
th
2
+
b
5
2
a
a,
−1
Substitute
Practice
of
r
4p
+
8p
1
=
0
2
+
3r
6
=
0
f
x
+
4x
+
7
=
0
11.2 More than one way to solve a problem
4 31
3
Solv
ths
quadrati
quations
2
a
giving
your
answrs
in
radial
2
=
x
x
+
5
b
form.
Tip
2
x
+
5x
=
+
6z
1
c
2x
x
=
1
For
2
d
7x
=
3
e
2z
=
Solv,
giving
your
0.6x
2.6
answr
to
3
x
2
+
=
0
b
3.2
+
x
1
Crat
rst
a
=
tabl
two
quation,
(if
if
0
e
as
th
3x
2.2x
sam
shown
possibl,
possibl),
th
sid
is
right
so
hand
+
0.3
=
0
c
4
2x
3x
=
qual
to
0.
0
=
3x
+
1.1
4
with
rows
that
2
1.2x
Exploration
rarrang
2
0.2x
2
d
3,
s.f.
2
a
quations
−95
in
4
th
2
x
olumn
hr,
using
omplting
thn
on
th
hadings
ntr
mthod
squar,
or
th
from
th
as
six
th
on
blow,
quations
th
thr
quadrati
blow
and
in
quivalnt
formula.
In
giv
th
it
rst
solution
th
8
rows.
Copy
olumn.
Solv
mthods:
fourth
olumn,
th
ah
fatorization
substitut
th
2
paramtrs
a,
b
and
c
of
th
quadrati
2
into
b
4ac
2
2x
+
5x
3
=
0
2
x
2
+
4x
4
=
0
2x
2
x
+
2x
2
=
0
+
x
+
1
=
0
2
3x
+
4x
+
1
=
0
2x
6x
+
5
=
0
Sketch
Can
of
the
Number
graphical
quadratic
2
Equation
Solutions
of
distinct
b
4ac
solution
be
solutions
(include
x-axis,
factorized?
but
2
+
2x
+
1
=
0
x
=
−1;
x
1
=
−1
1
2
4(2)(1)
=
1
Yes
2
2
x
Us
y-axis)
2
x
2
not
2
+
th
1
=
0
answrs
No
in
solutions
your
tabl
to
0
omplt
0
ths
4(1)(1)
=
−
4
No
sntns.
2
a
Whn
b
b
Whn
b
c
Whn
b
4ac
=
0,
th
quadrati
quation
has
_____
distint
solution(s).
4ac
>
0,
th
quadrati
quation
has
_____
distint
solution(s).
4ac
<
0,
th
quadrati
quation
has
_____
distint
solution(s).
2
2
3
Stat
at
how
many
points
th
quadrati
xprssion
intrsts
th
x-axiswhn:
b
2
4ac
=
may
look
bak
nd
at
quadratis
2
a
You
0
b
ar
2
b
4ac
>
0
c
b
4ac
<
to
othr
that
fatorizabl
0
in
Prati
1
and
2
4
Explain
a
how
quadrati
4 32
th
xprssion
xprssion
an
b
b
11 Equivalence
4ac
hlps
fatorizd.
you
dtrmin
whthr
or
not
2
valuat
b
4ac
A LG E B R A
2
Th
part
and
th
of
th
quadrati
symbol
w
us
formula,
for
it
is
b
th
4ac,
Grk
is
alld
lttr
th
dlta
Δ.
discriminant,
W
writ
2
Δ
=
In
b
4ac
820
th
ce,
mathmatiian
quadrati
Europ
around
by
today
By
for
Mohammad
positiv
mathmatiian
In
1545
quations.
Practice
1
th
1100.
quadrati
know
formula
in
Rn
onsidring
th
ii
whthr
th
numbr
or
of
disriminant
distint
not
th
quadrati
3x
+
2
=
0
b
+
7
=
rst
Géométrie
in
of
ah
x
is
of
th
th
you
4
e
valus
of
k
suh
+
16
+
8x
+
=
0
(1
form
on
w
stat:
quation
fatorizabl.
7x
+
6
=
4x
28x
+
0
40
that
th
c
=
quadrati
solution)
3x
+
17x
2
=
3
0
has
th
givn
numbr
of
solutions.
b
3x
●
How
●
Do
solving
did
or
+
kx
+
12
=
0
(1
solution)
2
k
=
0
(2
Solutions
squar,
th
solving
kx
4x
Whn
in
Spain
work
2
+
D
in
1637.
quadrati
2
c
livd
xisting
appard
quadrati,
2
a
th
to
2
−5x
Problem
Find
who
drivd
brought
2
x
2
2
Hiyya,
was
2
2x
3x
Al-Khwarismi
mthod
ompild
formula
solutions
2
d
La
His
bar
Cardano
quadrati
Dsarts’
Musa
6
i
a
Abraham
Girolamo
Th
bin
solutions.
do
systms,
quadrati
whih
th
you
of
th
quadrati
solutions)
of
d
kx
quadratic
dtrmin
modls
and
quations,
thr
a
‘bst
formula)
is
th
solv
for
solution
bst
+
5
=
0
(no
solutions)
equations
partiularly
of
3x
mthod’
mthods
mthods
+
among
quivalnt
problms
ral-lif
or
mthods?
rat
problms,
(fatorizing,
thm?
how
omplting
do
th
mthod?
ATL
Reect
and
discuss
●
of
thr
Whih
If
this
●
How
●
Whih
your
th
mthod
ould
did
you
mthod
solutions
mthods
not
nd
2
work,
out
would
to
6
if
you
would
whih
th
try
dimal
you
quation
rst
if
plas?
try
rst?
mthod
is
th
Explain
would
you
why.
try
nxt?
fatorizabl?
qustion
Explain
askd
you
to
giv
why.
11.2 More than one way to solve a problem
4 3 3
Example
7
2
A
gardnr
has
16
m
of
fning
to
stion
o
a
rtangular
ara
of
15
m
.
Tip
Dtrmin
th
dimnsions
of
onstraints
th
on
th
dimnsions
of
th
rtangl,
and
nd
th
rtangl.
‘Dtrmin
onstraints’
Primtr
=
2(lngth
+
width)
=
lngth
Lt
+
width
width
=
x.
=
8
Thn
for
lngth
=
8
1:
x
>
⇒
0
<
Ara
x
=
<
2:
th
inquality
valus’.
and
width
can
be
only
positive
values.
0
Write
Constraint
an
x.
Length
Constraint
mans
16
‘writ
⇒
th
8
x
>
0,
so
x
<
inequality
statements
to
reect
this.
8
8
x(8
x)
=
15
2
x
=
15
8x
+
15
8x
Calculate
the
discriminant
to
see
2
x
=
0
if
Hr,
a
=
1,
b
=
−8
2
⇒
and
c
=
4ac
=
(
The
8)
−
(4
×
1
equation
is
factorizable.
discriminant
is
positive,
15
2
b
the
×
15)
=
so
4
there
will
be
two
unique
solutions.
2
x
8x
x
=
3;
+
15
x
1
=
=
(x
3)(x
5)
5
Both
values
3
and
5
satisfy
Whn
x
=
3:
width
=
3,
lngth
=
8
3
=
5
Whn
x
=
5:
width
=
5,
lngth
=
8
5
=
3
Th
dimnsions
Practice
of
th
rtangl
ar
3
m
by
5
ths
Slt
x
<
8.
7
problms,
th
most
rat
int
a
mathmatial
mthod
to
solv
modl
th
to
solv
quadrati
th
problm.
A
quation.
skth
solving
gardnr
has
40
m
of
gomtri
modls,
you
Problem
1
<
m.
With
In
0
2
draw
to
a
hlp
visualiz
th
problm.
fning
to
mak
thr
qual
rtangular
plots.
2
Th
total
ara
dimnsions
2
50
m
an
xisting
of
of
of
th
th
fning
thr
plots
rtangls,
is
usd
to
is
and
mak
30
m
nd
thr
.
Dtrmin
th
th
dimnsions
sids
of
a
onstraints
of
ah
rtangular
on
th
rtangl.
ara,
using
2
Dtrmin
3
A
wall
th
rtangular
as
th
fourth
dimnsions
gardn
sid.
of
th
masurs
Th
ara
of
rtangular
24
m
by
32
th
rtangl
is
150
m
.
ara.
m.
A
path
of
uniform
width
x
is
10
built
all
around
th
outsid
of
th
gardn.
Th
total
ara
of
th
path
and
2
gardn
is
1540
m
.
Find
th
width
of
th
path.
6
4
A
rtangular
sids
of
th
swimming
pool
has
pool
width
x
masurs
mtrs.
10
Th
4
ara
is
of
th
pool
ara
on
its
3
4 34
11 Equivalence
own.
m
total
Find
x
by
6
ara
m.
A
pavd
inluding
ara
th
on
pavd
two
x
m
m
A LG E B R A
5
Th
of
6
primtr
th
A
studnt
th
sam
from
of
a
rtangl
is
42
m.
Its
diagonal
is
15
m.
Find
th
width
rtangl.
yls
hom
north
from
distan
and
to
hom
plus
shool
an
is
to
shool,
additional
17
km,
nd
7
du
km
th
north
du
for
ast.
distans
a
If
that
distan
th
th
x
dirt
km,
thn
distan
studnt
yls
ast.
3
7
A
box
Find
8
A
has
a
squar
bas
of
sid
x
m,
hight
4
m,
and
volum
289
m
.
x
box
with
utting
an
opn
squars
of
top
qual
is
mad
siz
from
from
a
ah
rtangular
ornr.
Th
pi
of
ardboard
ardboard
by
masurs
2
50
9
m
by
40
m.
a
Dtrmin
b
Find
Two
th
Th
th
sid
volum
numbrs
ara
hav
th
lngth
of
a
of
th
of
bas
th
is
875
squars
m
ut
from
ah
ornr.
box.
dirn
of
3
and
a
produt
of
88.
Find
th
numbrs.
10
11
12
Th
produt
Th
sum
Find
th
Whn
of
th
onsutiv
numbrs
is
9
odd
and
numbrs
th
sum
of
is
143.
thir
Find
th
squars
is
numbrs.
153.
numbrs.
squar
numbr.
Example
Whn
two
two
two
th
original
of
of
a
numbr
Find
th
is
drasd
by
1,
th
rsult
is
4
tims
th
numbr.
8
braks
ar
applid
in
a
rtain
mak
of
ar,
th
lngth
L
of
th
skid
2
in
mtrs
in
km/h
aidnt
tim
of
is
at
givn
th
by
tim
lavs
L
=
0.168s
whn
skid
th
marks
0.8s
braks
+
0.3,
wr
masuring
50
whr
rst
m.
s
is
applid.
Dtrmin
th
A
spd
ar
th
of
th
involvd
ar’s
spd
ar
in
an
at
th
braking.
2
0.8s
0.168s
+
0.3
=
50
Write
the
equation.
2
0.168s
0.8s
49.7
=
0
a,
Using
th
quadrati
b
and
c
quadratic
a
=
0.168,
b
=
−0.8,
c
=
=
−15.0;
1
s
=
be
so
the
factorized.
the
positive
value
makes
sense
19.7
2
in
Th
ar
Th
of
decimals,
cannot
−49.7
Only
s
are
formula:
was
travlling
distan
th
ar
distans
a
and
to
ar
at
skids
wathr
alulat
approximatly
dpnds
on
onditions.
travlling
20
svral
Car
spds,
fators,
aidnt
to
the
context
of
the
question.
km/h.
hlp
inluding
invstigators
disovr
th
th
us
spd
skid
aus
of
an
aidnt.
11.2 More than one way to solve a problem
4 3 5
Practice
8
Problem
1
Th
solving
amount
of
mony
in
bank
aount
A
is
givn
by
th
formula
t
A
=
P(1
+
r)
whr
ompoundd
invst
you
$5000
with
P
is
annually
and
your
th
and
would
rst
initial
yar
t
lik
of
is
it
invstmnt,
th
to
tim
b
in
is
yars.
worth
univrsity
r
intrst
Your
$6500
osts.
th
aftr
two
Dtrmin
rat
mothr
th
wants
yars,
to
intrst
to
hlp
rat
shnds.
2
An
intr nt
months
tr ms
ompany
bfor
of
th
going
was
listd
bankr upt.
numbr
of
on
a
Th
months
x
small
pri
that
th
stok
P
of
xhang
th
stok
for
ompany’s
tradd
on
svral
stok
th
in
xhang
2
isgivn
by
months
it
whn
3
A
=
sold
tikts
Price
−2.25x
for
th
organizs
From
9500
sold
Copy
=
+
40.5x
ompany
+
to
42.75.
go
Dtr min
bankr upt,
in
th
othr
numbr
of
words
0.
stadium.
thy
a
took
P (x)
harity
th
P(x)
a
past
tikts.
inrass
and
sor
omplt
Decrease
vry
rais
thy
$1
mony.
know
dras
that
in
ticket
Increase
price
15
15
15
=
0
14
15
14
=
1
ar
whn
tikt
th
pri,
15
000
tikts
th
sats
ost
in
$15,
numbr
of
tabl.
in
Total
($)
Thr
1000.
th
in
to
xprin
For
by
gam
attendance
Total
sales
0
1000
×
1
9500
=
1000
9500
+
1000
15
=
10
500
14
×
×
9500
10
=
500
142
=
12
b
From
of
c
15
th
tikt
Writ
an
nobody
d
p
last
row
pri
quation
would
Dtrmin
th
would
sll
you
e
Dtrmin
f
Considr
unsold
4 3 6
in
th
your
sats
if
th
tabl,
nd
a
funtion
for
sals
gnratd
in
trms
p
to
buy
a
tikt
at
nd
pri
that
tikt
tikt
Solv
that
it
pri
that
using
would
th
would
most
maximiz
b
so
xpnsiv
int
sals.
How
that
mthod.
many
tikts
pri?
pri
answrs
you
th
tikt.
sold
for
ndd
d
and
tikts
11 Equivalence
at
to
e.
ll
all
Did
th
pri
15
000
what
that
sats.
you
givs
ould
do
with
maximum
th
sals.
500
147
13
p
($)
attendance
000
A LG E B R A
4
A
swimming
tam
intrnational
shown
in
this
Price
in
a
Find
and
of
slls
t-shirts
omptition.
A
with
thir
survy
into
to
rais
and
mony
for
prditd
an
sals
is
tabl.
t-shir t
Euros
Predicted
(p)
number
t-shir ts
sold
5
165
10
150
15
130
20
118
25
97
30
85
35
65
40
56
45
42
50
28
th
logo
pris
lin
numbr
of
of
bst
t
t-shirts
that
bst
sold.
of
(s)
dsribs
This
is
th
th
rlationship
dmand
btwn
pri
Lins
funtion.
of
ovrd
5
b
Dtrmin
th
rvnu
c
Dtrmin
th
t-shirt
pri
that
would
maximiz
rvnus.
d
Dtrmin
th
t-shirt
pri
that
would
guarant
that
A
36
ski
rntal
rntals
rntal
pr
pri,
day,
th
and
hargs
day.
A
ski
avrag
vi
a
rntal
journal
ski
f
of
artil
rntal
$50
pr
stats
businss
pair
that
an
of
for
no
skis
t-shirts
and
vry
xpt
to
$5
los
two
b
Dtrmin
th
ski
rntal
pri
that
would
maximiz
rvnus.
c
Dtrmin
th
ski
rntal
pri
that
would
guarant
th
to
hav
xris
to
follow
numbr
a
Your
bik
a
of
will
linar
biks
a
nw
ost
how
ii
$120
and
many
to
mak.
is
th
c
Writ
a
funtion
for
th
total
d
Writ
a
funtion
for
th
prot
e
Calulat
what
th
maximum
th
biks,
to
pri
and
by
s
=
of
ould
bik
osts
Sals
pri
you
Us
on
shop
ould
not
of
will
suh
80
in
th
want
$800
to
000.
whr
sll
Eah
(dmand)
180p,
bik
at
you
b
biks
000
th
sll
whih
s
tnd
is
th
dollars.
pri
of
$500.
b
prot
xris
start-up
modld
p
iii
funtion
of
and
biks
$300
linar
kind
funtion
sold
i
th
in
rntals
funtion.
advrtising
Dtrmin
$100
sold.
skis.
dsignd
rtailrs.
4.3.
vrsa.
rvnu
You
ar
avrags
th
any
ar
inras
Dtrmin
out
t
topi
funtion.
a
rnt
6
shop
pr
bst
in
writ
of
us
th
osts
you
th
this
of
funtion
produing
would
bik
to
rvnu
mak
would
b
alulat
if
th
th
on
for
th
sal
biks.
biks.
produing
you
of
wr
pri
that
to
th
biks.
mak
would
no
nsur
a
prot.
11.2 More than one way to solve a problem
4 37
Objective:
ii.
slt
D.
Applying
appropriat
mathmatis
mathmatial
in
ral-lif
stratgis
ontxts
whn
solving
authnti
ral-lifsituations
Draw
and
method
label
to
suitable
solve
diagrams,
create
equations
and
select
the
most
ecient
them.
Activity
Do
1
th
opn
Suitass
ar
Eah
sht
sht
is
problm
mad
is
opnd
Draw
a
b
Exprss
all
c
Exprss
th
d
Dn
e
Us
up
th
your
and
Dtrmin
in
Prati
7
qustion
rtangular
half,
squars
shts
ar
8,
of
ut
bfor
lathr
from
th
this
60
ativity.
m
by
ornrs
92
and
m.
th
again.
diagram
th
in
from
foldd
a
as
2
box
and
labl
th
dimnsions
volum
of
th
variabl
in
trms
as
for
of
using
th
sid
your
ths
of
ah
squar
utout.
variabl.
dimnsions.
onstraints.
GDC
or
th
siz
th
siz
350
m
softwar
of
th
of
to
dtrmin
ut-out
th
squars
squars
to
th
that
ut
out
maximum
giv
to
this
giv
volum
maximum
a
as
of
th
volum.
with
3
volum
3
Crat
of
th
Us
18
a
mathmatial
squars
suitabl
maximum
4
Rsarh
to
b
suitabl
th
maximum
modl
ut
softwar
volum
alulat
giv
to
to
for
siz
siz
out
of
xplor
svral
ass
th
volum
whih
from
th
siz
siz
dirnt
for
a
your
st
of
st
and
of
you
to
did
on
th
sizs
rtangl.
th
starting
travl
rtangl
for
nabls
any
of
th
3
squar
giving
th
rtangls.
ass.
siz
of
Us
your
squars
to
modl
ut
out
to
ass.
Summary
2
Th
two
null
or
factor
mor
law
stats
numbrs
is
that
zro,
if
th
thn
produt
at
last
of
on
Th
fators
must
b
formula
to
solv
ax
+
bx
+
c
=
0,
of
a
th
quadrati
≠
0,
is
.
zro.
2
Th
For
two
fators
a
and
b,
if
ab
=
0
thn
a
=
0
par t
alld
b
=
0,
or
if
f (x)
f (x)
=
th
quadrati
formula,
b
4ac,
th
discriminant,
and
th
symbol
0
=
us
×
g (x)
h(x),
=
0
whr
or
h(x)
g
=
and
0,
or
h
ar
is
th
Grk
lttr
dlta
Δ.
W
writ
Δ
=
b
4ac
funtions,
●
If
Δ
<
0,
th
quadrati
has
no
●
If
Δ
=
0,
th
quadrati
has
on
ral
root
has
two
ral
roots.
ral
roots.
both.
2
Whn
a
quadrati
is
of
th
form
ax
c
=
0,
c
>
0,
(a
you
an
us
th
addition
prinipl
to
●
ax
c,
and
th
multipliation
rpatd
prinipl
to
If
Δ
Th
solution
is
>
0,
th
th
disriminant
is
a
prft
squar
(.g.
.
1,
4,
9,
16,
...),
fatorizabl.
4 3 8
quadrati
gt
Whn
.
root).
rarrang
2
to
for
2
g (x)
thn
is
w
both.
it
Whn
of
or
11 Equivalence
thn
th
quadrati
xprssion
is
A LG E B R A
Mixed
1
Solve
your
practice
ah
quadrati
answrs
by
fatorizing,
graphially.
Lav
and
your
hk
8
answrs
A
is
xat.
rtangular
lawn
surroundd
Th
by
ombind
a
masurs
bordr
ara
of
of
th
8
m
by
4
uniform
lawn
and
m
and
width.
bordr
2
2
a
is
2
x
− 2x
− 24
=
0
b
x
− 6x
=
2
c
=
5x
+ 4
d
17 x
2
2
Solve,
hk
+ 7x
if
=
2
by
answrs
omplting
graphially.
=
th
8x
− 59
=
0
b
x
=
bordr.
21
8
squar,
and
4
m
m
your
+ 12 x
2
=
−23
2
− 10 x
e
2x
+ 26
=
8
3x
d
+ 6x
=
7
9
2
Th
hight
Solve,
giving
8x
− 3
f
a
right-angld
triangl
is
5
m
lss
2
2x
− 5x
− 3
=
than
0
if
possibl,
answrs
using
to
2
th
d.p.,
quadrati
and
hk
its
answrs
10
2
+ 13
=
0
b
x
− 5x
= 1 + 3x
x
d
=
−
11
x
2
of
th
triangl
is
42
m
.
hight.
On
sid
of
a
Find
th
Find
two
triangl
and
sid
7
m
lngths
is
2
m
longr
of
th
shortr
than
than
th
th
third
sid.
triangl.
produt
onsutiv
is
odd
intgrs
whos
99.
2
+ 8x
= 1
f
−3 x
+ 2x
= 12
12
Solve,
and
ara
= 2
3
4 x
Th
7
1
2
2
3x
itsbas
hypotnus
2
+ 10 x
bas.
formula,
your
graphially.
x
of
2
=
Find
4
th
xat.
+ 6x
x
e
of
2
x
c
c
width
− 6
Lav
2
a
th
27
− 2x
2x
f
possibl,
your
answrs
3
Find
2
−5 x
a
.
m
2
6x
e
165
if
mthod.
possibl,
Lav
using
your
th
most
answr
appropriat
xat
and
Chk
your
answrs
squar
itslf
by
Find
two
of
72.
a
numbr
Find
th
xds
th
numbr
numbr.
ompltly
13
simplid.
Th
onsutiv
positiv
intgrs
suh
that
graphially.
th
squar
of
th
sond.
th
rst
lss
17
quals
4
tims
2
x
2
a
(x
5
2
+ 3)
+
x
− 9x
=
+
b
8
2
= −x
2
2
c
(x
2
− 2)
− 11 =
0
d
2x (x
− 1) − 5
=
14
Th
hight
into
th
h
in
mtrs
of
a
football
kikd
− x
air
an
b
modld
by
th
funtion
2
2
e
(2 x
− 6)
h (t )
= 12
a
f
(2 x
+ 5)( x
− 1)
=
(x
− 3)( x
=
−4.9t
Find
ths
nxt
problms
using
th
most
Round
your
answrs
to
5
Th
solving
lngth
of
a
rtangl
whr
aftr
bing
t
is
in
sonds.
kikd
it
taks
th
c
is
3
hit
th
ground.
Determine
how
long
it
taks
th
objt
to
appropriatly.
rah
Problem
long
+
int
b
mthod.
how
t
+ 8)
ball
Solve
+ 24.
m
gratr
than
its
15
For
Th
a
hight
how
long
prots
of
of
20
was
an
mtrs.
it
abov
this
intrnational
hight?
tikt
2
width.
Its
ara
is
108
m
.
Find
th
agny
dimnsions
an
b
modld
by
th
funtion
2
of
th
P (t )
rtangl.
=
−37t
numbr
6
Th
lngth
of
a
rtangl
is
2
m
mor
of
+ 1258t
tikts
− 7700,
sold
and
whr
P
is
in
t
is
th
dollars.
than
Determine
th
tikt
pri
that
would
lav
th
2
3
tims
its
width.
Its
ara
is
85
m
.
Find
th
agny
dimnsions
of
th
Th
lngth
of
a
rtangl
is
1
m
mor
than
If
th
lngth
of
th
rtangl
is
ara
of
th
loss.
A
squar
rtangl
inrass
by
a
pi
box.
of
Aftr
ardboard
5
m
is
squars
to
b
ar
formd
ut
from
ah
doubld,
2
th
or
its
into
width.
prot
rtangl.
16
7
no
30
m
ornr
and
th
hav
volum
sids
ar
foldd
up,
th
box
will
.
3
Find
th
dimnsions
of
th
original
a
of
400
m
.
Find
th
lngth
rtangl.
sid
of
th
original
pi
of
ardboard.
11.2 More than one way to solve a problem
4 3 9
of
a
Objective:
ii.
slt
ral-lif
In
the
and
D.
Applying
appropriat
mathmatis
mathmatial
in
ral-lif
stratgis
ontxts
whn
solving
authnti
situations
Review
select
the
Review
in
context
most
in
section,
eective
draw
method
to
and
label
solve
suitable
diagrams,
create
equations,
them.
context
Scientic and technical innovation
Quadrati
projtil
ground
that
is
quations
through
at
dirnt
proplld
kiking
a
sor
launhing
an
spa,
modl
and
tims.
with
ball,
A
for
th
its
path
hight
projtil
through
doing
th
is
th
high
of
a
abov
any
air,
jump
a
Writ
th
quadrati
b
Determine:
suh
or
as
i
th
ii
how
modls
of
projtil
motion
tak
maximum
iii
how
thr
initial
th
hight
initial
th
o
vloity
alration
th
A
long
with
whih
th
objt
=
objt
taks
to
rah
this
hight
th
objt
taks
to
strik
th
again.
du
to
is
launhd
gravity
that
upward
at
39.2
from
m/s.
ground
lvl
Determine
how
movs
ats
th
projtil’s
altitud
is
34.3
m
or
abov.
all
for
alulating
th
hight
abov
Suppos
NASA
sixth
of
2
(−9.8)t
+V
t
of
th
gravitational
is:
1
h (t )
rahs
wants
to
launh
a
prob
on
th
objts.
formula
ground
th
long
projtil
surfa
Th
objt
ground
4
falling
th
fators:
long
●
hight
into
dirtly
●
objt.
vn
3
th
this
objt
ground
●
for
rworks.
Quadrati
aount
modl
th
th
th
moon,
pull
of
whih
th
alration
launhr
is
0.5
has
Earth
du
to
on
(and
sixth
th
thrfor
gravity).
Th
on
hight
m.
+ h
i
i
2
a
whr
th
alration
du
to
gravity
(on
Earth)
On
th
moon,
how
long
would
it
tak
th
is
prob
to
rah
a
hight
of
60
m
if
its
initial
2
9.8
m/s
,
V
is
th
initial
vloity,
and
h
i
initial
is
th
i
vloity
is
b
If
th
initial
longr
Problem
will
A
m/s?
modl
th
rokt
ground
is
with
launhd
initial
from
vloity
1
quadrati
vloity
it
b
in
is
th
24.5
air
m/s,
on
th
how
muh
moon
solving
ompard
1
15
hight.
modl
h (t )
m
abov
(It
Use
bothplas.)
m/s.
th
is
a
launhd
similar
from
launh
th
on
surfa
Earth?
in
2
( −9.8)t
=
2.5
49
to
+ 49t
+ 2.5
to
c
Explain
how
you
think
sintists
tst
thir
2
determine
how
long
it
taks
for
it
to
hypothss
land.
th
2
An
objt
is
launhd
dirtly
upward
moon
m/s
from
a
platform
Reect
and
How
you
hav
30
m
(or
is
so
dirnt
high.
th
statmnt
of
inquiry?
Giv
spi
xampls.
Statement of Inquiry:
Rprsnting
modls
and
4 4 0
Mars)
discuss
xplord
pattrns
with
quivalnt
mthods.
11 Equivalence
th
bhavior
whn
th
of
objts
for
at
thr
24
about
forms
an
lad
to
bttr
systms,
than
on
Earth.
of
on
gravity
Seems
rational
to
me
11.3
Global
context:
Objectives
●
Solving
linear
Scientic
Inquiry
and
quadratic
rational
equations
and
technical
innovation
questions
●
What
●
How
●
How
is
a
rational
algebraic
equation?
F
algebraically
●
Using
and
graphically
equivalence
to
do
equations
Using
and
rational
solve
solve
equations
with
fractions?
are
the
dierent
methods
of
solving
C
rational
●
you
solve
equations
to
model
algebraic
equations
related?
situations
●
What
●
Is
makes
dierent
methods
equivalent?
problems
there
a
‘best
method’
for
solving
D
rational
●
Does
equations?
science
solve
problems
or
create
them?
ATL
Transfer
Apply
skills
and
knowledge
in
unfamiliar
situations
10.2
Statement of Inquiry:
Representing
helped
change
humans
apply
and
equivalence
their
in
a
understanding
variety
of
of
forms
scientic
has
principles.
11.3
E10.2
4 41
MROF
rational
transformations
Y
ou
●
should
apply
already
know
equivalence
transformations
linear
1
to
how
Solve
solve
a
2x
to:
using
+
3
=
equivalence
5x
9
b
transformations:
4(x
1) = 2x + 7
3x
equations
c
=
10
2
●
carr y
out
the
operations
four
on
arithmetic
2
Simplify :
x
rational
5
b
3
algebraic
7
3x
+
a
x + 2
4
x
expressions
2
2
2 x
6
c
x
2
16 x
+ 4 x + 3
x
d
×
14 x
÷
2
2
x
●
factorize
quadratic
3
9
4 x
5x + 5
12
2
a
2
x
16
b
3x
2
c
solve
quadratic
equations
4
5x
6
d
3x
b
x
c
What
●
How
is
a
do
2x
algebraic
rational
you
18
=
0
+
2x
=
3
2
+
7x
+
3
=
0
d
3x
4x
1
=
0
equations
algebraic
solve
12
2
2x
2
●
5x
Solve:
2
Rational
6x
2
x
a
F
49
Factorize:
expressions
●
x
equation?
equations
with
fractions?
p
A
rational
number
is
of
the
form
where
p
and
q
are
integers,
and
q
≠
0.
q
P (x )
Similarly
,
a
rational
algebraic
expression
is
of
the
form
where
Q (x )
Q(x)
≠
0.
A
algebraic
The
rules
algebraic
rational
algebraic
equation
is
an
equation
containing
rational
expressions.
and
procedures
expressions
Example
and
you
use
with
rational
numbers
also
apply
to
rational
equations.
1
There
is
no
standard
The
total
electrical
current
(16
amps)
used
by
two
appliances
is
given
by
voltage
x
= 16 ,
where
x
volts
is
the
voltage
used
by
each
10
countries.
voltage
used
by
each
most
appliance.
countries
200
it
is
and
x
+
= 16
240
10
volts
x
30
and
in
×
+
30
×
=
30
the
× 16
Americas
127
x
=
480
4x
=
480
x
=
120
Use
the
each
Each
120
of
it
is
30
between
+
most
x
10
3x
while
30
Japan
appliance
uses
120
multiplication
term
by
30
to
100
and
volts.
principle.
Multiply
simplify.
volts.
120
+
10
In
30
between
x
by
appliance.
all
the
used
x
+
equation
Find
mains
the
=
12
+
4
=
16
✓
30
4 4 2
11 Equivalence
Check
by
substitution.
A LG E B R A
Multiplying
eliminates
each
one
Practice
Solve
these
x
rational
5x
+ 3
− 3
3x
2x
4 x
+ 7
=
by
is
the
no
lowest
longer
2x
your
a
answers
− 4
common
rational
−1
x
1
−
3
7
+ 4
−
denominator
equation.
or
3x
− 5
+
f.
2x
4 x
4
−1
−
x
+ 2
=
3
2
=
5
5x
8
s.
x
2
8
3
2
+ 1
6
6
to
=
3
4 x
5x
=
3
1 − 5x
6
exact,
2x
=1
+
4
Problem
x
+1
−
8
6
12
solving
1
−
=
x
4
3
Reect
●
leaving
4
−1
+
3x
it
7
5
2
2
so
+
5x
=
5
2
− 9
2
2x
+
9
equation
6
6
7
the
equations,
5
4
x
in
denominators
1
= 2
4
x
the
3x
+
1
fraction
of
How
and
is
discuss
question
9
in
1
Practice
1
similar
to
and
dierent
from
the
other
questions?
●
How
did
that
When
C
●
How
aect
are
What
process
dierent
the
equations
●
the
dierent
you
is
used
the
methods
to
solve
the
equation?
same
of
solving
rational
algebraic
related?
makes
dierent
methods
equivalent?
Tip
Multiplying
solving
by
simple
denominators
or
equations
the
lowest
rational
are
have
common
algebraic
rational
denominator
equations.
numbers.
variables
in
the
What
The
is
a
useful
LCM
happens
is
method
easy
when
to
for
nd
rational
when
the
expressions
denominator?
Finding
lowest
denominator
same
as
lowest
Exploration
1
at
these
pairs
of
rational
expressions
and
their
lowest
two
is
the
nding
the
common
multiple,
of
Look
the
common
or
LCM,
more
common
numbers
or
two
denominators.
or
Rational
2
expressions
Lowest
common
more
algebraic
expressions.
denominator
3
x(x
,
x
x
x
+1
+
2)
+ 2
1
7
(x
,
x
5
+
1)(x
3)
− 3
1
x(x
,
7)
2
x
x
7x
Continued
on
next
page
11.3 Seems rational to me
4 4 3
10
6
,
(x
+
2)(x
2)
(x
+
1)(x
5)(x
1)
(x
+
2)(x
1)(x
1)
2
x
4
x
2
8
3
,
(x
+ 1)( x
2x
− 5)
5
(x
− 1)( x
+ 1)
3x
,
2
+
x
x
− 2
x
+ 2
−1
3x
1
x(2x
,
2
+
1)(x
1)
2
2x
+
x
x
2x
1
−
x
−1
4
2
(x
,
2
x
+
2
x
3)(x
+
3)
2
− 9
1
x
+ 6x
+ 9
Determine
the
rst
step
in
nding
the
LCM
of
two
algebraic
rational
expressions.
2
After
that
rational
rst
step,
algebraic
explain
how
you
would
nd
7
3
Find
the
LCM
of
the
LCM
these
three
x
3
,
expressions:
1
The
+1
2
x
6x
x
− 7x
+ 6
2
denominators
lowest
two
and
2
x
Practice
of
expressions.
common
of
two
or
more
denominator
of
algebraic
the
given
fractions
are
given.
2
a
x,
x
d
x,
x
2
g
x,
x
2,
j
x
b
2x,
e
x,
3x
2,
+
+
1
2
h
x,
k
x
x
x,
x
,
3x
f
x
2,
x
i
x
1,
x
1
l
x
1,
x
1,
n
x,
+
2
2
x
+
1
+
4,
1,
2x
+
2,
x
2
2
x,
x
2
m
x
a,
x
2
p
3x
Do
+
1,
x
8,
x
2x
and
you
see
+
use
Find
solve
the
+
x
1
x
x
6
o
2x,
2x
3,
x
+
1
any
4 4 4
2
potential
issues
with
having
variables
in
the
Explain.
value(s)
rational
same
2
discuss
any
Exploration
To
8
2
1
denominator?
●
+
2
,
Reect
●
the
4
c
2
x
Find
denominators.
of
x
that
are
not
allowed
in
each
expression
in
1.
algebraic
procedure
as
equations
for
with
rational
11 Equivalence
variables
equations
in
with
the
denominator
numerical
you
denominators.
A LG E B R A
Example
2
3
Solve
the
equation
8
and
=
x
LCM(x,
x
2)
=
x
x(x
− 2)
×
x (x
− 2)
=
x
2 )
(
x
2 )
=
8x
6
=
8x
2)
algebraic
each
x
This
only
3
x
2
x
2
= 1 for x ≠ 2
solution
is
excluded
valid
Check
−
−1.2 −
RHS
Why
are
these
check
your
you
✓
discuss
allowed
numerator
are
to
and
you
3
put
lines
through
the
factors
that
are
common
denominator?
really
multiplying
by
when
you
do
this?
3
rational
your
equations
answers
using
appropriate
algebraically.
Make
equivalence
sure
you
check
transformations,
for
extraneous
solutions.
4
9
1
10
=
x
4
4
2
5
+
3x
3
4(x
10
5
4
− 2)
5
=
x
5
+
3x
= 2 − x
x
8
x
+ 1 =
− 2
10
= 7 +
x
− 2
x
− 2
3
9
x
x
5
6
x
3
=
4
72
2
2 −
+ 1)
+
x (x
+
− 2
+ 4
3
x
6
=
x
=
4
3
7
x
15
2
x
2.
answer.
2
−2.5
number
Practice
and
=
and
both
What
Solve
and
= −2.5
2
Reect
●
the
0
8
=
to
are
= −2.5
8
●
since
values
1.2
RHS:
=
fraction.
3
=
x
LHS
equal
a
= −1.2
5
x
in
6
= −
LHS:
place
= 1 for x ≠ 1
,
8x
6
x
to
expression
x
×
2)
3x
5x
(
− 2) =
Rearrange
×
(x
x
3(x
algebraically.
8
=
x
3( x
answer
Find the lowest common denominator.
x
×
your
2)
3
x (x
check
2
x +
= 4
x
+1
11.3 Seems rational to me
4 4 5
In
rational
the
equations,
denominator
you
equal
to
must
exclude
any
3
For
example,
in
the
rational
to
x
zero.
=
eliminate
An
0
If
and
the
x
that
the
allowed
you
the
the
Reect
because
to
the
solution
original
in
have
2
the
is
one
equation.
equation
solutions,
and
you
●
How
●
What
could
●
What
is
●
What
values
easy
is
the
First
reject
variable
that
make
need
to
exclude
the
2
one
of
includes
any
the
denominators
either
0
or
2,
you
equal
need
to
algebraically,
determine
make
that
are
any
the
not
values
but
of
x
denominator
does
that
not
are
zero.
not
When
allowed.
3
=
2
− 4
to
you
nd
4
+
it
you
they
2
2
x
x
+ 2x
x
determine
do
rst,
to
the
LCM
make
it
of
the
easier
algebraic
to
fractions?
determine
the
LCM?
LCM?
of
x
are
not
allowed?
3
1
x
4
+
Solve
x
make
that
because
discuss
equation:
Example
they
equation
7
For
the
answer.
extraneous
satisfy
=
solution
of
8
=
equation
x
values
values
0.
=
and
check
your
answers
algebraically.
2
x
−
6
x
−
2
− 8x
x
+ 12
2
x
8x
+
1
12
−
(x
2)
6
x
−
− 6)( x
Factorize
the
denominator
on
the
right
side.
4
=
;
2
(x
−
6 )( x
−
1

(x
6)(x
x
+
x
=
x
− 2)
≠ 6,
x
≠ 2
Identify


= (x
+

− 6)( x
2)
x
6
+
any
values
of
x
that
are
not
allowed.


2 
x(x
4

6)
=
4
5x
2
=
4
5x
6
=
0
1)
=
0
Use
the
multiplication
principle
− 2)

 x
(x
x
2)

(x
−
6 )( x
−
2)
with

the
LCM
of
all
denominators.
2
x
2
x
(x
Since
x
≠
6,
6)(x
the
1
only
6
1
= −
−1 −
2
⇒
possible
1
+
LHS:
−1 −
+
Set
1
+
7
x
=
6,
x
solution
=
is
−1
x
=
−1.
Reject
LHS
=
RHS
4 4 6
any
and
to
0.
solve.
invalid
solutions.
original
equation.
4
21
=
2
− 8( −1) + 12
equal
Factorize
Substitute
( −1)
quadratic
=
3
=
RHS:
the
1 + 8 + 12
21
✓
11 Equivalence
the
solution
found
into
the
A LG E B R A
Practice
Solve
these
4
equations
using
appropriate
equivalence
transformations.
Make
7
5
10 x
+
1
2
x
=
2
2
x
+ 2
x
− 2
1
x
2
+
check
=
sure
for
you
extraneous
2
4
x
− 2
x
− 4
x
− 6x
+ 8
solutions.
2x
4
2
2
=
3
1
=
4
2
x
2
− 10 x
+ 16
2x
x
+ 2
x
−
x
x
−1
3x
=
5
+
2
2
x
Are
− 3
x
there
− 9
any
other
methods
Exploration
For
the
2
If
b
for
=
d,
a,
b,
and
c,
to
equations?
to
be
be
equal,
determine
what
conditions
d
what
must
be
true
about
a
and
c
in
order
Select
any
pair
of
equivalent
fractions,
for
example
true
about
across
these
the
equal
products.
sign,
so
Explain
3
if
×
24
you
and
6
think
×
12.
that
,
and
Generalize
your
result
for
any
pair
of
equivalent
it
Suggest
is
Prove
6
State
be
7
your
in
result
which
using
equivalence
examples
from
what
always
is
true.
c
fractions
=
.
d
b
5
multiply
24
a
4
the
12
=
6
diagonally
for
equal.
3
3
must
d
and
determine
fractions
rational
c
fractions
b
hold
solving
2
a
1
for
transformations.
Practice
3
and
Practice
4
this
result
would
useful.
State
the
condition
necessary
to
use
this
result
in
solving
rational
equations.
1
Consider
again
the
rational
equation
in
Example
x
4
+
3:
=
2
x
Instead
terms
so
of
in
you
multiplying
the
get
equation
an
both
with
equivalent
sides
the
by
the
LCM
as
LCM,
− 6
x
− 2
happens
denominator?
The
if
x
− 8x
you
LCM
is
+ 12
write
(x
all
2)(x
the
6),
equation:
1( x
2)
x (x
6)
4
+
(x
what
− 6 )( x
− 2)
=
(x
− 2 )( x
− 6)
(x
− 6 )( x
− 2)
Tip
which
simplies
to:
1( x
− 2) +
x (x
− 6)
Tr y
4
using
this
=
(x
− 6 )( x
− 2)
(x
− 6 )( x
method
− 2)
Practice
in
5,
before
2
x
− 5x
− 2
4
=
(x
− 6 )( x
− 2)
you
(x
− 6 )( x
decide
method
Since
the
denominators
are
equal,
the
which
− 2)
numerators
must
be
as
well,
so:
for
you
solving
rational
prefer
a
equation.
2
x
5x
2
=
4
11.3 Seems rational to me
4 4 7
Set
the
quadratic
equal
to
0
and
solve:
2
x
5x
(x
6)(x
x
From
the
original
denominators
So
x
=
6
is
an
Practice
Solve
sides
the
these
of
5
≠
2
and
solution.
equations
equation
equals
equation
4
−
x
x
to
by
1)
=
0
=
−1
or
≠
The
rst
sign.
x
6,
since
only
these
solution
exclude
any
10
x (x
sure
values
is
4
equal
that
extraneous
x
=
would
make
−1.
you
denominators
check
− 2)
your
10
3
x
both
in
4
−
2
− 2x
x
2
5
=
3
x
on
answers
answers.
=
2
x
obtaining
Make
=
− 2
x
0
zero.
extraneous
rational
the
to
6
=
5
original
1
equation
equal
=
+
6
x
− 2
2
2x
5
6
4 x
+
4
6
7
=
x
2x
+
5
= 2
x
x
4
+ 2
Reect
What
x
and
makes
x
2
discuss
dierent
rational
●
Is
●
Does
there
Choose
two
multiplying
2
State
any
3
Which
4
Solve
2
5
the
do
not
Justify
4 4 8
a
you
results
your
solving
Practice
think
using
the
or
rational
create
5
and
using
equations?
them?
solve
is
equations
better?
the
them
1
b
3 − x
− 1
about
be
the
your
same
answer.
11 Equivalence
2
the
your
again,
this
time
two
choice.
1
1
1
=
c
x
2
answers,
methods.
prefer:
=
x
notice
would
you
x
+
from
Explain
method
2
− 1
what
the
from
you
2
State
for
problems
between
=
a
problems
LCD.
equations
1
+ 1
method’
solve
questions
by
+
a
x
8
3
method
1
a
x
2
equivalent?
real-life
‘best
dierences
these
−
equations
science
Exploration
1
a
x
5
methods
Modelling
D
− 1
2
=
6
2
2
x
and
independent
+ 1
explain
of
the
x
−
4 x
− 5
whether
method
or
you
use.
x
A LG E B R A
Objective:
i.
solve
In
this
A.
Knowing
problems
practice
equations
in
Practice
set
that
you
rational
each
1
motorist
miles
On
for
in
brand
B,
same
given
she
b
Brand
a
various
A
below
Select
the
appropriate
problems
into
appropriate
rational
into
a
most
rational
ecient
equivalence
equation.
method
Make
for
sure
solving
transformations.
the
Interpret
context.
1080-mile
For
the
travels
the
3
journey
return
miles
costs
the
number
35
cents
of
trip,
fewer
modelled
of
a
miles
per
dierence
concentration
be
contexts
on
brand
she
per
uses
A
the
gallon,
of
gasoline
less
and
and
expensive
uses
4
averages
brand
gallons
B.
more
journey.
Determine
can
the
of
solutions.
described
use
gallon.
a
The
variety
correct
variables.
and
the
Calculate
2
all
makes
per
the
a
translate
the
situation
equation
answer
x
will
nd
in
solving
identify
your
A
to
understanding
6
Problem
Translate
correctly
you
order
and
per
gallon,
in
the
and
cost
medication
gallon
in
for
a
for
brand
the
the
B
initial
costs
1080-mile
patient’s
32
journey.
cents
per
gallon.
journey.
bloodstream
C
(in
mg/l ),
by:
Doctors
and
nurses
6t
C (t)
,
=
where
t
is
the
time
(in
hours)
after
taking
the
need
medication.
to
know
the
2
t
a
4
Determine
how
concentration
many
will
be
hours
2
after
taking
the
medication
concentration
of
medication
a
the
in
mg/l
patient’s bloodstream,
b
How
many
solutions
solutions
you
found
did
you
made
nd?
Explain
whether
or
not
all
so
the
a
sense.
they
safe
can
time
between
c
One
dose
equals
5
mg/l
and
the
concentration
should
never
be
more
decide
interval
doses.
than
1
6
mg/l.
Determine
a
safe
time
interval
between
doses.
8
3
On
a
and
mountain
then
faster
rate
4
To
than
at
stay
group
high
The
of
hikers
They
entire
walked
for
generally
excursion
7
miles
walked
lasted
4
3
on
a
level
miles
hours.
per
path,
hour
Determine
the
uphill.
divers
need
pressure
percentage
percentage
of
uphill.
hiked
underwater,
under
a
miles
hiked.
they
recommended
The
12
they
which
oxygen
trail,
hiked
oxygen
can
of
air
act
that
like
oxygen
(P)
in
contains
poison
the
air
recommended
enough
in
the
changes
at
depth
oxygen.
body,
as
d
so
you
meters
However,
the
dive
is
deeper.
calculated
1980
using
the
formula
P
=
d
a
At
sea
level
Explain
b
5
Find
Juan
same
the
drives
time
the
how
20
that
percentage
this
depth
+ 99
at
of
compares
which
km/h
per
Maria
the
hour
drives
oxygen
to
the
in
value
the
recommended
faster
75
km.
than
Find
air
is
approximately
calculated
using
amount
Maria.
their
Juan
driving
of
the
oxygen
drives
20%.
formula.
100
is
10%.
km
in
the
speeds.
11.3 Seems rational to me
4 4 9
on
6
Carl’s
water
Sareeta’s
long
7
A
it
would
scientist
water.
20m
pump
pump
take
to
studying
When
the
upstream
current)
removes
removes
in
70
10
the
remove
sh
river
seconds.
10
wants
speed
(against
the
000
same
of
000
to
is
l
water
amount
l
of
nd
0.5
current)
Determine
and
how
45
fast
fast
records
m
minutes.
minutes.
using
how
she
15
20
water
out
m/s
in
in
both
a
salmon
that
a
salmon
neighbor
how
pumps.
swims
salmon
downstream
the
His
Determine
(with
swims
in
still
swims
the
in
still
water.
Summary
p
A
rational
number
is
of
the
form
where
p
and
Or:
Rewrite
the
equation
so
that
all
terms
have
q
q
are
integers,
and
q
≠
0.
Similarly
,
a
the
common
denominator.
are
then
An
extraneous
The
two
numerators
rational
equal.
P (x )
algebraic
expression
is
of
the
form
,
where
Q (x )
Q(x)
an
≠
0.
A
rational
equation
algebraic
containing
equation
rational
that
is
the
algebraic
you
original
values
expressions.
of
because
To
solve
a
rational
equation,
rst
nd
common
denominator
(LCM)
x
solution
equation.
that
they
to
algebraically,
are
First
not
make
the
an
equation
but
does
determine
allowed
in
is
not
any
the
denominator
one
satisfy
equation
zero.
When
the
you
lowest
nd
of
all
have
the
solutions,
reject
any
that
are
not
the
allowed.
denominators.
Either:
LCM
Multiply
to
Mixed
Solve
both
eliminate
all
sides
the
of
the
equation
by
the
fractions.
practice
these
equations
equivalence
using
Problem
appropriate
transformations
and
the
6
ecient
method.
Make
sure
you
check
a
Determine
2x
3
solutions.
Leave
3x
x
answers
4 x
=
1
all
3
x
−
a
pair
of
consecutive
integers
5
whose
reciprocals
add
up
to
6
exact.
x
b
Determine
if
a
pair
of
consecutive
integers
=
2
+ 3
if
for
exists
extraneous
solving
most
4
x
−
4
x
3
− 3
exists
whose
reciprocals
add
up
to
4
2
3
4
+
10
3
+
4
x (x
−
4
=
2)
x
x
−
2
2x
+
20
5
=
c
2
x
+
2x
x
x
+
Determine
if
a
pair
of
consecutive
even
2
3
integers
2( x
+ 7)
5
exists
whose
reciprocals
add
up
to
4
− 2 =
x
4
2x
d
+ 8
Determine
if
a
pair
of
consecutive
odd
11
integers
exists
whose
reciprocals
add
up
to
60
ATL
Review
in
Scientic
context
and
technical
innovation
1
The
2
water
km/h.
for
15
On
km,
returns
takes
in
to
3.5
the
a
kayaking
takes
his
Danube
a
trip
half-hour
original
hours
Delta
Max
his
at
about
paddles
lunch
starting
(including
ows
break,
point.
lunch
His
Let
upstream
and
then
entire
x
Then
(x
km/h
the
2)
represent
kayak’s
km/h
and
the
average
trip
total
speed
distance
=
total
11 Equivalence
speed
downstream
break).
Average
4 5 0
kayak’s
time
average
upstream
is
(x
+
2)
speed.
is
km/h.
A LG E B R A
a
Justify
the
the
kayak
expressions
going
for
upstream
the
average
and
speed
of
‘speed
downstream.
12
means
b
Write
an
expression
for
the
time
taken
for
i
the
river’
upstream
x,
where
x
is
the
speed
of
the
is
current.
4
Lenses
are
images
on
used
to
a
to
look
at
objects,
and
to
project
screen.
downstream.
Use
your
expressions
from
part
b
to
write
an
The
equation
representing
the
whole
focal
parallel
Find
the
average
speed
of
the
length
of
a
lens
is
the
distance
journey.
between
d
to
travelling
travel:
upstream
ii
c
to
relative
speed
the
12
kayak
km/h
the
kayak
in
the
lens
rays
of
and
light
the
point
passing
where
through
the
still
lens
would
converge.
water.
2
When
resistors
are
connected
in
parallel,
the
The
total
resistance
is
R
given
distance
between
a
lens
and
to
the
focal
the
image
by:
tot
produced
d
is
related
length
f
and
i
1
1
1
1
the
distance
between
the
lens
and
the
object
d
o
R
R
1
R
2
R
1
tot
3
by
the
1
+
formula
=
d
d
o
Find
the
resistances
of
two
resistors
connected
The
focal
One
resistance
is
three
times
as
large
as
Total
resistance
is
12
resistance
is
3
ohms
Find
greater
than
An
object
Total
resistance
is
2
Find
in
a
lens
the
resistances
parallel
where
of
the
three
resistors
resistance
of
the
connected
one
is
is
than
the
smallest
one,
and
the
other
15
cm.
twice
as
large
away
as
The
the
image
original
how
far
the
object
is
from
the
lens.
is
2
placed
cm
in
more
front
than
of
the
a
lens,
focal
at
a
length.
The
asthe
smallest
appears
focal
to
length
be
of
4
cm
the
from
the
lens.
Find
lens.
The
human
eye
contains
a
lens
that
focuses
has
an
resistance
far
2ohms
c
greater
as
ohms.
image
c
of
twice
the
distance
other.
be
ohms.
b
One
to
the
object.
other.
b
length
where:
appears
a
f
i
in
a
parallel
1
image
on
the
retina
for
it
to
be
seen
clearly.
one.
An
object
is
at
a
distance
from
the
eye
that
10
The
total
resistance
is
ohms.
is
7
d
An
engineer
resistance
this
with
one
is
8
of
wants
3
two
to
ohms.
build
Show
resistors,
ohms
greater
a
circuit
how
where
than
he
the
the
with
can
(on
do
Lucy
makes
upstream
and
motorboat
12
km/h
speed
of
an
8-hour
45
relative
the
to
back
at
an
the
trip
of
45
downstream
average
river.
the
Find
of
from
other
the
than
the
side
focal
the
eye
of
to
the
length
of
focal
the
retina
lens
this
length.
from
is
The
20
the
and
How
you
eye.
An
airplane
km
in
on
headwind.
speed
a
in
of
Determine
the
same
still
air.
travels
time
The
910
that
it
speed
Determine
miles
with
travels
of
the
the
660
a
airplane
wind’s
tailwind
miles
is
with
305
speed.
the
discuss
explored
the
statement
of
inquiry?
Give
specic
examples.
Statement of Inquiry:
Representing
humans
change
apply
their
and
equivalence
understanding
mm
object).
current.
Reect
have
km
travelling
round
longer
other.
5
3
cm
distance
total
resistance
20
of
in
a
variety
scientic
of
forms
has
helped
principles.
11.3 Seems rational to me
4 51
m/h
12
Generalization
A general statement made on the basis of specic examples
In
mathematics
we
talk
about
two
different
types
of
‘All
generalization.
The
rst
involves
looking
at
generalizations
dangerous,
information
and
trying
specic
that
might
specic
problems
to
underlie
by
Hypernyms
hypernym
includes
a
it
The
to
a
specic
that
rather
cauliower
ask
if
they
4 5 2
use
ask
and
with
are
would
like
other
more
something
is
to
general
language
broad
words.
potato
if
with
meaning
So,
and
Alexandre
ones.
-
to
them
their
vegetables.
like
meal,
is
since
all.
It
would
generalize
they’d
that
vegetable
appropriate
individual
broccoli,
we
be
ideas:
carrots,
might
even
this
less
tackle
cabbage,
hyponyms.
hypernyms
beans
The
includes
called
someone
about
at
a
carrot,
which
out
Hyponyms
specic
onion,
words
we
than
word
term
in
and
more
for
general
say
a
other,
hypernym
is
is
it.
looking
Generalization
A
nd
are
specic
generalize
and
Dumas
one.’
Horse
Your
friend
school,
and
wearing
Look
from
think
on
sense
at
a
most
your
she
each
straw
the
of
says
passes
day,
hat
any
to
other
friend’s
feeding
general
specic
The
horse
●
The
farmer
feeds
●
The
farmer
is
●
Your
●
The
farmer
●
The
horse
●
Your
Is
To
the
If
if
a
The
not
a
digit
is
method
is
evenly
may
remove
the
likely
on
she
and
to
the
sees
brown
that
wears
to
but
only
to
a
to
farmer
horse.
rank
be
way
them
true.
could
be
Can
in
order
you
made
based
on
who
days
feeds
the
horse
same
time
horse
at
the
divisible
divisible
subtract
units
school
hat
school
to
carrots
school
a
divisible
need
least
person
evenly
and
a
statements
horse
racing
travels
number
result
3843
is
to
statements
walks
always
carrot
anything
only
evenly
removed
the
the
the
to
day
observations?
eats
never
friend
3843
see
never
friend
true
every
exception,
a
general
be
●
eld
without
following
likely
a
be
the
by
by
3,
7,
7,
then
remove
the
amount
the
several
leaving
day
7?
doubled
repeated
digit,
by
every
units
from
original
digit
the
number
of
the
number,
remaining
is
also
double
number.
divisible
by
7.
times.
384:
384
−
6
subtract
378
remove
the
the
doubled
units
number
digit,
8,
3
leaving
37:
37
−
16
subtract
21
this
is
a
evenly
This
the
●
is
a
Is
this
Try
doubled
multiple
divisible
specic
method
the
of
by
example
number
7,
so
8
3843
is
7.
that
works
with
described.
enough
some
to
other
say
that
numbers
the
for
general
method
always
works?
yourself.
4 53
Seeing
the
forest
and
12.1
the
trees
Objectives
●
Identifying
Inquiry
patterns
in
number
problems
questions
●
What
is
a
conjecture?
●
What
is
a
generalization?
●
How
F
●
Solving
a
●
more
complicated
general
Making
problems
by
looking
at
case
generalizations
from
a
given
pattern
C
MROF
●
D
ATL
Critical
Draw
Dierent
What
generalizations
be
used
to
solve
problems?
are
the
risks
of
making
generalizations?
thinking
reasonable
Conceptual
can
specic
conclusions
and
generalizations
8.1
understanding:
forms
can
be
used
to
generalize
and
justify
patterns.
12.1
12.2
E8.1
4 5 4
A LG E B R A
Y
ou
●
should
expand
already
brackets
algebraic
and
know
simplify
how
1
to:
Expand the brackets in each expression.
expressions
a
2
2x(3
4x)
Simplify
3
6 x
b
these
(4
+
x)(1
3x)
expressions.
2
3
y
12 x
a
5
y
12 x
b
2
2 x
●
understand
(lowest
and
the
terms
common
GCD
LCM
3
Find
multiple)
(greatest
a
or
HCF
In
of
in
●
What
is
a
conjecture?
●
What
is
a
generalization?
mathematics,
from
the
one
specic
information
underlying
LCM
of :
8
b
6
and
9
Find
the
12
GCD
and
(HCF)
54
b
of :
10
and
75
factor)
Generalization
F
the
and
3x
(highest
a
common
6
common
4
divisor)
3
4
y
to
the
and
trend,
of
the
general.
trying
rule
meanings
or
to
mathematics
of
Often,
make
a
generalization
it
means
statement
relates
looking
(a
at
a
to
reasoning
limited
conjecture)
about
amount
an
pattern.
Typically
,
Reect
and
discuss
a
Look
at
the
following
powers
1
4
of
4.
4
4
generalization
involves
some
7
=
=
2
information
trying
16
4
3
=
65 536
you
9
=
64
4
=
256
4
=
=
=
There
they
a
are
are
Suitable
1024
=
4
4096
general
several
all
even.
4
statement
things
You
pattern,
at
more
information.
=
4 194 304
=
16 777 216
power
●
Every
odd
●
Every
even
important
is
4
power
you
of
to
bear
will
4
of
in
in
a
4
or
a
with
that
hold
the
notice
notice
with
ends
mind
powers
of
about
4.
the
something
powers
about
of
the
4.
For
nal
example,
digits.
include:
with
always
discussed
also
ends
4
the
might
might
ends
power
generalizations
true
of
about
might
generalizations
Every
always
a
looking
12
●
It’s
nd
1 048 576
11
6
Make
you
10
5
4
patterns
If
262 144
try
4
any
notice.
don’t
4
4
to
8
=
describe
4
collecting
16 384
and
4
forming
1
a
4.
a
6.
you
true.
chapter
6.
don’t
yet
Proving
on
know
that
a
whether
particular
these
conjecture
is
justication.
12.1 Seeing the forest and the trees
4 5 5
A
conjecture
Usually
has
it
been
is
is
a
mathematical
consistent
proven
to
be
with
true,
statement
some
we
known
usually
which
has
not
information.
refer
to
it
as
a
yet
Once
been
a
proved.
In
1994
Wiles
conjecture
rst
theorem
conjecture
is
similar
to
the
idea
of
a
hypothesis
in
to
Fermat’s
science.
conjecture,
stated
that
positive
Practice
b
Write
down
the
rst
few
powers
of
and
the
5.
a
general
statement
about
the
last
digits
of
the
powers
of
a
5.
c
For
any
values
Write
number,
of
its
down
Suggest
a
the
digits.
some
general
digit
For
sum
of
example,
multiples
statement
of
the
number
the
the
about
digit
number
the
is
sum
digit
9.
found
of
List
sums
by
842
is
their
of
adding
8
+
4
digit
+
together
2
=
b
of
Look
at
the
multiplication
square
2
3
4
5
6
7
8
9
10
1
1
2
3
4
5
6
7
8
9
10
2
2
4
6
8
10
12
14
16
18
20
3
3
6
9
12
15
18
21
24
27
30
4
4
8
12
16
20
24
28
32
36
40
5
5
10
15
20
25
30
35
40
45
50
6
6
12
18
24
30
36
42
48
54
60
7
7
14
21
28
35
42
49
56
63
70
8
8
16
24
32
40
48
56
64
72
80
9
9
18
27
36
45
54
63
72
81
90
10
10
20
30
40
50
60
70
80
90
100
From
within
35
42
40
48
Find
the
diagonal
(35
+
square,
dierence
cells,
48)
Describe
the
between
inthis
(40
what
+
select
any
the
2
×
sums
2
of
square,
the
for
entries
example:
in
the
two
case:
42).
you
notice.
Suggest
a
suitable
general
statement.
1
4
The
arithmetic
mean
of
two
numbers
a
and
b
is
given
(a
by
+
b).
2
The
geometric
Choose
ve
arithmetic
the
mean
pairs
mean
arithmetic
4 5 6
of
of
numbers
dierent
and
mean
two
the
and
and
numbers.
geometric
the
a
For
mean.
geometric
12 Generalization
b
is
given
each
Form
mean
of
by
pair,
a
a
ab .
calculate
general
pair
of
the
statement
numbers.
c
for
value
n
greater
a
conjecture
had
below.
1
n
=
any
of
than
2,
that
mathematicians
9.
x
satisfy
the
been
prove
3
a,
14.
sums.
multiples
can
n
+
integer
2
three
integers
equation
n
Suggest
which
no
1
ATL
1
the
successful
proof
A
Andrew
found
about
for
trying
358
to
years.
A LG E B R A
5
Consecutive
For
integers
example,
a
Choose
b
For
56
four
each
are
and
57
pairs
pair,
of
whole
are
a
numbers
pair
of
consecutive
double
the
which
dier
consecutive
by
one.
integers.
integers.
smaller
number
and
add
it
to
the
larger
Tip
number.
Write
down
the
total
each
time.
You
c
Divide
each
total
by
3
and
describe
anything
you
notice.
Suggest
can
GCD(a,
suitable
general
use
a
b)
to
statement.
meangreatest
common
Problem
or
6
a
Choose
b
For
divisor,
solving
a
few
pairs
of
numbers
(a,
HCF(a,
mean
b).
the
common
each
pair,
i
their
product,
ii
their
lowest
iii
their
greatest
and
terms
ab
same
common
multiple,
common
divisor,
LCM(a,
factor
your
b.
The
mean
Describe
what
GCD(a,
b).
general
results.
you
notice.
Suggest
a
suitable
general
statement.
question
as
7
Observe
that
4
=
2
+
2,
6
=
3
+
3,
8
=
5
+
3,
10
=
7
+
3
and
12
=
7
+
the
formed
7
is
these
even
numbers
have
been
written
as
the
sum
of
two
Goldbach
prime
Show
that
14,
16,
18
and
20
can
also
be
written
as
the
sum
of
prime
rst
1742.
his
Determine
sum
it
whether
or
not
the
following
even
numbers
can
be
written
of
exactly
two
Although
claim
has
been
as
veried
the
posed
numbers.
in
b
Goldbach,
exactly
who
two
after
numbers.
Christian
a
in
known
5.
Conjecture,
All
two
the
thing.
statement
d
of
b)
The
Organize
to
nd:
a
c
b)
highest
for
numbers
primes.
18
as
i
30
ii
iii
128
iv
high
as
Form
a
Conjectures
collected
are
in
to
this
believe
way,
select
complex
In
this
take
based
the
based
series
that
of
it
a
on
your
signicant
evidence
conjecture
a
this
2
on
more
Investigating
apply
10
you
or
will
we
prove
always
observations.
amount
have
to
observations,
will
be
nd
prove
to
of
evidence
collected,
true.
that
more
Whenever
something
be
the
that
we
we
have
likely
we
generalize
they
have
in
true.
patter ns
mathematical
are
even
important
–
not
told
what
problem-solving
what
constitutes
information
maybe
,
unproven
day
.
techniques
to
discover
patter ns
practice
is
the
suggest
B.
generalization,
problem
usually
that
we
and
statement
studied;
and
Objective:
i.
are
and
common,
general
×
it
98
remains
c
4
some
of
you
the
enough
should
techniques
information
collect.
you
to
Choosing
used
in
form
how
Practice
a
you
1
start
will
the
help.
12.1 Seeing the forest and the trees
4 57
to
Practice
2
ATL
Problem
solving
n
1
Investigate
the
value
Generalize
and
Investigate
the
of
7
suggest
a
n
3
for
dierent
values
of
n,
where
n
∈
n
conjecture
regarding
the
value
of
n
7
Remember
3
n
∈
that
means
‘n
is
a
3
2
value
of
n
n
+
3
for
dierent
values
of
n,
where
n
∈
.
natural
number’.
3
Generalize
and
Investigate
the
Generalize
and
suggest
a
conjecture
regarding
the
value
of
n
n
+
3.
2
3
value
of
p
suggest
a
1,
where
p
is
a
prime
greater
than
3.
2
Sometimes
a
general
conjecture
approach
can
help
regarding
us
to
the
solve
value
of
numerical
p
1.
problems.
For
example:
2
Simplify
2014
This
very
is
a
2
2012
specic
problem
2
it
by
nding
working
Look
at
the
and
2014
without
a
2012
taken
Let
n
=
has
then
then
in
this
specic
nding
would
Example
answer.
You
could
solve
the
dierence,
prove
but
if
you
were
challenging.
1.
2012
2013
2
2012
2014
2
=
(n
=
(n
=
4n
=
4
+
2
1)
(n
1)
2
Reect
In
a
2
2014
2
Then
it
1
2
Simplify
and
calculator
approach
Example
and
2
and
Example
n
2013,
then
2014
2n
+
1)
(n
2n
+
1)
×
2013
=
8052
2012
n
the
Substitute
2
What
was
the
specic
problem?
●
What
was
the
general
problem
●
How
the
and
1.
brackets.
Simplify.
discuss
solving
1,
Expand
1:
did
n
2
+
●
general
that
was
problem
problem?
4 5 8
Since
12 Generalization
solved
make
it
instead?
easier
to
solve
the
specic
n
=
2013.
A LG E B R A
Generalization
C
●
How
can
in
generalizations
Exploration
mathematics
be
used
to
solve
specic
1
2
2
2003
1
By
letting
n
=
problems?
2000,
show
that
you
can
2000
6n + 9
write
as
2n + 3
2002 + 2001
2
2
2003
Simplify
this
expression
and
hence
nd
the
value
2000
of
2002 + 2001
1
2
By
letting
n
=
312,
show
that
you
can
write
(314
×
315
312
×
313)
as
2
2n
+
3.
1
Hence
nd
the
value
of
(314
×
315
312
×
313).
2
3
Find

a
suitable
value
of
2

to
simplify
this
expression:
Tip
2
(507

 507
n

1
1
+
508
−
1).
508 
Remember
Use
this
to
evaluate
the
‘Find’
2
6000
4
Find
a
suitable
this
to
value
Reect
●
In
evaluate
of
and
the
n
to
simplify
the
× 6000
expression
you
1,
how
must
that
show
.
working.
expression.
discuss
Exploration
means
2
× 6001 − 5999
17 999
Use
that
expression.
3
did
using
n
make
it
easier
to
evaluate
the
expressions?
●
Explain
how
your
work
2
in
Exploration
1
has
also
made
it
easier
to
2
3003
3000
evaluate
3002 + 3001
●
Did
it
matter
that
we
chose
n
2000
in
2
n
2003
to
nd
the
value
step
1?
Could
we
have
chosen
2
2003
2000
of
?
+
What
●
Is
dierence
there
than
Although
we
Exploration
way
In
it
can
for
of
to
with
to
the
which
are
pick
so
For
1500
previous
the
to
the
if
1500
1501
a
way
best
of
a
and
and
are
were
in
of
to
a
the
problem?
we
If
there
a
might
expressions
is
more
picked
a
To
5999
(e.g.
value
do
and
6000
in
expressions
understanding
(e.g.
another
the
algebraic
problem.
value
one
of
using
better
simplies
you
close
each
puzzles
of
attempting
1501,
all
generalize
provide
similar
multiples
to
way?
evaluated
number
expression
made?
calculator,
structure
are
you
have
easily
aid
generalize
close
that
1001,
and
have
it
one
there
the
which
example,
999,
only
is
simplify
numbers
n
way,
could
1
help
each
easy
always
one
would
this
of
of
and
n
for
6001)
a
in
which
made
yourself,
or
this
problem.
look
numbers
17 999).
Then
nicely.
question
pick
multiples
of
n
=
containing
500,
because
the
numbers
999,
1001,
500.
12.1 Seeing the forest and the trees
4 5 9
Practice
3
Without
using
a
calculator,
1
502
501
nd
the
value
of
each
×
these
expressions.
5002 × 5006 − 5003 × 5004
2
500
of
2
5
2001
2000
1
−
3
2000
+
4
2001
1001
×
2000
1000
×
2001
2000 × 2001
3001 + 3002
5
3001 × 3003 − 3000 × 3002
2
Generalization
D
●
What
are
the
risks
of
in
n
mathematics
making
n
+
n
+
1
43
2
47
3
53
4
61
5
71
6
83
7
97
8
113
9
131
10
151
41
generalizations?
2
Consider
the
expression
mathematicians
as
an
n
+
n
+
example
41.
of
It
is
one
quite
of
the
well
known
potential
among
pitfalls
of
careless
generalization.
2
The
table
Verify
here
that
shows
each
of
the
the
value
of
numbers
n
in
+
n
the
+
41
for
some
right-hand
values
column
is
of
n
prime.
This
leads
2
us
to
wonder
number.
On
Reect
whether
the
the
evidence
and
expression
here,
discuss
it
n
+
seems
n
+
like
41
a
always
sensible
generates
a
prime
generalization.
4
Prime
●
Read
about
RSA
encryption
to
the
right.
How
many
examples
are
of
2
n
+
n
+
41
generating
being
prime
primes
for
would
use
in
convince
secure
data
you
that
it
could
be
used
for
for
numbers
very
a
method
What
is
the
smallest
positive
integer
n
for
which
the
of
cryptography
transmission?
known
●
important
rule
as
RSA
fails?
encryption.
2
●
Investigate
●
Can
the
expression
n
+
n
+
11
in
the
same
way.
This
you
nd
any
other
expressions
with
similar
type
of
encryption,
properties?
relies
on
which
prime
factorization
Given
that
wonder
in
this
whether
Importantly,
case
generalization
generalization
generalization
is
is
a
has
created
useful
usually
skill
only
at
one
a
false
claim,
you
might
of
all.
part
of
a
mathematical
some
numbers,
underpins
our
process.
modern
Once
you
have
formed
a
conjecture,
it
is
important
that
you
then
proceed
your
claim.
In
this
case,
you
would
not
be
able
to
justify
the
claim
data
transmission.
+
n
+
essential
41
if
generating
is
secure
that
2
‘n
methods
to
of
justify
truly
enormous
not
is
we
are
prime
reliable
4 6 0
prime
for
to
all
discover
numbers,
until
it
n’,
has
because
new
but
been
it
is
not
true.
mathematics,
you
must
proved.
12 Generalization
always
Forming
such
bear
as
in
an
a
conjecture
ecient
mind
that
a
way
is
of
conjecture
A LG E B R A
Summary
●
A
conjecture
which
●
Once
we
has
a
is
not
a
mathematical
yet
been
conjecture
usually
refer
has
to
it
statement
proved.
been
as
a
proved
to
be
true,
●
theorem
If
the
creation
the
hope
you
Generalization
takes
two
creation
of
a
conjecture
based
on
Mixed
of
pieces
more
general
a
generalize
your
layout
to
problem
to
form
information
help
in
problem
you
a
in
a
identify
table
or
patterns.
If
you
are
of
trying
to
generalize
a
number
a
problem,
collection
to
gather
logical
a
forms:
●
the
of
simplifying
trying
conjecture,
other
●
are
of
look
at
numbers
which
are
similar,
evidence
or
are
in
the
close
to
being
factors
of
other
numbers
problem.
practice
1005 × 1995 − 995 × 2005
1
Consider
the
expression
n(n
+
1)(2n
+1)
for
c
1005 + 1995 + 995 + 2005
dierent
positive
Suggest
a
integer
values
of
n
542 × 536 + 8
d
value
of
suitable
n(n
+
general
1)(2n
statement
about
the
540
+1).
5
n+1
2
Consider
positive
the
expression
integer
Suggest
a
suitable
n+1
value
3
of
4
of
3
for
statement
about
10
grid
contains
the
numbers
1
to
100:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
the
3
are
×
n
general
numbers
10
dierent
2n
+
Mersenne
values
A
2n
+
4
numbers
of
the
form
n
2
1
Marin
where
n
>
2.
Mersenne,
a
French
mathematician,
n
observed
is
a
b
that
when
n
=
2,
3,
5
or
7,
then
2
1
prime.
Verify
this
Based
on
observation.
this
observation,
explain
why
a
n
reasonable
generalization
might
be
‘2
1
is
Pick
prime
whenever
n
is
any
set
of
ve
boxes
forming
a
cross
shape,
prime.’
example:
c
Show
that
this
generalization
does
not
hold
74
true
for
all
prime
values
of
n
83
4
Without
the
aid
of
a
calculator,
nd
the
value
84
85
of:
94
a
2002
×
3001
×
1003
−
2001
×
3002
Suggest
a
centre
the
general
rule
linking
the
number
in
the
2
b
1000
3000
(1001
+
1002)
boxes
Reect
and
How
have
you
Give
specic
Conceptual
Dierent
of
added
cross
to
the
total
of
all
ve
of
the
together.
discuss
explored
the
statement
of
conceptual
understanding?
examples.
understanding:
forms
can
be
used
to
generalize
and
justify
patterns.
12.1 Seeing the forest and the trees
4 61
for
Growing
predictably
12.2
Global
context:
Objectives
●
Finding
rules
●
and
and
Using
describe
Inquiry
justifying
formulae
explicit
Scientic
and
(or
for
proving)
formulae
sequences
technical
innovation
questions
●
What
is
an
●
What
is
a
●
What are the recursive and explicit formulae
arithmetic
sequence?
F
sequences
recursive
arithmetic
general
and
and
to
geometric
geometric
sequence?
for arithmetic and geometric sequences?
MROF
sequences
●
●
Recognizing
arithmetic
sequences
context
in
and
C
geometric
How
●
●
D
How
you
can
you
solve
and
How
the
does
sequence
●
recognize
sequences
arithmetic
the
to
vary
common
How
for
ATL
can
geometric
can
arithmetic
real-life
problems
geometric
behavior
involving
sequences?
of
depending
and
problems?
a
geometric
on
the
value
of
ratio?
you
arithmetic
predict
in
use
and
future
the
general
geometric
formulae
sequences
terms?
Critical-thinking
Identify
trends
and
forecast
possibilities
8.1
Statement of Inquiry:
Using
dierent
products,
forms
processes
to
and
generalize
and
justify
patterns
can
help
improve
solutions.
12.1
12.2
E8.1
4 6 2
A lg e b r A
Y
ou
●
should
use
already
notation
to
know
how
1
describe
to:
Write
each
sequences
a
down
b
rst
three
terms
of
sequence.
u
=
n
the
u
+ 1
v
+1
, u
n
=
5
=
2
1
= 2n − 1
n
c
w
=
n
●
write
recursive
and
2
explicit
the
laws
of
indices
for
3
positive
4,
9,
16,
…
Evaluate:
5
integer
1
formula for the sequence that begins
1,
use
, w
n
Write a recursive and an explicit
formulae
●
2w
+ 1
a
powers
3
2
×
5
3
b
6
3
÷
6
7
4
1
c
2
×
5
n
×
1
5
d
2
4
●
solve
simultaneous
4
equations
By
hand,
or
using
simultaneous
y
●
calculate
compound
5
interest
=
x
Find
a
General
and
formulae
geometric
●
What
is
an
●
What
is
a
●
What
are
the
geometric
and
the
for
y
value
3
GDC,
solve
the
=
3x
of
$150
interest
invested
rate
of
5%
at
per
years.
arithmetic
sequences
arithmetic
geometric
for
6
compound
year
F
+
a
equations:
sequence?
sequence?
recursive
and
explicit
formulae
for
arithmetic
and
sequences?
ATL
Exploration
1
Here
4,
are
7,
four
10,
1
dierent
13,
16,
19,
sequences.
…
23,
30,
37,
44,
51,
58,
48,
42,
36,
30,
24,
…
11,
42,
73,
104,
Copy
each
Describe
have
Suggest
sequence
any
been
135,
be
ways
…
…
and
similarities
write
down
about
the
anything
way
you
you
think
notice
the
about
it.
sequences
could
generated.
in
which
the
sequences
might
continue.
Continued
on
next
page
12.2 Growing predictably
4 63
2
Here
are
four
4,
12,
5,
−10,
96,
20,
25,
3
Discuss
4
The
ways
your
in
12,
…
…
sequence
and
anything
about
Consider
in
a
17,
c
1,
e
1,
g
64,
i
3,
19,
4,
9,
,
3,
of
the
21,
23,
25,
16,
25,
…
,
the
way
you
you
3,
sequences
notice
think
with
are
2
about
the
it.
sequences
have
arithmetic
are
geometric
sequences.
sequences.
sequences
geometric
3,
following
and
sequence,
…
32,
…
18,
d
5,
…
1,
if
it
is
an
neither.
9,
10,
13,
h
decide
or
b
f
40,
continue.
others.
following
a
might
…
48,
3,
the
1
step
sequence,
,
56,
step
in
each
arithmetic
which
conclusions
sequences
The
…
similarities
sequences
The
80,
1.5625,
each
any
…
generated.
Suggest
5
40,
6.25,
sequences.
324,
24,
down
Describe
been
108,
−48,
100,
Write
36,
more
4.5,
20,
39,
5.5,
2.25,
40,
117,
10,
…
80,
351,
14.5,
…
…
19,
…
j
recursive
formulae
each
dene
a
sequence.
A
In
each
case,
list
the
rst
few
terms
of
the
sequence
and
determine
is
it
is
an
arithmetic
sequence,
a
geometric
sequence,
or
a
n
c
e
=
n
is
an
a
When
is
d.
a
c
4
b
b
=
3
4
d
1,
e
=
1
f
g
,
n
arithmetic
g
=
1
h
=
the
dierence
d
+ 1
between
,
d
f
term-to-term
formula.
5
+
=
1024
1
2)(
n
=
n
sequence
(
=
1
n
+ 1
h
1
b
n
=
n
2,
+ 1
f
1
b
+ 1
d
n
n
+ 1
=
n
1
3e
=
=
n
geometric
working
for
the
sequence
with
term
Therefore
the
you
ratio
arithmetic
number.
could
between
sequences,
The
f
3),
f
n
h
6.4,
=
5
1
h
n
=
13.5
1
consecutive
constant
consecutive
we
typically
dierence
terms
use
a
between
terms
is
for
constant.
the
rst
consecutive
term,
terms
If
d
and
=
a
and
u
1
u
n + 1
=
the
second
=
a
and
1
The
dierence
4 6 4
d
is
usually
called
u
=
u
n + 1
the
12 Generalization
d
the
increasing,
is
negative
d
equation:
u
if
positive
is
n
it
rearranging
is
series
write:
u
or,
called
constant.
In
and
,
n
+ 1
g
3,
1
2c
+ 1
e
g
+
n
=
n
a
+ 1
c
n
In
=
formula
sometimes
neither.
a
a
recursive
if
common
+
d.
n
dierence
is
decreasing
A lg e b r A
ATL
Exploration
1
Use
the
2
recursive
formula
to
write
each
of
the
terms
u
,
u
1
an
2
arithmetic
Look
for
patterns
Describe
briey
arithmetic
3
Now
the
sequence
look
as
with
the
how
rst
term
sequence
these
a
and
moves
patterns
common
from
relate
to
one
what
,
…
u
2
of
6
dierence
term
you
to
the
know
d
next.
about
sequences.
for
formula.
conjecture
patterns
linking
Generalizing
for
a
formula
the
from
for
u
term
the
in
number
patterns
terms
of
to
you
n.
the
right-hand
observe,
This
is
an
write
explicit
side
of
down
a
formula.
An
is
explicit
formula
sometimes
called
n
a
4
Verify
that
your
explicit
formula
gives
=
u
a
+
6d.
7
5
Use
your
conjecture
to
nd
explicit
position-to-term
formula.
formulae
for
these
recursive
formulae:
a
s
=
n
b
s
+ 1
t
=
n
+
the
r
to
t
+ 1
2,
=
t
n
4
=
5
1
often
represent
term
s
1
Mathematicians
and
3,
n
the
number,
use
a
to
represent
constant
you
can
ratio
the
rst
between
term
of
a
consecutive
geometric
terms.
So
sequence
when
n
is
write:
u
n +1
u
= a
and
= r
1
u
n
or,
rearranging
the
second
expression:
u
=
a
and
u
1
r
is
usually
Looking
same
called
again
form
as
at
the
common
Exploration
the
formula
n
=
ru
+ 1
n
ratio
1,
step
5,
which
of
the
formulae
given
are
of
the
above?
ATL
Exploration
1
Use
the
of
geometric
3
recursive
formula
to
write
each
of
the
terms
u
,
u
1
2
a
By
sequence
considering
down
a
the
conjecture
with
pattern
for
an
rst
you
term
obtain,
explicit
a
and
and
formula
common
generalizing
for
u
,
the
,
…
u
2
ratio
nth
6
r
from
term
it,
of
write
a
n
geometric
3
Use
your
conjecture
sequences
a
=
c
n
b
+ 1
d
+ 1
described
2c
,
c
n
=
n
sequence
d
=
with
to
by
rst
nd
term
a
explicit
these
and
common
formulae
recursive
for
ratio
the
r
geometric
formulae:
3
1
,
n
d
=
1024
1
12.2 Growing predictably
4 6 5
An
arithmetic
recursive
sequence
formula
u
=
with
u
n + 1
u
=
a
+
(n
1)d
for
+
rst
d,
term
with
u
n
the
nth
a
=
and
a
and
common
explicit
dierence
d
has
Sometimes
formula
arithmetic
1
term
sequences
are
n
called
A
geometric
sequence
with
rst
term
a
and
common
ratio
r
n
formula
u
=
ru
n + 1
,
with
u
n
=
a
and
explicit
formula
u
1
=
has
recursive
progressions,
1
ar
arithmetic
for
the
nth
and
term.
n
geometric
sequences
called
Example
1
progressions.
The
An
arithmetic
Find
a
are
geometric
sequence
general
formula
has
for
rst
u
,
term
the
nth
7
and
term.
common
Simplify
dierence
your
3.
names
are
interchangeable.
answer.
n
a
u
=
7,
=
7
d
+
=
−3
(n
1)(
3)
Substitute
n
the
given
information
into
u
= a + (n
1)d.
n
=
u
7
3n
+
3
Expand
n
u
=
10
the
brackets.
3n
Simplify.
n
Example
A
2
geometric
Find
a
sequence
general
has
formula
rst
for
u
,
term
the
2
nth
and
common
ratio
5.
term.
n
a
=
2,
=
ar
=
2
r
n
u
=
5
1
n
n
u
×
n
1
Substitute the given information into u
5
1
= ar
n
n
Example
3
n
A
geometric
sequence
has
the
general
formula
u
=
3
×
6
n
n
a
Express
the
formula
in
the
form
u
=
1
ar
n
b
Hence
write
down
its
rst
term
(u
)
and
common
ratio.
1
n
a
=
u
3
×
n
6
You
n
are
given
the
formula,
but
it
is
not
in
the
form
u
1
= ar
n
1
u
=
3
×
=
18
n
6
×
1
6
Use
the
laws
of
indices.
u
= ar
n
n
u
×
1
6
n
This
n
is
now
in
the
form
1
n
b
The
rst
term
is
18
and
the
n
The
common
4 6 6
ratio
is
6.
12 Generalization
general
form
is
u
n
= ar
1
, so
a = 18
and
r = 6.
A lg e b r A
Example
4
n
A
geometric
sequence
has
the
general
formula
u
=
2
×
(
4)
.
n
Find
its
rst
term
and
common
ratio.
1
u
=
2
×
(
4)
=
−8
Using
the
given
formula,
you
can
nd
the
rst
term
directly.
1
The
rst
term
is
8.
Using
the
formula
again
gives
u
2
2
u
=
2
=
u
×
(
4)
=
32
2
The
common
ratio
r
is
equal
to
u
÷
2
r
÷
u
2
The
=
÷
−8
=
u
1
−4
1
common
Practice
1
32
Find
ratio
is
4.
1
explicit
formulae
for
the
nth
term
of:
a
an
arithmetic
sequence
with
rst
term
5
and
common
b
an
arithmetic
sequence
with
rst
term
4
and
second
c
a
geometric
sequence
with
rst
term
10
d
a
geometric
sequence
with
rst
term
3
e
a
geometric
sequence
with
rst
term
20
and
and
common
second
dierence
term
ratio
term
7
7
4
1
1
and
common
ratio
4
2
Find
the
rst
term
and
common
dierence
of
each
arithmetic
progression.
A
a
u
=
3
+
5(n
1)
b
u
n
c
u
=
u
n +1
3
Find
=
2n
4
=
−8
5n
4,
u
n
the
=
11
d
u
1
rst
and
common
ratio
of
each
to
is
as
often
a
progression.
n
term
sequence
referred
n
geometric
sequence.
1
n
a
u
=
4
×
1
n
5
b
u
n
=
×
1
7
n
2
8
n
c
u
=
3
×
5
d
u
n
=
n
n
3
1
n + 3
e
u
=
0.4
×
10
f
u
n
=
7,
u
1
=
u
n +1
n
2
4
A
Norwegian
retreating
the
at
length
l
glacier
a
was
constant
of
the
6.4
rate
glacier
km
of
n
long
80
m
years
in
per
after
1995.
year.
1995.
Since
Write
Use
then
an
this
it
has
explicit
formula
been
formula
to
for
predict
the
n
length
5
of
Moore’s
doubled
a
the
law
glacier
showed
every
Write
in
an
year.
2025
that,
In
explicit
if
the
retreat
between
1965
a
1965
continues
and
1975,
microprocessor
formula
for
the
number
at
the
the
power
contained
of
same
of
around
transistors
rate.
computers
60
in
t
transistors.
a
n
microprocessor
b
Moore
every
two
would
C
later
be
n
years.
in
Using
years
revised
a
How
How
How
law,
many
in
1965.
saying
can
you
in
that
transistors
after
did
1971
the
doubling
law
predict
would
that
occur
there
1993?
arithmetic
real-life
●
after
processor
sequences
●
his
and
geometric
context
recognize
arithmetic
and
geometric
sequences
in
problems?
can
you
geometric
solve
problems
involving
arithmetic
and
sequences?
12.2 Growing predictably
4 67
In
this
section
geometric
Example
An
you
will
explore
progressions
can
how
help
the
solve
general
arithmetic
=
=
u
for
involving
arithmetic
and
sequences.
5
sequence
has
terms
u
=
7
and
u
1
a
formulae
problems
=
18.
Find
u
2
10
7
1
The
d
=
u
u
2
=
18
7
=
common
between
Although
u
=
7
dierence
is
the
dierence
11
1
+
11(n
the
any
question
two
did
consecutive
not
ask
you
to,
terms.
it
is
useful
1)
n
to
u
=
7
+
11(10
=
7
+
99
1)
nd
For
the
the
explicit
10th
formula
term,
let
n
=
for
10
the
in
nth
the
term.
general
formula.
10
u
=
106
10
Example
An
6
arithmetic
sequence
has
terms
u
=
14.2
and
3
14.2
=
u
+
2d
(1)
+
4d
(2)
u
=
19.6.
Find
5
u
11
1
Use
u
=
n
19.6
=
u
u
+
(n
1)d to
write
two
simultaneous
equations.
1
1
2d
=
5.4
d
=
2.7
=
14.2
u
Subtract
2
×
2.7
=
8.8
Solve
the
(1)
from
simultaneous
(2)
and
rearrange.
equations
to
nd
u
1
1
u
=
8.8
+
2.7(n
1)
Write
n
a
general
formula
for
u
n
u
=
8.8
+
2.7(11
1)
Use
n
=
11
to
nd
u
11
11
u
=
8.8
+
27
=
35.8
11
Example
7
The
second
The
common
u
=
u
2
u
+
term
of
ratio
a
is
geometric
3.
Find
progression
the
rst
is
4
more
than
the
rst
term.
term.
4
u
1
=
is
4
greater
1
1
Since
the
common
ratio
is
3,
u
2
3u
=
u
=
2
1
u
u
3u
2
⇒
than
2
+
4
1
1
4 6 8
12 Generalization
is
3
times
u
1
A lg e b r A
Practice
1
An
arithmetic
Find
2
An
2
its
tenth
sequence
arithmetic
Find
its
has
rst
term
has
second
4
and
second
term
7.
term.
sequence
eighth
term
19
and
common
dierence
4.
You
term.
could
solve
simultaneous
3
An
arithmetic
sequence
has
third
term
11
and
fth
term
6.
Find
its
equations
rst
term,
tenth
term
and
common
or
4
An
arithmetic
second
5
An
term.
A
geometric
A
or
9
write
of
the
Find
rst
the
a
rst
has
hand,
The
second
term
which
term
8
third
and
15
and
term
and
term
third
term
value
second
three
times
the
terms.
ninth
has
is
GDC.
16.
Find
its
22.
term
24.
Find
its
third
term
48
and
fourth
term
96.
Find
its
term.
D.
rst
4th
term
or
second
and
of
of
each
by
a
of
the
must
sequence
the
constant
same
term
of
decide
By
the
The
equal
second
are
common
and
than
5th
to
its
both
term,
1000.
term
16.
rst
improvement,
of
The
equal
the
9th
to
rst
term
28.
dierences.
situations
modelled
the
Find
contexts
real-life
be
45.
trial
greater
term.
real-life
should
is
both
their
whether
term
using
that
rst
are
and
in
following
scale
term.
second
authentic
rst
third
the
mathematics
of
and
nth
the
7th
30
the
sequence
elements
you
in
have
the
each
term
for
term
term
Applying
whether
rate
has
sequences
the
sequence,
constant
the
relevant
identify
geometric
sixth
expression
sequence
rst
identify
To
nd
and
Objective:
i.
an
arithmetic
sequence
the
3.
your
term.
fth
sequence
down
otherwise,
Two
term
use
solving
geometric
and
has
has
fth
and
of
Determine
sequence
ratio
Problem
A
and
rst
value
sequence
geometric
has
sequence
ratio
common
8
the
dierence.
common
7
Find
arithmetic
common
6
sequence
by
dierence.
real
with
life
an
factors
arithmetic
lead
to
or
a
growth
at
factor.
ATL
Exploration
Sometimes
sequence.
described
In
1
each
You
you
3
real-life
Consider
using
case,
are
an
population
that
next
The
developers
users
existing
of
it
of
will
user
for
a
is
a
will
a
the
terms
save
the
is
introduce
by
are
new
a
and
every
by
to
believe
to
the
it
or
give
geometric
could
explain
money
be
why.
represent.
you
$10
each
to
get
month.
month.)
1.13%
factor
if
would
parents
pocket
trying
users
arithmetic
decide
sequence
Your
your
developers
3
the
increasing
website
The
of
an
and
sequence,
amount
increased
using
scenarios
trip.
half
same
Earth
new
modelled
geometric
summer
have
grow.
be
following
or
you
the
will
can
the
what
then
money
The
active
up
and
year
of
down
saving
pocket
problem
each
arithmetic
write
started,
(Your
2
a
4
of
per
year.
This
means
1.0113.
predict
that
how
every
its
number
month,
of
each
site.
Continued
on
next
page
12.2 Growing predictably
4 6 9
4
The
owners
customer
month,
5
A
20
minutes.
after
the
melts,
my
the
Reect
●
●
to
this
●
will
is
be
a
10
certain
has
a
in
has
so
a
out
how
think
their
that
each
business.
doubles
the
work
They
size
of
in
number
the
every
population
every
hour
15
you
each
deep
decreases
one
by
chose
either
of
scenarios
the
need
they
to
real-life
ii
a
be
can
examples
or
cm
morning.
15%
every
As
the
snow
hour.
1
that
the
the
resort,
coaches
starting
has
arrived,
Find
ski
full
coach
why
is
snow
sometimes
about
to
year.
their
bacteria
snow
of
describe
in
nd
trying
rst
in
arithmetic
made
be
in
in
order
successfully
Exploration
problem
that
geometric
or
geometric
Exploration
to
4.
make
real-life
modelled.
How
has
4?
means
i
sequence
an
will
arithmetic
be
sequence
appropriate?
8
minutes,
7th
it
will
are
the
records
discuss
appropriate,
Example
arrive
the
simpler
in
experiment.
depth
happened
What
the
others
to
shop
grow
Chloroexi
house
Assumptions
problems
At
of
total
sequences
coee
will
researcher
and
Explain
new
customers
of
A
start
Outside
a
new
population
27
6
of
numbers
at
arrived,
there
some
of
are
number
guests
equal
08:10,
there
920
size
ten
are
stay
the
minutes
785
skiers
overnight
during
at
skiers
the
after
at
and
day.
the
the
all
One
resort
resort.
the
other
coach
skiers
arrives
opens.
After
every
After
the
the
10th
coach
resort.
of
skiers
that
will
be
of
skiers
that
stayed
at
the
resort
once
the
12th
coach
arrived.
b
Find
a
Let
the
u
number
be
the
number
of
skiers
in
overnight.
the
n
th
resort
after
the
n
coach
has
arrived.
An
arithmetic
each
u
=
a
+
(n
=
785
coach
sequence
brings
the
models
same
this
problem
number
of
because
people.
1)d
n
u
=
a
+
6d
7
Form
u
=
920
=
a
+
two
equations.
9d
10
so
135
=
⇒
3d
d
=
u
=
45,
515
and
+
11
a
×
=
515
45
=
Solve
1010
Use
the
to
nd
formula
to
the
values
nd
the
of
a
and
twelfth
d
term.
12
Since
b
The
number
of
skiers
before
the
of
coach
is
given
by
u
=
515
45
the
number
of
overnight
guests
is
the
=
guests
when
0
coaches
have
arrived,
nding
tells
12 Generalization
u
0
470.
0
4 7 0
number
rst
you
the
number
of
overnight
guests.
A lg e b r A
ATL
Practice
1
Some
3
money
annually.
2
of
21
at
the
years
end
When
it
and
the
one
c
Find
the
d
Given
16
cm
Find
the
depth
a
u
215
Find
a
cm
of
a
rst
the
pays
is
compound
worth
amount
ladder’s
above
the
r ungs
r ung
20th
r ung
that
interest
$29 136.22.
of
money
in
At
the
the
end
account
11th
the
rung
is
170
cm
above
the
ground.
could
be
expected
to
form
is
the
at
ve
350
a
cm
ladder
the
ground
and
the
distance
the
pool
it
is
water
hours
it
above
the
ground,
determine
the
total
has.
constant
hours
deep
of
above
r ung.
swimming
rising
how
is
being
rate.
19.5
is
will
cm
after
take
relled
After
three
with
water.
hours
the
water
is
deep.
only
for
one
the
hour.
pool
to
ll
to
cm.
virus
spreads
computer
every
which
account
next.
the
top
after
25
infected
If
of
number
of
computer
computers
is
and
Determine
Each
the
the
the
r ungs
b
A
of
cleaned,
deep
is
heights
to
height
level
the
sequence.
r ung
a
a
4
being
water
account
upright,
rung
the
an
years
year.
vertically
height
that
in
20
$29 718.95.
30th
14th
of
of
worth
why
from
The
end
arithmetic
Find
After
invested
the
its
number
3
is
of
Explain
an
b
is
the
standing
ground
a
At
via
emails
from
successfully
an
infected
manages
to
computer.
infect
ve
new
day.
represents
the
number
of
infected
computers
at
the
star t
of
day
n,
n
explain
why
u
=
n
b
c
5
Given
number
the
star t
seats
19th
In
the
If
it
of
what
arena
by
the
95
computers
from
infected
which
the
at
vir us
the
was
star t
of
initially
day
5,
nd
launched
at
1.
an
has
2592
computers
day
on
n
are
day
are
same
seats.
the
vir us
arranged
number
so
of
Determine
will
infect
that
seats.
the
each
The
total
the
millionth
row
13th
number
is
longer
row
of
has
seats
computer.
than
71
in
the
seats
the
row
and
front
the
three
combined.
2014
doubling
a
in
of
row
rows
6
of
Determine
front
+ 1
there
the
The
in
that
6u
u
Ebola
every
outbreak,
month
represents
from
the
the
number
April.
number
of
In
of
June
deaths
n
deaths
there
from
were
months
the
252
after
disease
was
deaths.
April
explain
why
n
u
= 2u
n +1
b
Find
n
the
nding
number
the
c
Find
the
d
Explain
number
why
continued
of
number
this
of
deaths
of
deaths
model
number
of
in
deaths
April
in
from
does
deaths
and
any
Ebola
not
over
fully
a
write
month
in
an
after
formula
for
September.
explain
longer
explicit
April.
the
period
outbreak
of
and
the
time.
12.2 Growing predictably
4 71
D
When
do
geometric
sequences
defy
expectations?
●
How
the
●
does
value
How
the
of
can
you
sequences
behavior
the
to
of
common
use
the
predict
a
geometric
sequence
vary
depending
on
ratio?
general
future
formulae
for
arithmetic
and
geometric
terms?
ATL
Exploration
For
each
of
the
a
nd
the
b
describe
c
predict
5
sequences
rst
ten
what
what
described
terms
you
will
of
the
notice
happen
in
steps
1
to
4:
sequence
about
the
to
value
the
terms
of
u
,
the
nth
term,
n
as
n
becomes
very
large.
1
A
geometric
sequence,
rst
term
10,
common
ratio
2
A
geometric
sequence,
rst
term
10,
common
ratio
3
A
geometric
sequence,
rst
term
10,
common
ratio
4
A
geometric
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term
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happens
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it?
lot
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in
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sequences
If
large.
being
4 72
r
either
common
is
greater
When
r
positive
become
ratio.
than
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1
larger
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or
is
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or
negative.
is
12 Generalization
larger
between
less
the
and
than
terms
of
1
1,
a
or
and
the
has
geometric
not
smaller
1,
the
terms
geometric
and
values
will
smaller
will
get
become
sequence
alternate
as
geometric
of
the
but
normally
regarded
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sequences?
a
the
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because
problems
involved
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with
by
zero.
A lg e b r A
Summary
●
●
In
an
arithmetic
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the
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geometric
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consecutive
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for
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nth
term.
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sequence,
the
common
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formula
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nth
be
equal
to
0.
term
n
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a
geometric
consecutive
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1
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a
terms
is
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constant.
practice
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6
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down
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down
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term
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formula
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c
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explicit
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d
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the
value
of
the
fth
e
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the
value
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the
rst
for
the
nth
term.
term.
d
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the
value
e
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the
term
of
the
15th
number
of
term.
term.
the
term
term
to
exceed
with
one
million.
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7
2
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a
253
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term
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down
begins
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rst
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term
the
value
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down
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3
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term
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nth
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d
and
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down
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simultaneous
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terms
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value
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it
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product
value
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of
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the
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rst
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and
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term
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whether
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the
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term
c
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of
12.2 Growing predictably
the
sixth
term.
4 73
the
Review
in
context
Scientic and technical innovation
1
Consider
the
illustrates
ethane
following
the
and
chemicals
diagram,
chemical
propane,
known
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of
b
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methane,
examples
of
c
the
C
C
C
H
H
3
H
H
ethane
that
the
H
H
b
Let
u
pictured
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this
have
model
at
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how
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many
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of
members
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second
model
year.
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design
by
an
forms
number
of
produces
business
cards.
charge
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set
the
cards
in
boxes
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same
amount.
fee
for
of
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hydrogen
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propane
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carbon
4
atoms
in
A
with
142
hydrogen
machine
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developers
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of
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Reect
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number
over
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dierent
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4 74
components
made
in
of
components
made
in
nine-hour
discuss
explored
the
statement
of
inquiry?
Give
specic
examples.
Statement of Inquiry:
Using
of
sequence.
total
have
the
hour.
with
c
a
Find
forms
processes
to
and
generalize
and
solutions.
12 Generalization
justify
patterns
can
help
improve
working
day.
the
So,
what
do
you
think?
12.3
Global
context:
Objectives
●
Selecting
Inquiry
samples
and
making
inferences
●
about
F
populations
●
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the
purpose
of
taking
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sample
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it
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questions
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are
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sampling
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conclusions
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drawn
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data?
techniques
appropriate
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to
do
we
know
when
to
say
‘when’?
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generalize
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reliability
ATL
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sample
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your
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want
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reliability
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Recognize
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and
bias
4.3
Statement of Inquiry:
Generalizing
trends
you
and
among
representing
relationships
can
help
to
clarify
individuals.
12.3
E4.1
4 75
SPIHSNOITA LER
Using
relationships
methods?
C
●
and
Y
ou
●
should
represent
already
data
stem-and-leaf
using
how
both
diagrams
box-and-whisker
know
1
to:
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and
are
the
customers
diagrams
bar
over
Present
leaf
number
per
the
this
day
course
data
represent
data
frequency
using
cumulative
2
diagram.
54
77
71
73
89
55
17
53
70
75
88
29
34
33
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71
55
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Represent
the
people
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made
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note
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for
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and
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in
calculate
the
interquar tile
range
range
and
for
a
set
4
7
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<
t
≤
9
14
10
<
t
≤
14
10
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<
t
≤
20
3
21
<
t
≤
25
1
heights
school
of
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●
paper
on
4 7 6
we
we
background
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things
deductions
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meters)
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follows:
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the
mean
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the
range.
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example,
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quar tiles
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how
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on
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65
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graphs
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of
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classroom
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from
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for
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12 Generalization
is
and
that
while
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known
your
you
others
based
observation
as
inference.
teacher
were
are
has
having
an
on
and
out
infer
draw
For
laid
To
based
a
sheet
assessment.
or
means
to
conclusions
on
evidence
reasoning.
S TAT I S T I C S
Inference
The
in
in
Moon
8
Key :
7
two
High
at
for
shown
in
two
the
School
4
4
5
4
3
5
4
3
1
6
5
3
3
2
0
0
7
2
7
8
4
2
1
8
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3
5
5
8
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5
3
9
1
2
4
4
5
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1
10
4
4
6
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1
11
3
81
for
Crescent
discuss
stem-and-leaf
Moon
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diagram
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the
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the
performance
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and
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the
data
of
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data?
others?
did
not?
conclusions?
about
general
eAssessment
because
such
might
the
about
discuss
the
median
the
following
of
students
Crescent
The
In
this
as
you
time
explain
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spent
the
characteristics
scenario,
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performance
other.
conclusions
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at
on
conclusions
the
size
based
from
factors,
class
the
Calculate
at
score
quality
any
about
or
the
underlying
the
school,
nature
trends.
or
the
of
In
the
data,
this
students,
case,
it’s
you
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for
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Moon
teacher
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teacher
here.
unrepresentative.
Reect
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dier
from
conclusions
reached
draw
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school
draw
you
information.
inferences
about
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conclusions
pieces
make
information
●
for
compare
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same
1
How
unfair
85
the
diagram
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6
●
may
Bay
8
What
you
where
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from
schools
stem-and-leaf
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7
and
the
results
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schools.
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If
years
1 | 8 | 5 means
Reect
Look
eAssessment
consecutive
Crescent
P R O B A B I L I T Y
statistics
mathematics
taught
A N D
Bay
classes
do
from
you
Academy
the
think
are
more
two
are
schools.
justied?
intelligent
than
the
students
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better
job
of
teaching
the
class
at
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Bay
than
at
Moon.
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class
Bay
class.
at
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scored
lower
on
average
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12.3 So, what do you think?
4 77
Again,
it’s
dicult
performance,
data.
The
average,
even
you
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this
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surrounding
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students
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on
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that
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years
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inference
sample
consideration
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than
Specically,
under
the
eAssessment
tests
with
compare
condent
lower
intelligent
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of
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could
we
since
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subset
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act
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census
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representative
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when
questions
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know
skill
1
a
it
large
about
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factory,
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of
10 000
their
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12 Generalization
would
sta.
single
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4 78
It
In
contexts
assumptions
mathematical
the
would
real-life
knowledge
conduct
of
school.
real-life
authentic
general
a
sample
within
underlying
is
practical
representative
mathematics
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more
why
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is
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schools,
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Applying
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to
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whole
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students
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The
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concerning
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Objective:
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for
population.
example
Inference
a
the
as
cars
Japan
a
be
in
S TAT I S T I C S
2
The
managers
They
decide
brainstorm
They
could
ask
each
b
They
could
put
up
c
They
could
ask
one
d
They
e
They
it
They
would
times
be
it
of
the
There
are
rather
than
as
you
You
if
will
is
In
a
we
a
case
would
methods
all
●
A
●
A
other
A
stratied
the
department
respond
to
at
by
estimate
work.
email.
an
average
sta
list,
having
person
on
the
company’s
sta
list,
having
that
name
Explain
into
a
bag
how
useful
information
is
a
list
and
you
draw
think
about
to
sampling
sample
sample
an
of
sample
those
with
the
out
the
the
surveys
commute
above
If,
is
that
the
survey
and
can
idea
required
known
every
of
a
sample,
practicality.
arise
of
as
when
bias.
having
a
A
member
the
probably
a
There
taking
sample
of
it
the
is
is
be
of
all
the
frame.
population.
quite
company
not
list
sampling
employees,
however,
would
a
cost
population.
list
able
likely
wished
to
that
to
obtain
a
run
list
products.
member
available
a
to
sampling
of
us
the
population.
depending
frame.
The
on
whether
following
three
it
is
sampling
frame.
is
the
takes
sorted
do
surveying
every
likely
ordered
list
they
are
of
list.
their
obtain
to
identies
a
are
problems
This
are
conduct
which
samples
we
of
might
of
1,
are
not.
that
bought
equally
a
the
such
methods
from
from
ensures
to
arrived
company’s
two
these
did
list
member
systematic
person
they
information.
the
company
survey,
random
is
a
of
obtain
a
ideas.
where
had
require
any
as
employees.
the
on
representation
any
practical,
intervals
●
is
to
of
others
frame
population
of
the
person
Exploration
some
one,
who
simple
of
gathering
employees
each
representative
why
in
fair
sampling
or
sample
of
individually
employee’s
workforce,
satisfaction
possible,
a
employees.
Most
that
this
sampling
Dierent
a
able
everybody
A
seen
frame
be
customer
of
not
like
in
of
ways
department.
200th
these
out
reasons
whereas
sampling
for
asking
200th
every
of
whole
notice
survey
random.
each
have
employees,
A
put
at
however.
it
that
every
factory’s
the
notice
person
for
a
ideas
salary.
nding
good
sample,
biased
on
a
of
employee
every
ask
by
could
for
conduct
P R O B A B I L I T Y
alphabetically.
50names
Comment
ask
could
sorted
f
time
could
sorted
a
to
number
a
commuting
are,
a
A N D
a
to
sample
be
such
included
that
in
every
the
member
sample,
of
the
independent
population.
members
sampling
of
the
frame
population
–
e.g.
taking
at
regular
every
10th
alphabetically.
identies
groups
are
dierent
sections
represented
in
of
the
population
appropriate
and
proportions
in
sample.
12.3 So, what do you think?
4 79
Simple
When
make
or
by
random
creating
are
using
the
systematic
having
frame
rst
at
form
that
of
a
you
this
to
you
or
reected
a
is
formed
random
sample,
represent.
might
or
that
‘fairly
want
generator
by
taking
starting
important
done
on
an
point
to
the
the
make
age
sure
ranges
proportion
in
the
from
a
have
means
proportions
students
rst
example,
represented’
proportions
10
you
For
dierent
match
that
the
Sur veying
is
be
of
that
pulling
the
selections
names
from
a
you
hat
calculator.
ordered
sampling
–
taking
items
to
divide
the
of
if
that
frame
from
the
and
then
–
sampling
in
in
80
after
an
IB
exam
population
surveying
and
fairly
the
men
people
were
men
were
that
of
you
women
proportion
the
a
into
groups
group
were
fairly
represented.
of
men
population,
dierent
age
in
and
ranges
the
likewise
in
the
of
the
for
sample
population.
year
10
You
group
a
by
intervals.
stratied
should
women;
number
it
could
sampling
case,
sample
sample
picked
represented,
In
sample
This
sampling
want
people,
random
random.
random
regular
Stratied
To
simple
genuinely
Systematic
A
a
samples
to
are
1
=
sampling
80
see
8
students.
how
dicult
they
thought
it
was. The
1
From
students
were
taught
in
ve
of
16
2
20
of
the
cars
in
a
full
meet
cars
to
see
regulation
whether
their
were
requirements.
compacts,
250
cars
class,
2
you
need
of
pick
a
simple
random
students.
20
1
=
need
500
tires
of
the
cars.
25
1
100
So
cars
of
lot
You
500
16,
students.
each
sample
of
of
students.
From
Checking
class
8
16=
classes
each
dierent
you
need
of
100
=
4
compacts;
25
were
1
estates
and
150
cars
were
saloons.
of
250
=
10
of
150
=
6
estates;
25
1
saloons.
25
Pick
the
of
Reect
●
What
and
are
sampling
●
Suggest
the
discuss
advantages
described
some
a
simple
random
compacts,
the
10
of
sample
the
of
estates
4
of
and
6
saloons.
4
and
disadvantages
of
the
three
types
of
above?
situations
in
which
you
would
use
each
of
the
sampling
techniques.
●
Describe
4 8 0
the
diculties
you
might
12 Generalization
encounter
when
trying
to
apply
them.
S TAT I S T I C S
Example
Javier
The
going
are
60
to
year
and
how
a
simple
b
a
systematic
c
a
stratied
a
Method
up
students
provides
a
list
each
at
of
with
his
the
school.
500
100
total
students
students,
and
in
each
the
year
school.
group
girls.
could
create:
sample
sample
sample.
1
printed
its
Method
list
contents
to
separate
and
then
all
the
names
randomly
draw
and
out
place
50
them
in
a
bag.
names.
2
Number
number
40
random
a
Shue
50
groups,
Javier
a
Cut
sample
secretary
ve
boys
Explain
P R O B A B I L I T Y
1
school
There
has
is
A N D
each
student
generator
to
from
pick
1
50
to
500
in
numbers
alphabetical
between
1
order.
and
500
Use
a
random
(ignoring
any
duplicates).
b
List
the
students
generator
point,
to
take
starting
in
pick
every
a
alphabetical
random
10th
Each
group
of
You
are
You
can
students.
year
group
get
random
From
back
number
that
to
50 students out of a total of 500
the
means taking every 10th name.
of
the
school,
so
each
group
should
be
a
5
Within
take
a
each
simple
group
60%
random
are
boys
sample
of
and
6
40%
boys
girls,
and
4
so
girls.
1
concerned
contents.
a
10
each
Practice
1
point.
you
a
1
=
is
500
from
until
Use
point.
year
sample
starting
name
100
c
order.
tell
If
Explain
if
an
the
that
egg
egg
is
is
the
eggs
rotten
rotten,
why
you
might
how
you
could
in
by
there
wish
your
kitchen
breaking
will
to
be
it
a
sample
are
open
rotting.
and
pungent
the
eggs,
smelling
the
smell.
rather
than
conduct
a
census.
b
Explain
conduct
a
simple
random
sample
of
the
eggs.
Tip
2
Alex
lives
in
a
high-rise
apartment
building
containing
80
apartments,
There
ATL
numbered
1–80.
He
wants
to
conduct
a
survey
to
nd
out
how
many
is
relevant
on
average
live
in
each
information
people
to
situation
a
Explain
how
he
b
Explain
why
a
apartment
3
could
create
systematic
and
then
a
simple
sample
selecting
of
random
20
every
sample
apartments
fourth
of
20
starting
apartment
apartments.
Describe
any
factors
stratied
sample.
that
would
be
relevant
if
Alex
is
not
within
the
question.
the
design
Could
with
might
wished
that
contained
lead
to
to
take
a
of
the
bias.
building
c
the
apartment.
be
relevant
here?
12.3 So, what do you think?
4 81
3
A
teenager
He
sends
a
Determine
Comment
4
A
full
b
D
80
320
E
240
160
of
A
in
at
has
on
his
phone.
been
made
is
available
as
for
pay
grades.
follows:
a
survey
on
work.
the
company
could
construct
a
simple
random
the
company
could
construct
a
systematic
the
company
could
construct
a
stratied
sample
of
how
sample
of
how
and
of
of
women’s
students
it
would
scenarios
people
in
a
supporters
they
the
at
school
of
thought
voters
in
a
civil
training
a
be
it
sample
necessary
would
random
to
nd
big
your
to
of
combine
pay
be
from
out
a
the
possible
to
sampling
proportion
conduct
frame.
of
students
wish
of
have
know
could
of
views
to
you
people
people
is
in
as
to
who
hockey
was
town
to
doing
see
if
team
a
to
good
they
nd
out
job.
would
vote
in
mayor.
in
frame
what
decide
the
manager
an
American
cannot
sampling
as
ice
city
to
see
if
they
attended
regularly.
sample
subgroups
until
You
sessions
quota
particular
particular
town
sampling
responses,
a
the
servants
the
where
only
4 8 2
selecting
of
representative
questions
why
following
the
how
you
by
not
dierent
town.
the
re-electing
possible
Suppose
of
the
of
safety
into
clearly
B.
the
of
or
survey
decide
Explain
vegetarian.
survey
situations
be
how
sample
survey
guarantee
your
employees
which
are
to
means
the
provision
dierent
grade
30
In
split
on
pay
40
survey
re
rst
each
C
favour
A
employed
at
cartoons.
list
method.
15
whether
d
sampling
12
employees.
who
A
people,
women
watch
contact
is.
B
random
c
this
friends
the
2
list
Explain
A
900
and
this
in
1
grades
b
men
of
his
employees.
90
A
suitability
employs
of
sample
of
person
A
Describe
a
the
of
many
20th
employees.
45
a
sort
how
every
Women
Describe
c
know
to
Men
Describe
40
to
grade
childcare
5
on
number
Pay
a
what
company
The
A
wants
message
to
is
be.
local
who
as
live
equally.
12 Generalization
desired,
in
To
the
sampling.
needed
30
locally,
obtained,
used.
shoppers
survey
live
If
stratied
many
be
often
and
town,
want
to
is
a
you
possible
can
sample,
still
data
be
by
any
quota.
public
want
so
not
quota
collect
your
of
You
it
sample
Then
ll
think
people.
your
to
or
create
transport
your
are
represent
in
sample
going
men’s
to
to
ask
and
S TAT I S T I C S
An
appropriate
area
and
received
ask
way
the
to
rst
responses
conduct
people
from
15
the
that
survey
were
local
men
might
willing
and
15
to
be
to
talk
local
stand
to
in
you.
women,
the
P R O B A B I L I T Y
downtown
When
you
A N D
you
can
have
stop.
ATL
Reect
●
Why
and
might
important
●
A
What
It
sample
Identify
Give
as
a
b
You
want
c
You
want
botanist
group
in
school
An
A
One
and
event
is
Describe
a
A
to
are
a
as
her
d
a
conduct
on
the
taking
frame
enough
to
to
is
pieces
a
the
survey
quota
sample?
unknown,
of
be
survey)?
data
or
are
does
not
found.
Explain
assess
the
mark
of
to
in
your
want
nd
of
the
the
following
to
20
the
leaves
school
obtained
sample
scenarios.
by
collect
travel
a
in
30
time
a
type
school
in
everybody’s
and
average
to
students
employees
sta
on
tree.
each
your
day.
year
scores.
school
facilities
taken
of
by
which
sta.
runners
to
race.
the
on
and
satisfaction
plans
form
are
in
by
sample
the
with
their
latest
style
of
proposed.
the
form
random
to
box
for
every
product
that
is
sold
post.
of
customers
guarantee
disadvantages
estimate
and
a
a
He
records
She
of
to
based
registration
of
their
150
average
the
25
their
of
his
the
on
the
forms.
plans.
Determine
whether
her
heights
of
random
the
MYP
heights
height
members
of
of
students
his
correct
to
in
school
the
his
year
athletics
nearest
cm.
sample.
estimate
uses
suitability
simple
the
uses
suitability
wishes
the
group
how
at
administrative
dierent
a
length
customer
return
records
how
20
to
charity
lled
group.
on
mean
don’t
wishes
students.
on
each
samples.
student
year
but
wishes
call
for
students
feedback
sample
and
Explain
year
a
wishes
150
Comment
this
to
the
average
advantages
year
sample
c
when
sampling
until
method
how
test,
Some
put
is
nd
the
km
numbers
Another
in
10
to
Comment
b
the
data
to
possible.
teachers,
wants
student
team
to
organizer
the
of
as
nd
random
group
stand
inuence
encounter
when
out
recent
plan
telephone
3
nd
customers
Another
they
to
machine.
plan
ask
you
unwanted
sampling
detail
150
company
coee
used
principal
complete
2
where
you
collecting
wants
a
employs
e
of
have
might
suitable
much
A
A
5
2
a
d
choice
thus
is
involves
Practice
1
the
(and
problems
quota
exist.
discuss
average
History
correct
her
to
height
class
the
of
of
25
nearest
the
150
students
students
as
a
cm.
sample.
sample
of
25
students
could
be
taken
from
students.
stratied
sample
of
25
students
could
be
taken
from
this
group.
12.3 So, what do you think?
4 83
Generalization
C
●
If
a
How
sample
proper ties
you
can
is
use
sample
about
the
conclusions
representative
should
the
population.
the
are
In
reect
proper ties
doing
data
you
whole
average
She
thinks
than
The
a
The
188,
have
the
of
of
the
to
and
from
using
–
to
make
year
drawn,
means
about
its
that
the
whole
knowledge
broader
from
conclusions
each
16
have
181,
the
of
the
other
192,
the
8
boys
boys
from
heights,
each
181,
192,
following
194,
school
rowing
180,
174,
on
157,
of
the
rowers.
2
Calculate
the
mean
height
of
the
non-rowers.
3
Comment
claim
than
that
the
boys
rest
taller
on
of
who
their
make
year
team,
and
to
the
nearest
182,179.
170,
193,
height
her
are
group.
measured
mean
on
rowing
year
184,
heights:
167,
the
the
the
average
team
group.
Calculate
on
is
specic
make
1
taller
it
This
deductions
taking
it
data?
which
population.
make
generalizing
following
173,
187,
who
their
height
students
167,
are
of
sample
obtained
of
sample
centimeter:
other
190,
you
the
experimental
population
proper ties
of
boys
rest
the
random
rowers
whole
the
from
2
that
the
measures
takes
so,
have
drawn
inference
population.
Exploration
Antonia
of
the
and
154,
192,
165,
168,
165.
the
school
rowing
team
are
group.
ATL
Reect
●
Where
●
Is
●
In
the
and
have
discuss
you
generalized
generalization
this
problem
estimating
the
sensible
use
to
6
you
the
Exploration
2?
reasonable?
have
mean
in
of
used
the
range
of
the
mean
population.
the
sample
of
a
sample
Explain
to
why
estimate
as
it
the
a
way
might
range
of
not
of
be
the
population.
In
this
context,
information
from
which
could
these
draw
you
gained
it
should
from
came;
second,
conclusions
generalizations
4 8 4
have
the
that
about
are
made
sample
types
by
generalizations:
you
something
comparing
which
of
two
told
group
inference.
12 Generalization
the
means
contained
rst,
about
of
taller
that
the
two
the
population
groups
students.
you
Both
of
S TAT I S T I C S
Practice
1
Your
in
subject
2
says
that
your
Chemistry.
are
test
Your
presented
in
scores
results
this
in
(out
Physics
of
100)
are
for
better
the
than
last
six
your
tests
64
66
58
72
81
Physics
64
62
72
73
60
70
Calculate
your
b
Comment
c
Explain
on
how
suspect
mean
your
you
that
test
score
tutor’s
have
boys
in
Chemistry
and
in
Physics.
claim.
used
are
each
table:
56
a
test
in
Chemistr y
You
P R O B A B I L I T Y
3
tutor
scores
A N D
generalization
spending
more
in
this
money
problem.
than
girls
in
your
school
ATL
canteen.
You
spent
the
stand
by
the
cashier
one
lunch
time
and
record
the
amount
Tip
are
by
as
rst
ten
girls
and
by
the
rst
ten
boys.
The
amounts
you
record
Why
follows:
does
appear
Boys
Girls
in
the
so
$4.00
frequently
table?
Something
$3.12
$1.00
$4.00
stands
$4.00
$6.37
$1.37
$0.88
$4.00
$4.00
$4.00
$4.00
$5.16
$3.82
$4.00
$4.80
you
the
$2.32
Describe
the
type
of
sampling
Outline
one
way
in
which
this
of
sampling
might
have
Comment
d
Explain
from
have,
question
it
an
is
likely
impact
the
validity
of
bias.
the
c
you
have
on
introduced
infer
then
whether
used.
manner
what
$3.55
to
b
on
this
you
information
can
data
and
a
like
cause
reect
other
$3.65
out
should
to
$2.13
that
$2.51
on
the
where
Exploration
claim
you
that
have
boys
spend
generalized
in
more
your
than
girls
in
the
sample.
canteen.
answer.
3
ATL
GDP
GDP
per
capita
is
a
measure
of
the
average
income
per
person
in
Gross
country.
You
can
nd
this
data
in
annual
tables
in
the
World
stands
for
a
Domestic
Bank
Product.
database
(data.worldbank.org).
1
are
There
to
select
the
2
GDP
Calculate
whether
3
a
a
large
number
sample.
per
capita
the
it
Determine
mean
rose
if
the
whether
would
it
how
of
a
GDP
the
the
be
your
countries
years
per
better
results
of
to
2000
capita
use
a
compare
in
this
to
you
for
table
need
so
of
it
help
will
10
be
with
this.
necessary
countries
with
year
this
and
determine
2010.
made
the
stratied
using
you
2010.
each
2000
if
sample
and
for
from
countries
per
teacher
random
capita
period
GDP
your
simple
the
selection
about
see
for
over
generalize
and
Make
Ask
is
whole
fair
for
world.
sample.
Plan
you
to
Explain
how
to
do
this
method.
12.3 So, what do you think?
4 8 5
Exploration
A
factory
number
owner
of
components
they
each
the
output
produce
per
of
two
day
is
employees.
recorded
for
The
ten
days:
101
100
99
93
108
100
106
90
95
Employee
B
116
102
100
98
86
116
100
112
80
90
Present
each
3
If
the
factory
Why
●
What
ways
of
Bay
it
back
have
the
Crescent
at
Moon
(the
range
is
data
is
the
High
box-and-
keep
employees.
only
one
of
keeping.
the
two
Justify
employees,
your
decision.
to
comment
be
considered
are
you
on
the
in
this
mean
a
able
consistency
positive
to
of
his
context?
attribute?
calculate
that
describe
how
is?
the
range
a
and
set
of
interquartile
data
is
packed
range
are
together
appropriate
–
how
the
is
how
What
41
for
suggest
within
Crescent
be
Crescent
as
a
range
that
Moon
compare
range
at
and
and
High
School
the
location
interquartile
Oyster
marks
Bay
High
in
measures
together.
been
of
and
of
the
range
spread
12 Generalization
have
should
here
observe
consistent.
outliers,
–
so
that
was
Oyster
data
tell
you
IQR
rather
try
than
marks
at
Oyster
statistics
the
consistent,
or
variety.
than
whereas
smaller
of
that
greater
the
19.5.
Academy
say
more
always
the
pairs
Bay
might
showed
range
Both
Oyster
You
comparing
by
You
done
should
at
16.
Academy
School
aected
interquartile
range
interval.
interested
has
you
67
the
smaller
easily
your
smaller,
more
is
packed
data
might
the
interquartile
Moon
more
range
if
you
do
has
and
students
densely
of
at
School
IQR)
will
data
data?
range
the
the
were
4 8 6
the
however,
IQR
Academy
of
at
you
more
context
that
as
looked
the
of
because
general,
side-by-side
477).
together
results
general,
this
you
densely
median.
has
and
performance
the
two
7
eAssessment
the
spread
grouped
data
such
to
two
recommend
measures
of
how
(page
already
Academy
that
set
the
Bay
are
using
is.
examining
about
a
the
‘consistent’
consistency
spread
Academy
You
asks
does
of
able
would
owner
statistical
of
was
discuss
comparing
consistent
Look
you
What
might
consistent
Measures
owner
and
employees.
●
production
performance
which
Reect
The
the
factory
suggest
●
employee’s
diagrams.
Compare
the
compare
108
2
In
to
A
whisker
In
wishes
Employee
1
by
4
the
the
IQR
values,
to
range;
is
then
interpret
saying
Oyster
this
outlier
piece
the
simply
Bay
An
not.
lies
in
of
far
overall
pattern
of
the
is
data
a
that
outside
the
distribution
of
the
data.
rest
S TAT I S T I C S
Practice
1
A N D
P R O B A B I L I T Y
4
Joe
can
bus
journeys
travel
diagrams
to
work
and
40
by
bus
train
or
by
journeys
train.
are
The
total
illustrated
journey
in
the
times
for
40
box-and-whisker
below:
Bus
Train
0
20
30
40
a
Determine
which
mode
of
transport
is
b
Determine
which
mode
of
transport
has
c
Explain,
the
2
10
Two
bus
with
than
computer
messages.
The
speed
be
The
can
times
reference
the
these
a
statistics,
on
average.
more
why
predictable
Joe
might
journey
prefer
to
time.
take
train.
scientists
two
write
programs
assessed.
taken
to
faster
50
(in
Both
programs
(A
and
B)
programs
seconds)
are
as
designed
work
are
in
to
crack
dierent
timed
for
encoded
ways,
decoding
and
11
so
their
messages.
follows:
A
12.1
11.8
12.4
12.3
12.5
12.1
12.1
12.5
12.1
12.3
12.5
B
10.9
12.4
14.2
13.2
13.7
14.4
13.3
11.8
14.2
13.8
10.8
a
Determine
b
The
program
decodes
you
which
a
is
new
would
program
to
be
used
message
choose
to
is
in
the
use
faster
a
on
coding
fastest
under
average.
competition.
will
the
win
a
following
prize.
The
program
Select
scenarios.
which
Justify
which
program
your
answer.
i
ii
iii
3
In
to
You
are
12.5
seconds.
are
11.5
seconds.
You
are
a
factory,
them
taken
grams,
the
last
You
ll
are
the
the
Machine
A
Machine
B
rst
two
to
a
from
are
last
as
programmer
to
compete
and
the
current
best
time
is
programmer
to
compete
and
the
current
best
time
is
programmer
machines
weight
each
of
and
are
used
1000 g.
their
to
compete.
to
To
ll
sugar
compare
contents
bags.
the
weighed.
two
They
are
expected
machines, 12
Theresults
found,
bags
in
follows:
995
1009.5
999.6
1005.3
996.7
991.3
1022.3
1005.9
a
Find
the
mean
b
Find
the
interquartile
c
Compare
the
weight
output
for
each
range
of
the
for
two
997.1
1000
999.5
1005.4
1002
994.7
1002.4
1016.2
1005.2
1001.1
994.7
1000.7
996.2
1010.1
1009.6
1005.3
machine.
each
sample.
machines.
12.3 So, what do you think?
4 87
The
D
eect
●
How
●
Do
●
What
do
you
we
of
know
want
eect
to
sample
when
follow
does
to
say
the
sample
size
‘when’?
crowd?
size
have
on
the
reliability
of
our
generalizations?
In
by
spite
of
using
chance,
reliability
end
of
a
sampling
up
being
sample
Exploration
1
Use
a
integers
Microsoft
random
2
Select
3
Calculate
4
Collect
and
the
class
the
the
Predict
sample
is
a
Given
that
value,
why
the
the
sample
that
may
aects
still,
the
a
list
of
400
randomly
20.
Sheets
and
20
is
the
command
to
generate
‘=RANDBETWEEN(1,
from
this
a
20)’.
list.
sample.
that
have
stem-and-leaf
been
calculated
diagram
and
in
then
your
class
present
the
data
diagram.
using
a
sample
members
estimates
mean
size
of
the
of
class
estimate
we
20
instead
of
5,
gathering
have
found
dierent
values
explain
A
manufacturer
●
A
marine
wants
biologist
dierent
a
be
large
the
the
two
would
for
the
sample
mean
produce
small
taking
to
of
for
size
sample
have
on
sizes.
the
of
whole
or
the
the
set
of
mean
data.
of
a
small
population.
8
wants
areas
would
for
use
why
sample
found.
of
sizes
ever
diagrams
the
value
mean
sample
●
in
generate
numbers
means
a
discuss
would
and
5
varying
the
better
larger
the
factor
before.
eect
and
care,
mean.
whether
Reect
4 8 8
in
of
1
box-and-whisker
of
what
Suggest
and
your
dierent
what
consistency
crab
of
experiment
as
1
to
Google
sample
sample
data
why
Compare
scenarios
or
GDC
between
them
sample
Suggest
8
between
mean
the
your
box-and-whisker
Explain
for
7
a
Repeat
your
6
all
great
Another
size.
Excel
random
the
its
or
integer
organize
using
5
a
with
5
spreadsheet
selected
In
is
techniques
unrepresentative.
large
know
to
the
less
variation
samples?
samples
long
study
their
the
might
light
mass
Mediterranean
12 Generalization
in
the
Consider
of
be
bulbs
a
the
estimated
problematic:
last
before
particular
Sea.
mean
following
failing.
species
of
S TAT I S T I C S
A N D
P R O B A B I L I T Y
Summary
●
A
population
is
consideration
be
a
group
of
a
in
set
a
of
objects
statistical
people,
or
a
under
context.
collection
❍
It
could
of
A
stratied
sections
objects.
those
the
groups
proportions
●
A
census
member
●
A
is
a
of
a
sampling
the
collection
of
data
from
population.
frame
is
a
❍
list
of
every
●
member
A
quota
frame
of
It
population.
A
sample
❍
A
is
a
subset
of
a
is
representative
those
of
the
if
its
population
properties
from
which
is
●
simple
that
random
every
equally
sample
member
likely
independent
to
of
of
be
the
is
a
other
sample
population
included
any
A
sampling
Mixed
Tom
is
13
sample
in
the
❍
sample,
member
of
sample
takes
regular
intervals
at
the
the
data
is
are
biased
is
of
does
data
the
not
sampling
exist.
until
enough
found.
the
if
the
act
population
can
use
mean
Larger
members
from
it
is
not
a
fair
population.
of
drawing
based
on
conclusions
a
sample.
of
the
a
sample
mean
to
estimate
population.
samples
provide
more
reliable
of
the
the
Jo
are
comparing
the
IQR
The
for
their
IQR
Jo’s
marks
for
marks
in
Tom’s
is
18.
recent
3
marks
Tom
Paul
sh
because
his
IQR
is
smaller
he
has
in
the
assessments.
Jo
says
that
is
larger
she
has
performed
in
a
to
estimate
the
average
length
of
the
lake.
knows
that
roughly
40%
of
the
sh
in
the
performed
are
salmon
and
60%
of
the
sh
are
trout.
because
To
IQR
wants
says
lake
her
or
collecting
when
frame.
assessments.
and
better
a
You
He
that
used
practice
and
English
is
estimates.
systematic
population
that
appropriate
such
is
❍
A
of
Inference
population.
❍
ensures
in
sample.
unknown,
involves
about
A
sample
representation
it
comes.
❍
and
represented
the
dierent
population.
●
sample
reect
population
are
in
identies
every
pieces
1
of
sample
better
in
measure
the
sh,
he
will
catch
them
with
a
the
large
net
and
measure
them
on
the
shore
before
assessments.
returning
Comment
2
A
school’s
mean
of
on
ballet
height
21 cm.
their
of
The
claims.
class
has
1.51m
sample
11
and
school’s
a
students,
range
chess
club
of
with
a
a
heights
also
range
of
with
heights
a
mean
of
height
of
Comment
to
of
on
the
1.73 m
the
4
dierences
of
between
the
Explain
A
polling
to
chess
ballet
Explain
the
c
why
data
Claire
it
would
proposes
adult
clubs
suitable
sampling
method
for
how
he
would
take
such
a
a
Paul
sample.
company
would
the
like
to
conduct
extent
to
which
a
the
of
a
city
feel
that
advertising.
they
The
are
subject
company
to
proposes
range
be
preferable
rather
than
to
the
know
phone
a
to
use
the
concerning
chess
may
simple
directory
random
as
a
sample
sampling
using
the
frame.
range
the
company’s
plan.
sets.
predictions
and
take
to
club.
interquartile
the
to
and
Evaluate
of
plans
the
class
conduct
b
He
50.
determine
unwanted
the
water.
anda
43 cm.
members
a
the
use.
residents
heights
size
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survey
a
of
to
has
b
11students,
them
not
adult
players.
be
a
data
to
ballet
Explain
suitable
make
dancers
why
these
sample.
12.3 So, what do you think?
4 8 9
local
Review
in
Identities
1
The
and
average
neighboring
random
context
relationships
household
towns
sample
of
is
income
assessed
residents
in
by
two
a
taking
using
a
a
as
a
sampling
frame.
The
the
as
with
box-and-whisker
data
diagrams.
Describe
the
the
similarities
two
sets
of
and
dierences
data.
follows:
c
Income
range
($)
Town
A
Town
20 000
≤
x
<
30 000
4
1
30 000
≤
x
<
40 000
7
4
B
Comment
education
d
3
Evaluate
You
will
be
on
40 000
≤
x
<
50 000
11
≤
x
<
60 000
eectiveness
the
authority’s
able
to
of
nd
15
6
from
the
(www.ipu.org)
60 000
≤
x
<
70 000
5
12
70 000
≤
x
<
80 000
4
9
the
x
<
90 000
4
Take
5
≤
x
<100 000
0
3
0
1
a
from
≤
x
<
110 000
b
Present
of
this
information
cumulative
frequency
that
the
the
national
World
You
the
varies
random
countries
Compare
100 000
about
method.
parliaments
in
are
to
Bank
investigate
percentage
dierent
Union
of
women’s
continents.
7
a
90 000
in
Inter-Parliamentary
or
proposition
representation
≤
the
sampling
data
women
(data.worldbank.org).
80 000
of
intervention.
7
online
50 000
the
authority’s
representation
a
over-and-under
obtained
between
is
data
recent
b
census
Present
using
a
graphs
pair
on
the
Comment
stratied
from
Africa,
means
on
of
whether
representative
of
the
sample
Asia
these
the
data
of
and
Europe.
samples.
samples
in
this
are
case.
the
c
Describe
how
you
would
take
a
census
sameaxes.
from
b
Comment
earnings
on
of
the
dierences
households
in
between
towns
A
the
and
The
statistician
that
the
samples
towns
they
by
word
the
conducting
are
came
the
survey
representative
from.
Explain
representative
in
of
what
this
claims
the
is
data
to
meant
Elderly
people
number
with
of
were
hours,
volunteers
assistance
t,
To
in
assess
50
the
primary
authority
of
schools,
assesses
11-year-olds
after
impact
in
the
three
the
a
new
local
reading
of
implementation.
summarizes
a
the
The
the
the
three
surveyed
of
after
average
continents.
social
the
about
contact
the
they
introduction
of
have
a
new
scheme.
context.
Hours
2
compare
from
B.
4
c
this
representation
literacy
Before
After
strategy
education
age
of
schools
before
following
0
≤
t
<
2
4
0
2
≤
t
<
4
10
8
4
≤
t
<
5
7
9
5
≤
t
<
6
5
8
6
≤
t
<
8
6
5
all
and
table
data.
Before
After
Minimum
8.4
8.2
Q
9.2
10.3
10.1
10.8
10.5
11.1
13.2
12.8
8
≤
t
<
10
4
5
10
≤
t
<
12
3
2
12
≤
t
<
16
0
2
16
≤
t
<
20
1
1
1
Q
2
Q
3
Maximum
a
Interpret
not
4 9 0
12 Generalization
the
the
data
scheme
and
was
suggest
successful.
whether
or
S TAT I S T I C S
The
sample
details
waited
who
for
in
he
would
was
40
the
local
thought
be
obtained
elderly
bus
were
interested
by
collecting
people.
depot
over
in
The
and
75
such
a
contact
You
researcher
asked
whether
scheme
believe
depending
people
grew
they
up
country,
or
he
could
have
their
contact
the
success
of
the
Determine
c
Evaluate
d
Suggest
the
the
a
type
the
research
parents
sampling
used.
dierent
methods.
appropriate
way
b
You
want
more
to
nd
likely
sample.
out
to
if
to
in
local,
Outline
your
obtaining
reliable
data
a
whole
donate
You
to
is
year,
a
the
to
you
expect
relevant
to
form
easiest
give
which
money
give
take
that
data
family
the
country.
up
that
30%
locally,
of
45%
families
grew
country
and
25%
grew
up
in
you
would
your
use
sample
Reect
and
How
you
this
information
a
year,
you
with
receive
parents
40
that
responses
grew
up
from
locally,
from
those
that
grew
up
elsewhere
in
the
or
and
50
from
those
that
grew
up
in
a
to
to
country.
encounter
your
Calculate
way
people
they
can
to
collect
forms
ll
in
the
number
of
responses
you
inquiry.
to
in
keep
when
for
use
order
sample
to
from
each
obtain
the
category
largest
of
respondent
representative
possible.
they
charity.
to
every
student
in
your
school
discuss
explored
the
statement
of
inquiry?
Give
specic
examples.
Statement of Inquiry:
Generalizing
trends
and
among
a
representative.
home.
have
up
causes.
diculties
decide
the
make
should
You
in
school
national
c
in
in
to
dierent
a
dierent
vary
the
country.
how
try
country
international
a
elsewhere
suggests
grew
to
After
students
donate
in
in
of
75
are
that
Explain
families
5
up
will
parents
scheme.
researcher’s
more
constructing
of
grew
up
trends
to
elsewhere
b
grew
the
P R O B A B I L I T Y
and
details
had
evaluate
donation
whether
locally,
Separate
whether
that
on
A N D
representing
relationships
can
help
to
clarify
individuals.
12.3 So, what do you think?
4 91
13
Justication
Valid reasons or evidence used to suppor t a statement
Verifying
To
verify
need
You
to
a
statement
check
could
and
that
verify
it
the
justifying
(in
other
works
for
statement
words,
some
‘the
prove
the
truth
of
it),
you
examples.
difference
between
can
You
consecutive
square
numbers
is
odd’
by
giving
these
−
121
=
23
and
49
−
36
=
Give
some
more
justify
the
examples
statement,
you
to
verify
the
statement.
need
to
give
reasons
why
it
is
true.
by
can
justify
explaining
e
dierenc
●
Show
that
every
even
square
number
is
followed
by
an
number
and
vice
versa,
so
the
difference
square
numbers
must
always
be
an
High
Lift
cross-section
of
aircraft
number.
pressure
wing
Low
speed,
increased
4 9 2
odd
speed,
reduced
the
about
air
ies
pressure
and
b elow
its
wings,
between
and
consecutive
in
it
why
odd
above
square
g
observin
by
ying.
You
To
ies,
an
13
it
●
that
examples:
airplane
144
verify
pressure
how
this
gives
it
lift.
Justication
The
word
justify
‘justier’,
standard
The
If
meant
most
the
evidence
the
only
no
such
in
the
the
new
sale,
the
To
humans
negative
the
to
court
French
verb
proceedings’.
prosecution
must
The
prove
the
the
are
run
is
clinical
evidence
on
that
found
found
the
defendant
guilty.
not
trying
to
conditions,
Before
justify
is
If
there
guilty.
research
incurable
to
is
continually
effectively.
they
facts
defendant
medical
needs
collect
these
defendant
as-yet
more
of
that
a
new
the
trials
to
positive
or
to
drug
drug
test
produce
is
can
safe
the
results
treat
go
and
drugs
as
well
as
effects.
The
not
in
treat
this,
and
trials
committee.
will
to
do
then
then
company
side
clinical
can
that
companies
conditions
effective.
trial
submit
explanation
evidence,
medicines
curable
All
12th-century
case.
crime,
Pharmaceutical
on
the
facts
Justication
on
‘to
courtsis
logical
committed
is
from
evidence
prosecution
presents
or
comes
which
in
using
be
reasonably
have
to
be
approved
researchers
exposed
be
to
must
by
justify
unnecessary
expected
to
improve
a
scientic
that
risk,
the
and
patient
and
an
ethical
volunteers
that
the
in
new
the
drug
care.
4 9 3
Well-rounded
13.1
ideas
Global context: Personal and cultural expression
Objectives
●
Finding
angles
Inquiry
and
lengths
using
circle
theorems
questions
●
What
●
How
are
the
circle
theorems?
F
●
Proving
●
Examining
‘If
thetruth
their
results
of
using
...
circle
then
...’
theorems
statements
and
testing
do
we
justify
mathematical
C
converses
CIGOL
conclusions?
●
Is
the
opposite
of
a
true
statement
D
alwaysfalse?
●
ATL
Can
aesthetics
be
calculated?
Critical-thinking
Draw
reasonable
conclusions
and
generalizations
13.1
13.2
E13.1
Statement of Inquiry:
Logic
our
4 9 4
can
justify
appreciation
generalizations
of
the
aesthetic.
that
increase
G E O M E T R Y
Y
ou
●
should
name
the
●
These
A
case,
pass
of
tangent
Line
the
line
a
L
is
the
the
to:
Draw
a
circle,
then
draw
a
a
radius
b
a
diameter
c
a
chord
d
a
segment
e
an
f
the
center
g
the
circumference
basic
and
label:
arc
circle
theorems
theorems?
tangent
line
same
lines.
touches
the
curve
at
the
point
of
contact
and
does
gradient
(slope)
as
the
curve
at
the
point
where
the
curve.
tangent
the
the
T R I G O N O M E T R Y
curve.
the
the
1
circle
illustrate
the
how
circle
tangent
has
touches
intersects
are
all
through
tangent
L
What
diagrams
each
not
par ts
know
Discovering
F
In
already
A N D
circle
to
a
at
only
circle
at
one
point
point,
P
if
L
P.
P
ATL
Exploration
1
Using
a
GDC
1
or
dynamic
geometry
Q
A
software,
construct
a
circle
with
line
segment
center
connects
O
which
passes
through
a
second
point,
points,
Add
a
third
point
Q,
not
on
the
circle.
but
a
line
segment
joining
O
to
P,
a
line
P
extends
Draw
two
P.
in
both
and
directions.
add
the
line
PQ
O
Continued
on
next
page
13.1 Well-rounded ideas
4 9 5
2
Move
the
Repeat
and
a
point
few
nding
Q
freely
times
a
new
by
until
the
line
repositioning
position
for
Q
so
PQ
P
is
(and
that
a
tangent
hence
PQ
is
a
to
the
changing
tangent
to
circle
the
the
at
P.
circle)
circle.
Q
P
P
Q
O
3
Now
use
Describe
your
a
is
GDC
anything
the
circle.
the
relationship
P
is
a
point
circle,
the
a
the
tangent
perpendicular
dynamic
notice
your
between
of
on
or
you
Compare
circumference
If
O
to
geometry
about
ndings
a
angle
with
tangent
and
software
OPQ
others.
the
to
when
Form
radius
at
a
a
measure
PQ
is
a
∠OPQ.
tangent
conjecture
point
P
on
to
about
the
circle.
circumference
to
the
the
circle
radius
at
of
P
OP
P
O
ATL
Exploration
2
P
1
Using
draw
a
a
GDC
circle
or
dynamic
with
Draw
another
of
circle.
center
point,
P,
geometry
O
on
and
the
software,
diameter
AB.
circumference
O
the
2
Use
3
Move
the
the
Draw
software
point
circle.
4
Repeat
5
Compare
P
to
circles
your
line
segments
measure
around
Describe
with
the
the
you
dierent
ndings
with
notice
BP
A
B
13 Justication
of
about
∠APB
diameters.
others.
ndings.
4 9 6
and
∠APB
circumference
anything
of
AP
Form
a
conjecture
based
on
your
G E O M E T R Y
A N D
T R I G O N O M E T R Y
ATL
Exploration
1
Using
draw
a
a
chord
GDC
circle
AB.
3
or
with
Draw
circumference
segments
2
Use
the
3
Without
dynamic
AP
center
O
another
of
the
and
software
geometry
software,
P
and
point,
circle.
P,
Draw
on
the
the
line
O
BP
to
measure
∠APB
B
A
the
moving
circumference
anything
you
4
Now
5
Move
point
A
B.
6
A
of
P
A
Compare
B
around
your
move
(or
the
P
around
Describe
∠APB
both).
Describe
circumference
anything
ndings
point
circle.
about
or
Describe
B,
the
notice
reposition
and
or
you
with
notice
others.
the
of
the
about
Form
eect
a
of
circle,
this
on
without
∠APB
moving
∠APB
conjecture
based
on
yourndings.
A
cyclic
vertices
quadrilateral
all
lie
on
the
is
a
quadrilateral
circumference
of
a
B
whose
circle.
C
A
D
ATL
Exploration
1
Using
a
points
line
2
3
Use
GDC
P,
Q,
the
Now
or
R
segments
4
Tip
dynamic
and
so
on
that
software
reposition
S
to
any
geometry
the
PQRS
circumference.
is
measure
or
all
software,
of
a
cyclic
∠PQR
the
draw
Join
the
a
circle
four
and
points
mark
with
four
Make
straight
quadrilateral.
and
points.
points
S
are
sure
P,
in
the
Q,
R
order,
and
so
your
quadrilateral
does
not
∠RSP.
Describe
anything
cross
you
itself.
notice
about
the
quadrilateral
4
Compare
relationship
between
∠PQR
and
∠RSP
as
the
cyclic
changes.
your
ndings
with
others.
Form
a
conjecture
based
on
yourndings.
The
the
To
major
minor
show
point
on
arc
arc
the
the
is
is
the
the
long
way
around
and
short
way
around
a
direction
you
circumference
can
of
include
the
circle.
a
circle,
A
minor
arc
third
e.g.
ADB
major
arc
B
D
13.1 Well-rounded ideas
4 97
ATL
Exploration
1
Using
a
a
circle
on
the
GDC
with
5
or
dynamic
software,
O.
Mark
two
circumference.
Draw
another
the
major
AO
and
center
geometry
arc
AB.
Draw
the
line
points
A
and
point,
segments
P
draw
B
P,
on
AP,
BP,
O
BO
B
2
Measure
3
Move
∠AOB
points
Describe
∠AOB
A,
and
B
and
anything
and
4
Repeat
5
Compare
∠APB
the
∠APB
P,
you
as
keeping
notice
you
exploration
your
A
ndings
vary
with
the
a
with
P
in
about
the
the
major
positions
circle
of
others.
arc.
relationship
a
of
A,
dierent
Form
a
B
between
and
P
diameter.
conjecture
based
on
yourndings.
ATL
Exploration
6
Tip
C
1
Using
a
a
circle
GDC
with
or
circumference.
and
2
CA.
Also
Draw
∠ABC
Measure
the
tangent,
3
Move
4
Repeat
5
Compare
your
and
the
A,
the
line
A,
software,
B
and
segments
to
the
C
AB,
circle
draw
on
B
be
BC,
at
a
A
and
and
the
C.
ndings
and
tangent.
Describe
with
AB
a
with
anything
circle
others.
of
a
you
notice.
dierent
Form
a
diameter.
conjecture
based
on
ndings.
Activity
Did
you
2
6?
to
illustrate
the
nd
it
Below
them.
questions
Five
ve
Read
below
to
explain
ways
the
the
of
ve
the
writing
rules,
rules
you
these
look
at
discovered
rules
the
and
ve
diagrams,
in
Explorations
diagrams
and
then
to
answer
diagrams.
rules:
i
Thales’
ii
Opposite
iii
Angles
iv
Alternate
a
dicult
are
point
theorem:
in
An
angles
the
same
segment
is
in
equal
angle
a
cyclic
the
in
a
semicircle
quadrilateral
segment
theorem:
to
inscribed
subtended
the
angle
angle
are
by
between
subtended
by
is
a
right
angle.
supplementary.
equal
a
the
chords
chord
chord
are
and
in
equal
tangent
the
in
size.
at
alternate
segment.
Continued
4 9 8
13 Justication
software
able
to
should
tangent
construct
to
a
automatically
.
A
between
Y
our
O
its
∠ACB.
exploration
your
geometry
points
tangent
angles
AC
B
and
the
and
acute
points
O
draw
Measure
the
dynamic
center
on
next
page
circle
G E O M E T R Y
v
The
the
angle
angle
subtended
at
subtended
Five
a
the
by
by
a
chord
circumference
the
same
at
the
center
(inscribed
(central
angle)
in
the
angle)
same
is
A N D
T R I G O N O M E T R Y
twice
segment
chord.
diagrams:
x
+
y
=
b
180°
c
x
x
y
O
x
d
e
x
x
x
O
2x
Tip
Make
●
What
do
inscribed,
these
same
terms
mean?:
segment,
subtended,
alternate
segment.
supplementary,
Look
them
opposite
up
if
you
angle,
need
to.
Match
each
rule
to
its
you
understand
terms,
can
●
sure
so
all
that
understand
Match
You
can
each
use
Example
rule
these
to
ve
the
diagram.
information
●
these
you
one
of
results
the
to
Explorations
nd
angles
in
2
to
in
6.
geometric
a
given
question.
problems.
1
Find
the
value
of
your
reasoning.
angle
t,
and
B
justify
60°
D
O
t
50°
C
∠CDE
=
60°
because
∠CDE
and
E
∠CBE
‘Justify’ means ‘give clear reasons for your answers’.
are
angles
∠CED
=
in
90°
the
same
because
segment.
the
angle
inscribed
CD passes through O, so it is a diameter.
in
a
semicircle
∠BED
∠t
is
a
right
=
90°
−
=
40°
because
=
180°
=
80°
−
50°
60°
−
because
triangle
angle.
∠CED
=
90°
40°
the
sum
to
interior
angles
in
a
180°
13.1 Well-rounded ideas
4 9 9
Example
2
A
Calculate
∠BOD
128°
D
B
O
C
∠BCD
=
180°
−
128°
Opposite
angles
quadrilateral
∠BOD
=
52°
=
2
=
104°
×
The
52°
the
Example
Prove
that
are
in
a
cyclic
supplementary.
central
angle
inscribed
is
twice
angle.
3
∠BOD
=
C
64°
O
D
B
∠BCD
∠BOD
=
=
32°
64°
Practice
by
the
alternate
because
the
segment
central
32°
theorem.
angle
is
twice
the
inscribed
angle.
1
Tip
1
Calculate
the
size
a
of
the
marked
angles
in
each
b
diagram.
Calculate
c
and
nd
e
both
mean
that
41°
you
should
show
a
O
O
relevant
27°
c
your
working,
5
0
°
you
to
do
not
justify
e
f
h
28°
130°
n
O
m
g
j
36°
5
8
°
k
5 0 0
13 Justication
but
need
all
reasoning.
d
in
f
d
b
stages
your
G E O M E T R Y
Problem
2
Find
In
the
each
A N D
T R I G O N O M E T R Y
solving
size
case,
of
the
marked
justify
your
angle
in
each
diagram.
answer.
H
D
C
E
B
K
40°
c
O
O
a
28°
b
A
J
F
G
I
P
A
B
U
e
70°
Q
F
O
70°
C
f
78°
V
X
d
T
W
20°
D
R
S
E
Justifying
C
●
In
this
How
section
Objective:
iv.
do
you
C.
we
will
the
justify
use
circle
theorems
mathematical
direct
(deductive)
conclusions?
proof
to
justify
the
circle
theorems.
Communicating
communicate
complete,
coherent
and
concise
mathematical
lines
of
reasoning
The
following
sure
your
two
proofs
proofs
are
Exploration
1
Copy
and
complete
concise
and
the
that
proof
they
of
the
make
cyclic
quadrilateral
theorem.
Make
sense.
7
complete
this
skeleton
proof.
Theorem
Opposite
angles
in
a
cyclic
quadrilateral
are
supplementary.
Proof
Consider
a
cyclic
quadrilateral
ABCD
inscribed
in
a
circle
with
centerO
A
B
A
diagram
makes
it
easier
for
readers
O
understand
your
proof.
D
C
Continued
on
next
page
13.1 Well-rounded ideas
5 01
to
OA
=
OB
=
Therefore
OC
=
OD
ΔOAB,
because
_______,
____________
_______,
What
and
_______
Therefore
_______
because
∠OAB
are
all
∠OAB
=
=
_______
∠OBA,
_______,
and
type
of
triangle
_______
_______
=
=
Use
the
’triangle
fact‘
from
+
∠OBA
2(∠OAB
+
____
+
____
+
∠OAD
+
+
⇒
+
____+
____
The
D
the
are
proof
is
Bydrawing
that
the
angles
at
A
and
+
_______)
_______)
_______
Therefore
2
_______
2(∠DAB
and
the
line
above.
____________
⇒
B
ΔOAB?
_______,
_______
+
____+
____
=
360°
⇒
⇒
is
triangles.
+
C
=
=
means
’implies‘
360°
360°
_______
are
=
Notice
how
∠OAB
+
∠OAD
=
∠DAB
180°
supplementary,
and
the
angles
at
supplementary.
incomplete
suitable
opposite
as
you
diagrams
angles
are
need
and
to
consider
modifying
supplementary
in
two
the
other
proof,
cyclic
cases.
prove
quadrilaterals
where:
a
one
b
the
edge
of
center
the
of
Exploration
Copy
and
quadrilateral
the
circle
is
not
passes
inside
through
the
the
center
of
the
circle
quadrilateral.
8
complete
this
skeleton
proof
of
Thales’
theorem.
Theorem
Any
angle
inscribed
in
a
semicircle
is
a
right
angle.
Proof
A
O
B
C
Consider
point
on
ΔOAC
a
circle
the
and
with
center
circumference
ΔOCB
Hence
∠OCA
∠ACB
=
=
∠OCA
are
_______
_______
+
O
of
and
where
the
AB
is
a
diameter.
Let
C
be
any
circle.
triangles.
∠OCB
=
_______
∠OCB
Exterior
∠AOC
=
_______
+
_______
and
∠BOC
=
_______
+
interior
Continued
5 0 2
13 Justication
angle
=
sum
of
_______
on
next
page
opposite
angles.
G E O M E T R Y
⇒
∠AOC
=
2∠OCB
⇒
∠AOC
+
∠BOC
=
2(_______
⇒
∠AOC
+
∠BOC
=
2
⇒
2_______
⇒
_______
=
=
and
∠BOC
=
T R I G O N O M E T R Y
2∠OCA
+
_______)
_______
180°
∠OCA
+
∠OCB
=
∠ACB
90°
Exploration
Copy
and
A N D
9
complete
this
skeleton
proof.
Theorem
Angles
in
the
same
segment
are
equal.
Proof
A
P
R
B
Q
Consider
the
points
chord
∠PAQ
+
∠PBQ
because
Hence
through
∠PRQ
+
∠PRQ
∠PAQ
circle,
=
and
P
B
and
that
Q.
lie
in
Point
the
R
same
lies
in
segment
the
of
opposite
a
circle
divided
by
segment.
_______
=
_______
_____________________________________
Example
A
A
=
_______
4
center
circumference.
O,
has
RST
is
points
a
S
and
straight
T
line
on
R
its
segment.
S
Prove
that
∠SOT
=
2
×
∠RSO
180°
O
T
∠OST
=
180°
−
∠OST
=
∠OTS
∠SOT
=
180°
−
∠RSO
(base
(angles
angles
∠OTS
−
∠SOT
=
180°
−
2∠OST
⇒
∠SOT
=
180°
−
2(180°
⇒
∠SOT
=
2
∠RSO
an
∠OST
⇒
×
in
on
−
a
straight
isosceles
(angles
in
a
line)
triangle)
triangle
sum
to
180°)
∠RSO)
180°
13.1 Well-rounded ideas
5 0 3
Practice
2
Problem
1
Points
such
A,
that
solving
B
and
A,
B
C
lie
and
C
on
all
a
circle
fall
with
within
center
the
same
O
semicircle.
1
Prove
that
∠ABC
=
180°
−
O
∠AOC
2
A
C
B
2
Points
The
Let
P,
X
and
tangents
O
be
the
Y
to
lie
the
center
on
the
circle
of
at
the
circumference
X
and
circle.
Y
meet
Prove
of
at
a
P
circle.
Q.
that
O
Y
1
∠XPY
=
(180°
−
∠XQY).
2
X
3
Points
a
B,
tangent
D
and
to
the
E
lie
on
circle
a
at
circle.
B.
DE
ABC
is
is
parallel
to
ABC
E
D
Prove
that
ΔBDE
is
isosceles.
O
C
B
A
4
A,
B,
and
C
and
when
extended
Prove
D
lie
the
line
they
that
on
circumference
segments
meet
ΔACX
the
is
at
X,
AB
and
outside
similar
to
of
DC
the
a
D
circle,
are
circle.
ΔDBX
A
Tip
C
B
Remember
that
two
X
triangles
5
A,
B,
C,
D,
E
and
F
lie
on
the
circumference
of
a
if
with
AB
=
BC
and
DF
perpendicular
to
their
BE
and
CE
meet
DF
at
X,
Y
and
Z
respectively.
that
∠EXY
=
B
∠EZY.
‘Bisect’
cut
D
The
D
The
converse
●
Is
●
Can
theorems
the
opposite
aesthetics
you
have
statements.
For
and
diameter,
be
AC
split
is
a
into
5 0 4
‘If
then
parts,
of
be
proved
example,
two
are
same.
Y
Z
Prove
angles
A
E
BE.
the
AE,
similar
F
X
circle,
are
the
a
of
the
true
circle
statement
theorems
always
false?
calculated?
are
points
∠ABC
all
A,
is
premise
a
of
B
right
(the
13 Justication
the
and
‘if ’
form
C
lie
angle’.
part)
‘If
on
p,
Y
ou
and
then
the
can
the
q’,
where
p
circumference
see
that
conclusion
the
and
of
a
q
are
circle
statement
(the
‘then’
can
part).
in
means
half
to
exactly.
G E O M E T R Y
●
If
ABC
●
If
x
●
If
x
is
is
a
a
triangle,
whole
then
number
∠ABC
which
+
∠BCA
ends
in
a
+
∠CAB
zero,
then
=
180°
x
is
not
A N D
T R I G O N O M E T R Y
Tip
prime.
2
●
If
x
=
+
4,
3
then
=
7,
x
=
then
2.
x
You
=
the
4.
p
then
This
Reect
and
discuss
The
the
of
the
converse
opposite
These
‘If
of
a
If
∠ABC
●
If
x
is
+
not
then
…’
statement
order.
statements
●
…
So
are
the
∠BCA
prime,
the
+
statements
takes
its
converse
then
x
is
=
a
two
of
converses
∠CAB
also
is
q’
as
p
read
write
‘If
⇒
q.
as
2
‘p
Which
can
statement
‘If
of
above
then
four
then
whole
q’
parts
‘If
q
is
a
and
then
previous
ABC
number
is
q’.
true?
constituent
p
the
180°,
are
implies
places
them
in
p’.
statements.
triangle.
which
ends
in
a
zero.
2
●
If
x
=
2,
then
x
●
If
x
=
4,
then
x
+
4.
3
=
7.
Reect
and
discuss
●
Which
of
preceding
●
If
the
original
statement
is
true,
●
If
the
original
statement
is
false,
You
should
working
not
with
of
theorem
and
circumference
what
the
is
be:
of
‘If
a
converse
a
its
once
∠ABC
circle,
is
and
means
–
2
two
points
Draw
a
A
and
third
then
a
and
then
then
is
then
their
a
that
‘If
also
converse
converses
you
an
‘If
have
…
can
to
false?
be
be
then
true?
true?
also
true
or
careful
…’
is
angle
A,
a
then
diameter’.
it
is
B
and
right
points
It’s
not
a
C
lie
angle’.
A,
little
obvious
on
The
B
false
when
statement
the
does
circumference
converse
and
C
harder
lie
to
whether
on
see
or
of
the
the
exactly
not
it
is
true.
Tip
geometry
software
to
C
draw
C
point
are
true.
points
certainly
statements
converse
the
proving
is
…’
the
is
and
∠ABC
right
is
…
means
more:
AC
dynamic
or
This
diameter,
1
GDC
‘If
converse
10
a
of
statements
Exploration
Use
list
justication;
that
theorem
AC
would
that
3
another.
and
mean
Thales’
circle
seen
one
proof
necessarily
Consider
a
the
have
independently
of
=
A
D,
and
the
line
segment
Y
our
be
a
AD
software
able
to
should
construct
perpendicular
to
a
B
line
3
Draw
a
line
perpendicular
to
AD
through
through
a
point.
C.
D
Let
point
B
be
perpendicular
You
part
a
have
of
right
stays
4
Use
the
point
5
the
B.
Explain
a
converse
By
same
Trace
Vary
if
intersection
through
created
angle.
the
the
set
of
of
AD
and
the
C
of
three
Thales’
moving
point
points,
A,
theorem:
D,
you
B
and
three
should
C
that
points
see
satisfy
such
that
as
the
that
B
‘if’
∠ABC
moves
is
∠ABC
size.
facility
the
your
of
your
position
diagram
of
GDC
B
by
supports
or
software
moving
the
to
track
the
position
of
D
converse
of
Thales’
theorem.
13.1 Well-rounded ideas
5 0 5
Reect
Is
and
Exploration
suciently
discuss
10
the
a
GDC
other
others,
is
that
the
tricky
or
four
the
–
that
dynamic
same
the
you
converse
believe
geometry
theorems
you
size
know
as
of
the
opposite
angles
this
in
one
how
a
software
also
another.
so
tackle
of
Thales’
theorem?
converse
to
be
If
true
not,
is
it
anyway?
11
circle
provided
that
proof
convincing
Exploration
Use
a
4
hold.
to
use
the
Exploring
cyclic
to
Some
verify
are
software
the
that
to
converses
produce
converse
quadrilateral
the
signicantly
are
of
the
easier
an
of
than
angle
theorem
supplementary
is
quite
last!
Summary
Line
a
L
is
circle
the
at
tangent
point
P
L
to
●
if
Thales’
theorem:
inscribed
in
a
An
angle
semicircle
P
L
intersects
at
only
one
the
circle
point,
is
a
right
O
●
Opposite
cyclic
A
cyclic
angle.
P
quadrilateral
is
B
a
angles
in
a
quadrilateral
x
+
y
=
180°
are
supplementary.
x
quadrilateral
whose
vertices
y
all
of
lie
a
on
the
circumference
circle.
C
A
●
Angles
in
the
same
D
segment
subtended
by
x
x
equal
The
major
arc
is
the
long
way
around
the
a
minor
major
arc
arc
is
you
the
short
way
around.
can
circumference
of
the
A
circle
include
circle,
a
third
e.g.
To
point
to
a
size.
on
the
ADB.
Alternate
The
at
a
point
P
on
segment
angle
theorem:
between
a
chord
its
and
circumference
equal
indicate
●
tangent
are
circle;
in
the
chords
tangent
at
a
point
x
is
is
equal
to
the
angle
P
perpendicular
to
its
subtended
radius
by
the
chord
OP
in
the
alternate
segment.
x
0
●
The
angle
subtended
by
x
a
chord
twice
at
the
the
center
angle
at
is
the
O
circumference
in
the
same
subtended
2x
by
5 0 6
13 Justication
segment
the
same
chord.
G E O M E T R Y
Mixed
1
Find
A N D
T R I G O N O M E T R Y
practice
the
size
of
the
marked
angles.
a
b
c
If
it
d
If
a
it
is
is
a
bird
then
polygon
a
has
it
has
four
wings.
sides,
then
square.
b
40°
2
e
If
c
=
f
If
a
9
then
c
=
81.
O
42°
65°
c
right-angled
triangle
is
drawn
with
a
itsvertices
on
circlethen
its
ofthe
2
Find
the
Justify
size
your
of
the
marked
the
circumference
hypotenuse
is
a
a
diameter
circle.
B
A
angle.
4
answers.
a
of
In
the
cyclic
ABCD,
b
quadrilateral
∠DAB
≅
∠ABC.
a
Prove
that
AD
=
BC
67°
D
O
132°
B
5
A,
B,
C
and
D
lie
on
a
b
circle
DOB
3
For
each
statement,
Determine
which
write
down
statements,
the
and
center
bisects
O,
and
∠ABC
converse.
O
which
Prove
that
ΔABC
A
of
a
their
If
a
converses,
=
7
then
are
3a
2
true.
=
is
isosceles.
19.
C
D
b
If
b
=
a
Review
then
in
a
b
=
0.
context
Personal and cultural expression
1
A
circular
end
the
of
it.
auditorium
When
a
auditorium,
patron’s
eld
the
of
has
patron
a
stage
sits
stage
at
across
O,
occupies
the
88°
Find
the
center
of
of
the
wall
total
projectors
so
vision.
the
length
projected
their
are
on
of
the
when
the
positioned
beams
overlap
by
stage
20
cm.
17°
Find
eld
the
of
amount
vision
of
that
the
the
O
stage
would
occupy
viewed
from
edge
the
P,
at
P
when
the
very
P
2
of
Three
auditorium.
projectors,
used
to
display
images
Reect
and
How
you
have
wall
of
mounted
each
a
on
have
one
above
The
circle
a
circular
the
performance
opposite
beam
width
another
at
a
wall.
of
The
17°,
single
space,
diameter
6
statement
of
Give
specic
examples.
are
are
placed
Statement of Inquiry:
P
Logic
has
the
projectors
and
point
explored
on
inquiry?
the
discuss
can
justify
generalizations
that
increase
m.
our
appreciation
of
the
aesthetic.
13.1 Well-rounded ideas
5 07
It
13.2
strikes
a
chord
Global context: Personal and cultural expression
Objectives
●
Using
circle
Inquiry
theorems
to
nd
lengths
of
chords
questions
●
What
●
How
can
you
●
How
can
a
is
the
intersecting
chords
theorem?
F
●
Finding
lengths
using
the
intersecting
chord
apply
the
theorem?
theorem
theorem
have
dierent
CIGOL
C
cases?
●
Can
aesthetics
be
calculated?
D
ATL
Critical-thinking
Test
generalizations
and
conclusions
13.1
13.2
E13.1
Statement of Inquiry:
Logic
can
increase
5 0 8
justify
our
generalizations
appreciation
of
the
that
aesthetic.
G E O M E T R Y
Y
ou
should
already
know
how
A N D
T R I G O N O M E T R Y
to:
2
●
solve
quadratic
equations
1
Solve
the
equation
2x
x
cm
has
3
=
0
for
x
by
factorizing.
●
apply
the
circle
theorems
2
A
circle
point
the
how
where
Intersecting
●
What
●
How
is
th
an
the
5
a
tangent
circumference.
is
far
13
is
cm
this
tangent
from
If
a
the
point
center
from
touches
at
point
the
the
a
on
of
the
point
circle?
chords
intrsting
you
radius
the
tangent
circle,
F
of
on
apply
th
hords
thorm?
thorm?
ATL
Exploration
1
Draw
a
irl
dynami
Draw
1
of
radius
gomtry
four
points
5
using
a
GDC
A
or
softwar.
A,
B,
C
B
and
D
on
th
O
irumfrn
Draw
hav
2
Not
in
lin
a
Stat
and
hords
a
so
that
suital
CD
th
will
Find
th
valus
Vary
th
position
rul
5
Vrify
of
It
that
hang
you
hav
Th
rsult
irl
B,
hav
dout.
gnral
and
D.
radius
of
th
points,
modify
your
intrst.
points
and
A,
of
lngths
CX
×
B,
C
and
D
intrstion
AX,
BX,
suh
of
CX
AB
and
that
and
hords
CD
as
AB
X
DX.
DX
Dsri
anything
you
noti,
and
rul.
of
is
known
ass,
holds
tru
th
of
as
Dtrmin
of
mt
point
th
disovrd
numr
not
CD
th
th
BX
C
D
position
points.
discuss
you
thorms.
any
×
th
do
and
masur
should
on.
th
on
You
irl.
as
th
whih
you
intersecting
this
is
ontinu
whthr
it
Dsri
th
to
vary
ontinus
any
chords
rst.
th
to
positions
hold
limitations
to
whn
th
rul
osrvd.
that
you
AB
thorm
C
th
and
intrsting
a
this
Lal
AX
of
hav
has
th
A,
you
Reect
of
suital
you
points
to
CD
CD
ondition
softwar
theorem.
to
and
intrst.
4
Th
and
varying
hords
Us
a
AB
AB
By
3
suggst
irl.
similar
diagram.
diagram
th
sgmnts
diagram
that
this
of
In
hords
sn.
Do
A
hav
th
just
1
found
Exploration
thorm
y
think
that
provd
you
hav
in
proof
th
must
dynami
th
just
dmonstrating
mathmatial
you
is
don,
that
show
Extndd
it
th
gomtry
you
haptr
hav
works
in
all
rsult
to

approah
on
justid
th
tru
dos
th
situations
yond
this?
13.2 It strikes a chord
5 0 9
Theorem:
If
A,
irumfrn
hord
AX
×
CD
BX
at
=
a
B,
of
C
a
point
CX
×
and
irl
X
D
ar
suh
points
that
intrior
to
on
AB
th
C
th
mts
irl
A
th
thn
X
B
DX
D
Example
In
th
1
irl
shown
hr,
hords
AB
and
A
C
x
CD
mt
at
P.
Find
th
valu
of
3
x
B
P
4
12
D
Make
AP
×
BP
=
CP
×
sure
that
you
correctly
DP
identify
the
sides
involved.
⇒
Example
In
this
2
irl,
hords
PR
and
QS
mt
at
T.
Q
Find
th
valu
of
x
and
hn
nd
th
possil
x
valus
of
th
lngths
of
PT,
RT
and
1
QT
P
x
x
+
1
R
T
6
S
PT × RT = QT × ST
x
× (x + 1) = (x − 1) × 6
Substitute
in
the
known
values.
2
x
+ x = 6x
6
2
x
− 5x + 6 = 0
(x −
So
PT
2) (x − 3) = 0
x
=
=
2
2
or
or
51 0
x
3,
=
Factorize.
3.
RT
There
=
3
or
4
and
QT
=
13 Justication
1
or
2
are
two
possible
values
of
x
G E O M E T R Y
Practice
1
Find
th
A N D
T R I G O N O M E T R Y
1
valu
of
x
in
ah
diagram.
Hn
nd
th
missing
b
a
lngths.
c
5
x
4
9
x
+
1
4
10
3
8
2x
x
x
e
d
1
f
4
x
x
x
3
6
x
x
−
1
3
10
x
+
x
3
x
x
Chords
C
●
How
that
an
a
intersect
thorm
2
1
hav
outside
dirnt
the
circle
ass?
ATL
Exploration
Not
2
that
ontinus
1
Draw
a
irl
using
a
GDC
or
softwar.
dirtions,
four
points
A,
B,
C
and
D
on
th
Draw
lins
AB
Draw
point
X,
of
th
and
line
segment
B
whih
X
irumfrn
oth
unlik
A
a
Draw
line
in
dynami
C
gomtry
a
irl.
stops
at
its
ndpoints.
CD.
whr
AB
and
CD
and
intrst.
D
You
2
should
Mov
CD
is
th
mt
th
hav
points
diagram
so
xtrior
sond
a
as
that
to
lins
th
of
similar
AB
irl.
th
to
this.
and
A
This
B
intrsting
X
C
hords
Using
thorm.
th
softwar,
masur
th
lngths
D
AX,
BX,
CX
and
rlationship
AX
th
and
lins
Can

you
AB
DX.
×
possil
to
BX
CD
dsri
say
Vrify
=
CX
that
×
intrst
any
that
th
DX
onditions
AX
×
still
xtrior
BX
=
holds
to
undr
CX
×
whn
th
irl.
whih
it
would
not
DX ?
13.2 It strikes a chord
51 1
Th
intrsting
point.
th
You
hav
hords
now
thorm
sn
that
rfrrd
th
to
thorm
hords
is
that
idntial
intrst
if
point
P
at
is
an
intrior
xtrior
to
irl.
Example
Points
J,
K,
3
L
and
M
li
on
th
irumfrn
J
of
a
LM
irl.
Th
intrst
at
lins
P.
that
Find
pass
th
through
valu
of
JK
and
10
K
x
2
P
3
x
M
L
JP
×
KP
=
LP
×
MP
⇒
Substitute
in
the
known
values.
⇒
⇒
⇒
Practice
1
Us
th
Hn
2
intrsting
nd
th
hords
missing
thorm
lngths
a
in
to
th
nd
th
valu
of
x
in
ah
diagram.
diagrams.
b
c
x
x
+
1
1
2x
x
6x
x
9
1
2
1
x
x
D
When
●
Can
a
chord
asthtis

3
2
becomes
a
tangent
alulatd?
ATL
A
Exploration
3
You
may
nd
to
B
mov
1
Rall
th
diagram
from
Exploration
sur
stp
2
whr
th
lins
AB
and
CD
xtrnally
at
X
C
D
Continued
512
13 Justication
to
that
mak
Xrmains
X
on
mt
A
2
on
next
page
th
pag.
G E O M E T R Y
2
Us
th
softwar
onrm
that
AX
to
×
masur
BX
=
CX
th
×
lngths
of
AX,
BX,
CX,
and
DX
A N D
T R I G O N O M E T R Y
and
DX.
A
B
X
C
D
3
Mov
th
th
othr
and
hk
4
What
5
Rpat
as
6
do
stps
In
3
your
th
it
gts
rmain
aout
4
a
th
fw
losr
to
stationary.
intrsting
and
A.
and
th
a
3,
th
hords
two
mor
th
mak
your
what
point
onjtur
intrsting
Us
Tst
discuss
Exploration
Mak
●
that
osrvations,
approahd
●
so
noti
approahs
Reect
●
that
you
B
points
A.
You
thorm
lngths
tims,
should
R-masur
AX
still
and
allowing
four
sur
that
lngths
applis.
BX
B
mak
th
as
and
B
A
approahs
to
gt
as
A?
los
possil.
From
B
point
thr
for
softwar
to
onjtur
a
aout
th
limiting
as
as
2
happnd
A?
hord
a
onjtur.
What
to
th
othr
lin
lin
tangent-chord
AX
dos
theorem
as
this
that
th
point
B
rsml?
is
th
limiting
as
of
thorm.
onstrut
a
tangnt
to
a
irl
and
onrm
that
your
Just
onjtur
is
as
in
tru.
Explorations
and
2,
usd
The
If
tangent-chord
TX
PQX
is
a
tangnt
mts
th
to
theorem:
a
irl
irl
at
P
T
at
and
T
Q,
and
fully
2
PX
×
QX
=
and
tst
onjtur.
thn
to
a
To
justify
th
onjturs
TX
1
hav
softwar
form
X
you
that
Q
you
hav
proof
If
P
is
you
hav
writtn
using
mad,
similar
might
prov
sor
playr
running
down
th
sidlin
tak
a
shot
with
th
st
han
of

you
al
th
to
thorms
wants
in
to
alrady
proofs
triangls,
A
a
ndd.
this
stion
for
soring
G
yourslf.
a
goal.
It
turns
out
that
th
st
han
for
To
gt
a
O
startd,
sussful
shot
happns
whn
th
playr
irl
that
is
at
th
an
point
th
of
goal
tangny
posts
and
of
is
th
tangnt
to
th
passs
through
s
if
you
A
xplain
why
L
triangls
PTX
TQX
similar.
and
sidlin.
13.2 It strikes a chord
ar
513
Summary
●
Th
If
intrsting
A,
B,
C
and
hords
D
ar
●
thorm:
points
on
Th
If
th
tangnt-hord
TX
lis
tangnt
thorm:
to
a
irl
at
T
and
PQX
mts
2
irumfrn
th
hord
thn
AX
CD
×
of
at
BX
a
a
=
irl
point
CX
×
suh
X
that
intrior
AB
to
th
mts
th
irl
at
P
and
Q,
thn
PX
×
QX
=
TX
irl
DX
T
X
C
A
A
B
B
X
Q
X
C
D
D
P
Mixed
practice
1
has
A
irl
points
irumfrn.
A,
B,
Chords
C
AB
and
D
and
on
CD
its
3
hav
lngth
Find
x,
y
th
and
valus
of
z
5
x
11
and
irl
a
7
at
rsptivly,
point
Sketch
C,D
th
and
X
and
mt
intrior
to
th
A,
B,
X
irl
on
and
th
indiat
points
5
6
y
skth.
z
b
Givn
that
AX,
BX,
CX
and
DX
ar
intgrs,
4
nd
th
possil
valus
that
thy
an
In
th
A
tak.
diagram,
2
Find
th
valu
of
th
unknown(s)
in
ah
BC
diagram.
is
a
hord
x
3
D
and
a
mts
b
x
O
5
a
7
diamtr
8
AB
at
B.
AO
=
BO
y
3
40
x
+
=
3
5
8
and
BC
=
C
8.
B
Find
x
and
y
R
x
5
15
Find
th
valus
of
x
c
x
d
and
2
y
Q
x
5
+
1
x
4
y
y
14
8
x
5.5
P
2
y
4
T
2y
10
x
1
6
x
e
x
f
In
this
a
show
3x
=
diagram:
2x
3
that
4y
11
4
x
3x
16
b
hn
valus
nd
of
x
th
and
y
3x
2
4
3y
6
16
51 4
13 Justication
2
G E O M E T R Y
Objective:
ii.
select
D.
Applying
appropriate
mathmatis
mathematical
in
ral-lif
strategies
A N D
T R I G O N O M E T R Y
ontxts
when
solving
authentic
real-life
situations
For
each
of
determine
the
Review
A
version
in
Personal
1
questions
which
fragmnt
the
consider
the
intersecting
geometry
chords
of
the
theorem
real-life
you
of
a
cultural
owl,
dats
Arhaologists
owl
th
A
situation
to
aring
an
anint
from
th
7th
ntury
liv
th
original
2
bce.
owl
Hug
in
was
and
AB
midpoint
B
ar
is
hosn
masurd
of
AB,
and
on
to
MC
th

is
dg
5 m.
of
M
th
hallngs
is
spially
prpndiular
whr
C
lis
on
th
irumfrn
of
th
is
masurd
to

hav
East.
to
nd
out
hundrds
has
as
n
som
n
disovrd
Arhaologists
to
of
mor
of
aout
mtrs
thm
ar
ths
aross,
masur
in
hav
thm
Jordan
n
irls.
on
of
th
auratly,
inomplt.
to
points,
A
and
B
ar
markd
on
th
irl
and
owl.
th
MC
irls
Middl
attmpting
Two
AB
ston
th
Masuring
and
and
use.
expression
irular.
Points
need
context
and
insription,
below,
of
distan
twn
thm
is
masurd.
Th
mid-
0.25 m.
point
of
AB
is
point
to
th
narst
Survyors
distan
a
Use
found
from
this
sond
mak
and
a
that
th
of
of
From
straight
masur
a
distan
on
AB
th
was
to
from
irl
114
was
9
is
m
this
masurd.
and
th
m.
calculate
th
irl.
arhaologists
point
lin
th
th
midpoint
th
group
thniqu.
and
point
information
diamtr
A
found
outsid
whih
distan
is
to
a
us
th
a
irl
tangnt
th
dirnt
point
to
of
thy
th
irl
ontat.
C
Thn
A
thy
original
masur
point
to
th
th
shortst
distan
from
th
irl.
M
B
a
Sketch
th
sgmnts
omplt
AB
and
owl,
inluding
th
Th
b
lin
distans
Calculate
ar
th
150
and
diamtr
masurmnts
MC
m
and
52
m
from
thn
rsptivly.
ths
compare
th
tworsults.
b
Show that
th
c
ntr
Use
th
th
th
of
sgmnt
owl
intrsting
diamtr
Reect
and
How
you
hav
lin
th
of
th
MC
whn
hords
passs
through
xtndd.
thorm
to
nd
owl.
discuss
xplord
th
statmnt
of
inquiry?
Giv
spi
xampls.
Statement of Inquiry:
Logi
an
of
asthti.
th
justify
gnralizations
that
inras
our
appriation
13.2 It strikes a chord
51 5
14
Models
Depictions of real-life events using expressions, equations or graphs
Modelling
The
length
The
area
of
a
a
problem
rectangular
garden
is
twice
as
long
as
its
width.
2
Find
the
Drawing
a
of
the
garden
dimensions
a
diagram
of
can
is
72
the
help
m
.
garden.
you
write
an
equation
to
represent
situation.
2x
x
●
For
to
the
nd
x,
Translating
equation
adapt
●
a
rectangular
a
and
word
creates
model
Adapt
then
to
your
rectangular
a
garden,
nd
the
problem
model
solve
garden
eld
the
similar
model
and
length
into
of
whose
write
a
51 6
and
has
an
area
problem.
equation
width.
expression
Sometimes
you
or
can
problems.
to
nd
length
of
and
an
mathematical
is
the
7500
dimensions
three
2
width,
solve
m
times
as
of
a
long
as
its
Estimating
Using
a
model
reasonable
The
●
human
you
calculate
body
is
about
water.
Modelling
calculate
volume
●
help
estimates.
average
60%
can
How
an
of
to
body
estimate
water
could
model
more
your
you
make
in
as
a
for
your
improve
your
cylinder,
the
body.
your
estimate
accurate?
Should
you
take
an
umbrella?
account
of
the
atmosphere,
thermodynamics
and
the
uncertainty
formation
equations
in
a
of
chaotic
of
the
motion
and
system.
51 7
The
14.1
power
Global
context:
Objectives
●
Recognizing
of
exponentials
Orientation
Inquiry
exponential
functions
in
space
and
time
questions
●
What
●
How
are
exponential
functions?
F
●
Using
exponential
functions
to
model
real-life
problems
●
Identifying
and
using
translations,
do
you
transform
exponential
MROF
C
reections
functions?
and
dilations
with
exponential
functions
●
How
of
do
you
recognize
exponential
transformations
functions?
●
Do
exponential
●
Do
patterns
models
have
limitations?
D
ATL
lead
to
accurate
predictions?
Critical-thinking
Draw
reasonable
conclusions
and
generalizations
10.1
Statement of Inquiry:
Relationships
predict
model
duration,
patterns
frequency
of
and
change
that
can
help
clarify
and
variability.
14.1
51 8
A LG E B R A
Y
ou
●
should
evaluate
already
expressions
know
how
with
1
to:
Write
down
the
0
exponents
●
identify
a
and
reections
apply
and
translations,
dilations
2
value
of :
4
2
b
Here
is
1
2
the
c
graph
to
of
2
4
y
d
=
3
f(x).
y
functions
6
4
2
x
4
2
2
4
2
4
Sketch
a
y
c
e
Exponential
F
●
What
are
Exploration
1
A
wealthy
Option
of
1:
b
y
=
f(x
−f(x)
exponential
oers
one-time
d
y
=
f(
money
to
a
y
2f(x)
f
y
=
f(2x)
charity
donation
in
Option
day
four
2:
Now
a
nd
Copy
the
and
any
total
calculations,
amount
complete
Day
Amount
(cents)
Amount
as
the
donated
b
3
Write
Use
your
days
4
1
Graph
to
a
of
day
each
3,
day,
option
Option
of
on
and
for
gives
one
day
so
20
the
cent
2,
on,
days.
most
money.
2.
1
2
3
4
5
1
2
4
8
16
a
GDC
your
under
on
cents
2
2
function
20.
which
donation
two
cents
ways:
table:
2
power
predict
two
A
1,
doubling
2
2)
x)
functions?
on
doing
3
functions
$5000.
Without
+
of :
y =
=
f(x)
graph
1
donor
A
=
the
or
Find
a
for
the
amount
spreadsheet
the
function
total
from
in
to
US
step
on
day
work
dollars
2b
for
1
x
out
for
≤
x
the
all
≤
amounts
20
for
days.
20.
ATL
Explain
which
option
gives
the
most
money.
14.1 The power of exponentials
51 9
x
In
a
function
increase,
the
y
y
=
b
,
where
values
Exploration
b
is
a
increase
positive
very
integer
greater
than
1,
as
the
x
values
rapidly.
2
x
1.1
1
Graph
a
the
slider
the
function
for
slider
the
to
y
=
b
and
parameter
explore
b.
what
insert
Move
y
happens
6
x
to
the
than
graph
for
values
of
b
greater
y
=
b
5
1.
4
a
State
of
y
what
as
b
happens
to
the
value
3
increases.
2
b
Determine
remains
if
the
the
y-intercept
same
for
all
b
3.0
0.0
1
10.0
values
0
of
c
b.
Determine
an
Explain
of
e
if
the
x-intercept,
function
d
3
Explain.
x
get
will
Suggest
ever
what
less
is,
be
equal
an
to
x
1
1
2
3
4
has
whether
happens
and
why
function
that
2
to
the
the
0.
value
of
y
for
x
<
0,
as
the
values
less.
exponential
function
with
b
>
1
is
called
an
ATL
exponential
growth
function.
1.1
2
Now
redene
the
slider
so
that
0
<
b
<
1.
y
6
Answer
parts
a,
b
and
c
in
step
1
for
x
y
=
b
5
this
function.
4
a
Explain
what
value
y
happens
to
the
3
of
for
x
>
0,
as
the
values
b
of
x
get
larger
and
0.3
larger.
0.0
1.0
1
b
Suggest
why
function
an
with
0
exponential
<
b
<
1
is
called
3
exponential
Explain
what
decay
happens
x
0
3
an
2
1
1
2
3
4
function.
to
the
graph
if
b
<
0,
and
justify
your
answer.
x
An
exponential
The
function
independent
Exploration
1
By
choosing
a
is
variable
of
x
the
is
the
form
f (x)
=
a
×
b
,
where
a
≠
0,
b
>
0,
b
≠
1.
exponent.
3
xed
value
for
b
where
b
>
1,
graph
the
function
x
y
=
a
×
b
and
y-intercept
2
Repeat
3
Explain
step
1,
how
exponential
5 20
insert
a
slider
for
the
parameter
a.
Describe
how
the
changes.
but
the
this
value
time
of
function.
14 Models
a
choose
is
a
related
xed
to
value
the
of
b
such
y-intercept
for
that
any
0
<
b
<
1.
A LG E B R A
x
The
exponential
any
x-intercepts.
function
y
=
b
does
not
have
y
4
x
y
For
b
>
1,
as
x
gets
less
and
less,
b
2
function
approaches
0,
but
=
b
the
never
equals
3.0
0.
0.0
10.0
x
0
4
For
0
<
b
<
1,
as
x
gets
larger
and
2
2
4
larger,
y
the
function
approaches
0,
but
never
equals
0.
4
x
x
For
any
value
of
b
>
0
the
function
y
=
y
=
b
b
b
has
a
horizontal
asymptote
at
y
=
0.
0.3
2
0.0
1.0
x
0
4
A
horizontal
gets
larger
Reect
asymptote
and
is
the
line
that
the
graph
of
2
f (x)
2
approaches
4
as
x
larger.
and
discuss
1
x
●
Here
is
the
Deduce
graph
the
of
value
exponential
of
b
from
the
function
y
=
b
y
graph.
(1, 4)
Check
the
of
your
answer
exponential
b
you
chose.
by
graphing
function
Explain
with
how
the
you
value
found
b
(
●
Identify
the
value
of
b
from
the
1, 0.25)
second
(0, 1)
0
x
x
graph
of
y
=
b
.
Explain
how
you
found
b
y
●
Given
the
standard
form
of
the
exponential
x
function
the
is
value
y
of
=
a
a
is
×
b
1.
,
for
these
Explain
two
how
Tip
graphs
you
(
could
1, 4)
Look
identify
the
value
of
a
from
a
graph
y
=
a
×
b
you
where
a
≠
at
anygraphs
of
x
drew
in
(0, 1)
1.
Practice
1.
(1, 0.25)
x
0
Population
growth,
radioactive
substances,
can
all
be
modelled
bacterial
and
with
and
viral
nances
growth,
such
exponential
as
spread
interest
of
and
epidemics,
decay
credit
payments
card
of
functions.
14.1 The power of exponentials
5 21
Example
There
are
128
eliminated
a
Write
b
players
after
a
round
1
each
table
of
in
a
tennis
tournament.
Half
of
the
players
are
round.
values,
and
a
function
for
the
number
of
players
after
x
Determine
the
number
of
rounds
needed
to
declare
a
winner.
a
Round
number
Number
(x)
at
end
of
of
0
players
round
(y)
128
1
2
3
Look for a pattern
4
in the y values.
5
6
7
y
=
128
×
7
rounds
After 7 rounds only one
b
are
needed.
player, the winner, is left.
1
In
Example
1,
a
=
128
(the
initial
number
of
tennis
players)
and
b
=
variable
x
2
(the
the
factor
round
by
which
number,
the
and
number
y
of
players
represents
the
reduces).
number
of
The
players
left
after
represents
round
x
In
the
standard
form
of
the
exponential
function
●
the
parameter
a
represents
the
initial
●
the
parameter
b
represents
the
growth
●
x
is
the
independent
●
y
is
the
dependent
5 2 2
variable
variable
14 Models
(for
(for
y
=
a
×
b
:
amount
or
reduction
example:
example:
(decay)
time)
population)
factor
x
A LG E B R A
Practice
1
One
2
1
bacterium
a
Write
down
b
Use
c
Graph
your
the
ii
when
The
a
divides
function
function
your
i
cell
formula
number
there
half-life
to
of
of
for
produce
the
to
be
the
of
cells
cells
number
every
after
of
x
cells
minute.
minutes.
after
15
minutes.
determine:
after
1
carbon-11
two
number
calculate
cells
will
to
half
an
million
is
20
hour
cells
(after
how
many
minutes).
minutes.
The
a
A
sample
of
carbon-11
has
10 000
nuclei.
Make
a
table
of
values
for
a
20-minute
intervals,
and
write
down
a
function
for
the
number
of
half-life
radioactive
nuclei
material
after
x
number
of
20
minute
Find
the
number
of
nuclei
is
the
intervals.
time
b
of
ve
after
3
taken
number
hours.
of
for
the
nuclei
halve.
c
Determine
less
than
Problem
3
Drug
dose
of
trials
body
the
Determine
By
the
it
takes
for
original
show
reduces
medicine
how
long
Exploration
1
long
of
the
number
of
nuclei
to
reduce
to
number.
solving
testing
person’s
how
10%
choosing
a
that
by
the
one
when
after
a
amount
quarter
there
dose
is
of
of
a
every
less
400
hour.
than
mg
pain-relieving
it
It
is
200
mg
will
be
safe
in
to
the
safe
medicine
take
in
a
another
body.
to
take
another
dose.
4
xed
value
for
b
such
that
b
>
1,
graph
the
function
x
y
=
a
×
b
and
insert
a
horizontal
asymptote
asymptote
of
slider
for
changes,
the
parameter
and
write
an
a.
Describe
equation
for
how
the
the
horizontal
x
2
Do
the
that
0
same
<
b
<
any
as
1.
exponential
in
step
Write
1,
an
but
equation
this
equation
time
for
y
a
×
b
select
a
xed
the
=
when
horizontal
b
>
value
1.
of
b
such
asymptote
of
any
x
exponential
The
parameter
fraction.
So
if
an
To
b,
the
increase
initial
End
equation
of
=
growth
an
amount
day
y
1
×
b
,
factor,
initial
y
a
may
amount
increases
End
for
of
by
day
0
<
be
by
b
a
<
percentage
10%,
10%
each
2
1.
End
you
day
,
of
2
1.1y
To
So
decrease
if
an
an
initial
End
of
1.1
initial
amount
amount
day
1
y
y
by
of
5%,
by
day
0.95
day
1.1
2
0.95y
then
3
by
you
of
1
an
+
y
2
you
5%
End
End
of
day
1.1
by
1
then
day
x
3
0.05
you
=
y
0.95.
have:
End
of
day
3
0.95
or
1.1.
have:
y
day,
of
=
x
multiply
each
integer
0.1
3
decreases
End
instead
multiply
y
x
x
0.95
y
14.1 The power of exponentials
5 23
to
x
Exponential
decimal
growth
form
(the
is
modelled
percentage
by
y
=
a(1
expressed
+
as
r)
a
where
r
is
the
growth
rate
in
decimal).
x
Exponential
decimal
modelled
by
y
=
a(1
r)
where
r
is
the
decay
rate
in
2
population
population
has
a
Determine
b
Write
c
Use
a
in
a
the
to
the
factor
b
=
estimated
year
factor
model
to
year
was
each
growth
function
in
Growth
town
increased
function
your
people)
a
is
form.
Example
The
decay
b
by
of
future
estimate
to
be
38 000
the
population
population
the
in
approximately
2015.
Since
then
the
2.4%.
in
the
town.
growth.
population
(to
the
nearest
hundred
2030.
1
+
r
=
1
+
0.024
=
1.024
Change 2.4% to a decimal and add 1.
x
b
y
=
38 000
years,
c
x
=
×
1.024
y
=
and
2030
,
total
2015
=
where
x
=
number
population
after
x
of
x
years.
a
=
initial
population,
y
=
a(1
+
r)
15
15
y
y
=
38 000
×
Example
Sarah
54 234.4
Find
x
and
substitute
into
the
model.
3
bought
a
approximately
a
Find
b
Write
c
Use
the
a
=
1.024
54 200
≈
the
a
for
$15 000.
each
depreciation
function
your
to
function
nearest
Decay
car
15%
(decay)
model
to
Its
value
depreciates
(loses
value)
by
year.
factor.
the
future
estimate
the
value
value
of
of
the
car.
Sarah’s
car
after
three
years,
to
dollar.
factor
b
=
1
r
=
1
0.15
=
0.85
Change
15%
to
a
decimal
and
subtract
from
1.
x
b
y
=
15 000
and
y
=
×
0.85
value
of
,
where
the
car
x
=
after
number
x
of
years,
Dene
years.
3
c
y
y
=
≈
15 000
$9212
Practice
1
For
each
ii
the
a
y
×
in
0.85
=
three
9211.88
years.
2
exponential
growth/decay
function,
rate
r,
and
identify
iii
the
i
the
initial
growth/decay
value
factor
b,
a
x
x
=
2
x
1.43
b
y
=
3
×
0.62
y
c
=
4
×
(
x
d
)
y
=
0.8
×
3
2
A
jeweler
estimates
a
Determine
b
Write
£300
c
Use
a
in
the
growth
function
to
value
factor
model
of
the
of
gold
the
is
value
future
increasing
of
value
by
12%
per
function
to
estimate
14 Models
the
value
of
year.
gold.
of
a
gold
bracelet
valued
2015.
your
5 24
the
that
the
bracelet
in
2030.
at
1.2
the
variables
x
and
y
A LG E B R A
3
An
electrical
decreasing
a
Write
b
Use
sell
4
In
a
there
were
of
the
cell
year.
that
In
model
to
the
2010
future
estimate
3014
number
they
of
sold
DVD
the
initial
The
population
registered
phone
number
The
a
to
function
number
Estimate
5
per
DVD
470
players
DVD
sold
is
players.
sales.
number
of
DVD
players
the
store
will
2018.
2015,
The
calculates
22%
function
your
in
store
by
of
population
grows
users
cell
of
at
an
a
is
phone
ant
rate
cell
users
colony
of
phone
estimated
12%
in
in
a
increase
small
town.
by
42%
approximately
600.
per
year.
2025.
was
per
users
to
week.
Find:
i
the
ii
a
iii
the
growth
factor
of
the
population
of
ants
In
function
to
model
the
population
growth
of
the
ants
y
approximate
number
of
ants
in
the
colony
after
5b,
on
15
weeks
and
the
=
b
A
a
b
Graph
your
population
Write
after
3
ii
after
10
the
your
to
9
An
less
the
it
in
and
graph
decreases
use
to
it
determine
to
reduce
invest
$5000
annually.
to
estimate
by
9%
calculate
how
to
in
a
long
less
it
than
long-term
Determine
estimate
decreasing
take
a
how
there
rate
population
destroys
for
million
that
at
10%
the
of
to
how
many
weeks
the
ant
every
the
year.
number
of
insects
left
would
half
its
take
for
present
size.
money
many
fund
years
it
oering
will
take
4%
for
interest
this
be
antibiotic
How
do
you
transform
How
do
you
recognize
A.
are
80
3.5%
wolves
per
year.
in
a
forest
Estimate
and
how
that
the
long
it
will
halved.
to
present
reduce
in
50
one
hour.
million
Determine
bacteria
how
down
to
bacteria.
●
Knowing
appropriate
of
bacteria
●
unfamiliar
to
size.
Transformations
Objective:
select
1
the
insects
formula
present
would
than
C
i.
is
antibiotic
long
x
double.
population
for
read
of
solving
Conservationists
take
and
value
years.
parents
amount
10 000
a
Use
double
population
compounded
8
of
to
formula
insect
Sahil’s
the
years
Problem
7
function.
takes
down
i
Use
1200,
point
where
weeks.
population
6
the
graph
after
o
30
nd
and
of
exponential
exponential
functions
functions?
transformations
of
exponential
functions?
understanding
mathematics
when
solving
problems
in
both
familiar
and
situations
In Exploration 5, use the rules f or transf ormations of
graphs to justif y your conjectures.
14.1 The power of exponentials
5 25
Exploration
5
x
1
Graph
the
y
=
3
and
symmetr y
on
between
the
the
two
same
set
g raphs.
of
axes.
Make
Describe
and
justify
a
and
justify
conjecture
x
about
the
symmetr y
of
all
x
2
Graph
y
=
of
g raphs
of
the
for m
y
=
b
and
x
3
symmetr y
pairs
and
y
=
between
−3
the
on
two
the
same
graphs.
axes.
Make
Describe
and
and
justify
a
justify
the
conjecture
x
the
3
symmetr y
Graph
such
the
the
that
all
pairs
functions
a
value
of
>
of
0.
a
of
graphs
below
Describe
varies,
and
what
and
state
of
the
insert
a
type
y
slider
happens
the
form
to
b
for
the
of
=
about
x
and
the
shape
y
=
−b
parameter
of
the
transformation
a,
graph
as
this
represents.
x
a
y
=
a
×
b
x
,
for
b
=
2
b
y
=
a
×
b
,
for
b
=
0.5
x
4
Graph
By
the
function
changing
the
y
=
value
x
function
y
=
2
+
2
of
k,
k,
and
state
y
=
2
function
y
+
=
k.
Describe
Graph
By
the
function
changing
the
y
=
value
x
function
y
=
2
for
the
y
,
of
=
2
function
y
ab
+
Explain
how
takes
k.
the
the
transformation
to
from
any
k.
h,
and
state
insert
the
a
slider
for
the
transformation
parameter
that
takes
h.
the
h)
.
=
Describe
ab
the
(x
to
y
=
transformation
transform
the
from
any
h)
Tip
ab
x
6
parameter
that
h)
2
(x
to
to
x
exponential
slider
x
ab
(x
5
a
transformation
x
to
x
exponential
insert
the
function
y
=
(x
ab
to
y
=
−
h)
ab
+
You
k
may
better
the
is
Transformations of exponential functions
y
by
Reection:
For
the
exponential
function
f (x)
=
a
×
b
looking
:
the
graph
of
y
=
−f (x)
is
a
reection
of
the
graph
of
y
=
f (x)
in
the
x-axis.
●
the
graph
of
y
=
f (
is
a
reection
of
the
graph
of
y
=
f (x)
in
the
y-axis.
the
right.
x
Translation:
●
●
●
y
=
f
(x
For
h)
the
exponential
translates
y
=
f
function
(x)
by
h
f (x)
units
=
in
a
the
When
h
>
0,
the
graph
moves
in
the
positive
When
h
<
0,
the
graph
moves
in
the
negative
(x)
+
k
y
=
f
translates
y
=
f
(x)
by
k
units
in
k
>
0,
the
graph
moves
in
the
positive
When
k
<
0,
the
graph
moves
in
the
negative
=
k
units
f
(x
h)
in
+
the
k
translates
y
=
f
(x)
by
h
b
:
x-direction.
x-direction,
the
left.
up.
y-direction,
in
to
y-direction.
y-direction,
units
to
x-direction,
the
When
y
×
the
down.
x-direction
and
y-direction.
Continued
5 2 6
14 Models
−
on
next
page
y
what
2
=
at
h)
2
(x
●
x)
=
of
a
of
transformation
(x
x
get
idea
2
−
h)
instead
A LG E B R A
x
Dilation:
●
y
=
af
●
y
=
f
For
(x)
the
is
(ax)
a
is
a
exponential
vertical
function
dilation
horizontal
of
dilation
f
f
(x)
(x),
of
f
=
a
scale
(x),
×
b
:
factor
scale
a,
parallel
factor
,
parallel
af (x)
a
>
y
af(x)
0
<
a
<
dilation
f(x)
horizontal
f(x)
dilation
af(x)
x
0
Example
x-axis.
x
y
vertical
the
0
x
y
1
to
y-axis.
f(ax)
f(x)
0
vertical
the
f (ax)
y
1
to
f(x)
f(ax)
0
x
horizontal
dilation
dilation
4
x
Start
with
the
a
In
words,
b
Write
c
Without
the
a
i
the
describe
(x)
the
using
a
factor
0.5,
functions
2,
as
program
on
the
0.5f (x)
parallel
=
factor
2
transformation
graphing
transformation
scale
=
function
transformation
scale
The
f
transformed
transformed
The
ii
function
to
is
a
is
parallel
or
=
(x),
0.5f
GDC,
set
vertical
of
and
(x),
i
c
i
g (x)
=
0.5
Graph
of
×
(0.5x).
and
sketch
dilation
horizontal
the
ii
both
h (x)
the
=
f
(0.5x)
function
and
of
the
graph
of
f (x),
dilation
of
the
graph
of
f (x),
x-axis.
0.5x
2
f (x)
f
axes.
x
b
ii
y-axis.
a
to
0.5f
g (x)
same
the
f (0.5x)
i
i
→
Graph
of
g(x)
ii
h (x)
=
2
ii
Graph
of
(0,
1)
→
(0,
0.5)
(0,
1)
→
(1,
2)
→
(1,
1)
(1,
2)
→
(2,
4)
→
(2,
2)
(2,
4)
→
f (x)
→
(0,
1)
(4,
4)
Graph
of
h(x)
y
y
x
f(x)
=
2
5
5
x
f(x)
=
0.5x
2
h(x)
=
2
4
4
3
3
2
2
1
x
0.5
g(x)
=
0.5
×
2
0
0
3
2
1
1
2
3
4
x
3
2
1
1
2
3
4
x
14.1 The power of exponentials
5 27
Example
Given
the
5
functions
a
Describe
b
Write
c
Sketch
a
i
g
i
is
the
g,
a
h
in
same
horizontal
ii
h
b
i
g (x)
c
i
Graph
(0,
ii
the
negative
is
transformations
and
on
:
a
x
1)
f (x
of
set
of
of
g,
and
ii
h
of
1,
axes
translation
i
f
of
and
g,
f
4
by
and
ii
units
+
dilation
of
f,
scale
factor
f
and
in
4)
f (x)
→
Graph
of
g(x)
0.5,
f.
h
the
parallel
the
h (x)
ii
Graph
of
(
2,
4)
→
(
2,
2)
(
1,
2)
→
(
1,
1)
(0,
1)
→
(0,
→
=
to
ii
→
2)
function
f
→
(
the
direction.
vertical
=
terms
i
0.5f (x)
f (x)
→
=
(
2
x+4
x
=
(
2
)
5
4
2
3
3
1
(
h(x)
1
f(x)
4
=
of
y
y
)
5
g(x)
Graph
0.5)
x
1
f(x)
y-axis.
)
2
2
x
1
h(x)
=
0.5
1
4
3
Practice
2
1
2
)
1
x
0
0
5
(
1
2
3
5
x
4
4
3
2
1
1
2
3
4
3
x
1
You
are
given
i
Describe
ii
Write
iii
Sketch
a
f
(x
e
3
+
f
2f
5 2 8
(x)
(x)
=
function
f
and
f
f
(x)
(2x)
solving
j
3
below
function
b
1)
f
transformation
the
2)
(x
function
transformed
both
Problem
i
each
the
+
the
f
(
14 Models
2x)
as
in
g(x)
words.
=
___.
transformed
c
g
f
function
(x)
2f
−
(x)
2
on
the
same
d
f
(
x)
h
f
(0.5x)
set
of
axes.
A LG E B R A
2
For
each
pair
i
Describe
ii
Write
iii
Sketch
g
of
the
in
functions
and
g:
transformation(s)
terms
both
f
of
of
function
f
to
function
g
f
functions
on
the
same
set
of
axes.
x
x
a
f
(x )
= 1.5
1
x
,
g (x )
f
b
= −1.5
(x )
=
x
(
,
)
g (x )
=
2
2
x
c
f
(x )
= 3
e
f
(x )
=
x +2
,
g (x )
x
= 3
− 3
d
f
(x )
2 x
= 0.5
,
g (x )
= 0.5
g (x )
= 3 ×
x
x
2
g (x )
=
x
1
0.5 x
,
2
f
f
(x )
=
(
1
,
)
(
)
3
Problem
solving
x
2 x
1
g
f
(x )
=
(
1
,
)
g (x )
=
(
)
2
3
On
the
same
2
set
of
transformations.
axes,
Label
x
f
(x )
=
sketch
each
the
graph
graph
and
2 x
2
;
3
f
(x )
=
of
any
the
1
0.5 x
2
;
f
1
(x )
=
function
intercepts
−2 × 2
;
f
2
(x )
f
and
with
its
the
three
axes.
2x
=
× 2
3
2
Modelling
D
with
●
Do
exponential
●
Do
patterns
Exploration
exponential
models
lead
to
have
accurate
functions
limitations?
predictions?
6
Bank
In
this
rate
exploration
of
100%,
you
when
will
the
investigate
interest
is
how
much
compounded
$1
at
can
grow
dierent
at
an
interest
accounts
usually
intervals.
pay
compound
per
1
Find
the
amount
$1
is
worth
at
the
end
of
one
year
at
100%
interest.
2
Find
the
amount
$1
is
worth
at
the
end
of
one
year
at
100%
interest
means
is
interest
annum,
the
which
interest
calculated
and
x
compounded
half-yearly.
investment
$1,
is
the
In
the
growth
function
rate
is
=
y
=
a(1
50%
=
+
r)
0.5,
,
the
and
x
initial
=
2,
as
added
interest
2
is
3
now
Find
compounded
the
amount
two
$1
is
times
worth
per
at
year.
the
Therefore,
end
of
one
y
year
=
(1
at
+
0.5)
100%
once
a
to
the
account
year.
2
or
y
=
1.5
interest
4
compounded
quarterly
(every
3
months).
In
this
case,
y
=
(1
+
0.25)
or
4
y
4
=
Write
of
5
$1
Use
the
6
1.25
down
at
100%
your
end
an
exponential
interest
function
of
one
to
year
compounded
nd
at
function
the
100%
interest
monthly
b
weekly
d
hourly
e
each
$1
your
function.
investment
number
of
can
times
at
in
100%
times
one
if
your
it
is
per
if
interest.
if
it
is
is
Justify
of
$1
is
worth
at
compounded:
f
there
investment
year.
c
year
the
investment
minute
Determine
grow
x
amount
a
Graph
representing
a
limit
daily
each
to
the
compounded
your
second.
amount
an
the
innite
answer.
14.1 The power of exponentials
5 2 9
The
two
limit
to
decimal
just
π,
like
Leonhard
Y
ou
and
have
Is
and
What
that
model
model
Sketch
a
the
number
non-repeating
model
the
who
for
in
e,
one
which
number.
rst
growth
appropriate
for
grow
all
It
year
is
is
of
will
graph
a
discovered
of
of
population
that
are
bacteria
limit
period
●
What
of
growth
this
more
environmental
Sketch
a
humans
Is
your
Why,
●
over
a
human
why
Research
the
What
graph
or
after
it.
of
people,
population
bacteria,
growth?
growth
necessary
population?
growth?
accurately
The
of
growth
the
bacteria
over
rate
population
accelerates
reects
a
T ime
long
time.
population?
●
of
number
‘e’
0
population
to
2
factors
a
$2.72,
special
populations
phases
is
called
one.
growth
factors
could
mathematician
environmental
sustain
$1
actually
discuss
this
the
noitalupoP
to
a
is
innite
problems
like
What
It
the
such
exponential
look
an
that
ezis
●
is
Euler,
Reect
can
amount
places.
and
seen
insects.
An
the
the
growth
that
long
more
are
necessary
factors
accurately
period
growth
of
graph
to
sustain
limit
reects
growth
human
of
a
human
population
population
growth
growth?
of
time.
dierent
from
your
bacteria
more
accurately
growth
graph?
not?
types
of
factors
environmental
of
human
functions
used
populations
to
over
an
extended
model
period
of
and
predict
time.
Summary
x
An
exponential
function
is
of
the
form
The
exponential
function
y
=
b
does
not
x
f (x)
=
The
a
×
b
,
where
independent
a
≠
0,
b
variable
>
x
0,
is
b
≠
the
1.
have
exponent.
For
any
b
>
x-intercepts.
1,
as
approaches
In
the
standard
form
of
the
exponential
x
0,
gets
but
less
and
never
less,
equals
the
function
0.
function
x
y
=
a
×
b
:
For
0
<
b
function
●
the
parameter
a
represents
the
initial
●
the
parameter
b
represents
the
growth
<
1,
as
x
gets
approaches
larger
0,
but
and
never
larger,
equals
the
0.
amount
x
For
any
value
horizontal
reduction
●
●
x
is
the
(decay)
is
the
time
independent
dependent
variable
variable
x)
53 0
b
>
0
the
function
y
=
b
asymptote
at
y
=
has
a
0.
factor.
(e.g.
time)
A
of
y
of
or
14 Models
(e.g.
amount
at
horizontal
f (x)
asymptote
approaches
as
x
is
the
gets
line
larger
that
and
the
graph
larger.
A LG E B R A
x
Exponential
where
r
is
growth
the
percentage
is
growth
expressed
modeled
by
y
rate
in
as
decimal).
a
=
decimal
a(1
+
form
When
r)
h
<
0,
x-direction,
(the
the
to
graph
the
moves
in
the
negative
left.
Exponential
●
y
=
f
(x)
+
k
translates
y
=
f
(x)
by
moving
it
k
x
decay
is
modeled
by
y
=
a(1
r)
where
r
is
the
units
decay
rate
in
decimal
in
of
y-direction.
form.
When
Transformations
the
exponential
k
>
0,
the
graph
moves
in
the
positive
functions
y-direction,
up.
x
Reection:
For
the
exponential
function
f (x)
=
a
×
b
:
When
The
y
=
graph
f (x)
in
of
y
the
=
−f (x)
is
a
reection
of
the
graph
of
y
=
graph
f (x)
in
of
y
the
=
f (
x)
is
a
reection
of
the
graph
y
=
f
the
of
=
a
×
b
y
=
f
moves
in
the
negative
(x
h)
+
k
translates
and
k
y
units
=
f
in
(x)
the
by
h
units
For
the
exponential
For
the
exponential
function
f
(x)
=
a
translates
y
=
f
(x)
by
moving
it
=
af (x)
is
a
vertical
dilation
of
f (x),
scale
factor
to
the
y-axis.
h
y
the
=
f (ax)
is
a
horizontal
dilation
of
f (x),
scale
x-direction.
1
factor
,
parallel
to
the
x-axis.
a
When
h
>
0,
x-direction,
the
to
graph
the
moves
in
the
positive
right.
af (x)
a
>
f (ax)
y
1
y
af(x)
f(x)
0
vertical
0
<
a
<
1
×
b
function
y
h)
in
in
y-direction.
:
(x
units
graph
x
parallel
●
the
down.
y-axis.
x
f (x)
0,
x-direction
Dilation:
Translation:
<
x-axis.
●
The
k
y-direction,
y
x
dilation
f(x)
vertical
y
x
0
dilation
f(x)
0
horizontal
af(x)
f(ax)
x
dilation
f(x)
f(ax)
0
horizontal
x
dilation
14.1 The power of exponentials
5 31
a,
:
Mixed
1
Xixi
It
practice
bought
a
depreciates
a
Write
for
b
down
this
Write
car
ve
years
ago
approximately
the
for
15%
depreciation
6
$18 000.
every
year.
(decay)
factor
Gina
invests
1.4%
interest
a
State
the
b
Write
down
down
a
function
of
the
car
to
model
over
a
savings
growth
factor
account
that
pays
annually.
on
this
investment.
of
the
the
value
of
her
car
her
a
function
to
model
the
growth
savings.
time.
c
Find
in
compounded
problem.
depreciation
c
$400
now,
to
the
nearest
Determine
how
her
to
savings
many
years
it
will
take
for
double.
$100.
7
Write
these
transformations
in
terms
of
the
x
function
Problem
2
A
strain
every
5
of
the
3
end
The
bacteria
one
isotope
At
doubling
midday
the
radium-226
nuclei.
in
there
number
a
left
are
The
after
1600
has
years.
Determine
150
pesticide
of
is:
of
reected
b
translated
bacteria
at
c
reected
a
A
half-life
in
its
and
the
y-axis
d
dilated
by
scale
factor
3
parallel
to
the
x-axis
e
dilated
by
scale
factor
4
parallel
to
the
y-axis.
3
units
to
the
left
and
4
units
in
the
x-axis
sample
how
many
has
nuclei
will
solving
years.
DDT
is
banned
long-lasting
animals.
in
most
On
the
same
toxic
The
eects
half-life
of
set
of
axes,
sketch
contains
of
years.
A
sample
graph
f
and
DDT
graph
its
and
three
any
transformations.
intercepts
area
of
with
the
DDT.
Determine
(x)
=
DDT
in
the
soil
after
70
Label
the
axes.
2x
4
f
(x)
=
4
1
2 x
0.5x
amount
f
(x)
=
−2
×
4
f
(x )
=
× 4
3
2
of
the
is
soil
1
g
of
on
x
15
the
countries
f
approximately
100
down
of
each
humans
f
a
function
because
when
20
8
4
2
laboratory
Problem
be
=
hour.
approximately
10 000
is
Determine
of
(x)
solving
minutes.
bacteria.
f
2
years.
x
9
5
The
population
of
a
town
in
2010
was
a
Rewrite
was
at
an
estimated
annual
population
rate
size
that
of
in
its
population
2.68%.
the
year
will
Determine
2025,
to
function
f
(x)
=
9
with
a
base
how
you
of
38 720.
3
It
the
the
instead
of
9.
Then
describe
can
grow
graph
this
function
in
form
the
using
the
base
of
3,
ax
the
y
=
3
nearest
x
b
100
Rewrite
the
function
f
(x)
=
9
with
a
base
of
81.
people.
Then
describe
how
you
can
graph
this
ax
function
532
14 Models
using
81
as
a
base,
in
the
form
y
=
81
A LG E B R A
Objective:
iii.
In
apply
this
fossils
D.
the
Applying
selected
Review
using
in
Review
in
mathematical
context
carbon
mathematics
you
will
in
real-life
strategies
apply
contexts
successfully
exponential
functions
to
to
reach
a
determine
solution
the
age
of
dating.
context
Write
Orientation
in
space
and
an
exponential
function
relating
time
,
N
x,
and
N
0
Ötzi
was
German
an
found
on
tourists,
elevation
of
3210
Austrian–Italian
clothing
Simons
and
19
were
the
and
meters
border.
tools
thought
September
Helmut
1991
Erika
(10 530
Because
so
man
well
had
by
two
Simon
ft)
the
on
at
the
University
of
you
use
your
solve
function
the
from
following
the
Problem
solving
Innsbruck
When
it
dies,
an
organism
contains
carbon
dating
to
estimate
died
about
5300
years
ago,
nuclei
14
C.
Calculate
the
number
of
C
nuclei
in
the
that
organism
Ötzi
30 000
in
14
used
to
recently.
of
Austria
1
problems.
body,
2
Scientists
can
help
the
preserved,
died
You
at
making
after
11 400
years.
him
14
the
oldest,
best
preserved
mummy
in
the
3
world.
The
amount
0.25
times
Calculate
Carbon
dating
of
fossils
is
based
upon
the
decay
of
the
the
C
in
a
amount
fossil
when
approximate
is
calculated
the
age
to
organism
of
the
be
died.
fossil.
of
14
C,
a
radioactive
isotope
of
carbon
with
a
relatively
4
A
fossil
bone
is
approximately
16 500
years
old.
14
long
half-life
of
about
5700
years.
Estimate
the
fraction
of
C
still
in
the
All
living
organisms
get
fossil.
14
14
C
from
the
5
atmosphere.
Only
6%
of
the
original
amount
of
C
remains
14
When
an
which
begins
organism
to
dies,
decay
it
stops
absorbing
exponentially.
in
C,
Carbon
a
ago
dating
fossil
it
bone.
Estimate
how
many
years
died.
14
compares
the
amount
of
C
in
fossil
remains
with
14
6
the
amount
in
the
atmosphere,
to
work
out
Calculate
in
much
has
decayed,
and
therefore
how
long
ago
died.
1
=
approximate
percentage
a
fossil
bone
sample
after
35 000
Analysis
on
an
archeological
animal
site
bone
reveals
fossil
that
14
Let
N
the
initial
amount
of
C
at
the
of
C
left
years.
the
7
organism
the
how
the
at
an
bone
has
lost
14
time
between
90%
and
95%
of
its
C.
Determine
an
0
of
death.
approximate
inter val
for
the
possible
ages
of
the
bone.
Let
x
=
the
half-life
is
number
5700
of
half-lives,
where
each
8
years.
Using
than
Let
N
=
the
amount
present
after
x
number
your
50 000
function,
years
explain
may
have
why
an
fossils
older
undetectable
of
14
amount
of
C.
half-lives.
Reect
and
How
you
have
discuss
explored
the
statement
of
inquiry?
Give
specic
examples.
Statement of Inquiry:
Relationships
predict
model
duration,
patterns
frequency
of
and
change
that
can
help
clarify
and
variability.
14.1 The power of exponentials
53 3
Like
gentle
ocean
waves
14.2
Global
context:
Objectives
●
Graphing
sine
Scientic
Inquiry
and
cosine
functions
and
technical
innovation
questions
●
What
is
●
What
are
a
periodic
function?
F
●
●
Understanding
periodicity
graphs
Transforming
translations,
sine
and
reections
cosine
and
functions
SPIHSNOITA LER
Recognizing
cosine
transformations
graphs,
and
nding
Modelling
cosine
real-life
=
main
characteristics
a sin bx
and
y
=
of
the
a cos bx?
using
of
sine
equations
of
problems
using
sine
How
do
you
translate,
dilate
and
reect
C
and
periodic
functions?
graphs
●
●
the
y
dilations
●
●
of
and
How
does
function
functions
performed
●
Does
are
the
the
on
What
●
Have
real-life
the
of
a
sinusoidal
transformations
it?
order
performed
●
equation
represent
in
which
transformations
matter?
phenomena
are
periodic?
D
scientic
provided
ATL
more
models
and
answers
or
methods
questions?
Critical-thinking
Draw
reasonable
conclusions
and
generalizations
7.4
Statement of Inquiry:
Generalizing
better
models
relationships
and
between
measurements
can
lead
to
methods.
14.2
534
A LG E B R A
Y
ou
●
should
use
the
sine
already
trigonometric
and
know
ratios
how
for
1
to:
Here
is
a
right-angled
triangle.
cosine
13
5
x
12
●
nd
values
of
sine
and
cosine
2
Write
down
a
x
sin
Use
a
●
transform
functions
translations,
using
reections
3
a
calculator
sin
Here
b
45°
is
cos
to
b
the
nd
cos
graph
x
of
the
value
of
37
f(x).
and
y
dilations
5
4
3
2
1
0
5
4
3
2
x
1
1
2
3
On graph paper, sketch f(x) and
a
f(x)
2d
f(x)
1
b
3f(x)e
f(
x)
2
c
f(x)f
Make
f(2x)
sure
your
GDC
is
in
degree
mode.
The
F
●
What
is
●
What
are
y
You
sine
have
triangles.
=
used
You
a
a
Draw
the
can
a
cosine
periodic
the
curves
function?
main
characteristics
of
the
graphs
of
y
=
a
sin bx
and
cos bx?
trigonometric
use
Exploration
1
a
and
ratios
GDC
to
sine,
nd
cosine
these
and
ratios
tangent
for
in
dierent
right-angled
angles.
1
table
θ
your
0°
of
values
10°
for
angle
20°
θ
=
0°,
30°
10°,
40°
20°,
50°
…,
360°,
60°
and
…
sin θ
360°
sin θ
b
Find
sin0°,
sin10°,
sin20°,
etc.
using
your
GDC.
Round
the
values
to
Tip
2decimal
c
On
graph
x-axis
and
places
paper,
the
and
plot
sine
write
the
them
graph
values
on
of
the
in
your
sin
θ,
y-axis.
table.
with
Join
θ
from
your
0°
to
points
360°
with
a
on
the
Make
smooth
curve.
GDC
sure
is
on
your
degree
mode.
Continued
on
next
page
14.2 Like gentle ocean waves
53 5
2
Follow
the
3
Check
4
For
your
the
sin θ
b
cos θ
Reect
●
similar
What
is
dierent?
360°
how
to
Describe
sine
the
gives
left
step
1
to
draw
≤
θ
≤
360°,
a
table
of
values
and
plot
the
the
think
gives
the
graphs
each
symmetry
to
of
draw
curves
cosine
write
cos θ
and
down
the
on
your
range
GDC.
of
1
❍
GDC
sin θ
graphing
720°
cosine
sine
by
about
you
the
your
and
the
in
discuss
is
Use
to
0°
What
❍
The
and
Predict
●
graphs
domain
a
●
instructions
ofcos θ
graph
curve
0°
each
the
graph.
sine
will
to
and
cosine?
continue
from:
360°
curve.
graphs
have
for
to
check
same
your
shape.
Translating
the
predictions.
Translating
cosine
the
graph
sine
90°
to
graph
the
90°
right
graph.
Sine
The
amplitude
is
the
height
from
the
mean
value
of
the
function
to
and
graphs
maximum
or
minimum
value.
The
graphs
of
y
=
sin x
and
y
=
cos x
are
also
sine
and
have
called
amplitude
1.
cosine
The
period
is
the
horizontal
length
of
one
complete
cycle.
The
graphs
waves.
of
Waves
y
=
sin x
cosine
its
and
y
=
cos x
have
period
have
amplitude
y
y
=
y
period.
sin x
1
1
amplitude
y
amplitude
=
=
180°
360°
360°
of
y
=
The
graph
of
y
=
360°
180°
1
period
graph
x
180°
1
The
1
0
x
180°
=
cos x
1
0
360°
sin x
=
360°
passes
cos x
has
a
period
through
(0,
maximum
0)
at
and
(0,
has
a
maximum
=
360°
at
(90°,
1).
1).
ATL
Exploration
1
Use
to
what
predict
your
2
you
the
GDC
Predict
GDC
the
to
to
2
know
shape
check
shape
check
of
your
about
of
the
your
the
transformations
graph
of
y
=
of
2 sin x.
functions
Graph
from
the
f (x)
function
af (x)
on
of
y
=
sin x.
Graph
the
function
on
your
prediction.
Continued
53 6
to
prediction.
graph
14 Models
an
360°
on
next
page
and
a
A LG E B R A
3
4
Find
the
a
y
=
2 sin x
b
y
=
sin x
c
y
=
a sin x
Predict
check
5
6
amplitude
Find
the
your
the
shapes
amplitude
y
=
2 cos x
b
y
=
cos x
c
y
=
a cos x
y
=
The
of
y
of
generalizations
a sin x
graph
and
of
the
=
graph
2 cos x
of:
and
y
=
cos x.
Graph
the
functions
to
predictions.
a
Make
of
y
y
=
=
the
graph
about
the
of:
eect
of
parameter
‘a’
on
the
graphs
of
a cos x
a sin x
and
the
graph
of
y
=
a cos x
both
have
amplitude
a
y
The
amplitude
distance
of
a
between
minimum
curve
the
is
half
maximum
the
and
4
values.
3
4
0
360°
x
180°
180°
360°
4
Amplitude
=
=
2
2
Example
Write
down
1
the
function
represented
by
this
graph:
y
3
2
1
0
x
90°
180°
270°
360°
1
2
3
This
is
a
sine
graph.
The
amplitude
⇒
=
y
is
Compare
to
the
sine
and
cosine
graphs.
This
passes
through
(0,
3.
3 sin x
14.2 Like gentle ocean waves
537
0).
Practice
1
Sketch
1
the
graph
of
these
functions.
Label
all
the
x-intercepts
on
your
sketchgraphs.
a
y
=
4 sin x
Problem
2
Write
a
b
y
=
2 cos x
solving
down
the
function
shown
in
each
graph.
y
b
y
5
3
2
1
0
0
x
x
180°
360°
1
2
5
3
c
y
d
y
2
0.5
1
0
0
x
x
90°
180°
270°
180°
360°
360°
1
2
0.5
3
Four
functions
function(s)
are
shown
at
the
a
intersect
the
y-axis
at
(0,
b
intersect
the
x-axis
at
(180,
c
intersect
the
x-axis
at
(90,
d
has
the
smallest
Exploration
1
Describe
2
On
y
=
the
the
for
geometry
Add
the
4
Write
When
graphed,
choose
the
on
axes,
0
0)
0)
and
and
(360,
(270,
0)
≤
x
the
graph
sketch
≤
the
360°.
of
y
=
f (x)
graphs
Check
of
with
when
the
your
it
is
transformed
functions
GDC
or
y
=
to
sin x
with
y
=
f (2x).
and
dynamic
software.
graph
down
of
the
y
=
2 sin 2x
amplitude
y
=
to
and
sin x
your
sketch.
period
y
=
of
each
sin 2x
graph,
y
=
in
a
table
like
this:
2sin 2x
Period
Predict
y
=
sin
GDC
how
x.
or
the
graph
Predict
geometry
the
of
y
=
period
sin x
of
changes
the
graph
when
of
y
=
it
is
sin
transformed
x.
Check
to
with
your
software.
Continued
53 8
=
2 cos x
y
=
0.5 sin x
y
=
4 cos x
y
=
2 sin x
0)
amplitude.
Graph
5
y
2)
3
eect
same
sin2x
3
right.
that
14 Models
on
next
page
A LG E B R A
6
Add
7
Repeat
8
Conjecture
y
Sine
=
column
steps
a sin bx
and
A
a
y
=
for
the
rule
for
nding
y
=
function
of
are
The
period
the
The
parameter
b
of
the
the
and
a
of
y
y
=
cos x,
period
of
a sin bx
frequency,
or
of
a
why
y
=
cos 2x
function
your
y
rule
and
=
y
=
sin bx,
makes
x
y
=
cos bx,
sense.
functions.
pattern
=
table.
explain
pattern
the
functions
is
your
periodic
repeats
repetition
to
graphs
a cos bx,
functions
complete
x
1–6
and
cosine
periodic
One
a
for
y-values
is
called
and
y
number
=
of
at
a
regular
intervals.
cycle
a cos bx
cycles
is
.
between
0°
and
360°.
y
2
cycles
frequency
=
2
4
3
2
The
graph
y
=
3 sin 2x
has
1
360°
=
period
180
2
0
x
The
frequency
is
2
and
180°
the
360°
1
amplitude
is
3.
2
3
y
1
4
period
Practice
1
State
the
graph
a
y
=
period
State
sin 3x
the
graph
=
180°
and
frequency
0°
≤
x
≤
of
the
360°.
graph
Then
of
each
check
function.
your
graphs
Sketch
with
a
the
GDC
or
software.
b
Problem
2
3 sin 2x
2
forthedomain
graphing
=
cycle
=
cos 4x
solving
period
for
y
the
and
frequency
domain
0°
≤
x
≤
of
the
720°.
graph
Then
of
each
check
function.
your
graphs
Sketch
with
a
the
GDC
or
graphingsoftware.
1
1
a
y
=
sin
x
b
y
=
cos
3
Write
the
x
3
2
equation
of
a
each
graph.
b
y
y
1
1
0
360°
0
180°
x
x
180°
180°
360°
c
360°
1
1
d
y
y
1
1
0
0
360°
180°
1
x
x
180°
360°
180°
360°
1
14.2 Like gentle ocean waves
53 9
Problem
4
Match
solving
each
equation
to
its
graph.
x
y
0.5 cos 3x
y
3 sin
y
3 sin 2x
y
0.5 cos 0.5x
2
a
y
b
y
1
3
2
0.5
1
x
0
180°
360°
0
180°
360°
x
360°
180°
1
0.5
2
1
3
c
y
d
y
3
1
2
0.5
1
x
0
360°
0
360°
180°
180°
360°
x
180°
180°
1
360°
2
0.5
3
1
Example
Sketch
the
2
graph
a
y
=
0.5 sin 3x,
b
y
=
3 cos 0.5x,
a
amplitude
=
of
each
for
0°
for
≤
0°
≤
function
x
x
≤
≤
on
the
domain
given.
360°
720°
0.5
State
frequency
period
=
=
and
3
=
Draw
120°
y
The
1
are
y
=
the
amplitude,
frequency
period.
axes
for
the
maximum
0.5
and
domain.
and
0.5
minimum
y
values
respectively.
0.5 sin 3x
0.5
Frequency
=
3,
so
there
are
3
complete
0
x
90°
180°
270°
sine
curves
between
0°
and
360°
360°
0.5
Period = 120°, so there is a complete
1
sine curve between 0° and 120°
Sine
Continued
5 4 0
14 Models
on
curve
next
passes
page
through
(0,
0).
A LG E B R A
b
amplitude
frequency
period
=
=
3
0.5
=
=
720°
y
3
Frequency
=
0.5,
so
there
is
half
a
2
y
=
3 cos 0.5x
complete
cosine
curve
between
1
0°and
360°.
0
x
180°
360°
540°
720°
Period
=
cosine
curve
720°
so
there
is
a
complete
1
between
0°
and
720°
2
Cosine
curve
has
a
maximum
at
3
x
Practice
1
Write
the
graphing
a
y
=
c
y
=
the
amplitude,
graph
2 sin 4x,
for
the
for
0°
≤
x
be
for
0°
y
=
720°
period
Check
of
your
each
function,
graphs
on
a
then
GDC
or
360°
b
y
=
4 cos 3x,
1080°
d
y
=
0.7
How
do
●
How
does
Does
≤
you
the
x
≤
translate,
the
from
function.
can
be
For
is
a
of
cos
of
a
performed
in
which
any
on
obtained
by
reect
,
0°
for
≤
x
≤
360°
0°
≤
x
≤
1440°
functions
periodic
function
functions?
represent
the
it?
transformations
function
the
and
sinusoidal
that
transformation
example,
periodic
dilate
equation
order
function
obtained
cos x
for
4
●
sine
at
x
,
sinusoidal
the
of
≤
Transformations
can
and
domain.
3
transformations
A
frequency
given
x
sin
3
●
and
software.
1
C
0°
3
down
sketch
=
of
are
performed
matter?
y
1
y
=
sin x
graph
translating
0
x
the
graph
Soy
=
of
cos x
y
is
=
a
sin x
by
90°
sinusoidal
to
the
90°
left.
180°
270°
360°
function.
y
=
cos x
1
14.2 Like gentle ocean waves
5 41
Exploration
Transforming
1
Use
to
what
4
f(x)
you
predict
to
f(x)
know
how
the
+
d
about
transformations
transformation
of
y
=
of
sin
x
graphs
to
y
=
from
sin
x
f (x)
+
d
to
f (x)
+
d
1.1
will
d
y
aect
the
graph.
0
5
5
1
2
Graph
the
function
y
=
sin
x
+
d
on
the
domain
0°
≤
of
d
x
≤
360°
for
dierent
0
x
values
of
d.
Investigate
how
changing
the
value
aects
the
graph.
1
Was
3
your
Verify
that
aected
4
prediction
Repeat
Use
the
steps
Transforming
5
the
by
what
=
sin x
f(x)
6
Verify
7
Repeat
and
that
steps
Transforming
8
Use
y
what
9
= sin (
Verify
10
that
Repeat
and
period
3
y
of
=
y
a sin
5
and
on
the
6
y
of
y
graph
8
and
=
f (x)
in
y
=
the
the
and
y
the
sin x
steps
of
of
y
=
function
cos x
transformations
to
y
=
−sin x
domain
=
will
0°
−a sin bx
graphs
of
≤
+
y
=
a sin bx
+
d
is
sin (x)
+
d
not
d
of
≤
have
y
=
graphs
aect
x
the
from
f (x)
thegraph.
360°
cos x
transformations
to
domain
9
Transformations
The
graph
about
=
a sin bx
the
the
=
to
check
same
and
y
to
your
amplitude
=
f (x)
Graph
to
the
predict
how
the
how
the
functions
prediction.
andperiod.
−cos x
f(−x)
=
For
on
for
y
Reection:
the
sin x
bx
know
of
of
f1(x)
d
about
=
−sin x
to
correct?
−f(x)
=
f(x)
of
for
know
you
x)
amplitude
to
transformation
y
1
to
transformation
y
step
value
1
you
from
and
for
y
is
a
=
x)
≤
360°
x
graphs
≤
bx)
of
and
functions
reection
x-axis.
sin (
a sin (
sine
sinusoidal
−f (x)
=
360°
the
of
y
of
y
of
will
to
have
=
graphs
aect
check
the
cos x
f (x)
the
=
sin
graph
x
your
y
f (x)
graph.
same
and
cosine
from
the
=
to
f (
x)
Graph
to
the
predict
functions
y
= sin x
and
prediction.
amplitude
cos(
andperiod.
x).
functions
and
f (x)
The
of
y
=
cos
graph
=
f (x)
x:
of
in
y
=
the
f (
x)
is
a
reection
of
the
graph
y-axis.
y
y
y
=
y
sin x
=
sin x
1
1
0
0
x
180°
360°
360°
x
180°
180°
360°
1
1
y
=
y
−sin x
=
sin (
x)
y
y
=
y
cos x
=
cos x
1
1
0
0
x
x
180°
360°
360°
180°
180°
360°
1
1
y
=
y
−cos x
=
cos (
x)
Continued
5 4 2
14 Models
on
next
page
A LG E B R A
Translation:
For
the
sinusoidal
functions
f (x)
=
sin
x
y
and
f (x)
=
cos
x,
y
=
f (x)
+
d
translates
y
=
f (x)
by
d
units
in
the
y-direction.
y
=
sin x
+
2.5
4
When
d
>
When
d
<
0,
the
graph
moves
in
the
positive
y-direction
(up).
y
Dilation:
0,
the
For
graph
the
moves
sinusoidal
in
the
negative
functions
f (x)
=
y-direction
a sin bx
and
=
sin x
(down).
f (x)
=
a
x
y
cos bx:
=
sin x
1.5
4
y
=
af (x)
is
a
vertical
dilation
y
=
f (bx)
is
a
horizontal
of
f (x),
dilation
of
scale
f (x),
factor
scale
a,
parallel
factor
,
to
parallel
the
to
y-axis.
the
a
is
x-axis.
the
bis
af(x)
a
>
the
frequency.
f(bx)
b
1
amplitude.
>
1
y
y
2
2
y
=
2 sin x
y
=
sin 2 x
1
1
0
0
x
180°
x
360°
180°
1
y
=
sin x
y
2
<
a
<
stretch
=
720°
sin x
horizontal
0
1
540°
2
ver tical
0
360°
1
<
b
<
compression
1
y
y
1
y
=
=
sin x
1
sin x
0
0
x
x
180°
1
=
ver tical
the
easy
to
help
you
graphs
of
recognize
nd
y
by
that
=
540°
720°
0.5 sin x
y
While
360°
1
y
sin
compression
x
looking
=
and
at
a
y
=
cos
specic
sin(0.5 x)
horizontal
x
share
point
many
on
the
similarities,
graph.
they
stretch
are
Exploration
5
will
point.
ATL
Exploration
1
Sketch
2
In
Reect
sine
and
state
You
the
can
values
3
the
Use
graphs
and
x
cosine
=
=
y
=
1
graphs.
similarities
your
y
of
Discuss
recognize
at
form
5
the
sin x
you
and
=
using
of
y
=
noted
proper
dierences
graphs
cos x
informally
Now,
and
y
mathematical
between
sin x
and
dierences
y
the
=
sine
cos x
and
from
between
the
terminology,
cosine
graphs.
their
0.
GDC
a
to
cos(bx)
graph
+
d.
several
cosine
Generalize
how
functions
to
nd
its
with
equations
y-intercept
of
from
the
the
equation.
4
Repeat
step
3
with
the
function
y
=
a
sin(bx)
+
d
14.2 Like gentle ocean waves
5 4 3
The
y
=
y-intercept
a sin bx
For
y
=
+
d,
of
the
a cos(bx)
y
sine
a
+
function
is
y-intercept
d,
the
the
is
y-value
the
when
average
y-intercept
is
the
x
value
=
of
y
y
=
5
Specically,
the
maximum
graph
5
0.
function:
value
cosine
for
of
y
graph
4
max
y-intercept
y-intercept
is
4
is
4
3
5
+
1
=
3
3
2
2
2
1
y
1
=
1
min
0
x
1
0
2
Practice
2
3
4
5
=
sin x
x
1
1
2
Describe
how
y
=
cos x
c
y
=
cos x
Then,
+
the
write
a
4
5
4
a
Identify
3
to
transform
the
graph
of
y
=
cos x
to:
7
b
y
=
−cos x
2
d
y
=
cos(
transformation
down
the
of
each
equation
of
graph
each
from
either
y
or
y
=
cos x.
graph.
y
b
y
x)
2
5
1
4
3
0
x
90°
270°
1
2
2
1
0
x
90°
90°
270°
450°
1
c
y
d
y
1
2
1
0
90°
x
90°
270°
450°
1
0
x
90°
270°
450°
2
3
2
4
5
6
3
Draw
on
the
your
graph
graph
a
y
=
sin x
c
y
=
cos x
e
y
=
cos x
5 4 4
–
+
of
the
each
function
amplitude,
2
the
for
0°
<
x
≤
y-intercept
360°.
and
Be
the
b
y
=
−cos x
11
d
y
=
−sin x
8
f
y
=
sin x
14 Models
+
sure
to
period.
1
clearly
indicate
A LG E B R A
4
The
graph
of
y
=
sin x
was
transformed
to
produce
the
following
graph:
y
a
Find
the
amplitude,
frequency
and
period
of
6
the
function.
5
b
Write
down
c
Dennis
the
equation
of
the
graph.
4
wants
to
transform
the
graph
so
3
that
its
minimum
value
is
y
=
0.
Describe
2
the
transformation
that
will
allow
him
to
1
accomplish
d
Write
this.
down
the
equation
for
Dennis’
0
graph.
x
90°
Problem
5
Jawad
and
90°
270°
450°
solving
Aaron
disagree
about
the
equation
of
the
following
graph:
y
2
1
x
90°
270°
1
2
Aaron
State
says
who
Example
By
the
is
the
the
the
is
y
=
−sin x
justify
your
while
Jawad
thinks
it
is
y
=
sin(
x).
answer.
the
y
transformations
sinusoidal
determine
and
3
describing
on
equation
correct,
2
graph,
function
1
for
graph.
0
180°
x
90°
90°
180°
270°
360°
1
2
3
4
This
is
a
cosine
curve,
form:
y
=
a
cos (bx)
+
d
Cosine
The
max
The
amplitude,
and
min
a,
is
half
the
distance
between
the
Find
frequency,
and
360°
=
so
amplitude
the
graph
amplitude
b,
is
the
number
of
cycles
the
is
is
3
but
the
translated
maximum
down
1
value
unit.
is
3 cos
x
and
x
=
frequency
0.
b
Hence
1
amplitude
unit
is
3,
the
downward,
graph
hence
d
has
=
−1.
2,
d
=
−1
into
=
a
at
between
Substitute
y
maximum
1
shifted
The
has
points:
Since
0°
the
curve
y
=
a
your
values
cos (bx)
+
for
a,
b
and
d.
1
14.2 Like gentle ocean waves
5 4 5
d
Example
Write
a
4
function
that
describes
this
y
For the sine function,
sinusoidal
graph.
2
a
1
0
reection
in
x-axis
gives
graph
as
in
y-axis.
a
the
the
same
reection
x
120°
120°
240°
360°
the
1
2
This
is
a
sine
curve,
reected
in
either
the
x-axis
or
y-axis.
This
x =
The
are
3
y-intercept
cycles
c
=
from
0°
to
360°,
so
b
=
y
=
−2 sin(3x)
or
Exploration
y
=
2 sin(
3x)
but
of
this
y
=
sin(x)
graph
Determine
which
b
Start
with
the
graph
of
y
=
sin
x
and
apply
the
two
transformations.
c
Start
with
the
graph
of
y
=
sin
x
and
apply
the
two
transformations
the
opposite
Do
you
Repeat
get
step
1
for
y
=
−3 sin 2x
b
y
=
3 sin(
c
y
=
−3 sin
Does
the
3
a
Repeat
b
Write
1
Describe
a
y
=
same
the
graph
graphs
each
y
=
you
step
carry
1
down
for
the
your
out
the
equation
ndings
is
these
graph
by
transformations
of
of
y
all
=
and
3 cos
the
stating
important,
x
to
y
=
3 sin
2x
in
x
+
graphs
when
when
it
1
aect
and
you
the
=
graph?
3 sin
x
+
1.
produce.
order
doesn’t
y
the
you
carry
out
matter.
5
how
to
transform
4 sin x
the
b
graph
y
=
of
y
sin
2 sin x
x
to:
9
c
y
=
0.5 sin x
+
4
1
x
sin
Describe
sin
of:
+
3
e
y
=
x
sin
how
f
y
=
sin (
2x)
+
1
3
2
2
=
time?
1
d
y
x
transformations
Practice
transform
2x)
order
Summarize
transformations
order.
the
a
two
to
transform
the
graph
of
y
cos
x
to:
1
a
y
=
3 cos x
b
y
=
2 cos x
+
1
c
y
=
x
cos
+
2
2
1
d
y
=
0.5 cos x
1
e
y
=
cos 3x
f
y
=
x
cos
2
5 4 6
14 Models
increases
decreases
6
a
d
4
curve
because
average
at
value.
0
ATL
2
sine
has
3.
Graph
1
a
it
amplitude,
There
⇒
is
0
3
from
over
0°
that
to
90°
range.
A LG E B R A
Problem
3
Each
For
graph
each
function
solving
is
a
transformation
graph,
that
identify
describes
a
the
the
of
either
y
=
sin x
transformations
on
or
y
the
=
cos x.
graph
and
write
the
graph.
y
b
y
3
3
2
2
1
1
x
x
180°
90°
90°
360°
180°
180°
180°
540°
2
2
3
3
c
360°
1
1
d
y
1
5
4
x
540°
180°
180°
360°
540°
1
3
2
2
3
1
4
x
180°
180°
360°
5
1
e
f
y
y
4
2
3
1.5
2
1
0.5
x
x
180°
90°
90°
180°
120°
120°
240°
360°
480°
270°
1
y
y
4
5
3
4
2
3
1
2
1
x
180°
90°
90°
180°
270°
360°
450°
1
x
180°
180°
360°
540°
1
Modelling
D
●
What
●
Have
or
In
the
In
and
sinusoidal
show
other
real-life
phenomena
scientic
models
situations
are
and
periodic?
methods
provided
more
answers
questions?
sinusoidal
degrees,
real-life
the
graphs
y-axis
curves
units,
you
have
shows
the
representing
such
as
time
worked
sine
or
real-life
on
the
with
so
cosine
far,
function
situations,
x-axis
and
the
the
height
x-axis
for
x
each
and
on
has
the
y
shown
angle.
axes
may
y-axis.
14.2 Like gentle ocean waves
5 4 7
The
graph
wrecking
shows
ball
the
from
distance
its
(in
original
meters)
position
of
as
Distance
a
(m)
it
6
swings
left
and
then
right.
4
From
●
the
The
graph:
amplitude
is
5,
so
the
maximum
distance
0
2
the
●
wrecking
The
negative
when
it
ball
reaches
y-values
swings
to
the
from
its
represent
left,
and
original
the
position
is
5 m.
distance
the
positive
the
right.
4
T ime
2
4
values
represent
6
the
distance
●
The
period
To
nd
the
when
is
6
it
swings
seconds
to
(the
time
needed
for
one
cycle).
Tip
equation
of
the
graph:
When
The
graph
is
a
sine
curve.
It
has
been
reected
in
the
x-
or
graphing
y-axis.
sinusoidal
Frequency
b
=
=
=
with
60°
the
a
=
function,
=
0
So,
y
=
5 sin (
60°x)
Exploration
A
Ferris
lowest
above
the
1
wheel
seat
the
against
your
time
2
Find
the
3
Use
4
Identify
5
Determine
y
=
the
on
the
arranged
the
at
ground.
the
ground
(sec)
to
of
as
in
a
table.
for
0
≤
t
period
write
≤
the
changes
for
the
that
the
seats.
diameter
the
an
Ferris
wheel
Use
a
to
through
Graph
height
h
(m)
m
and
your
the
height
drawing
of
motion.
rotates
passes
75
scale
its
ruler
is
wheel
accurate
describes
wheel
so
that
it
takes
determine
each
above
your
position.
the
ground
180.
and
down
that
and
the
the
transformations
order
of
Its
ride
Draw
function
bottom
20°.
you
continuously.
kind
positions
results
every
As
of
frequency
the
the
equation
graph
would
relationship
of
need
to
be
y
to
your
of
=
graph.
your
graph.
cosx
be
made
described
to
by
the
Ferris
transformations
6
you
lungs
can
Find
1
seats
sin x .
After
i
get
t
5 sin 60°x
changes
results
the
in
Practice
a
18
amplitude,
your
wheel
=
above
between
above
Record
of
m
y
7
has
you
seconds
height
2
or
Determine
Suppose
5
1
is
ground
wheel.
exercise,
be
the
s
b
Explain
c
Find
5 4 8
the
in
make
5
sure
c
degrees
6
period
Amplitude
curves
360°
360°
the
modelled
velocity
ii
of
2.5
what
you
period
of
velocity
by
air
the
iii
think
the
air
y
=
(in
2
liters
sin
per
second)
90°t
at:
4
s
negative
function,
14 Models
ow
equation
ow
s
of
velocities
and
what
represent.
it
represents.
into
your
to
type
the
degree
symbol
the
line.
y
=
on
(s)
A LG E B R A
2
d
Find
e
Draw
The
the
the
graph
with
the
amplitude
graph
here
of
the
shows
waves.
and
Find
maximum
function
how
a
a
in
buoy’s
sinusoidal
velocity.
the
rst
8
distance
function
seconds.
from
that
the
ocean
models
the
oor
changes
graph.
d
3
s retem
2.5
2
ni
ecnatsiD
1.5
1
0.5
t
0
1
2
3
4
T ime
Problem
3
Find
a
5
in
6
7
8
9
10
seconds
solving
sinusoidal
function
that
models
the
relationship
y
shown
in
each
graph.
y
2
5
1
4
0
3
x
90
1
2
2
1
3
0
4
1
x
10
2
3
y
y
8
1
7
0
x
20
40
60
80
100
120
1
6
2
5
3
4
4
3
5
2
6
1
0
x
10
4
The
height
upand
is
in
down
measured
the
meters
can
be
(from
its
modelled
position
by
the
at
20
rest)
function
30
of
a
h(t)
40
50
spring
=
60
as
70
it
bounces
2 sin (60°t)
+
1,
where
t
inseconds.
a
Find
amplitude
b
Sketch
c
Determine
if
position
rest.
the
at
graph
the
of
of
h(t).
the
function.
amplitude
is
the
same
as
the
maximum
height
from
its
14.2 Like gentle ocean waves
5 4 9
5
When
you
changes.
the
time
ride
The
for
sinusoidal
a
a
Ferris
wheel,
relationship
complete
graph
like
your
between
revolution
vertical
your
of
height
height
the
above
above
wheel
can
the
be
the
ground
ground
modeled
and
with
a
this:
h
8
7
6
s retem
5
ni
4
thgieH
3
2
1
0
t
4
8
12
T ime
6
a
State
the
height
b
State
the
maximum
c
Determine
how
ground
the
for
Determine
your
e
Determine
how
f
The
g
Find
The
Bal
a
height
in
number
of
h
long
is
hours
when
of
the
Draw
c
Determine
the
time
d
Determine
the
height
e
A
cross
graph
can
level.
tide
the
day,
was
next
a
Find
b
Draw
5 5 0
tide
3.3 m,
high
modelled
a
rise
the
for
on
to
the
wheel.
Ferris
wheel.
approximately
ground
one
this
by
5 m
above
the
its
8
seconds.
revolution
of
the
wheel.
radius.
graph.
above
the
after
complete
Determine
tide
12,
the
of
the
in
and
tide
with
is
a
function
a
reach
Ferris
the
mean
function
h(t)
sea
=
level
one
3 sin(30°t),
day
at
where
t
is
the
18
and
function
high
of
for
tides,
the
harbor
the
24
tide
if
a
24
and
at
the
times
hours.
4
their
is
it
at
can
cycle.
maximum
o’clock
tide
when
hour
in
least
the
2
cross
heights.
afternoon.
m
the
above
the
average
harbor.
solving
high
be
of
6,
Determine
Problem
One
0,
the
midnight.
b
ship
takes
modeled
=
to
above
models
Find
the
takes
circular.
that
after
t
it
a
sea
7
be
it
24
seconds
board
you
20
time.
height
meters
can
you
height
long
wheel
function
Harbor
which
rst
d
Ferris
at
in
16
graph
Venice,
later
12
low
hours
this
function.
14 Models
the
and
at
midnight.
water
that
level
the
function.
models
your
was
tide
later
sinusoidal
that
of
at
Italy
situation.
The
was
height
water
0.1 m.
of
the
level
at
Assume
water
high
that
level
can
A LG E B R A
Summary
The
of
amplitude
the
function
value.
The
is
the
to
its
graphs
height
from
maximum
ofy
=
sin
x
the
or
and
mean
value
minimum
y
=
cos
x
Reection:
f (x)
=
period
complete
the
f (x)
sinusoidal
=
is
the
cycle.
horizontal
The
graphs
length
of
y
=
of
sin
=
graph
f (x)
one
x
in
of
y
the
=
−f (x)
is
a
reection
=
cos
x
of
the
graph
x-axis.
y
y
y
and
=
sin x
y
1
y
functions
cos x.
1.
y
The
For
and
have
The
amplitude
sin x
have
period
=
cos x
1
360°
0
0
x
180°
y
y
=
x
360°
180°
360°
sin x
1
Amplitude
=
1
1
y
=
−sin x
y
=
−cos x
1
The
0
of
y
=
f (
x)
is
a
reection
of
the
graph
x
180°
360°
graph
180°
360°
of
y
=
f (x)
in
the
y-axis.
1
y
=
Period
360°
y
=
sin x
1
y
y
=
cos x
Amplitude
=
0
1
1
360°
x
180°
180°
360°
1
y
0
360°
=
sin (
x)
x
180°
360°
180°
1
y
=
cos x
1
Period
=
360°
0
x
The
graph
of
y
=
a
sin x
has
amplitude
a.
360°
180°
180°
360°
1
The
graph
of
y
=
a
cos x
has
amplitude
a
y
Translation:
A
periodic
function
repeats
a
pattern
of
regular
y
=
period
a
cos
=
sin
x
and
of
bx
the
functions
y
=
a
sin
bx
and
y
=
f (x)
+
d
=
cos
translates
When
is
d