MYP
Mathematics
A
David Weber • Talei Kunkel
Rose Harrison • Fatima Remtulla
concept-based
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Contents
Introduction
1
Number:
iv
discoveries
4
38
Unit
review
40
Triangles:
assessment
principles,
and
solutions
50
89
Unit
review
90
Linear
assessment
relationships:
97
impact
of
human
decision-making
102
Unit
summar y
139
Unit
review
141
Summative
assessment
3D
products,
shapes:
148
processes
and
solutions
154
Unit
summar y
192
Unit
review
193
Bivariate
assessment
data:
what
it
199
means
to
be
human
204
Unit
summar y
242
Unit
review
244
Geometric
assessment
252
transfor mations:
expressing
beliefs
and
values
256
Unit
summar y
298
Unit
review
300
Summative
7
processes
summar y
Summative
6
45
Unit
Summative
5
4
summar y
Summative
3
developments
Unit
Summative
2
and
Linear
assessment
systems:
social
311
entrepreneurship
316
Unit
summar y
350
Unit
review
351
Summative
assessment
355
Answers
358
Index
400
i i i
MYP
This
is
(kind
you
Mathematics
not
of
your
like
develop
a
active
reect
you
who
and
your
only
math
on
and
Kunkel:
paper),
you
fact,
will
you
my
two
motivation
your
will
you
and
David Weber:
to
never
Rose
I
stop
learn
writing
would
To
my
including
forward
MYP
me
to
Talei
in
path
this
to
as
and
to
Y
ou
do
my
and
as
of
pushes
and
learn
an
part
but
apply
,
mathematical
doing
with
of
a
what
new
this
is
to
algorithms
be
of
a
before
supposed
Y
ou
thereby
generation
help
perform
yourself,
mathematics.
means
will
will
rules,
text
peers,
information
text
you
concepts
your
it
this
philosophy
,
classroom,
by
present
will
to
then
allowing
math
student,
mathematician.
husband,
as
James,
they
for
are
his
my
suppor t
and
inspiration
patience
and
father
and
Poppy
,
for
Bobby
,
who
and
outside
showing
push
Beatrice,
star ting
to
on
men
and
this
much
I
strive
incredible
zone.
who
Thank
have
inspired
Brenda
colleague,
comfort
who
always
Russell
me
how
two
me
admired
my
me
the
for
for
me
more.
your
jour ney
.
has
you
learned.
always
for
I
look
you.
Scientic
How
to
use
this
be
authors:
students,
me
from
and
mathematical
the
respected
for
MYP
thoughts
are
textbooks
practice
MYP
textbook…
textbooks.
my
my
a
the
math
most
the
Matthew
,
Marlene
–
then
of
formulate
math,
David,
Kunkel
and
like
from
thank
to
project
continuing
to
series
well
and
your
thank
family;
suppor t
Remtulla:
this
like
my
to
you
mathematics
discuss
Kathr yn
believing,
never-ending
guided
like
children,
Har rison:
Fatima
would
Whereas
Following
Much
how
type
generate
understanding.
understands
for
which
discover
them.
where
learning
I
new
textbook.
Acknowledgements
Talei
a
mathematician.
applying
deepen
not
In
resource,
on
to
a
where
procedures.
practicing
an
lecture
into
investigations
and
average
3:
Digital
The
Digital
computer
1
book
The
Number
modern
over
Every
queries
The
study
of
very
large
and
very
small
numbers
can
lead
you
on
through
will
in
some
interesting
discoveries
and
developments,
as
see
work
this
with
unit.
these
Globalization
However,
types
and
of
there
are
numbers
other
is
just
contexts
as
where
or
40
1980,
had
years,
when
early
only
that
16
the
personal
versions
000
number
of
bytes
has
of
soared
bytes.
one
total
In
desktops
than
search
searches,
in
innovation
1970s
which
every
engine
means
day.
In
manages
over
addition
3.5
to
over
billion
this,
40
000
search
the
tweets
and
3
billion
snaps
are
sent
500
every
day.
you
What
to
the
a
million
journey
in
introduced.
less
billion
second,
internet
Each unit in this book explores mathematical content
was
In
16
technical
began
laptops
memory.
to
and
Age
Age
did
people
do
with
this
time
before
the
internet?
ability
fundamental.
sustainability
through a single global context. However, a dierent
Population
Large
global context could have been selected as a focus of
the
and
study
of
resources
In
study.
1960,
people.
What would the study of this content look like
In
The chapter opener gives you
All
own.
water
it
feed
for
is
million
there
can
grown
billion
of
over
to
us
in
of
almost
and
and
if
so,
are
billion.
over
and
land.
food
of
how
understand
8
goats,
kilometers
and,
billion
there
pigs
water
cubic
to
3
agricultural
need
enough
help
part
consumption
just
population,
1
planet
4
the
was
of
fresh
long
and
plan
will
for
future.
there
sucient
population,
encouraging you to explore and discover these options
just
Numbers
than
kilometers
the
important
sustainability.
since
growing
more
of
With
has
an
Earth,
of
population
our
square
play
on
analysis
number
available,
last?
the
Is
an
cows,
million
quantities
global
inhabitants
their
just a small taste of the endless possibilities that exist,
to
billion
consumption
populations
and
the
That
order
1.5
150
in a dierent context?
and
small
making
Microbes
become
our
land
are
land
crops
more
for
and
arable
microscopic
more
fertile
the
and
growing
other
human
animals?
involves
organisms
produce
One
method
microbes.
that
more
can
help
soil
crops.
on your own.
The
internet
electrons,
before
2
i v
itself
each
writing
is
with
the
made
a
up
mass
rst
of
so
tiny
small
non-zero
particles
that
number!
it
known
would
as
take
electrons.
30
zeros
Electricity
after
the
is
the
decimal
ow
place
of
T
opic
Key
opening
concept
The Approaches to Learning
Y
ou
(ATL) skills developed
already
know
throughout the unit.
how
–
to:
these
for
are
the
should
page
skills
that
unit.
will
be
applied
N U M B E R
in
the
you
The
1
identied
Research:
A
TL1
Number
literacy
Discoveries
and
developments
Use
memory
long-term
Global
Practise
develop
positive
skills
already
thinking
have
an
memory
understanding
CONCEPT: FORM
Y
ou
explored
1
Related
concepts: Quantity,
Representation,
should
throughout
already
Solve
simple
Solve
the
your
know
how
to:
4
equations
following
answers
as
equations.
integers
or
Apply
3(2
28
b
72
Global
unit,
189
context:
you
your
will
explore
some
understanding
of
amazing
the
global
human
context
discoveries
of
and
orientation
developments
in
space
and
the
property
of.
following
of
will
see
that
being
able
to
manipulate
quantities,
simplify
them
and
as
you
represent
a
variety
of
ways
is
one
tool
that
can
be
used
to
uncover
everything
from
to
the
tiniest
of
environments,
and
all
that
lies
in
a
car
travels
how
far
If
Olympic
9(
3
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may
want
4)
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34
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goats,
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agricultural
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4
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and
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for
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animals?
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organisms
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on
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developments,
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population,
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population
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cows,
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it
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Number
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concepts:
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In
this
unit,
expand
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operations
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●
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developments
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great
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it
take
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following
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each
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a
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4
2
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3
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4
c
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5
×
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circle
with
the
a
a
radius
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b
a
diameter
following
measurements.
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of
10 cm
5
Introducing
How
far
is
really
number
far?
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close
without
really,
What
this
basic
to
skill:
the
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very
discoveries
how
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far
new
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of
and
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perform
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6
of
other
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operations
ability
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to
Number
these
with
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with
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integers
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make
you
to
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know
describe
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calculate
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exist
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of
manipulate
variety
results
discovery!
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natural
real
numbers,
numbers
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rational
can
or
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also
be
irrational
something
it,
even
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on
invisible
dimensions
how
represented
to
kinds
to
that
are
very
to
touching
eye?
the
the
get
small?
things
requires
describe
world’s
and
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number
or
need
irrational
with
numbers.
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this,
to
questions
necessary
may
you
you
numbers.
ways.
are
planet
bacteria,
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work
numbers
new
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are
of
to
all
developments.
a
Your
accurately,
able
large,
away
ways.
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answering
you
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really
human
Interestingly,
can
both
classied
to
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if
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Did
you
Our
understanding
history.
count.
to
know...?
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started
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break
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since
throughout
we
to
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the
triangle
to
shocking
prove
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he
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the
is
was
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need
of
Greek
mathematician.
that
number.
that
the
discovery
Pythagorean
irrational
needed
discovered
pieces.
attributed
who
was
as
smaller
often
developed
numbers
developed
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discovery
for
measure
rumoured
sentenced
to
it.
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said
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were
hypotenuse
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death
an
numbers
things
numbers
philosopher
of
with
that
his
a
number
Hippasus
discovery
be
used
of
a
rational
right
irrational
triangle
numbers.
or
like
In
irrational?
the
this
one
shown
isosceles
right
45º
triangle,
the
sides
next
to
the
right
angle
measure
1
unit
each,
2
1
while
the
What
hypotenuse
kind
of
number
measures
is
1?
2
How
units.
about
2
?
What
makes
a
45º
number
rational
or
irrational?
The
activity
below
will
help
you
1
see
the
dierence
Activity
Some
1
–
examples
between
What
of
these
does
rational
and
two
it
types
mean
irrational
Rational
of
to
number.
be
numbers
numbers
rational?
are
given
Irrational
0.25
in
the
table
below.
numbers
0.43256931…
1.4
−1.559824…
−3.141414…
23.057419…
25.659
56.1416278…
4.5555555…
−423.745782309…
—
105.37
72.2290004681…
−11.258258258…
1
Based
an
on
these
irrational
examples,
number?
how
would
Compare
your
you
2.71828182845…
dene
denition
a
rational
with
a
number?
How
would
you
dene
peer.
Continued
on
next
page
7
2
Copy
of
and
the
complete
following
rational
or
this
table.
numbers
in
Use
a
calculator
decimal
form
and
to
represent
classify
each
each
as
To
either
show
decimal
irrational.
itself
over
Number
Decimal
form
Rational
or
that
a
repeats
over
and
again,
or
irrational?
recurs,
one
of
two
2
representations
3
can
be
used:
0.888888….
3
also
be
_
can
written
π
as
0.8
3
17.63636363…
4
can
17
also
written
be
as
17.63
99
8
2
2
36
3
Based
on
decimal
Abdisa
5
Research
says:
and
types
7
Write
down
irrational.
Reect
not
of
an
and
as
zero”.
more
of
how
can
recognize
roots
the
are
you
an
to
recognize
irrational
irrational
words
used
your
numbers
examples
rational
number?
numbers.”
“nite”,
describe
a
Do
you
“terminating”,
decimal
agree?
“innite”,
numbers.
Write
can
ratio
of
do
are
rational?
(decimal,
What
fraction,
types
etc.)
are
Is
●
Jeremiah
7
a
rational
written
8
1
or
says:
as
a
be
of
each
Justify
it
is
not
your
written
in
answer.
“repeating”,
down
type
1
dened
two
the
as
integers,
examples
irrational
“An
“any
as
number
long
you
number?
irrational
fraction.”
Number
if
the
“periodic”
denition
of
irrational?
as
have
the
seen
that
can
be
denominator
so
far
t
that
denition?
●
number
Explain.
choices.
discuss
How
2,
example.
number
the
you
when
decimal
two
step
can
square
Explain
rational
written
in
meaning
give
What
is
“All
the
6
A
How
“non-periodic”
each
●
results
form?
4
and
your
Is
he
Justify
number
correct?
is
a
your
answer.
number
Explain.
that
cannot
be
of
number:
rational
and
N U M B E R
Representing
If
a
rational
where
the
rational
number
can
denominator
converting
a
decimal
Investigation
is
to
1
–
be
a
numbers
written
not
as
zero,
a
ratio
how
do
of
you
two
go
integers
about
fraction?
Rational
numbers
in
dierent
forms
t
e ri
c
Take
a
look
at
the
following
decimal
o
n
r
i
numbers:
B
0.5
1
2.7
Classify
37.4
0. 3
each
number
as
either
each
number
as
a
9.12
2.4567
“terminating/nite”
or
“innite”.
Justify
your
answer.
p
2
Express
fraction
in
the
form
,
where
p
and
q
are
both
integers
q
and
3
q
Write
≠
0.
down
a
general
rule
for
how
to
represent
a
nite
decimal
number
as
a
fraction.
Take
a
look
at
the
following
fractions:
1
5
34
52
8
68
9
9
99
99
9
99
4
Represent
each
fraction
in
decimal
form.
Classify
each
number
in
as
many
ways
as
possible.
5
What
6
How
7
patterns
would
to
verify
a
2.4…
8
Verify
9
Justify
you
that
you
write
each
notice
each
answer
b
Summarize
periodic
do
your
decimal
each
why
of
rules
each
for
of
rules
your
of
is
your
the
answers?
following
c
how
as
for
numbers
as
fraction?
Use
a
calculator
correct.
0.26
numbers
your
in
to
represent
23.125125125…
nite
decimal
numbers
and
innite,
fractions.
two
rules
other
numbers.
works.
9
Reect
●
Explain
and
why
decimal
●
You
●
a
How
innite
Based
on
is
impossible
2
to
write
a
non-repeating,
innite
fraction.
perform
screen.
an
as
it
discuss
a
calculation
do
you
periodic
the
on
know
decimal
patterns
your
calculator
whether
you
or
an
found
that
innite
in
and
number
step
the
is
a
answer
nite
non-periodic
5
in
lls
the
decimal,
decimal?
Investigation
1,
how
9
should
be
represented
as
a
decimal?
How
does
that
conict
with
9
9
the
value
of
?
9
So
far,
kinds
you
of
innite
does
have
periodic
decimal
A
decimals
number
to
in
has
represent
fraction
a
part
nite
form.
that
decimals
What
repeats
and
certain
happens
and
a
part
if
the
that
1
Represent
a
how
not?
Example
Q
seen
each
of
the
following
as
a
fraction
with
integer
numerator
0.0353535…
a
If
x
=
b
0.0353535…
=
35.353535…
=
the
dierent
the
−10x
=
x
=
number
powers
same
of
decimal
2.13789789789…
by
10
so
par t
If
both
that
100 000x = 213 789.789789…
99
900x
=
x
=
35
for
x
198
your
answer
if
possible.
x
=
213 576
17 798
Simplify
=
=
8325
198
÷
0.0353535…
Verify
your
answer
by
dividing.
17
798
÷
8325
=
2.13789789…
=
=
198
10
1
Number
2.13789789…
17 798
7
0.0353535…
213.789789…
99 900
7
7
=
213 576
Solve
990
x
2.13789789789…
numbers.
Subtract.
35
=
is
−100x
in
x
two
0.35353535…
repeating
990x
denominator.
b
Multiply
1000x
and
8325
N U M B E R
Reect
●
Show
and
how
0.7777…
●
Show
you
to
how
Create
a
a
can
use
3
the
procedure
in
Example
1
to
convert
the
procedure
in
Example
1
to
convert
fraction.
you
1.242424…
●
discuss
to
two-
can
a
or
use
fraction.
three-line
rhyme
that
will
help
you
remember
how
ATL
to
1
&
convert
decimals
to
fractions.
Share
your
rhyme
with
a
few
peers
2
and
●
give
Write
positive
down
feedback
what
is
going
to
each
well
so
other.
far
in
this
unit
in
terms
of
your
ATL2
learning.
Practice
1
Classify
1
each
of
the
following
numbers
as
either
rational
or
irrational.
Justify
your
answer.
—
a
− 4
g
2
Represent
b
c
h
i
each
of
the
14.23
d
−19.34862147…
following
rational
e
j
104.25
f
2.089999…
0
numbers
as
a
fraction
in
simplied
form.
—
a
0.38
b
0.21
c
1.6
d
2.4
e
12.874
—
f
3
0.1333…
g
a
Represent
b
Show
0.3
and
5
h
0.6
as
−4.1392
i
0.9
equivalent
c
Repeat
d
Find
that
1
by
fractions
steps
your
=
a
and
own
0.999...
=
adding
which
b
with
example
j
9.03555…
fractions.
_
that
5.672
together
you
0.25
of
0. 3
found
and
two
in
_
and
step
0. 6
and
then
that
can
adding
together
their
a.
0.74.
rational
numbers
be
used
to
demonstrate
1.
Exponents
x
An
“x”
exponent
is
used
the
to
is
written
exponent
simplify
or
in
the
power.
longer
form
You
a
,
have
expressions.
It
where
“a”
already
would
is
seen
be
the
base
how
dicult
and
this
to
can
be
write
10
2×
2
×
2
×
2
×
2
×
2
×
2
×
2
×
2
×
2
over
and
over
again:
2
is
much
10
simpler.
2
is
a
power
of
2,
with
a
base
of
2
and
an
exponent
of
10.
10
2
base
exponent
11
Activity
2
–
The
T
ower
of
Hanoi
t
er
i
r
i
n
c
B
Mathematical
Egypt.
More
puzzle
the
disks.
another,
world
the
a
to
bottom)
Simulate
can
i
use
n
disks
top.
to
all
a
smallest
the
puzzle
coins
to
Lucas
on
is
larger
the
(on
a
one
temple
of
developed
versions
there
are
The
placing
when
been
Édouard
legend,
the
on
never
ends!)
by
the
All
smallest
have
contemporary
invented
According
gold
puzzles
Hindu
post,
disk
top).
using
1883
priests
disks
on
are
the
ordered
must
to
a
centuries,
Cube
based
that
in
move
of
size
smaller
moves
start
and
on
on
with
disks
one.
Sudoku.
You
three
the
as
far
The
at
a
back
as
Tower
to
draw
in
posts
disk
time
puzzle
post
take
can
large
largest
one
The
it
dating
of
Ancient
Hanoi
is
a
legend.
another
would
with.
a
contains
the
stacked
many
disks
is
temple
top
the
Rubik’s
that
again
How
three
represent
include
in
over
is
at
the
from
from
complete
posts
the
and
64
dierent
bottom
one
post
completed
order
the
and
and
to
(and
largest
the
(on
puzzle?
disks
or
you
disks.
k
b
e
You
can
game
by
website
Hanoi.
the
play
going
and
The
gradually
b
What
the
is
last,
the
as
to
the
of
will
minimum
the
and
the
the
of
of
the
Tower
track
you
number
of
of
can
of
puzzle.
number
rule
for
keep
moves
solve
version
mathisfun.com
searching
increase
you
following
online
game
number
disks
an
of
moves
never
you
placing
a
need
to
larger
transfer
disk
on
three
top
of
a
disks
from
smaller
the
rst
Continued
12
1
Number
post
to
one?
on
next
page
N U M B E R
c
Gradually
table
like
increase
the
Number
one
of
the
number
of
disks
as
you
solve
the
puzzle
and
record
your
results
in
a
below.
disks
Minimum
number
of
moves
3
4
5
6
d
Based
on
priests
e
Use
If
move
n
there
all
were
aect
to
What
the
to
no
64
your
of
write
disks,
solve
examples
integer.
all
in
the
a
table,
disks
rule
for
what
to
the
the
is
the
last
minimum
pole?
minimum
Write
number
number
your
of
of
moves
answer
moves
using
required
required
an
to
for
the
exponent.
solve
the
puzzle
disks.
necessary
In
pattern
exponents
with
f
to
the
you
the
of
the
2
your
rule
correctly
predicts
the
minimum
number
of
moves
if
met
the
before,
base
is
the
base
negative?
has
been
How
a
does
positive
that
quantity?
–
Positive
and
negative
i
bases
t
er
i
r
Investigation
that
puzzle.
have
happens
value
show
Find
the
value
2
of
3
2
2
4
3
2
3
2
2
Generalize
positive
3
Verify
your
4
Justify
why
bases,
exponent
4
results
negative
rule
for
your
0
or
5
rule
−1
(−3)
and
base
two
write
to
more
an
down
a
rule
related
to
the
result
of
raising
a
exponent.
examples
of
your
own
choosing.
works.
exponents
of
5
(−2)
(−3)
your
and
5
4
(−3)
6
5
(−1)
(−2)
3
(−3)
5
4
(−2)
6
4
5
(−1)
3
(−2)
5
4
(−1)
6
3
4
5
2
(−1)
5
4
3
6
2
3
4
5
B
5
2
3
4
5
following.
4
3
4
the
2
3
3
Like
each
2
2
2
of
n
c
1
can
also
be
integers.
But
what
would
an
mean?
13
Zero
The
and
typical
negative
denition
multiply
a
integers,
decimals
powers
t
base
into
of
number
and
this
by
exponent
itself”
.
even
very
3
an
–
is
However,
fractions.
familiar
Zero
“the
How
number
of
exponents
do
these
times
can
you
be
unfamiliar
denition?
and
negative
i
exponents
t
er
i
r
Investigation
powers
n
c
0
1
What
do
you
think
6
2
What
do
you
think
4
3
Copy
and
is
equal
to?
Explain
your
B
thinking.
−1
powers.
Write
allowed!
can
see
complete
If
in
each
you
the
are
is
the
equal
table
answer
unsure
powers
by
as
of
above
to?
Explain
nding
either
the
a
your
the
value
whole
value
of
a
thinking.
of
each
number
power,
or
a
simply
of
the
following
fraction.
follow
No
the
decimals
pattern
you
it.
5
=
2
4
2
4
=
3
=
3
=
3
=
3
=
3
3
2
3
2
2
5
=
5
=
5
=
5
−1
3
=
3
=
3
=
3
−2
2
5
=
5
=
5
=
5
=
−2
−3
−4
2
=
−1
=
−2
−3
2
=
0
−1
=
=
1
0
2
=
2
1
0
2
3
=
2
1
2
=
=
−3
−4
=
−4
=
−5
2
=
4

4
Repeat
the
same
process,
starting
with


3
1

2

and
ending
with

1



2
.

5
Based
on
your
results,
what
conclusion
can
you
draw
about
an
6
Based
on
your
results,
what
conclusion
can
you
draw
about
negative
7
Write
your
numeric
8
Verify
the
9
14
Justify
as
general
rules,
using
variables
instead
of
of
zero?
exponents?
specic
examples.
your
ones
1
conclusions
exponent
rules
for
two
more
examples
above.
why
each
Number
of
your
rules
works.
of
your
own,
using
a
pattern
similar
to
N U M B E R
Reect
●
Explain
and
how
denition
of
discuss
the
an
results
of
4
Investigation
3
challenge
the
typical
exponent.
2
 2
●
What
do
you
think
is
the
value
of


●
ATL1
Create
and
a
memory
negative
Example
Q
Find
the
aid
to
help
to
Explain
your
answer.

remember
your
rules
for
zero
exponents.
2
value
of
each
of
−3
a
you
?

3
the
following:
−2
4
b
(−3)
−2
c
−5
−3
A
a
4
3
A
3

1



4
negative
exponent
indicates
that
you
take
the
1
reciprocal
or
of
the
base.
3

4
1
1
1
×
1 × 1 × 1
×
4
4
Two
or
fractions
that
multiply
to
1
are
called
4 × 4 × 4
4
reciprocals.
For
example,
and
are
1
reciprocals
since
×
=
1
64
−2
b
(−3)
A
negative
exponent
indicates
that
you
take
the
2
1


−

reciprocal
of
the
base.


3

1
−
1
−
×
3
1
As
you
found
before,
a
negative
base
raised
to
an
=
3
even
9
exponent
produces
a
positive
answer.
−2
c
−5
2
−

1




5
A
negative
the
base
1
−
1
×
5
5

is
5,
not
of
the
indicates
base.
that
Note
you
that
the
take
original
−5.
1


exponent
reciprocal
=
−
25
15
Practice
1
Find
or
a
the
2
value
of
each
of
0
6
b
(−3)
g
7
−1
f
Write
your
answer
−3
c
as
either
a
whole
d
number
−2
q
e
(−8)
0
4
i
m
−2
2
1
−3
h
l
−12
−5
10
3
(−9)
k
p
following.
fraction.
−2
a
the
−6
5
j
n
o
s
t
(−2)
2
(−12)
r
0
b
2
Rewrite
a
4
×
4
each
×
4
of
×
4
the
×
4
following
×
in
the
4
e
Order
the
quantities
a
,
where
−2
5
in
each
set
of
−7
four
(−7)
(−7)
from
(−7)
lowest
to
−2
3
c
−2
7
b
integers
(no
fractions).
Show
your
working.
0
−1
2
3
2
−3
2
are
highest.
2
−1
b
(−7)
−1
1
4
and
c
−2
0
a
a
b
d
3
form
0
1
d
−1
1
1
3
0
0
4
Ricardo
says:
anything.
5
Talei
the
6
“5
You
says:
have
says:
“
of
zero
zero.”
“Negative
denominator
Anna
equals
a
because
Explain
exponents
fraction,
must
be
4
any
make
then
since
you
it
you
have
faults
zero
in
fractions.
makes
have
an
half
his
If
5s,
which
you
don’t
have
thinking.
the
negative
integer.”
of
means
8.”
Is
exponent
Talei
Explain
correct?
why
this
is
already
in
Explain.
thinking
is
faulty.
7
Thomas
A
states:
negative
Explain
“A
positive
exponent
your
is
the
exponent
number
is
of
the
number
times
you
of
times
divide
by
you
the
multiply
base.”
Do
the
you
base.
agree?
answer.
Continued
16
1
Number
on
next
page
N U M B E R
8
The
development
of
18thcentury.
Basic
were
often
created,
example,
could
be
1
liter
of
created
kilogram.
Some
the
current
units
derived
water
by
of
for
the
system
measurements
from
has
using
metric
a
the
mass
prexes,
prexes
like
properties
of
1 kg.
such
are
of
as
given
of
used
table
in
lengths,
natural
Multiples
the
began
angles,
those
in
units
or
in
France
mass
objects
the
below.
and
such
divisions
units
Copy
of
in
as
the
capacity
water.
these
units
millimeter
and
For
and
complete
thetable.
Prex
Exponential
form
Expanded
form
9
giga
10
mega
1
000
000
3
kilo
10
1
deci
10
1
centi
100
1
milli
1000
6
micro
10
1
nano
1 000 000 000
12
pico
10
1
femto
1 000 000 000 000 000
18
atto
n
10
k
b
e
A
set
you
of
to
where
14
year
see
how
you
can
largest
of
old
scroll
celestial
Multiplying
twins
large
created
dierent
to
see
the
Scale
powers
everything
of
from
already
or
it
possible
know
dierent
to
to
that
2
investigations
rules
about
multiply
the
to
the
smallest
application,
site
of
which
allows
http://htwins.net/scale2/
subatomic
particles
to
the
4
means
2
to
will
and
look
nd
the
2
×
2
and
2
means
2
×
2
×
2
×2.
4
2
multiplication
you
how
the
Universe
Go
powers
2
Is
the
are.
objects.
2
You
of
10
for
and,
you
if
do
patterns
product
of
so,
how
already?
and
two
is
In
this
the
determine
or
more
similar
following
general
powers.
17
4
–
Product
i
rule
t
er
i
r
Investigation
Copy
and
complete
this
Question
2
B
table.
Expanded
form
Simplied
4
2
×
2
×
3
×
2
×
y
5
form
6
2
×
2
×
2
×
2
×
2
×
2
2
3
3
4
5
2
9
2
y
6
w
×
2
h
w
3
×
3
h
×
2
Describe
any
3
Describe
a
4
Generalize
h
patterns
that
you
see
among
your
answers
12
rather
rule
that
your
than
you
rule
specic
to
could
use
to
multiplying
numeric
multiply
powers
4
with
in
each
row.
20
×
4
any
base.
Be
sure
to
use
in
many
variables
examples.
6
5
You
know
ways
or
6
(2
as
×
that
2
possible
2)
Compare
×
(2
×
your
means
by
2)
2
×
2
inserting
×
(2
×
×
can
you
2
×
2
×
brackets.
2
×
For
2.
Rearrange
example,
2
×
this
(2
×
2
as
×
2
×
2
×
dierent
2)
2).
representations
representations
with
those
of
a
peer.
How
many
dierent
nd?
6
7
Knowing
that
all
representations
8
Verify
other
9
your
than
Justify
Can
you
representations
follow
from
the
step
rule
4
for
that
two
should
you
all
found
more
equal
in
step
examples
of
your
and
use
expression
limitations
rule
18
Can
●
What
1
you
,
show
that
discuss
the
like
on
rule
you
3
4
your
own,
2
×
the
3
?
5
discovered
Explain
bases
your
involved
in
in
investigation
answer,
the
use
your
limitations,
Number
rule
if
with
any,
are
an
there
on
4
with
describing
an
any
multiplication.
expression
the
like
4
−2
×
4
exponents?
your
4.
works.
5
●
2
2.
why
Reect
●
rule
your
n
c
1
?
Explain.
Explain.
with
a
base
N U M B E R
Activity
3
–
Reinforcing
zero
and
negative
exponents
Pairs
1
Dene
What
the
term
happens
inverse?
“multiplicative
when
you
Demonstrate
inverse”
multiply
with
an
a
–
research
number
by
its
this
if
necessary.
multiplicative
example.
3
2
Write
down
3
Rewrite
the
multiplicative
inverse
of
7
b
a
and
b
the
are
multiplicative
both
inverse
from
step
2
in
the
form
a
,
where
integers.
3
4
Multiply
product
and
7
rule
its
you
multiplicative
have
just
inverse
from
step
3
using
the
established.
6
5
Repeat
the
same
procedure
for
6
Repeat
the
same
procedure
for
5
4
1

2
How
zero
do
your
and
Dividing
While
a
it
is
general
more
results
negative
reinforce
powers?

what
you
have
already
learned
about
Explain.
exponents
always
rule
possible
allows
eciently.
Is
there
5
–
to
a
expand
and
accomplish
similar
rule
Quotient
multiply
the
for
same
powers,
operation
dividing
nding
much
powers?
rule
i
t
er
i
r
Investigation
you
to
o
7

Copy
and
complete
this
n
c
1
table.
B
Question
5
2
2
÷
2
Simplied
form
Exponential
÷
5
form
2
2
×
2
2
4
6
5
form
3
÷
7
2
Expanded
2
6
4
÷
4
5
t
4
÷
t
÷
m
8
m
4
Continued
on
next
page
19
2
Describe
any
3
Generalize
variables
4
Verify
5
Justify
your
can
●
Show
Write
for
your
how
be
the
used
how
to
the
that
than
and
negative
●
rule
why
Show
rule
rather
Reect
●
a
patterns
that
you
see
could
specic
two
rule
you
among
use
to
numeric
other
your
divide
answers
powers.
in
Be
each
sure
row.
to
use
examples.
Remember:
if
exponent
not
explicitly
then
examples.
it
is
is
the
written,
1.
works.
discuss
quotient
derive
rule
the
quotient
6
that
zero
rule
you
discovered
exponent
can
be
in
Investigation
5
rule.
used
to
derive
the
rule
for
exponents.
down
what
is
going
well
so
far
in
this
unit
in
terms
of
your
ATL2
learning.
●
Identify
the
strengths
that
you
have
as
a
student.
How
have
they
ATL2
helped
you
Example
Q
Simplify
succeed
in
this
unit?
3
the
following
using
the
laws
of
exponents.
The
Write
your
answer
with
positive
exponents
laws
here
3
2
m
5
a
4n
9
b
m
5
a
9
exponents
the
product
that
rule
you
and
need
the
quotient
rule,
both
of
which
you
c
6
A
are
4
p
−8
×
9
of
only.
1
10n
3
have
p
discovered
in
this
unit.
−8
×
9
When
5+(−8)
9
−3
=
9
base,
multiplying
simply
add
expressions
the
with
the
same
exponents.
1
Rewrite
your
answer
with
positive
exponents
3
9
only.
3
m
b
6
m
3−(−6)
When
dividing
expressions
with
m
simply
subtract
the
exponents.
9
m
Continued
20
1
Number
on
next
page
the
same
base,
N U M B E R
2
4
4n
p
Simplify
the
coecients
as
you
would
simplify
a
c
1
10n
3
fraction.
p
2 − ( −1)
2n
−4 − 3
When
p
dividing
simply
expressions
subtract
the
with
the
same
base,
exponents.
5
3
2n
7
p
Simplify.
5
3
2n
Write
your
answer
with
positive
exponents
only.
7
5 p
Practice
1
Simplify
6
a
5
3
the
following.
3
×
2
5
b
Write
−6
6
×
answers
−6
6
c
with
positive
exponents
×
8
d
e
g
h
i
j
k
l
m
n
o
p
q
u
v
With
the
divide
5x
−5
×
r
2ab
4
c
2
×
3a
3
c
s
w
development
quantities
Prex
2
6x
of
because
the
of
Exponential
metric
the
form
use
only.
2
8
f
−2
2
your
(−2pq
3
) (7p
q)
easier
to
−3
t
(−6y
z
−3
−1
) (−
y
z)
x
system
of
it
became
compare,
multiply
and
prexes.
a
How
many
picograms
are
b
How
many
nanobytes
c
How
many
femtometers
d
What
in
a
kilogram?
9
giga
10
are
in
a
gigabyte?
6
mega
10
kilo
10
are
in
a
millimeter?
3
fraction
of
a
megameter
of
a
decigram
is
an
1
deci
10
centi
10
milli
10
micro
10
nano
10
attometer?
2
3
e
What
fraction
is
a
microgram?
6
9
12
pico
10
femto
10
atto
10
15
18
Continued
on
next
page
21
3
Which
quantity
3
a
4
2
2
or
−2
5
3
In
the
physics,
Daviet
James
In
c
Write
b
7
correct
Clerk
1784,
in
×
the
1761,
d
answers
c
often
was
later
e
or
−2
÷
This
−2
8
integers
3
used
answer.
and
−2
or
−3
2
is
4
as
5
4
for
−3
10
your
analysis
units
Foncenex
or
−2
×
answer.
−1
5
0
6
each
−2
4
dimensional
the
de
or
following.
÷
Justify
3
3
−2
2
produce
a
b
−3
×
bigger?
4
3
Evaluate
a
is
to
9
6
−5
or
−4
2
f
of
6
verify
process
d
that
was
developed
by
the
units
in
a
formula
rst
published
by
other
scientists,
for
François
example
Maxwell.
Charles-Augustin
de
Coulomb
discovered
that
the
attraction
between
two
electric
charges,
q
and
q
,
calculated
using
the
formula
,
where:
the
r (the
charge
distance
is
measured
between
the
in
charges)
2
the
constant
Substitute
correct
b
The
k
has
these
units
electric
of
the
units
units
into
force,
the
(V)
is
coulombs)
at
a
(C),
measured
in
meters,
C
formula
and
show
that
the
formula
produces
the
(N).
created
distance
r
by
a
charge
q
(measured
(in
metres)
from
the
charge
to
be
given
by
the
formula
,
that
was
4π
discovered
and
simplify.
Remember
in
into
−2
Nm
newtons
potential
coulombs
given
formula
then
electric
the
can
2
units
each
4
2
÷
1
be
or
fractions.
Substitute
force
−2
2
where
is
and,
ε
a
constant
therefore,
o
has
2
is
measured
potential
in
C
−1
N
no
units.
−2
m
.
What
should
the
units
of
electric
be?
assessment
i
t
er
i
r
Formative
who
can,
n
c
“Those
teach.”
C
In
this
lesson
of
task,
to
you
teach
possible
will
work
your
topics
is
in
peers
given
small
one
on
of
the
groups
the
to
develop
remaining
next
an
laws
of
investigation
exponents.
and
The
list
page.
Continued
2 2
1
Number
on
next
page
N U M B E R
Topic
Example
m
Power
of
a
(x
product
y)
m
x

Power
of
a

quotient



y 
a
Power
of
a
(x
power
b
)
?
Roots
as
x
=
x
x
=
x
exponents
?
3
You
must
hand
in
a
detailed
lesson
plan
which
includes:
●
lesson
●
an
●
examples
●
practice problems (but no word problems or problems in a global context are needed).
objectives/goals
investigation
that
you
are
going
to
use
Objectives/goals:
●
What
are
the
desired
●
What
are
the
main
end
of
the
outcomes
concepts
of
that
the
you
lesson?
would
like
the
students
to
understand
by
the
lesson?
Investigation:
●
Create
to
an
write
it
problems
activity
that
in
own
so
their
that
the
will
allow
words.
pattern
your
Make
is
peers
sure
to
the
discover
the
investigation
rule
for
themselves
includes
at
least
and
ve
clear.
Examples:
●
●
You
must
guide
necessary
to
Start
a
with
questions
Handout
●
Prepare
It
a
should
appear
the
through
type
one-step
lesson
handout
your
class
each
multiple
contain
in
worked
basic
with
for
the
teach
to
at
examples
your
question
(or
as
many
examples
as
topic).
and
move
into
multi-step
variables.
help
the
notes):
students
follow
questions
Leave
and
two
within
variable)
(scaolded
lesson.
least
question
(single
headings,
examples
of
blanks
key
and
for
your
lesson
examples
students
concepts
that
to
they
in
ll
and
the
in
record
order
the
discover
their
that
answers
during
learning.
they
to
the
will
your
lesson.
Practice:
●
Create
and
with
●
a
worksheet
your
any
Provide
lesson
group
to
help
members
reinforce
should
questions/diculties
an
so
answer
that
your
key
and
teacher
that
submit
can
the
walk
your
this
check
concepts
around
peers
at
least
that
while
may
one
you
the
have
class
is
just
taught.
working
and
You
help
have.
day
before
you
teach
your
it.
Continued
on
next
page
23
Activity/Game:
●
In
order
to
keep
the
class’s
attention
and
The
check
their
understanding,
plan
a
student-created
presented
or
activity
must
for
submit
this
lesson
Every
the
your
group
prepared
the
the
activity
for
understand
from
for
at
least
of
is
and
to
lesson.
any
to
the
class
using
could
the
day
prior
to
your
check.
and
and
to
should
all
Instead
be
of
student
questions
class.
where
and
a
whole
work
each
can
class
be
student
teaches
it
to
activity,
used
in
as
a
all
a
group
his/her
peers.
to
4
of
jigsaw
learns
is
the
the
activity
one
no
laws
exponents.
Q
Simplify
the
may
following:
the
3
3

1

j
use
rule
order
of
You
them
order
that
in
you
2
2
3i
app.
You
There
Example
be
ShowMe
questions
expected
material
the
the
one
teacher
answer
rest
of
details
member
the
to
end
activities
game

a
4t
6

u
wish,
as
long
as
b

2
4
i

jk



 18t
−2
−4
u

you
apply
them

correctly.
3
3

A
2
3i

j
Use
the
power
of
a
power
rule
to
simplify
the
a


2
4
i
jk

outer

exponent.
exponent
3
9
6
the “3” has
an
1.
j
3
i
that
6
i
3
of
Remember
j
Use
12
the
quotient
rule
to
simplify
again
by
k
subtracting
3
exponents.
3
27i
NOTE:
j
rule
12
Alternatively,
rst
inside
the
you
could
use
parentheses
the
and
quotient
then
use
the
k
power
of
a
power
rule.
2
1

4t
6

u
b

−2
 18t
−4
u


2
10


2tu
Simplify

using
the
quotient
rule.

9


2
9


A


10
negative
exponent
indicates
that
you
need


2tu
work
with
the
reciprocal.
81
Use
2
4t
24
1
20
u
Number
the
power
of
a
power
rule
to
simplify.
to
N U M B E R
Practice
1
Evaluate
each
simplied
2
4
of
the
following.
Write
your
answer
as
either
b
c
d
e
f
g
h
i
j
the
following.
Write
your
answers
−1
a
b
d
e
g
h
j
k
m
n
p
q
s
t
(9
−6
×
9
with
positive
−3
4
)
c
(a
2
)
exponents
f
2
(−9y)
a
bookmark
with
the
(2t)
a
a
(3t)
2
(2y)
i
only.
2
×
3
×
−(−3x
)
2
5
(−y
)
l
3
Create
or
−3
×
3
3
integer
fraction.
a
Simplify
an
(4g
2
h)
o
r
laws
of
exponents
on
it.
Make
sure
that
all
the
laws
t
on
ATL1
a
small,
rectangular
Enhance
your
piece
bookmark
of
paper.
through
Include
the
use
of
the
name
colors,
of
each
pictures,
rule
and
arrows,
the
rule
itself.
etc.
25
Scientic
Did
In
you
216
notation
know...?
BC,
Archimedes
Sand Reckoner,
an
upper
sand
that
in
way
to
Greek
for
had
name
a
quantities
Descartes
in
the
that
we
most
such
could
likely
rst
by
a
the
dicult
It
wasn’t
represent
the
today.
was
large
using
it
of
time
invent
eciently
use
The
grains
rst
to
quantity.
more
notation”
1900s,
had
developed
book
of
the
made
humans
René
“scientic
was
However,
large
large
exponents
number
It
system
that
his
established
considered
them!
such
1637
the
he
Archimedes
number
write
until
so
wrote
which
universe.
humans
numbers
to
limit
the
in
when
system
The
of
term
used
computer
scientists.
The
17th
numbers
of
the
century
were
at
telescope
Leeuwenhoek's
later,
humans
distance
such
new
small
A
water
275
be
Are
2 6
by
now
which
of
Galileo
see
with
to
called
large
the
and
that
Being
required
scientic
were
the
as
where
one
approximately
picometer
0.000 000 000 001m.
of
water
really
represent
measures
of
or
these
is
1
water
followed
supposed
is
there
these
Number
contains
a
to
A
roughly
molecules.
by
18
write
more
quantities?
can
zeros!
all
of
ecient
way
very
with
Antonie
roughly
able
a
to
50
van
years
great
describe
development
(or
small
development
either
notation
really
and
the
microscope
small.
and
large
With
Galilei
things
ease
very
science.
incredibly
now
really
zeros
1
is
molecule
drop
you
1610
were
this
quintillion
those
to
or
quintillion
One
in
improvements
picometers,
single
time
quantities
written
1.7
in
a
forefront
numbers
system:
Writing
the
could
away
diverse
was
of
standard
a
form).
N U M B E R
6
–
Developing
scientic
notation
I
i
t
er
i
r
Investigation
1800,
a
group
searching
Titius.
rst
of
1
a
Around
asteroid
rocks
Mars
415
for
and
and
000
of
the
in
a
astronomers
whose
same
debris
left
Its
another
now
over
known
from
average
calling
existence
time,
location
Jupiter.
000
25
planet
the
themselves
was
predicted
astronomer,
as
the
from
the
of
Celestial Police
the
astronomer
Giuseppe
asteroid
formation
distance
the
by
belt.
the
Sun
is
Piazzi,
The
universe,
were
belt
located
B
Johann
discovered
asteroid
n
c
In
is
a
the
ring
between
approximately
km.
415 000 000 can be represented in a variety of ways. Copy the table and ll in the
missing values.
415
000
using
000
415
products
using
000
000
powers
of
10
1
41
4
500
150
000
000
______
______
×
×
×
×
___
41
500
000
×
10
___
1000
10
000
5
4150
×
10
6
415
2
41.5
×
_____
4.15
×
_____
×
10
Piazzi named that rst asteroid Ceres and it is now known to be the largest asteroid
in the asteroid belt. It measures approximately 950 000 meters in diameter. Use the
same procedure as in step 1 to represent this quantity in a variety of ways.
950
using
000
950
products
using
000
powers
of
10
?
95
000
9500
×
×
___
95
000
×
10
___
?
9.5
3
Look
at
both
examples
a
the
quantity
powers
of
Write
5
The
Celestial
and
Vesta.
down
a
Police
Vesta
Justify
in
why
in
scientic
rule
for
a
mass
meters
rule
of
from
two
259
the
10
in
the
last
row
standard form.
of
each
Describe
table.
the
These
are
components
of
notation.
writing
discovered
has
scientic
your
representation
scientic notation or
general
297 000 000 000
6
10
represented
4
quantities
of
×
a
large
large
number
asteroids
quintillion
Sun.
Verify
in
grams.
your
in
scientic
the
At
rule
asteroid
its
by
notation.
belt:
closest,
writing
Juno
Juno
each
of
is
these
notation.
works.
27
A
quantity
represented
component,
called
the
in
scientic
coecient,
notation
multiplied
has
by
a
a
decimal
power
of
10.
4
2.75
×
10
coecient
Example
a
A
a
of
10
5
Represent
Q
power
the
following
2139
b
quantities
4
980
using
scientic
000
c
notation.
310
billion
2139
The
2.139
×
decimal
number
needs
to
be
between
1000
1
and
10.
3
2.139
×
10
Represent
b
4
980
×
of
10
using
an
exponent.
1000 000
6
×
Create
10
and
310
billion
310
×
A
9
2
×
10
3.10
×
10
decimal
the
an
third
number
power
of
equivalent
way
equivalent
10
3.10
a
nd
create
c
power
000
4.980000
4.98
the
is
as
to
a
ten
between
that
is
one
represent “billion” using
power
of
10.
10
Represent
310
using
scientic
notation.
11
Simplify
Reect
●
and
discuss
Describe
two
advantages
scientic
notation.
of
7
representing
quantities
12
●
Which
quantity
Explain
2 8
1
how
Number
is
you
greater,
know.
using
7.32
×
10
14
or
4.2
×
10
?
using
the
to
expression.
9
×
and
required
laws
of
exponents.
its
ten
N U M B E R
Did
you
Using
know...?
the
bacteria
microscope
in
plaque
organisms
that
he
developed,
scraped
measure
7
–
on
his
Antonie
own
average
Van
teeth.
0.000005
Developing
He
Leeuwenhoek
called
meters
these
discovered
small
“animalcules”.
scientic
notation
II
i
t
er
i
r
Investigation
from
Using
the
0.000006
same
using
procedure
products
as
of
in
10
the
and
0.000006
using
0.
products
using
previous
powers
investigation,
of
represent
n
c
1
B
10.
000006
powers
of
10
−1
0.00006
0.0006
6.0
×
×
×
0.1
0.00006
×
10
___
___
2
Write
0.000000304
3
Write
down
a
in
scientic
general
rule
for
notation.
writing
a
small
number
in
scientic
notation.
4
Van
to
Leeuwenhoek
see
cells,
rule
sperm
whose
by
cells,
made
which
minimum
writing
each
of
other
have
discoveries
a
length
thickness
these
is
0.8
of
as
well.
He
0.0047 cm,
micrometers.
quantities
in
meters
was
and
red
Verify
using
the
rst
blood
your
scientic
notation.
5
Justify
why
Reect
●
and
Compare
small
your
and
rule
works.
discuss
contrast
8
scientic
notation
for
very
−4
●
large
and
very
numbers.
Which
quantity
is
greater,
3.8
×
10
−7
or
9.2
×
10
.
Explain
how
you
know.
●
When
tell
if
a
it
number
is
less
is
than
represented
or
greater
in
than
scientic
1?
notation,
how
can
you
Explain.
2 9
Practice
1
5
Represent
a
these
23 500
b
numbers
in
365 800
scientic
c
notation.
210 000 000
d
3 650 000
e
569 000
f
7 800 000 000
2
2
Represent
these
numbers
as
numbers
6
a
3
1.45
Order
×
b
2.807
following
×
c
numbers
4
1.42
Important
in
×
on
a
9.83
discoveries
scientic
9.8
×
in
(e.g.
number
×
d
line.
physics
3.7
×
7.8
are
×
listed
1.034
×
Explain
302
the
Quantity
e
your
5.06
×
−8
10
f
1676
Speed
table
×
of
Acceleration
1835
Earth’s
1850
light
Average
word
in
is
said
that
“googol”,
a
Write
b
Research
with
1
1
represented
ordinary
air
299
792
14
2.876
Represent
ATL1
Research
d
How
3 0
in
do
have
you
an
he
atom
1
by
should
name
each
10
quantity
as
Quantity
number
in
represented
scientic
km/s
notation
km/s
the
call
2
cm/s
cm/s
microteslas
000
000
teslas
m/s
m/s
nanometer
meters
mathematician
the
“Google”
planned
number
was
to
an
1
Edward
followed
accidental
make
Kasner,
by
100
large
asked
zeros.
misspelling
incredibly
who
of
the
amounts
word
of
people.
quantity
the
think
learned
that
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of
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225
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gravity
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2
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notation.
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3
10
write
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10
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10
4
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the
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and
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10.
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six-word
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notation.
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aid
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move
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form
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N U M B E R
The
development
large
and
ability
to
small
of
scientic
numbers
perform
notation
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mathematical
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numbers?
scientic
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decimal
subtracted
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develop
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Activity
4
1
table
this
the
Do
–
of
for
scientic
for
to
operations
scientic
adding
be
added
become
notation?
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more
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subtracting
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this
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quantities
notation.
Addition
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numbers
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allows
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and
in
scientic
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127
+
345
5212
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3158
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33
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subtract
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distributive property
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column.
evaluate
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expressions
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step
2.
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your
answers
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discuss
you
the
the
9
distributive
about
the
represented
in
property
conditions
scientic
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Activity
necessary
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4?
add
or
notation?
6
●
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express
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this
do
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if
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get
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14.5
×
10
?
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will
you
notation?
Continued
on
next
page
31
4
You
are
them
asked
from
to
add
scientic
the
following
numbers
2
(4.11
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will
you
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×
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3
)
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able
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5
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6
the
sure
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numbers
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same
scientic
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acronym
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)
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×
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)
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your
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subtracting
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few
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of
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notation.
letters
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Share
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discuss
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to
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subtracting
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subtracting
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scientic
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In
1676,
light
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travels
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moons
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at
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large
planet.
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be
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eclipses
speed
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of
light
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8
approximately
how
far
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multiplying
32
1
it
3
×
have
these
Number
10
m/s.
If
traveled?
large
light
Can
numbers
travels
for
scientic
easier?
1
million
notation
seconds,
make
N U M B E R
Activity
5
–
How
far
can
light
travel?
8
Pairs
How
1
far
will
Provide
light
a
travel
reason
or
in
1
million
justication
seconds
for
Step
×
10
×
×
(1
×
10
1
×
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the
of
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following
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10
m/s?
steps.
1)
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)
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6
10
×
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speed
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)
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3
of
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each
at
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6
×
10
)
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14
3
×
10
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14
Light
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will
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travel
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3
How
350
a
4
far
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eye)?
light
from
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5
Generalize
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in
63
correct
million
scientic
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in
quantities
notation.
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●
answer
travel
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11
Explain.
what
your
will
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down
light
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quantities
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to
seconds.
takes
dividing
scientic
than
to
million
working.
does
of
1
calculate
sure
seconds
procedure
and
or
your
far
to
in
in
Sun
480
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(the
represent
scientic
easier
travel
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the
multiplying
●
years).
division
to
meters
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notation
and
written
10
Show
approximately
travel?
2
×
microseconds
human
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3
you
represent
numbers
your
went
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well
in
quantities
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in
numbers
scientic
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product
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example.
in
terms
of
your
learning.
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this
in
mathematics
did
you
develop
or
enhance
in
unit?
3 3
Practice
6
6
1
a
=
1.3
×
10
Evaluate
a
−2
,
b
each
=
of
5a
4.9
the
b
×
10
4
,
c
following.
7b
c
=
8.32
×
10
Represent
b
+
−3
,
d
your
d
d
=
7.6
×
answers
2e
−
10
in
5
,
e
=
scientic
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e
ae
k
d
q
(ce)
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10
notation.
f
bd
2
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h
m
n
i
2
2
In
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Tower
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bc
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de
j
l
2
e
−
−
5c
+
2e
2
c
Hanoi
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o
puzzle
p
(see
pages
2abd
12–13),
you
found
that
it
would
take
the
64
priests
2
−
1
moves
to
transfer
the
64
disks
to
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dierent
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their
order.
a
Using
a
correct
b
calculator,
to
Calculate
3
write
signicant
the
number
down
the
number
of
moves
using
scientic
notation,
gures.
of
seconds
in
one
year.
Represent
your
answer
in
scientic
notation.
c
Legend
new
the
has
post.
3
a
of
However,
to
world
each
complete
one
accident.
sample
the
will
move
the
end
takes
puzzle
if
when
1
the
second.
there
are
monks
Find
64
move
the
disks.
the
number
Show
nal
of
your
disk
years
to
it
the
will
working
take
using
notation.
Considered
by
that
Suppose
monks
scientic
it
In
of
the
1928,
bacteria
where
world’s
Alexander
had
the
greatest
been
mold
Fleming
left
had
discoveries,
out
returned
and
grown,
the
had
penicillin
to
his
become
bacteria
had
was
actually
laboratory
to
contaminated
been
found
nd
by
destroyed.
that
mold.
A
single
−7
bacterium
has
a
diameter
of
8
×
10
meters
and
a
penicillium
mold
spore
has
a
−6
diameter
a
How
of
a
of
3.5
many
×
10
times
meters.
larger
is
the
diameter
of
a
penicillium
mold
spore
than
that
bacterium?
b
Assuming
a
roughly
circular
shape,
c
If
there
are
5
million
bacteria,
d
If
there
are
1
million
bacteria
nd
and
nd
the
1
the
area
area
they
million
of
one
bacterium.
occupy.
mold
spores,
nd
the
total
area
they
occupy.
e
Find
the
number
of
mold
spores
that
would
equal
the
area
of
1
million
bacteria.
Continued
34
1
Number
on
next
page
N U M B E R
4
The
development
over
and
2000
models
objects
into
Goddard
paper
on
extreme
for
the
His
the
years
air.
in
It
and
development
basis
of
eventually
the
the
to
propel
Robert
published
could
paved
the
modern
way
rockets.
formed
programs
humans
a
reach
experiments
space
took
was
of
began
experiments
steam
1919,
rockets
and
rocket
with
used
altitudes
designs
and
the
how
the
ago
that
who,
of
into
that
outer
space
Moon.
5
a
The
average
distance
from
the
Earth
to
the
Moon
is
3.84
×
10
km.
4
A
rocket
can
standard
of
b
fuel
tanks
At
closest,
its
at
to
the
the
return
a
when
How
many
now
or
used
500
the
the
form
each
would
is
50
5.8
on
one
quantity
have
million
×
closest
times
for
to
km
104 km/h,
tank
and
take
of
to
to
fuel.
nd
Use
the
ensure
it
number
could
from
how
Earth.
long
If
will
a
it
rocket
take
can
to
reach
distance?
further
is
but
so
from
Earth
A
6375
also
in
the
a
way
more,
satellite
above
one
on
as
signals
kilometres.
button
with
developed
much
kilometres
satellite
pi
was
television
destination.
Earth
by
of
technology
broadcasting
orbit
km
journey.
its
communication,
new
10
is
Mars
(at
its
closest)
than
moon?
Satellite
are
×
represent
Mars
at
3.60
rocket
speed
Mars
the
5
form
make
travel
c
travel
coecient
and
of
spy
even
Earth’s
the
1960s
on
rounded
the
the
Use
you
Earth
two
the
your
means
people.
collecting
maximum
Write
to
a
surface.The
Earth.
calculator.
as
other
helping
around
Calculate
orbit
your
to
the
including
travels
the
in
Satellites
weather
navigate
in
a
of
distance
answer
of
in
data,
to
a
circular
radius
value
decimal
of
π
the
traveled
as
3.14,
standard
places.
3 5
assessment
i
t
er
i
r
Formative
idea
that
all
development.
as
the
idea
elements:
dierent
how
the
that
materials
and
current
nucleus
orbit
all
earth,
discoveries
The
Before
phases
atom
matter
of
of
will
the
contains
Neutrons
atomic
re
react
water.
with
the
that
and
people
of
Atomic
make
up
includes
protons
had
an
a
and
variety
helps
it
However,
over
an
of
to
allows
even
time,
important
beliefs,
of
basic
explain
you
the
owing
to
the
predict
theory
to
D
such
of
important
atom.
three
and
incredibly
combination
theory
gas)
other.
is
some
developed
atom
neutrons
atoms
liquid,
each
has
of
made
(solid,
particles
of
theory,
were
and
matter
structure
model
around
composed
objects
air,
its
is
n
c
The
is
types
of
particle:
surrounded
by
the
electrons
that
it.
and
protons
have
roughly
equivalent
masses
that
can
be
written
28
as
16.6
×
10
kg.
Using
the
same
power
of
10,
electrons
have
a
mass
of
28
0.00911
Atomic
×
10
kg.
particles
a
Represent
b
Which
each
particle
of
has
these
the
masses
smallest
using
mass?
correct
Explain
scientic
how
you
notation.
know.
Continued
3 6
1
Number
on
next
page
N U M B E R
Current
For
elements
each
of
the
following
notation and show
c
How
d
Oxygen
is
atom
oxygen
kg,
e
many
of
of
The
times
an
one
your
element
greater
has
of
perform
all
of
your
calculations using scientic
working. Express your answers in correct scientic notation.
element
atom
questions,
8
is
the
that
mass
was
carbon-14,
(the
which
a
proton
discovered
electrons,
oxygen
of
8
atomic
is
in
protons
used
in
1772
and
mass).
than
8
by
mass
Carl
your
very
old
of
an
Wilhem
neutrons.
Show
dating
the
electron?
Scheele.
Calculate
the
One
mass,
in
working.
objects,
was
discovered
−26
in
1940
one
in
f
by
atom
one
Martin
of
carbon-14
atom
Because
Kamen
of
the
in
a
has
6
element.
individual
together
and
atoms
larger
Sam
electrons
Show
are
amount,
Ruben.
so
and
your
small,
called
a
Its
6
atomic
mass
protons,
nd
is
2.324728
the
×
number
10
of
of
a
substance
is
6.02
×
they
mole.
are
The
often
grouped
number
of
atoms
in
a
You
,
10
which
is
referred
to
as
If
one
mole
of
water
has
a
mass
of
18 g,
nd
the
can
number
the
metric
of
prexes
atoms
in
1000
grams
(1
liter)
of
The
distance
from
Earth
to
Mars
is
54.6
million
kilometres.
A
has
a
length
of
approximately
280
picometers.
produced
helium
in
molecule
table
water.
you
g
refer
Avogadro’s
to
number.
If
working.
23
mole
kg.
neutrons
Find
Practice
question
many
helium
molecules
could
t
between
these
two
h
new
(see
16).
discovery
Imagine
an
4
planets.
page
A
2
how
you
have
info-graphic
calculations
that
using
of
•
the
name
•
the
number
•
the
properties
•
the
mass
•
the
number
of
discovered
the
of
element
protons,
of
one
of
includes
standard
the
an
the
atoms
in
following
and
its
any
other
information,
on
the
showing
planet.
all
Prepare
necessary
symbol
neutrons
of
unlike
form:
element
atom
element
the
25
and
that
make
element
grams
electrons
of
in
the
it
so
unique
and
valuable
grams
substance.
37
Unit
A
rational
ratio
of
To
the
and
cannot
a
down
the
until
Subtract
then
dened
long
written
as
is
two
is
and
If
0.0353535…
=
=
the
990x
=
35
=
the
and
can
number
For
be
which
the
as
containing
following
quantities
laws
of
raised
example,
it
by
one
decimal
shown
to
an
a
m
with
same
base:
Product
rule
with
same
exponent:
×
a
n
=
a
×
b
n
a
n−m
rule
with
same
base:
Quotient
rule
with
same
exponent:
n
a
power
rule:
(a
m
)
nm
=
0
power
rule:
a
=
1
−m
Negative
power
Fractional
as
the
Irrational
place
0.324
or
is
“three
or
more
are
value
and
hundred
.
powers
exactly
of
the
10
same.
below:
exponent
rule:
exponents:
Number
=
a
a
=
+
m
n
rules:
Quotient
of
its
parts
rules:
Quotient
written
zero”.
as
exponents.
rule
1
using
written
multiply
n
3 8
not
be
fraction:
Product
Zero
is
can
or
Expressions
Power
that
35.353535…
0.35353535…
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a
equation,
=
the
number
denominator
fraction.
in
−10x
x
“any
read
periodic,
numbers
solve
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to
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thousandths”
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fractions.
number
them
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have
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you
be
be
decimal
decimal
twenty
can
integers,
convert
write
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number
two
numbers
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summary
a
n
=
(ab)
can
be
simplied
Quantities
standard
by
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represented
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power
have
of
10.
a
in
scientic
coecient
For
notation
between
1
(also
and
known
10
as
multiplied
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the
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well
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include
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drawing
caption
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describes
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photo.
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the
and
next
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these
great
questions
discovery?
–
Are
What
great
does
it
take
discoveries
to
make
planned
or
accidental?
4 7
2
Many
in
scientic
this
have
Triangles
unit.
However,
taken
aesthetic
your
and
Personal
you
some
of
desired
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the
bring
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to
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completely
and
solutions
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dierent
rely
on
the
important
direction,
triangle,
roles
such
in
as
as
other
our
you
will
contexts
discover
that
appreciation
of
could
the
infrastructure.
expression
great
careful
of
triangles
can
artists
Mona
was
is
also
100%
in
could
enhance
Leonardo
the
art
planning
be
our
realized
da
Vinci
Lisa’s
inspiration
order
a
lesson
power
used
face
in
to
the
his
and
produce
on
appreciation
the
supposed
to
of
of
the
the
Golden
an
the
how
famous
produce
creativity,
aesthetic.
triangle
Triangle
painting.
aesthetically
portrait.
picture?
appear
work.
Triangle
Photographers
in
that
very
study
attention
Golden
stability
think
principles
a
a
have
and
aesthetic
Renaissance,
composing
to
the
A
in
planning
cultural
involves
mathematical
During
of
processes
triangles
study
urban
and
Appreciation
While
principles,
that
triangles
work.
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Is
Can
does
there
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to
you
it
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see
make
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to
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to
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work.
accomplish
that
a
How
in
Globalization
Urban
Cities
planning
are
often
rectangle
However,
population,
becoming
their
A
use
study
you
on
a
and
housing
are
with
space.
could
some
take
very
structures
the
shape
that
three
citizens
streets.
the
use
a
streets
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bordering
interior
well
as
a
Ayala
the
space
roads
to
alternative
alongside
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oer
as
living
space.
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an
a
provide
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views
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walking
also
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Philippines,
between
utilizes
to
gathering
Triangle
oer
development
Spain
exterior
central
the
limited
of
a
(radial).
planners
creative
urban
northern
spaces
to
circle
triangles
tour
triangular
In
a
using
designs.
This
in
infrastructure
ever-increasing
urban
of
innovative
and
or
an
more
of
and
sustainability
planned
(grid)
with
and
the
and
busy
paths
route
than
roads.
4 9
2
Triangles
Principles,
KEY
CONCEPT:
Related
Global
How
was
energy)
unit,
of
was
your
use
the
that
engine
how
a
used
invented?
basic
to
will
and
lead
ideas.
threaten
the
of
Generalizing
How
new
the
you
on
a
These
environment
in
their
ideas
or
of
Solving
problems
(the
of
how
law
can
plague
●
then
our
be
to
These
and
solutions.
and
and
applied
In
to
new
solve
problems
lives.
Inquiry
between
processes
and
measurements
involving
right
help
questions
triangles
What
is
a
What
are
relationship?
theorem
whether
or
not
two
the
three
fundamental
triangles
ratios?
similar
Using
the
properties
of
similar
triangles
How
do
we
missing
trigonometric
ratios
to
measurements?
How
involving
right
much
proof
is
“enough”?
solve
Do
problems
relationships
measurements
D
Using
generalize
to
between
nd
●
can
solutions.
C
●
this
mathematicians
trigonometric
are
of
technical
formulate
everyday
are
conservation
Inquiry
Pythagoras’
Determining
of
scientists
F
using
Measurement
work?
scientic
disciplines
Objectives
●
power
processes
context
relationships
principles,
solar
principle
journey
new
of
does
products,
global
principles
Statement
develop
Generalization,
scientic
create
foundational
methods
solutions
RELA
TIONSHIPS
concepts:
exploration
innovation
and
context:
the
examples
processes
scientific
principles
lead
to
good
triangles
solutions
or
do
good
scientific
principles?
solutions
lead
to
G E O M E T R Y
A
TL2
Thinking:
A
TL1
Critical-thinking
Test
Y
ou
1
generalizations
should
and
already
Find
the
areas
Find
the
area
of
Communication
Give
conclusions
common
how
and
receive
each
shapes
4
Use
properties
meaningful
feedback
of
sizes
parallel
of
lines
to
angles
shape.
Find
a
skills
to:
determine
of
T R I G O N O M E T R Y
Communication:
skills
know
A N D
the
size
of
the
indicated
angles.
b
Justify
each
step.
9 cm
5 cm
4 cm
5 cm
a
E
B
a
5 cm
12 cm
c
b
A
c
e
d
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f
C
F
b
j
l
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Solve
multi-step
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m
k
p
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equations.
s
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c
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h
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Plot
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problems
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ratios
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points.
c
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51
Introducing
What
do
kitchens,
common?
The
of
to
work
Since
activity,
between
the
these
minimize
L-shape
While
limited
range
in
photos
be
of
There
a
distance
a
clear
of
the
planet.
principles
the
as
to
right
a
right
those
photos,
apply
produce
amount
of
detail.
Even
to
solving
determine
that
like
are
very
Mount
the
the
In
a
Everest,
unit,
an
you
in
triangles
to
develop
principles,
the
2
of
Triangles
tools
humans
clever
and
may
the
measures
solutions.
make
discovery
this
easily
discover
pretty
processes
great
investigate
5 2
will
you
be
right
allowed
some
locations
triangles.
between
has
knows,
next
of
how
access,
can
the
of
of
to
understanding
relationships
Who
with
knowledge
this
how
height
dicult
accomplished
and
problem
as
common
you
shape.
stove
even
kitchen
This
using
such
aerial
of
and
when
all
have
in
is
designing
are
the
based
optimal
between
Earth,
view
used
kitchen
are
U-shape
above
Everest
triangles!
refrigerator
walking
Cameras,
with
Mount
concept
design
calculated
taking
triangle
sink,
a
kitchen
portion
can
is
appliances.
the
have
triangles.
used
the
and
enough,
triangle
optimal
orbiting
satellites
satellites
Interestingly
kitchen
kitchens.
triangles
these
main
on
a
centres
triangle
measurements
areas.
Island
kitchen
G E O M E T R Y
Reect
●
How
easy
your
home?
what
●
Give
Do
is
kind
What
●
and
it
to
two
you
an
true
mathematician
This
of
be
lead
are
may
many
hand,
is
equal
seem
other
a
guesses)
a
sink
kitchen
and
work
refrigerator
triangle?
If
in
so,
have
you
seen
that
use
triangles?
to
a
until
scientic
or
obvious
that
has
a
needs
are
solutions
or
do
Explain.
statement
no
also
But
evidence
A
proved,
start
out
equal
is
but
that
For
such
to
it
with
conjectures
does
it
as:
one
on
take
is
example,
“Things
another”.
forms
theorem,
often
as
what
proof.
axioms,
statement,
been
proved.
much
is
principles.
Theorems
are
good
several
thing
mathematical
How
postulate
stated
same
rather
they
to
principles?
therefore,
Euclid
the
lead
proof
and,
axioms.
something?
have
principles
axiom
statement
established
to
and
mathematics,
which
stove,
it?
solutions
scientic
Theorems
the
the
examples.
think
to
or
kitchen
is
T R I G O N O M E T R Y
1
between
your
triangle
solutions
assumed
move
Does
of
processes
good
In
discuss
A N D
the
the
the
basis
other
help
of
(educated
to
prove
enough?
Proof
Mathematics
coming
Proofs
that
In
it
up
are
is
is
based
with
how
new
is
court
the
statements
reasoning,
based
on
construct
true
which
means
statements.
knowledge
and
know
correct.
to
of
deductive
mathematicians
mathematics,
goal
on
proof
convince
law,
proof
a
is
people
“reasonable
must
be
a
proved
logical
that
something
doubt”
to
be
sequence
is
true
not
for
is
of
arguments
true.
Unlike
acceptable.
every
case,
whose
in
Every
a
step
without
of
any
doubt.
53
Activity
1
–
Is
a
pattern
enough
proof ?
1
3
2
2
1
Looking
at
the
Number
of
diagrams
points
on
above,
copy
the
and
complete
this
table.
Number of dierent regions in the
circle
circle
1
2
3
4
5
2
What
do
you
predict
will
be
the
number
of
regions
if
there
are
six
A
points
on
the
circle?
What
about
seven
points?
Explain
a
reasoning
in
each
conjecture
conclusion
Write
down
a
conjecture
predicting
how
many
regions
there
will
any
circle
as
compared
with
the
number
in
the
previous
Do
your
step
5
ATL1
3?
Look
at
results
in
this
activity
prove
the
statement
you
wrote
in
regions
in
you
the
when
diagrams
there
are
below
six
and
points
count
and
the
when
number
there
are
of
seven
have.
the
drawing
points.
two
rows
to
your
table
for
these
copies
results.
of
these
two
diagrams,
can
count
Was
7
Is
your
establishing
Activity
is
Proving
5 4
1
“always
that
conjecture
are
2
is
a
a
pattern
good
true”
in
Triangles
Explain.
enough
example
of
and
to
be
true
accepted
proof?
why
mathematics
something
known
true?
you
based
has
to
facts.
Explain.
cannot
on
start
just
from
a
say
few
basic
something
examples.
principles
so
number
regions
6
on
Explain.
larger
Add
based
information
that
Try
circle
is
circle.
the
4
guess
be
that
in
or
case.
educated
3
is
your
as
you
the
you
them.
G E O M E T R Y
What
A
is
theorem
true.
did
now
As
bears
go
and
statement
you
are
circles
young
study
still
was
homework!
through
Activity
1770,
to
math
the
that
theorems
related
theorems
In
mathematical
new
her
you
a
T R I G O N O M E T R Y
theorem?
mathematics
but
theorem
she
is
The
work,
a
A N D
that
is
being
unit,
you
later
will
be
on
proved
by
an
proved
name:
In
2016,
Israeli
her
be
of
a
new
teenager
theorem,
“Tamar’s
encounter
to
centuries
developed.
mathematician’s
this
based
established
She
can
as
which
theorem”.
examples
of
both
proofs.
2
–
Joseph
Four-square
Louis
would
eventually
“Every
positive
be
Lagrange
called
integer
can
the
be
theorem
published
the
proof
of
a
theorem
Lagrange four-square theorem.
written
as
the
sum
of
the
It
squares
that
states,
of
four
integers.”
2
For
1
example,
17
2
+
1
b
Select
four
3
=
3
2
+
3
2
+
2
4
or
11
=
2
3
+
1
2
+
1
2
+
0
Write the following numbers as the sum of the squares of four integers.
a
2
35
any
30
positive
c
integer
43
and
d
write
it
as
the
103
sum
of
the
squares
of
integers.
With
a
peer,
number
play
that
the
the
following
other
player
game.
must
Each
write
person
as
the
selects
sum
a
2-digit
of
the
squares
of
solve
the
problem
Pairs
four
and
integers.
one
solved.
Add
Reect
●
Have
●
The
the
Greek
to
like
discuss
the
for
the
to
every
time
create
make
a
you
problem
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game
that
more
cannot
be
challenging.
2
four-square
area
to
about
the
of
the
area
right
circumference
know
dicult
this
for
you
restriction
mathematician
already
How
point
time
theorem
with
your
examples?
answer.
equal
sides
the
you
●
is
one
every
time
proved
your
formula
circle
the
a
for
and
you
Justify
Score
point
one
do
by
of
circle
is
actually
Archimedes.
of
a
think
is
equal
circle.”
the
it
area
is
Archimedes?
It
to
to
How
of
a
a
theorem
states,
right-angled
angle
the
about
you
a
“The
triangle
the
does
in
radius,
this
proved
area
of
which
and
relate
any
one
the
to
by
of
other
what
circle?
prove
mathematically
a
statement
Explain.
5 5
One
of
the
most
Pythagoras’
the
famous
theorem.
measurements
of
It
theorems
that
establishes
the
sides
in
a
you
will
learn
relationship
any
right
A
is
between
triangle.
hypotenuse
leg
In
a
right
side
the
triangle,
across
from
hypotenuse.
the
the
The
sides
right
other
have
angle,
two
specic
the
sides,
names.
longest
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The
side,
form
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right
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Reect
●
Explain
longest
and
why
discuss
the
side
Would
you
●
Where
is
say
the
that
the
found
activities
by
his
Did
you
Samos
he
in
most
angle
has
to
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have
side
in
“proven”
a
triangle
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result?
located?
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Be
as
specic
in
this
section,
and
you
examine
will
explore
some
of
the
the
as
relationship
proofs
associated
know...?
was
a
around
famous
lengths
never
of
did
principles.
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Greek
569
sides
down
later
Triangles
born
on
the
Island
in
any
explained
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right-angled
of
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relationship
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of
triangles.
ndings,
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between
Although
followers
and
on.
theorem
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mathematician
BC.
theorem
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wrote
Pythagoras’
2
right
theorem.
students
5 6
the
theorem
Pythagoras
Pythagoras
the
from
can.
with
His
you
shortest
Pythagoras’
In
across
3
side.
●
you
legs
is
possibly
proved
the
both
most
visually
proven
and
of
all
algebraically.
G E O M E T R Y
Activity
Your
l
3
teacher
i
n
–
Pythagorean
will
give
you
the
A N D
T R I G O N O M E T R Y
puzzles
Pythagorean
puzzle
cutouts.
k
b
e
You
can
also
download
and
print
www.jamieyorkpress.com
“Grade
7
Downloads”,
–
where
this
Select
you
puzzle
“Free
will
nd
from
the
following
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the
and
website:
scroll
“Pythagorean
down
Theorem
the
Cutout
Puzzle”.
1
Cut
the
2
What
on
3
ATL2
out
the
pieces
from
each
of
the
smaller
squares
and
rearrange
them
to
form
a
square
on
hypotenuse.
generalization
each
With
a
of
sides
partner,
questions
Follow
the
to
that
of
look
clarify
with
a
can
a
at
you
deduce
right
each
the
relationship
between
the
area
of
the
squares
triangle?
other’s
anything
about
that
suggestion
for
generalizations.
isn’t
quite
clear.
improvement,
Take
Then,
if
turns
give
giving
at
least
feedback
one
by
positive
rst
asking
comment.
appropriate.
Pairs
The
previous
relationship
is
activity
between
Pythagoras’
smaller
ones,
discover
is
this
a
the
theorem?
as
you
good
sides
of
a
Breaking
will
famous
rst
see
in
step
right
up
the
in
triangle.
the
next
generalizing
squares
activity,
But
into
will
the
what
exactly
much
help
you
to
relationship.
Investigation 1 – Discovering Pythagoras’ theorem
t
er
i
r
i
can
use
following
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i
n
paper
and
pencil
or
dynamic
geometry
software
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perform
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the
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activity.
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use
for
on
middle
6
of
units
link
the
a
applet
on
the
investigation.
term
the
choose
measure
the
this
“Exploring
that
size
piece
and
8
the
appears
of
of
learnalberta.ca
In
squared
“Find
the
you
right
website
Resources”
Pythagorean
on
triangle
the
of
instead
section
Theorem”
the
page.
in
the
Select
of
of
drawing
the
“Enter
the
home
the
triangles
page,
Keyword”
“Interactive”
enter
box.
Then
option
want.
paper,
draw
a
right
triangle
whose
legs
units.
Continued
on
next
page
57
2
Construct
and
the
3
squares
width
of
the
on
these
square
is
two
the
legs.
(The
length
of
length
the
side
y
of
30
triangle.)
Find
the
number
squares.
What
of
does
small
this
squares
in
these
two
new
represent?
20
4
Construct
5
Count
a
the
square.
new
total
You
square
on
number
may
need
of
to
the
hypotenuse.
squares
count
in
“half
this
larger
squares”
10
together.
6
What
do
square?
smaller
you
How
notice
does
about
it
relate
the
to
area
the
of
this
area
of
larger
the
two
squares?
x
0
10
7
Repeat
legs
steps
that
1
through
measure
5
6
and
for
12
a
right
triangle
with
triangle
with
20
30
units.
y
8
Repeat
legs
9
If
that
the
you
steps
1
through
measure
measures
calculate
of
the
8
6
and
the
area
for
15
legs
of
a
right
30
units.
were
the
a
and
square
b,
how
formed
would
on
20
each
10
leg
Write
of
the
down
between
squares
drawn
Pythagoras’
the
areas
of
on
theorem
the
three
these
as
a
sides?
relationship
squares.
10
ATL1
11
Verify
that
your
formula
works
for
two
more
right
triangles:
12
a
a
right
b
a right triangle with legs of length 20 and 21 units.
Justify
triangle
why
Reect
your
and
with
legs
formula
of
length
7
and
24
units
x
0
10
20
30
works.
discuss
4
Answer these questions on your own and then discuss them in a group.
●
Explain
how
between
●
Show
not
Pythagoras’
theorem
generalizes
a
relationship
measurements.
that
Pythagoras’
right-angled.
theorem
Explain
your
does
not
work
for
triangles
2
●
Most
people
2
do
the
a
triangle
remember
2
,
is
b
that
are
method.
Pythagoras’
theorem
as
a
2
+
b
2
=
c
.
What
2
and
c
isolated
represent?
in
the
How
theorem?
do
you
know
which
side
of
the
A
term
2
Triangles
is
isolated
is
itself
one
of
5 8
that
on
the
by
side
equation.
G E O M E T R Y
There
are
algebra
over
while
Activity
A
visual
l
i
350
proofs
others
4
proof
–
of
A
are
of
Pythagoras’
visual
visual
Pythagoras’
theorem!
Some
A N D
T R I G O N O M E T R Y
involve
proofs.
proof
theorem
of
Pythagoras’
using
two
squares
is
theorem
shown
below.
n
b
e
For
an
animation
“Pythagoras’
showing
Theorem
this
Proof
visual
proof,
Animation”
go
and
to
youtube
select
the
Explain
large
how
you
squares
know
have
that
the
both
same
of
search
by
for
Maths
a
b
1
and
video
Whenever.
a
b
the
area.
a
c
2
Write
the
down
area
of
diagram
an
the
on
expression
green
the
to
square
b
calculate
in
c
the
c
b
right.
c
3
Write
the
down
area
of
an
expression
each
of
the
to
two
calculate
blue
a
squares
a
in
4
the
How
diagram
do
square?
5
How
area
you
With
a
this
the
remaining
6
know
the
b
a
b
left.
that
the
sum
of
the
areas
of
the
blue
squares
equals
the
area
of
the
green
Explain.
does
of
on
prove
square
two
partner,
Pythagoras’
on
the
theorem?
hypotenuse
is
(The
equal
theorem
to
the
sum
states,
of
the
“In
any
areas
of
right
the
triangle,
squares
the
on
the
sides.”)
look
at
each
other’s
proofs.
Take
turns
giving
feedback
by
rst
asking
ATL2
questions
Follow
to
that
clarify
with
a
anything
that
suggestion
for
isn’t
quite
clear.
improvement,
if
Then,
give
at
least
one
positive
comment.
appropriate.
Pairs
Reect
and
discuss
5
In small groups, discuss the following questions.
●
●
What
do
have
in
Look
on
the
common?
the
Pythagoras’
●
Which
of
Explain
dierent
ways
of
showing
Pythagoras’
theorem
all
Explain.
internet
for
a
video
showing
the
water
proof
of
theorem.
the
your
proofs
that
you
have
seen
gives
the
clearest
evidence?
reasoning.
5 9
Applying
Pythagoras’
sides
a
of
side
of
the
two
a
in
theorem
any
the
sides
objects
tallest
Hyperion,
tree
in
A
is
of
a
tree
to
measure
height
is
to
a
in
the
its
of
a
is
relationship
allows
you
provided
especially
measure,
the
to
but
nd
you
useful
the
between
the
know
in
the
length
the
is
of
lengths
situations
third
three
where
not.
tall
by
rangender.
named
redwood
USA.
A
surveyor
from
laser
tree
He
of
the
base
rangender
is
beam
top
very
easier
world,
height.
the
laser
to
laser
the
away
uses
of
made
coastal
40 m
the
surveyor
easy
It
triangle
This
height
been
tree
of
long
are
California,
standing
the
sides.
the
has
invention
The
triangle.
the
theorem
1
Measuring
the
generalizes
right-angled
two
Example
Q
Pythagorean
right-angled
other
of
the
nds
116 m.
from
the
that
How
the
tree?
y
Construct
right
a
right-angled
triangle.
Be
sure
to
indicate
angle.
Label
the
legs
a
and
b,
and
the
hypotenuse
c
c
a
x
2
c
2
=
a
=
40
2
+ b
Apply
2
c
2
Pythagoras’ theorem,
using
the
values
that
2
+ 116
you
know :
a
=
40 m,
b
=
116 m.
2
c
= 1600 + 13 456
2
c
= 15 056
Do
c
the
The
6 0
not
forget
= 123 ( 3  s.f .)
length
2
of
the
Triangles
laser
is
123 m.
value
of
c.
to
take
the
square
root
in
order
to
nd
the
G E O M E T R Y
Practice
1
Find
the
A N D
T R I G O N O M E T R Y
1
value
of
x
in
each
diagram.
Round
your
answers
to
the
nearest
tenth
where
necessary.
a
x
b
c
1.6 m
x
7 mm
66 cm
8.2 m
x
6 mm
40 cm
d
e
f
26 m
10 m
5 m
24 m
25 cm
x
x
x
7 cm
2
Use
Pythagoras’
theorem
to
determine
a
whether
the
following
triangles
b
contain
right
angles.
9 cm
c
13 cm
12 cm
5 cm
21.6 cm
12 cm
15 cm
6 cm
23.4 cm
3
The
length
rectangle
of
is
the
length
b
the
area
The
5
In
To
try
base
a
If
A
of
to
of
a
the
the
in
a
solve
ladder
How
ladder
How
is
other
a
long
are
problem,
reach
be
a
should
the
10 cm
industry,
should
to
up
rectangle
is
50 cm.
If
the
length
of
one
side
of
the
side
square
measuring
high
a
rectangle.
the
ladder
of
calculate:
construction
wall.
b
of
diagonals
the
diagonal
30 cm,
a
4
the
from
the
the
of
the
to
Calculate
of
all
it
5.6 m,
be
in
have
ladder
is
been
base
order
to
base
length
of
happen
introduced
leaning
the
its
the
accidents
have
wall
ladder
needs
can
third
regulations
height
4 m
wall
one
long.
the
sides
while
that
of
using
specify
the
a
square.
ladder.
how
far
the
against.
must
meet
placed
be
at
this
least
safety
97 cm
1.4 m
from
the
regulation?
from
the
wall.
reach?
Continued
on
next
page
61
6
Surveying
is
the
constructing
maps
measurements.
cartographers
heights
and
dierent
through
device
sight
form
a
right
calculate
the
top
stick
and
Suppose
on
away.
surveyor
to
A
a
the
process
that
the
between
(a
is
a
telescope’s
and
is
a
line
stick
the
height
horizontal
stick
at
known
Since
length
the
looks
measuring
measuring
the
that
calculate
telescope)
angle.
the
on
surveyor
The
knows
of
a
distances
based
distance
is
use
of
based
theodolite
measuring
of
It
points.
a
science
of
the
stick
steepness
distance
90 m
and
of
a
and
hill
between
the
stick
the
or
the
horizontal
distance
to
it,
she
can
slope.
theodolite
measures
90 m
and
1.5 m.
1.5 m
How
stick?
7
The
the
far
is
the
Show
heights
tops
of
top
your
of
the
of
the
theodolite
from
the
bottom
of
the
working.
two
vertical
towers
is
towers
61 m,
are
how
far
75 m
and
apart
are
65 m.
the
If
the
middle
shortest
of
the
distance
bases
of
between
the
Continued
6 2
2
Triangles
towers?
on
next
page
G E O M E T R Y
8
Over
the
last
technology
wider
they
has
range
television
are
few
of
decades,
improved,
screen
screen
measured
to
using
T R I G O N O M E T R Y
television
allowing
sizes.
sizes
A N D
To
be
much
enable
easily
the
a
compared,
diagonal
of
the
television.
Suppose
and
the
height
Will
9
you
cabinet
of
the
Groups
have
13
you
inches
television
of
three
“Pythagorean
Pythagorean
a
3,
4,
b
5,
12,
…
c
8,
…,
17
d
…,
bought
want
and
t
in
whole
triples”.
triples.
a
a
42-inch
to
put
width
the
it
in
of
has
39
a
inches.
cabinet?
numbers
Find
They
television
the
are
that
satisfy
third
given
Pythagoras’
number
in
that
numerical
theorem
would
make
are
these
called
into
order.
…
A
10
You
40,
41
proved
squares
on
Pythagoras’
each
of
the
theorem
sides.
by
Does
constructing
the
formula
work
b
with
If
other
shapes,
semicircles
triangle
are
whose
for
example
constructed
sides
measure
=
4
c
=
5
circles?
on
the
3 cm,
sides
4 cm
of
a
and
right
5 cm,
B
C
show
l
i
n
that
Pythagoras’
theorem
still
a
=
3
applies.
k
b
e
The Pythagorean theorem is so famous, it has been quoted in several movies! Unfortunately,
not all of them have used the theorem correctly. If you search for “Scarecrow Pythagorean
Theorem” on teachertube.com, you will see one of the more famous blunders!
The
of
Pythagorean
triangles.
coordinate
However,
geometry
Putting
The
it
as
is
has
a
distance
plane
can
very
also
important
formed
the
tool
basis
in
of
the
study
solutions
in
well.
Pythagoras
shortest
coordinate
theorem
between
always
on
any
be
the
two
map
points
calculated
on
using
the
Cartesian
Pythagoras’
theorem.
63
Investigation
2
– The
distance
formula
t
er
i
r
i
Draw
the
shortest
distance
between
the
coordinates
below.
Measure
this
distance.
B
y
10
9
Point:
(3,
6)
8
7
6
5
4
Point:
(6,
2)
3
2
1
0
x
0
1
2
2
Create
3
Show
4
Repeat
c
5
(2,
a
right
(–1,
4
you
activity
triangle
and
–2)
5
6
triangle
can
the
a
5)
Using
3
how
create
a
(5,
and
to
7
use
for
the
Verify
the
with
the
segment
pairs
illustrate
,
y
formula
drew
to
in
step
calculate
coordinates
below.
1
as
the
Plot
the
hypotenuse.
distance
each
you
pair
of
measured.
points
and
ndings.
b
(–13,
d
(15,
) and( x
1
between
for
2)
and
(–1,
–1)
and
(3,
y
,
2
works
you
theorem
of
your
–8)
distance
your
10
Pythagoras’
points ( x
shortest
that
9
1)
(–7,
general
8
with
1
6
),
7)
–6)
generalize
your
ndings
to
nd
a
formula
2
any
two
two
points.
more
pairs
of
points.
Show
that
you
obtain
ATL1
the
same
result
7
Justify
why
8
With
partner,
a
as
your
when
you
formula
look
at
plot
works
each
in
the
points
every
other’s
and
create
a
right
triangle.
case.
distance
formulae.
Take
turns
giving
feedback
by
ATL2
rst
asking
positive
questions
comment.
to
clarify
Follow
that
anything
with
a
that
isn’t
suggestion
quite
for
clear.
Then,
improvement,
Pairs
Reect
●
Does
it
and
matter
discuss
which
point
6
you
use
rst
in
your
distance
formula?
Explain.
●
Explain
how
the
distance
formula
is
an
application
theorem.
●
6 4
n
c
1
Do
2
you
need
Triangles
to
memorize
this
formula?
Explain.
of
Pythagoras’
give
if
at
least
one
appropriate.
G E O M E T R Y
Example
Q
When
and
per
distances
measured
speeds
hour).
in
in
knots
Because
y
are
nautical
(nautical
one
9
miles
8
miles
7
nautical
Robin
mile
is
equal
latitude,
T R I G O N O M E T R Y
2
boating,
typically
A N D
to
this
one
solves
minute
the
Hood’
s
Bay:
(2.5,
5
problem
Boat:
of
having
to
to
latitude
convert
4.6)
6
of
(7.7,
2.5)
4
distances
3
and
longitude
when
2
navigating
on
open
water.
1
Using the map below, calculate
0
x
the shortest distance between the
0
1
2
3
4
5
6
7
8
9
boat and Robin Hood’s Bay. (Each
distance is measured in nautical miles.)
y
A
9
8
7
Robin
Hood’
s
Bay:
(2.5,
4.6)
Construct
6
a
right-angled
triangle.
Be
sure
to
Ruswarp
indicate
the
right
angle.
Hawaker
5
Robin
Hood’
s
Bay
4
Boat:
(7.7,
2.5)
Ravenscar
Fylingdales
3
Staintondale
2
Cloughton
Harwood
Dale
Newlands
Cloughton
1
Burniston
Langdale
End
Hackness
0
x
0
1
2
2
d
3
4
2
= ( x
−
2
x
)
5
6
2
+ ( y
1
−
2
y
7
Let
2
d
with
the
coordinates
(2.5,
4.6)
be
( x
,
1
the
boat
with
the
coordinates
(7.7,
2.5)
be ( x
,
2
2
y
)
and
1
y
)
2
2
+ ( −2.1
)
Do
2
d
Bay
+ ( 2.5 − 4.6 )
2
= ( 5.2 )
Hood’s
)
2
= (7.7 − 2.5)
Robin
9
1
let
2
d
8
not
forget
to
take
the
square
root
to
nd
the
distance.
Because
= 27.04 + 4.41
you
are
dealing
with
distances,
only
the
positive
root
makes
sense.
2
d
= 31.45
distance
=
distance
=
31.45
5.61
The
shortest
Bay
is
5.61
nautical
distance
nautical
miles
between
the
boat
and
Robin
Hood’s
miles.
6 5
Practice
1
Find
the
Show
a
2
(0,
is
is
it
an
where
parts
in
of
it
space,
should
to
use
each
your
b
(–7,
d
(5,
electronically,
be
of
the
answers
–11)
–2)
for
to
and
and
Pythagoras’
the
(4,
(13,
example
missing.
following
pairs
nearest
of
points.
tenth.
–25)
–2)
when
Computer
theorem
you
download
programmers,
to
make
sure
a
song,
who
that
the
imagine
data
be.
original
it
round
3)
sent
want
and
between
10)
(4,
points
where
Given
(3,
and
data
as
distance
working
and
don’t
data
is
4)
–6)
When
you
shortest
your
(–1,
c
2
point
should
be
and
(a
its
nal
distance
d
location,
from
its
determine
original
whether
location).
or
Copy
not
and
a
piece
of
complete
data
this
table.
Original
Final
location
location
(2,
6)
(–1,
(1,
–1)
(0,
3
In
3)
It
then
b
For
6
7
–5)
13
the
which
i
Player
ii
Player
iii
Player
iv
Player
Establish
of
this
(1,
C
E
value
(3,
the
4)
(–2,
and
location
Justify
easily
with
a
for
D
F
two
is
calculate
game
is
2
was
travel
the
would
a
using
that
of
“hit”
her
be
data
end
supposed
coordinates.
between
have
to
the
was
distance
programed
units
Did
( d)
tracked
distances
within
(2,
been
register
a
up
where
to?
it
(Y/N)
The
characters.
set
for
“hit”
certain
by
a
opponent.
registered?
2)
(0,
(4,
Player
1)
1)
H
(–1,
players
0)
(other
than
those
above)
where
a
“hit”
would
be
answer.
coordinate
the
is
B
Player
Player
your
record
to
scenarios
Player
and
character
minimum
character
and
–3)
each
role-playing
and
2)
2)
use
a
of
theorem
following
(–1,
G
a
if
the
A
Archeologists
to
location
Pythagoras’
sword
In
them
5)
example,
registered.
4
–1)
uses
character’s
a
(2,
compares
actions.
data
to
10
games,
program
Distance
supposed
14)
(–5,
video
distance
traveled
(8,
(–4,
7)
Actual
grid
location
system
of
when
artefacts
at
a
excavating
site,
which
a
site.
This
simplies
allows
future
excavations.
Continued
6 6
2
Triangles
on
next
page
G E O M E T R Y
If
the
coordinates
skeleton
nearest
5
The
are
(6.2,
tenth
distance
of
(in
cm)
22.3)
a
of
and
the
(9.1,
centimetre.
between
(–2,
6)
top
and
bottom
of
the
11.8),
nd
the
length
(Assume
that
the
arm
and
(4,
y)
is
10
units.
arm
of
bone
Find
bone
the
is
the
A N D
of
arm
T R I G O N O M E T R Y
this
bone,
perfectly
values
of
dinosaur
rounded
to
the
straight.)
y.
Show
your
working.
6
A
ship
is
transporting
whose
coordinates
(2152,
1470)
total
length
to
of
in
refuel
the
goods
from
nautical
and
voyage
miles
then
in
China
are
to
United
(–2140,
continues
nautical
the
on
to
1320).
Los
States.
It
It
stops
Angeles
leaves
o
in
(5120,
Shanghai,
Honolulu
2016).
Find
the
miles.
NORTH
AMERICA
Los
Angeles
(5120,
2016)
Shanghai
(–2140,
1320)
Honolulu
(2152,
1470)
67
assessment
–
How
can
“the
same”
i
t
er
i
r
Formative
The
“not
screen
diagonal.
have
The
the
Do
:
9.
or
smaller
aspect
Find
the
screen
and
c
the
Repeat
ratio
d
of
Explain
televisions,
with
the
is
the
same
measure
screen
of
size
also
cell
are
:
is
in
a
the
ratio
phones
ratio
values.)
of
of
the
rectangular
typically
16
:
However,
9,
have
but
some
an
they
aspect
may
newer
side
be
ratio
larger
models
have
9.
phones
is
given
as
127 mm,
but
dierent.
of
aspect
a
rectangular
ratio
of
16
:
9
cell
phone
whose
diagonal
of
your
18.5
:
screen.
to
calculations
Show
the
for
a
measurement,
all
of
nearest
cell
but
your
working
hundredth.
phone
with
an
with
the
aspect
9.
phone’s
and
this
answer
screen
has
discuss
degree
of
a
larger
area?
7
accuracy
of
your
solution
to
the
cell
phone
problem.
●
Does
●
Why
your
solution
would
a
make
company
sense
change
in
the
the
context
aspect
6 8
Which
2
phone
Triangles
would
you
rather
have?
of
ratio
Explain.
●
their
127 mm.
area
the
and
lengths
diagonal
Which
like
advertised
screen
two
is
an
these
of
Reect
●
18.5
side
with
round
same
a
those
ratio
measures
Find
of
of
size
aspect
phones,
phones
sides
than
ratios
D
area?
(Their
screen
their
cell
Televisions
16
b
of
cell
same
of
a
size
aspect ratio
lengths.
The
equal”?
n
c
be
Explain.
the
of
its
problem?
phone?
G E O M E T R Y
Relationships
Pythagoras’
lengths
of
between
theorem
the
relationship,
sides
it
establishes
in
only
between
relationships
exist
there
and
Regardless
each
other
of
or
Activity
Below
are
the
type
similar
–
some
the
right
side
to
pairs
of
Are
right
in
is
an
important
there
other
triangles?
other
types
the
Do
of
triangle?
triangles?
triangles
other
Congruent
in
between
this
triangles.
dierent
triangle,
each
While
lengths
between
of
relationship
lengths
congruent
the
5
to
T R I G O N O M E T R Y
triangles
triangle.
side
between
relationships
Similar
right
applies
relationships
Are
a
A N D
two
or
triangles
can
be
congruent
to
neither.
triangles
congruent
triangles.
P
K
J
R
5 cm
9.2 cm
9.2 cm
10.5 cm
10.5 cm
L
5 cm
Q
A
D
30°
30°
A
Q
P
3 cm
6 cm
6 cm
R
60°
60°
B
1
Use
your
observations
of
these
diagrams
to
help
3 cm
you
C
write
a
denition
of
“congruent
triangles”.
2
How
much
information
congruent?
do
you
think
would
be
enough
to
prove
that
two
triangles
are
Explain.
Continued
on
next
page
6 9
3
Do
you
think
the
following
pairs
of
a
triangles
are
necessarily
congruent?
your
answer.
F
b
T
Justify
D
B
25 m
E
20°
O
35 m
A
125°
C
S
20°
35 m
25 m
35°
15 m
35°
M
125°
15 m
N
U
R
c
S
B
d
C ′
O
B ′
A
P
l
i
n
A ′
C
Q
k
b
e
On
the
this
illuminations.nctm.org
interactive
triangles
Similar
other,
triangles
although
Knowing
further
range
7 0
also
this
two
problems.
Triangles
resource
to
discover
the
search
for
“congruence
conditions
necessary
congruent.
have
is
a
particular
dierent
triangles
mathematical
of
2
that
are
website,
than
are
principles
relationship
that
similar
as
well
of
each
congruent
allows
as
to
you
to
solutions
triangles.
develop
to
a
wide
theorems”.
to
prove
two
Use
G E O M E T R Y
Investigation
3
–
Proper ties
of
similar
A N D
T R I G O N O M E T R Y
triangles
t
er
i
r
i
you
can
can
draw
use
1
Draw
2
Measure
sides
two
of
the
Compare
4
Draw
5
Do
6
In
three
a
two
of
this
the
investigation
software
dierent
of
sizes
sides.
to
that
What
by
hand
achieve
both
do
the
have
you
a
notice
or
same
50°
angle
about
with
triangles
those
of
of
a
dierent
peer.
sizes
results
table
like
from
the
and
steps
2
following,
similar
and
3
still
What
that
do
have
you
70°
lengths
angle.
of
the
the
notice?
same
angles
and
measure
generalize
hold?
the
relationships
between
between
Relationship
angles
measures
in
between
sides
triangles
Congruent
7
the
a
triangles.
Relationship
ATL1
and
sides.
congruent
Similar
B
results.
triangles?
results
right
in
geometry
lengths
your
two
your
triangles
triangles
the
3
the
the
dynamic
n
c
You
triangles
Verify the relationship between the sides of similar triangles for another pair of
triangles.
8
Justify
why
Reect
●
Are
●
Can
all
that
right
two
are
not
triangles
have
triangle
are
the
corresponding
order
in
each
of
similar
triangles
exists.
Explain.
same
angle
measures
but
side
lengths
Explain.
is
∼.
The
notation
for
showing
that
two
is
ΔABC
The
sides
similarity
“similar”
similar
the
8
similar?
proportional?
for
between
discuss
triangles
symbol
triangles
relationship
and
Proving
The
the
angles
and
∼
ΔDEF
sides
must
be
written
in
the
same
triangle.
71
Reect
If ΔABC
●
and
discuss
8
∼ ΔDEF ,
write
down
the
angles
that
are
equal
Your
be
●
write
down
the
ratios
of
the
sides
that
are
Investigation
two
triangles.
If
3,
you
two
developed
triangles
are
the
the
should
form
proportional.
“
In
ratios
in
principle
similar,
of
then
similarity
the
”
between
following
are
true:
●
●
In
their
corresponding
angles
their
corresponding
sides
order
that
be
one
true,
to
prove
of
these
then
Activity
–
two
know
are
equal
proportional.
triangles
conditions
you
6
that
are
that
is
true.
the
Similarity
are
similar,
Once
other
one
you
of
statement
need
them
is
is
also
to
show
proved
to
true.
postulates
ATL2
Perform
this
feedback
one
activity
by
rst
positive
with
asking
comment.
a
peer,
taking
questions
Follow
to
that
turns
clarify
with
a
to
describe
anything
your
that
suggestion
for
thinking.
isn’t
quite
Also
clear.
improvement,
if
take
Then,
turns
give
at
giving
least
appropriate.
Pairs
1
Each
(e.g.
diagram
ΔABC
Justify
shows
∼ ΔDEF ).
your
a
pair
Which
of
similar
principle
triangles.
of
Identify
similarity
allows
the
triangles
you
to
prove
that
are
them
similar
to
be
similar?
answer.
a
b
P
63°
Q
39°
10 cm
40 cm
C
R
39°
B
63°
A
Continued
72
2
Triangles
on
next
page
G E O M E T R Y
c
A N D
T R I G O N O M E T R Y
d
A
E
A
C
5
13
B
D
C
B
D
E
D
e
B
f
A
C
A
D
E
Q
g
A
h
P
D
B
C
2
One
of
the
Angle”.
It
triangle,
angles
3
show
4
A
the
it
two
are
sides
is
Applications
Once
that
This
you
their
have
are
information
some
of
which
are
an
help
more
similar.”
What
you
“If
example
to
are
are
similar
congruent
Explain
why
you
to
is
two
only
called
“AA”
angles
need
to
of
for
“Angle-
another
prove
that
two
do
you
think
this
refers
to?
Use
an
example
to
two
of
sides
angle
two
in
of
similar
demonstrate
one
both
triangle
triangles
triangles
what
this
are
is
and
proportional
congruent,
indicate
postulate
to
then
the
two
the
measures
of
means.
triangles
two
to
triangles
similar.
included
and
critical
are
states,
angle
that
two
triangle
three.
“SSS”.
the
congruent
can
all
similar
established
angles
are
which
Draw
that
one
triangles
and
included
of
of
of
is
two
“SAS”
triangle
the
prove
triangles
instead
that
similar.”
and
to
angles
postulate
proves
another
used
two
congruent
postulate
of
triangles
two
“If
similarity
how
third
sides
then
are
Another
postulates
states,
triangles
that
solve
than
their
a
you
are
similar,
sides
wide
are
range
might
you
know
proportional.
of
problems,
think.
73
Example
Q
When
3
using
important
the
patient’s
which
can
spinal
cord.
to
solve
how
far
should
The
far
to
is
of
4.8 cm
apart
too
it
much
triangles
and
sources
is
exposing
radiation,
cells
can
and
be
the
used
determine
of
radiation
placed.
radiation
A
the
avoid
damage
problem
above
avoid
to
Similar
sources
cord
body
apart
therapy,
doctors
severely
this
be
120 cm
the
radiation
that
radiation
the
below
the
extends
should
are
patient
the
a
the
patient’s
distance
sources
overexposure
placed
and
to
of
spinal
skin.
of
If
20 cm
radiation
on
be
the
patient’s
placed
from
back,
each
how
other
radiation?
A
120 cm
Draw
a
diagram
of
the
situation,
including
all
the
measurements.
20 cm
B
C
4.8 cm
D
The
E
two
triangles
postulate.
They
and
A
angle
∆A
∆A
AB
BC
is
are
each
the
similar
have
same
in
a
by
the
right
AA
angle
Establish
120
AE
in
fact,
similar.
triangles
are
similar,
down
the
their
ratios
sides
that
are
are
equivalent.
20
Fill
in
the
measures
that
you
have
and
solve.
DE
120DE
=
20(124.8)
120DE
=
2496
20.8 cm
Each
2(20.8)
=
large
triangle
has
a
base
of
20.8 cm,
which
means
the
41.6 cm
sources
The radiation sources should be placed
at least 41.6 cm away from each other to
avoid radiation overexposure.
2
the
propor tional. Write
=
124.8
74
are,
AC
DE
=
triangles
=
AD
DE
the
E
Because
=
that
each.
Triangles
of
radiation
need
to
be
41.6 cm
away
from
each
other.
G E O M E T R Y
Practice
1
Indicate
that
you
A N D
T R I G O N O M E T R Y
3
which
used
of
to
the
following
establish
are
pairs
of
similar
triangles
name
the
postulate
similarity.
B
a
and
30°
D
b
E
D
A
89°
89°
7.5 cm
5 cm
60°
3 cm
A
F
C
E
4.5 cm
F
A
c
D
6 cm
4 cm
D
d
20 m
A
36 cm
72 cm
E
F
58 m
104 m
40 m
B
48 cm
C
E
E
192 cm
B
D
e
C
C
f
A
B
E
E
F
D
A
2
Indicate
which
similarity
postulate
measurement.
where
pairs
of
triangles
you
Show
used
your
to
in
each
diagram
establish
working
and
the
are
similar
relationship.
round
your
and
Then
answers
to
write
nd
the
down
the
the
missing
nearest
hundredth
appropriate.
M
a
A
b
4 cm
c
P
5 m
Q
5 cm
R
x
O
L
N
3 m
x cm
11.2 cm
10 m
8 cm
B
E
x cm
T
S
14 cm
D
K
C
10 cm
Continued
on
next
page
75
A
d
y
e
S
R
2 ft
20
12
cm
cm
x
C
x
D
B
E
P
Q
8 ft
16
cm
24 ft
3
Just
as
in
Fermat’s
light
mathematics,
principle
behaves
from
this
at
angle,
an
states
when
will
that
reected
principle.
it
principles
The
law
reect
o
light
o
of
a
physics
travels
surface
reection
that
a
in
surface
According
Show
the
c
that
was
a
supposedly
when
they
is
tall,
measured
of
a
know
that
b
Find
c
Thales
Try
the
this
this
George
has
a
has
of
to
this
not
of
the
would
have
be
Find
be
reection)
light
hits
a
time.
can
be
How
derived
reective
surface
of
reection,
on
two
a
which
copy
of
similar
angles
the
will
be
diagram.
triangles
and
state
information
could
be
used
to
nd
the
lamppost.
the
tall
to
the
measuring
height
and
solve
At
would
a
the
the
of
know
in
Pythagoras.
of
Cheops
himself.
Thales,
same
to
order
to
lamppost?
Pyramid
problem
shadows.
exactly
of
need
teacher
Great
the
you
who
time,
he
One
was
He
was
and,
method
176 cm
measured
the
Indicate
the
similar
triangles
and
justify
how
you
similar.
until
Pyramid
to
his
helped
use
him
and
cm.
as
to
triangles
was
the
determine
compare
The
calculated
similar
shadow
isosceles
the
this
how
160 cm.
least
used.
philosopher
out
the
results.
13 320 cm.
have
question,
of
triangle.
to
are
right-angled
hypotenuse
set
involved
Great
actually
waited
he
useful
angle.
produces
calculate
priests
of
when
these
measurements
situation.
triangles
have
Triangles
used
pyramid
height
a
him,
shadow
challenge
George’s
2
his
the
did
tell
have
diagram
could
how
to
the
Draw
He
7 6
said
to
Egyptian
other
requires
law
same
you
the
mathematician,
the
refused
he
a
Pairs
asked
that
shadow
5
Greek
this
height
actually
Thales
that
law
to
that
the
Indicate
how
What
the
the
Describe
d
4
at
lead
path
states
postulate
of
a
(called
to
congruent?
b
in
often
your
triangle
area
dimensions
of
of
by
to
same
the
Prita
Prita’s
nd
of
and
has
triangle
George’s
the
length
height
answer
and
Thales.
a
is
height
as
his
the
similar
triangle.
the
height.
pyramid.
Explain
pyramid.
working
four
of
with
a
triangle
times
the
peer.
which
area
of
G E O M E T R Y
Trigonometric
Understanding
angled
triangles
established
The
is
relationships
very
This
also
is
useful
sine
exist
where
T R I G O N O M E T R Y
ratios
Pythagoras’
Relationships
angles.
the
A N D
between
in
sides
mathematics.
theorem
and
between
a
the
the
the
right
principles
of
in
You
principle
triangle’s
right-
have
of
similarity.
sides
trigonometry
already
and
its
begin.
ratio
B
In
triangles,
while
with
the
sides
the
angles
are
same
are
named
letter
generally
using
as
the
named
lower
angle
case
that
is
using
upper
letters.
across
A
case
side
from
is
it,
letters
labeled
as
shown
in
diagram.
Understanding
triangles
even
is
more
a
the
very
skills.
relationships
important
You
have
between
technique
already
the
and
sides
leads
established
in
to
right-angled
developing
Pythagoras’
b
C
theorem
and
Activity
1
Using
a
having
2
the
7
The
protractor
one
Measure
Record
–
principle
of
the
your
the
a
acute
lengths
Triangle
sine
and
results
of
of
in
draw
angles
the
the
two
right
A
triangles.
right-angled
measuring
hypotenuse
rst
two
of
side
with
ratio
ruler,
Measure
opposite
similarity
rows
60
and
of
Measure
(O)
side
table
opposite
like
of
hypotenuse
of
dierent
sizes
but
both
degrees.
the
a
triangles
O
this
+
the
60°
angle
in
your
triangles.
one.
H
O
−
H
OH
O
H
(H)
1
2
3
4
3
Share
your
4
Explore
results
the
with
a
peer
relationships
so
that
between
you
the
have
data
hypotenuse
for
and
four
the
right
triangles.
opposite
side
by
carrying
out
the
Pairs
5
dierent
calculations.
Examine
your
results.
Add
your
Which
results
to
calculations
the
table.
produce
observable
patterns?
What
patterns
do
you
observe?
6
Draw
two
right
triangles
each
of
which
has
a
45°
angle.
Then
repeat
steps
1
through
4.
Continued
on
next
page
77
7
What
8
With
patterns
a
do
partner,
you
notice
discuss
the
in
your
patterns
two
you
sets
of
both
data?
found.
Explain.
Take
turns
giving
feedback
by
rst
ATL2
asking
questions
comment.
to
Follow
clarify
that
anything
with
a
that
isn’t
suggestion
for
quite
clear.
Then,
improvement,
if
give
at
least
one
positive
appropriate.
Pairs
Reect
and
discuss
9
O
●
Explain
why
the
value
of
the
ratio
is
always
between
0
and
1.
H
●
What
mathematical
principle
underlies
the
reason
why
the
ratio
O
of
is
the
same
in
some
of
your
triangles?
Explain.
H
Did
you
know...?
Thankfully, you do not have to draw triangles in order to calculate
the sine ratio for an angle. Technology can be used instead,
although the technology has changed dramatically over the years.
The slide rule (pictured right) was invented in the 17th century.
This tool could perform calculations involving many mathematical
operations, including nding the sine ratio.
In
the
were
use
You
1970s,
more
an
algorithm
can
triangles
use
to
popular
string
to
the
but
was
the
that
the
replaced
more
sine
missing
used
it
is
that
to
kite
the
kite?
a
the
slide
user-friendly.
rule.
They
Calculators
ratio.
sine
ratio
sides
kite
lightning
unlikely
wasn’t
get
Franklin,
the
Assuming
was
Franklin
belief,
Suppose
fact
principle:
lightning,
needed
nd
much
is
the
and/or
same
in
similar
right
angles.
4
Benjamin
scientic
the
to
calculators
and
calculate
Example
Q
scientic
powerful
his
string
Round
it
that
high
a
at
key
form
his
in
height
ew
was
a
for
the
was
an
answer
to
discover
electricity.
was
the
air
actually
during
of
taut,
the
an
important
Contrary
struck
experiment.
1.75 m,
angle
completely
your
to
of
kite
necessary
kite
whose
and
and
is
a
above
how
high
nearest
by
Franklin
thunderstorm.
attached
70°
All
to
16 m
the
o
of
horizontal.
the
ground
hundredth.
Continued
78
2
Triangles
on
next
page
G E O M E T R Y
A N D
T R I G O N O M E T R Y
A
Construct
Label
the
the
a
measurements
length
Label
the
right-angled
of
the
side
string
that
you
triangle.
that
and
Be
you
sure
know :
Benjamin
want
to
nd
to
the
indicate
angle
Franklin’s
with
a
the
of
right
the
kite
angle.
string,
height.
letter.
k
16 m
70°
1.75 m
opposite
sin 70° =
hypotenuse
k
sin 70° =
Write
the
sine
ratio
with
the
information
that
you
know.
16
16 sin 70° =
k
Isolate
Make
k
sure
calculator
degree
15.04
= k
Use
your
calculator
to
nd
the
value
of
mode!
some
calculators,
enter
+
1.75 m
=
16.79 m
Do
not
forget
to
add
on
Franklin’s
height
of
the
kite
o
is
16.8 m
to
3
then
to
“sin 70”
of
the
nd
side
the
length
to
nd
relates
opposite
length
of
the
the
of
of
a
one
an
length
given
either
other
size
the
of
angle
angle.
these
and
of
the
the
You
sides
size
given
hypotenuse
of
the
can
as
use
long
the
the
as
angle.
lengths
of
and
sine
you
Is
the
it
these
length
ratio
know
to
would
your
nd
ratio.
others,
ratio
press
s.f.
sine
sine
rst
the
“sin”
The
number
degrees
and
ground
the
you
height.
of
The
in
k
On
15.04 m
your
is
the
On
you
like
type
you
write
it
in
notebook.
the
possible
two
sides
instead?
79
Example
Q
Planes
to
5
use
achieve
actually
principles
ight.
ying
considerable
Pilots
can
A
plane
amount
of
while
involves
a
mathematics.
some
of
computers
the
do
most
them.
plane
is
10 000 m.
is
physics
However,
the
approximate
calculations
of
of
ying
It
at
begins
approximately
from
the
angle
of
an
altitude
its
descent
of
when
it
191 000 m
runway.
What
is
the
plane’s
descent?
A
Construct
sure
to
a
right-angled
indicate
the
right
triangle.
angle.
Be
Label
191 000 m
the
height
of
the
plane
and
its
distance
10 000 m
from
be
Runway
the
runway.
Indicate
the
angle
to
found.
A
opposite
sin A =
hypotenuse
10
000
Write
the
sine
ratio
with
the
information
that
you
know.
sin A =
191 000
1
=
A
=
sin

10


191 000
plane’s
angle
Practice
Use
In


order
to
nd
the
angle
when
you
know
the
ratio,
use
–1
the
arcsin,
or
sin
,
command
on
your
to
the
calculator.
3.0°
The
1
000

your
of
descent
is
3.0°
above
the
horizontal.
4
calculator
to
nd
the
given
ratio.
Round
your
answers
nearest
hundredth.
a
d
sin 58°
sin 30°
b
sin 12°
c
e
sin 45°
f
sin 74°
sin 62°
Continued
8 0
2
Triangles
on
next
page
G E O M E T R Y
2
Use
your
answers
a
d
3
sin A
sin A
Find
calculator
to
the
to
nearest
=
e
ratios
sin B
and
B
a
the
tenth
b
=
the
nd
size
of
a
sin A
=
sin A
sin C
=
in
of
the
angle
with
the
given
A N D
sine
T R I G O N O M E T R Y
ratio.
Round
your
degree.
0.64
each
of
the
sin A
=
f
sin A
=
following
4 cm
A
b
c
0.15
triangles.
B
Simplify
all
fractions.
B
c
52
20
3 cm
C
5 cm
A
48
15
17 cm
C
8 cm
A
d
C
B
e
B
f
15.9 cm
B
A
22 cm
35 m
4 m
C
A
30 m
C
A
C
2 m
4
Find
the
missing
a
measurements.
Round
your
answers
b
x
to
the
nearest
tenth.
c
10 cm
17 cm
35°
8 cm
20°
a
20 cm
15 cm
a
d
e
25°
θ
x
f
90 cm
54 cm
44 cm
5 cm
8 cm
?
72 cm
Continued
on
next
page
81
5
a
Mesn
safety
75
bought
6-meter
regulations
degrees
reach
a
with
with
his
state
the
ladder
that
ground.
to
are
If
you
up
want
Give
other
trigonometric
a
ratios
Activity
1
Use
O
the
generalizes
given
two
–
like
the
Triangle
A
the
marked)
pairs
The
right
(opposite),
table
(or
between
8
of
to
paint
be
some
murals
placed
maximum
at
height
an
at
school.
angle
that
of
Ladder
no
Mesn
more
will
be
than
able
to
triangles
were
the
the
are
reach
to
reach
These
they
to
to
developed
a
their
height
regulations,
length
nearest
of
high
ladders
half
the
tenth
to
of
places
can
4.5 m
a
but
a
when
using
length
ladder
the
reach
length
what
of
solve
need
compact
range
fully
a
of
ladder
closed
ladders
enough
heights.
extended.
extension
when
for
that
is
ladder
and
set
do
round
you
your
meter.
ratios
relationship
angle
and
between
the
the
side
hypotenuse.
Are
there
sides?
cosine
(adjacent)
one
the
anywhere.
closed,
answer
opposite
to
should
enough
according
need?
ratio
is
ladders
long
store
When
sine
ladder
What
Extension
that
The
order
ladder?
b
Other
the
in
and
with
and
H
a
60°
tangent
angle
that
(hypotenuse).
ratios
you
constructed
Measure
and
in
record
Activity
these
8.
side
Label
the
lengths
sides,
in
a
below.
Length
opposite
of
side
Length
(O)
of
hypotenuse
(H)
Length
adjacent
of
(A)
A
O
H
A
1
2
3
4
Continued
8 2
2
Triangles
on
next
page
G E O M E T R Y
2
Share
60°
your
results
with
a
peer
so
that
you
have
data
for
four
right
A N D
triangles,
T R I G O N O M E T R Y
each
with
a
angle.
3
Calculate
4
Repeat
drew
the
steps
in
ratios
1
What
patterns
6
Research
Reect
the
the
through
Activity
5
in
for
two
right
columns.
triangles
What
do
you
notice?
containing
a
45 °
two
data?
angle.
You
may
use
the
ones
you
8.
do
you
names
and
4
last
notice
of
between
these
discuss
two
your
sets
of
Explain.
ratios.
10
A
●
Explain
why
the
value
of
is
always
between
0
and
1.
H
O
●
Describe
●
What
any
limitations
on
the
value
of
A
are
mathematical
the
same
in
principle
certain
right
underlies
triangles?
the
reason
why
these
ratios
Explain.
Some
use
people
the
acronym
SOHCAHTOA
remember
to
the
three fundamental
The
three
trigonometric
ratios
relate
the
side
lengths
in
any
right
trigonometric
triangle,
where
angle
A
is
not
the
right
angle.
ratios.
opposite
sin A
=
hypotenuse
adjacent
cos A
=
hypotenuse
opposite
tan A
=
adjacent
You
can
solve
problems
with
all
these
generalized
relationships.
Feel
If
you
know
relevant
two
ratio
to
measures
solve
for
a
in
a
right
third
triangle,
you
measurement.
can
use
the
up
free
with
to
come
your
memory
own
aid!
83
Example
Q
6
In the past, it was extremely dicult
for people in a wheelchair or
pushing a stroller to access buildings
with steps up to the entrance. To
solve this problem, wheelchair
ramps were developed. Many
countries now have strict guidelines
for wheelchair ramps so that the
angle of the ramp is not too steep.
The
most
states
common
that
for
regulation
every
horizontal
distance
maximum
of
angle
of
a
1
12
the
meter.
wheelchair
meters
ramp
What
of
can
is
the
rise
a
maximum
ramp?
A
Sketch
A
a
right-angled
triangle.
Be
sure
to
indicate
the
right
1 m
angle.
Label
the
lengths
of
the
sides.
Indicate
the
angle
to
12 m
be
opposite
found.
You
have
the
lengths
of
the
opposite
and
the
adjacent
sides
tan A =
adjacent
to
the
angle
you
want
to
nd,
so
use
the
tangent
ratio.
1
tan A =
Write
the
tangent
ratio
with
the
information
that
you
know.
12
−1
tan
−1
( tan A) =
tan
1


Remember



12
same

that
operation
solving
on
equations
both
sides. The
involves
per forming
opposite
the
of “tangent ” is
1
“arctan”
.
1
A =
tan
The
is
a
the
enough
angle?
In
protractor.
theodolite
of
to
lengths
to
do.
measure
measure
angles
You
build
8 4
the
in
Triangles
of
sides
class,
both
use
order
your
summative
2
wheelchair
of
own
right
use
might
a
tool
have
a
the
clinometer,
for
seems
measure
used
level
and
called
task
you
transit
calculate
assessment
triangle
do
horizontal
a
to
a
how
you
surveyors
professionals
in
a
However,
geometry
Land
is
denoted
as
tan

Make
angle
arctan

12
Forestry
can
calculator
4.8°.
Measuring
simple
a
4.8°
maximum
ramp
an
=
On



A
1

or
a
vertical
angles.
clinometer
height
which
this
of
you
unit.
a
to
tree.
will
use
sure
your
calculator
is
in
degree
mode.
G E O M E T R Y
Activity
You
can
9
–
you
T R I G O N O M E T R Y
Clinometer
download
thisapp,
A N D
view
a
free
the
clinometer
desired
app
object
instead
along
the
of
making
edge
of
the
your
one
phone
in
this
and
activity.
read
the
Using
angle
on
thescreen.
To
a
make
small
your
own
weight
clinometer,
and
a
you
photocopy
will
of
a
need
the
following
materials:
cardboard,
string,
a
straw,
protractor.
tape
straw
0
6
0
08
tape
0
8
tape
back)
0
4
(on
0
6
2
0
0
4
0
2
0
weight
string
weight
1
Search
2
Glue the protractor onto the cardboard and then cut around it so there is no excess cardboard.
3
Pierce
0°
4
Tie
6
Tape
To
is
of
the
Read
to
eye
the
to
bottom.
the
below
protractor
the
the
the
end
and
of
hole
and
Print
it
and
cardboard
tie
protractor
the
cut
at
it
the
out.
midpoint
of
the
line
connecting
a
knot
when
in
you
it
or
hold
stick
it
it
with
in
place
the
with
tape.
horizontal
Make
edge
at
sure
the
top.
string.
protractor
clinometer,
The
whose
angle
your
how
trigonometry
10
the
protractor”.
along
the
line
between
0°
and
180°.
is
the
It
will
cover
the
hole
previously.
object
Measure
your
“printable
through
hangs
straw
between
9
string
your
the
the
a
through
weight
made
use
at
the
string
the
you
hole
for
180°.
Thread
5
8
a
and
some
7
online
on
to
weight
height
the
angle
far
hold
nd
so
that
should
you
are
clinometer.
and
your
it
90°
feet
the
–
are
from
height
of
hang
trying
To
take
the
horizontal
vertically
to
obtain
the
the
the
edge
down.
at
Look
top
and
through
the
the
curved
straw
at
edge
the
top
calculate.
the
absolute
edge
of
object.
angle
value
the
of
of
the
this
object.
Remember
object,
the
dierence
dierence.
Draw
to
nd
add
a
diagram
on
your
and
own
use
height
up
to
level.
Compare
your
value
with
that
of
a
peer,
especially
if
he/she
used
the
clinometer
app.
8 5
Practice
1
Find
the
5
value
of
each
trigonometric
ratio.
Round
your
answers
to
the
nearest
thousandth.
a
2
3
cos 62°
Find
the
a
tan A
a
For
b
size
of
=
b
this
tan 17°
the
angle
cos A
triangle,
fractions
where
sin B
c
sin 88°
with
=
c
nd
the
given
cos A
each
of
d
e
trigonometric
=
the
tan 34°
d
tan A
following
cos 29°
f
sin 44°
ratio.
=
e
cos A
trigonometric
=
ratios.
necessary.
0.41
f
tan A
Simplify
=
3.42
your
A
cos A
tan B
24 cm
sin A
cos B
tan C
C
4
b
Explain
A
ladder
wishes
ruler.
is
to
leaning
nd
She
cos B
the
=
sin C
against
vertical
measures
a
She
measures
a
From
this
information,
Show
your
you
use
step
a
verify
Janine
She
is
that
her
worried
its
base
eyes
head.
between
20
is
to
that
angle
the
top
The
the
ladder
which
15 cm
of
the
determine
B
30 cm
cos C
the
up
a
known
ladder
the
string
the
has
reaches
ladder
to
vertical
be
length
and
but
drops
of
3 m.
she
a
Cassandra
only
piece
of
has
a
15 cm
string
to
the
10.6 cm.
height
to
which
the
ladder
reaches.
her
hit
from
be
method
you
didn’t
use
in
her
to
home
measure
if
it
falls
the
in
height
a
big
of
the
storm,
tree
and
near
has
her
house.
measured
top
of
distance
and
Janine’s
to
the
tall.
the
clinometer
tree
Use
from
1.80 m
eyes
trigonometry?
clinometer
may
away
is
or
answer.
horizontal
from
of
it
18 m
Janine’s
the
use
10 cm
The
meters.
length
=
The
to
of
triangles
your
Janine
are
wall.
height
the
similar
going
house.
Her
her
is
a
sin B
working.
Did
to
and
distance
ground.
b
5
why
the
is
measures
eye
40
tree
level
40°
to
degrees.
20 m
10 cm
a
Find
the
b
Should
height
of
the
tree.
1.80 m
the
tree
your
Janine
hitting
be
worried
her
house?
about
Justify
answer.
Continued
8 6
2
Triangles
on
next
page
G E O M E T R Y
The
put
in
a
of
pile.
repose
this
conical
of
pile
radius
In
is
of
easier
to
break
component
of
physics
Suppose
25 m/s
Find
the
object’s
can
an
at
and
in
of
as
What
is
the
object
up
the
then
be
travels
angle
of
horizontal
steep
40
and
as
angle
travels
at
into
to
with
is
of
a
dry,
a
the
this
it.
material
material
science
make
of
loose
will
is
slide.
experiment,
The
pile
is
stable
you
11 cm
In
when
snowy
create
high
a
and
has
sand?
it
horizontal
both
The
25 m/s
principles
ver tical
component
directions.
with
vertical
can
angle,
velocity
degrees
For
repose
an
a
which
exceeded,
you
component.
applied
at
avalanche.
motion
vertical
angle
repose
an
is
object
an
angle
that
an
a
steepest
result
sand
when
the
the
can
16 cm.
physics,
is
When
mountains,
a
7
angle
T R I G O N O M E T R Y
(speed)
the
of
40°
ground.
components
of
horizontal
component
the
velocity.
assessment
–
The
creation
of
ags
i
t
er
i
r
Formative
c
Do
you
know
the
story
behind
your
country’s
ag?
A
ag
can
reveal
n
6
A N D
much
C,D
about
a
process
analyse
design
The
country’s
or
the
one
Union
How
do
already
United
scientic
Union
of
you
solve
their
the
the
or
(the
culture.
maybe
ag
of
a
How
are
combination
the
United
Irish
Union
designed?
of
both?
Kingdom)
a
the
of
the
Cross
problem
own
of
of
individual
three
of
English
St
the
rather
a
of
(The
than
St
Jack,
with
some
for
them
that
George,
Welsh
a
ag
Add
countries
Cross
Patrick.
principality
creating
ags?
ag
In
Is
this
this
before
three
all
a
creative
task,
you
attempting
countries
together!
came
the
was
separate
+
The
ags
will
to
own.
ags
Kingdom:
considered
one
Jack
your
and
Jack
have
combines
and
a
history
The
together
Scottish
not
to
form
Cross
included,
that
Union
of
as
Jack
the
St
Andrew
Wales
was
kingdom.)
+
of
its
dimensions,
is
shown
on
the
next
page.
Continued
on
next
page
87
6 cm
10 cm
2 cm
6 cm
2 cm
10 cm
6 cm
2 cm
25 cm
a
Using
your
and
angles.
Design
Did
your
you
b
You
that
will
current
the
an
that
now
ags
your
of
the
similar
25 cm
triangles,
measurements
your
of
Pythagoras’
the
blue
theorem
triangles.
and
Give
side
lengths
working.
ag
the
Olympic
rings
include
colours
from
the
ag
of
every
participates?
Olympic
Use
nd
Show
own
know
country
6 cm
knowledge
trigonometry,
2 cm
design
that
a
you
rings
and
knowledge
aesthetically
ag
of
have
it
of
your
seen.
should
own.
Your
have
Pythagoras’
pleasing
ag.
Show
Feel
ag
a
free
must
height
to
take
contain
that
is
inspiration
colours
one
theorem
and
all
dimensions
of
the
half
of
trigonometry
and
from
that
its
to
all
are
in
length.
achieve
of
your
working.
c
Explain
the
elements
d
With
a
story
of
your
partner,
behind
your
ag
and
why
you
have
chosen
the
colours
and
ag.
look
at
each
other’s
ag
and
the
supporting
work.
Take
ATL2
turns
giving
quite
clear.
suggestion
Then,
for
Pairs
8 8
2
feedback
Triangles
give
by
rst
at
least
improvement,
asking
one
if
questions
positive
to
clarify
comment.
appropriate.
anything
Follow
that
that
with
isn’t
a
Unit
summary
Pythagoras’
theorem
states
that
b
c
a
where
the
A
is
and
length
triangle
an
and
A
a
an
the
the
is
said
copy
angles
exact
are
is
the
lengths
are
Similarity
to
of
be
the
congruent
the
other.
exactly
said
to
must
the
be
be
of
be
same
the
to
established
Two
AA
Angle–Angle
two
SAS
Side–Angle–Side
the
to
of
a
Side–Side–Side
the
another
have
right
triangle
triangle
exactly
another
the
one
The
same
and
c
is
if
the
one
same
triangle
length
triangle
if
one
corresponding
size.
The
sides
triangle
angles
of
is
in
similar
another.
using
any
triangles
are
one
lengths
of
lengths
two
and
of
all
of
similar
corresponding
proportional
SSS
legs
size.
other.
exactly
Postulates
to
Sides
similar
proportional
can
of
hypotenuse
enlargement
triangles
triangles
are
of
exact
triangle
b
angles
the
if
three
…
are
congruent.
corresponding
the
postulates
included
sides
angle
corresponding
is
sides
are
congruent.
are
proportional.
If
The
then
three
between
fundamental
the
sides
of
a
trigonometric
right
ratios
establish
relationships
triangle.
B
hypotenuse
opposite
A
C
adjacent
8 9
t
e ri
o
c
n
r
i
Unit
Key
to
review
Unit
review
Level 1–2
1
Explain
A
question
levels:
Level 3–4
how
Pythagoras’
the
Level 5–6
diagram
below
can
be
Level 7–8
used
to
prove
theorem.
a
c
b
2
Are
the
down
following
the
similar
pairs
of
triangles
triangles
and
the
similar?
If
similarity
D
are,
postulate
A
a
they
write
used.
A
b
E
74°
58°
55°
B
C
D
C
B
55°
47°
E
F
A
c
D
d
A
21
B
12
35
25
C
E
C
15
B
E
F
D
9 0
2
Triangles
20
A
e
A
3
6
2
5
C
C
4
B
8
B
3
Find
the
your
answers
a
d
4
value
of
to
the
the
nearest
tan 50°
sin 8°
Find
ratio.
the
measure
Round
of
your
following
sin 38°
e
cos 71°
angle
answers
to
with
the
5
a
cos A
5
tan A
Find
b
a
cos 75°
f
tan 26°
given
size
of
e
the
Round
nearest
trigonometric
tenth
of
a
degree.
tan A
4
=
c
sin A
=
3
=1.15
the
the
c
4
=
8
d
ratios.
hundredth.
b
the
trigonometric
sin A
=
unknown
9
0.89
side
in
f
each
cos A
=
0.71
triangle.
b
C
x
8
21
15
B
15
x
A
c
x
d
12.2
7
15
17.3
x
91
6
Write
in
down
each
Round
pair
your
the
are
similarity
similar.
answers
to
postulate
Then
3
nd
the
proves
missing
the
triangles
measurement.
s.f.
B
a
that
B
b
E
8
24
16
x
21
A
C
D
x
F
16
A
C
18
c
z
8
40°
12
10
110°
40°
5
x
y
7
Find
the
value
right
triangles.
of
the
indicated
measure
in
each
b
a
of
the
following
c
d
x
18
x
9 cm
K
8 cm
34°
28
53°
θ
45
5 cm
9 2
2
Triangles
8
In
ancient
Egypt,
boundaries
process
Land
they
surveyors
lines
Find
Their
length
of
Find
distance
c
(0,
(4,
–2)
4)
and
and
(–2,
is
underneath
a
In
deep
very
old
volcanic
heavier
the
and
ooded,
An
to
it
often
accurate
re-establish
Pythagoras’
a
the
theorem
drawn
by
to
erased
the
surveying
property
verify
calculating
lines.
that
how
long
plate
process
line
is
in
the
diagonal
if
the
where
pair
the
(–5,
d
(–10,
a
shape
of
of
rectangle.
sides
the
of
the
a
property
a
a
directly
depth
two
of
plates
points.
1)
plate.
on
and
–3)
heavier
depending
zones,
of
b
continental
formed
of
the
each
8)
reaches
boundary
that
is
6)
shallow,
is
line
property
subduction
arc,
property
55 m.
lighter
or
a
between
(–3,
Subduction
either
order
property
48 m
the
in
correctly
occupy
measure
a
10
the
Nile
be.
farmers
rectangle.
used
were
the
landowners.
necessary
should
Two
9
between
was
property
when
The
the
100 km.
also
(–7,
A
of
range,
point
1)
plate
process
angle
mountain
is
and
–8)
oceanic
the
above
(3,
can
be
subduction.
called
where
deep
plunges
a
the
trench
along
formed.
c
r
a
c
i
n
l
a
c
o
V
h
c
n
e
r
T
Continental
crust
Oceanic
A
volcanic
the
arc
is
located
a
horizontal
crust
distance
of
250 km
from
trench.
a
Draw
b
Find
a
diagram
the
nearest
angle
tenth
and
of
of
a
indicate
subduction.
any
relevant
Round
your
information.
answer
to
the
degree.
9 3
11
An
isosceles
5 cm
12
long.
triangle
has
the
perpendicular
b
the
area
to
must
and
make
Find
the
of
back
ground,
a
of
owners
their
base
of
6 cm
and
its
equal
sides
are
Calculate:
a
The
a
a
an
how
of
the
triangle
triangle.
house
porch
to
height
want
with
comply
angle
long
replace
ramp.
with
of
the
a
to
4.8
The
slope
of
the
steps
porch
building
degrees
the
is
1
leading
meter
regulations,
with
the
ramp
the
up
o
the
ramp
ground.
will
be.
Show
your
working.
b
Find
start
13
Airport
the
of
cloud
horizontal
the
ramp.
of
The
is
airplanes.
cloud
visible.
allowed
to
distance
Show
meteorologists
safety
ceiling.
not
the
If
the
take
your
keep
One
ceiling
or
an
is
eye
the
the
porch
and
the
working.
thing
cloud
o
between
on
they
is
weather
watch
lowest
ceiling
the
is
altitude
too
low,
the
at
to
ensure
cloud
which
the
solid
planes
are
land.
Weather
office
Angle
Searchlight
One
at
process
night
from
the
her
shine
from
the
how
The
spot
to
is
take
of
Triangles
into
to
located
the
height
on
land
the
height?
the
of
use
that
the
to
is
nd
of
from
light
cloud
a
Then
light
the
on
the
located
clouds.
spot
40 m
spot
cloud
or
can
searchlight
oce
the
o
light
minimum
2
is
minimum
safely
9 4
the
angle
high
a
vertically
searchlight
If
b
to
meteorologist
oce
angle
The
a
is
a
on
xed
she
the
ceiling
distance
measures
cloud.
oce.
the
cloud
is
38
degrees,
ceiling?
of
is
the
cloud
305 m.
cloud
when
ceiling
What
the
is
for
the
cloud
planes
angle
ceiling
to
is
to
the
at
the
14
During
and
a
solar
Earth
eclipse,
and
considerably
covers
smaller.
the
the
Moon
Sun
How
is
passes
almost
this
between
entirely,
the
Sun
despite
being
possible?
150 000 000 km
d
3476 km
m
384 400 km
a
Label
the
similar
b
On
triangles.
and
indicate
average,
the
150 000 000 km
is
3476 km
Use
this
almost
15
A
the
Earth
from
while
the
information
all
television
Write
of
the
Sun
down
similarity
is
the
Sun.
of
an
made
from
the
Sun
why
that
that
the
diameter
determine
during
manufacturer
The
triangles
postulate
384 400 km
diameter
to
the
is
the
are
you
Moon
of
the
used.
and
Moon
1 400 000 km.
Moon
covers
eclipse.
the
following
claim
in
an
advertisement:
“Our
of
a
a
80-inch
55-inch
Show
Show
an
TV,
that
double
b
AQUOS
the
that
aspect
for
an
an
TV
delivers
amazing
80-inch
screen
this
ratio
of
viewing
television
area
claim
more
of
is
16 : 9.
a
than
can
even
(Aspect
the
screen
area
experience.”
55-inch
true
double
if
deliver
more
than
television.
both
ratio
is
televisions
have
length:width)
9 5
16
In
order
to
describe
astronomers
our
Sun
at
coordinate
a
Write
use
the
in
a
the
down
Find
the
answers
the
c
9 6
i
Polaris
ii
Alpha
iii
Sirius
Find
2
stars
to
the
Triangles
(0,
form
the
0,
(x,
0).
y,
of
celestial
All
bodies,
coordinate
other
objects
system
are
with
given
a
z).
distance
formula
for
a
pair
of
coordinates.
distance
following
locations
three-dimensional
origin
3-dimensional
b
the
between
(distances
nearest
(99.6,
(–3.4,
distance
given
in
and
light
each
of
years).
the
Round
376.0)
(–1.8,
–3.1,
Sun
tenth.
28.2,
Centauri
our
0.0,
3.9)
7.3)
from
Polaris
to
Alpha
Centauri.
your
i
e
t
r
i
c
Summative
a
r
assessment
C, D
Measuring
In
this
task,
be
measured
dierent
your
The
an
base
You
and
and
of
will
then
the
interest
the
height
calculate
reect
on
usefulness
the
is
so
to
Your
around
of
of
its
object
height
how
of
an
the
that
using
accurate
cannot
three
you
think
methods.
it
that
whose
chosen
and
object
you
you
which
you
can
object
must
is
height
be
needs
able
directly
measure
would
to
to
have
access
under
the
be
its
dicult
some
a
at,
point
maximum
horizontal
distance
to
object.
objects
mountain
highest
is
that
the
calculate
directly.
This
Suitable
It
are
object
ground
height.
A
easily.
answers
the
the
will
methods
measure
level
at
you
immeasurable
object
Find
to
the
would
include
not
work
a
tree,
since
a
you
building,
can’t
be
a
agpole,
directly
etc.
under
its
point.)
very
the
might
important
points
same
level
from
as
that
the
which
the
base
object
you
of
is
take
the
not
on
your
a
sloping
surface
measurements
are
so
all
on
object.
97
Method
a
1
Standing
measure
the
b
you
c
a
the
Use
the
the
angle
you
Draw
took
Use
meters
to
the
the
the
a
away
clinometer
object.
object
you
the
and
are
a
using
are
object
Next
the
to
measure
clinometer.
standing
and
indicate
from
the
the
place
where
measurements
on
from
of
the
two
are
the
to
to
the
its
the
of
the
object.
object
points
not
angle
to
the
top
of
the
base.
object.
Be
from
allowed
and
sure
measure
to
which
to
the
measure
you
took
the
your
measure/know
how
object.
represent
to
add
from
towards
measurements
method
the
and
the
the
calculate
the
Reect
in
and
●
Describe
●
Explain
height
object,
places
measurements
the
of
the
height
the
of
you
themselves.
the
clinometer
where
object.
to
your
result.
How
height
the
if
you
could
may
similar
use
one
triangles
of
the
to
calculate
methods
that
the
has
unit.
discuss
or
not
accuracy
answers?
applies
You
this
whether
the
of
that
object.
described
How
height
3
third
answer
the
measure
away
the
trigonometry
Method
to
You
diagram
the
of
calculate
top
between
Remember
●
far
of
represent
to
point
measurements.
been
of
measurement,
a
few
distance
height
to
clinometer
a
a
top
use
2
Move
Use
object,
the
how
trigonometry
from
new
c
to
bottom
diagram
object
far
the
diagram.
Method
b
angle
measure
took
Use
a
to
from
object.
Draw
the
the
angle
Finally,
the
away
close
of
do
object?
have
you
your
your
you
(Feel
access
improve
answers
results.
think
free
to
to
that
your
make
Did
you
all
are
methods
to
compare
sense.
nding
your
give
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results
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results?
Continued
9 8
2
Triangles
on
next
page
●
How
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the
process
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heights
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9 9
3
Modeling
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Earth
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8
15
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4
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2
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5
x
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Continued
on
next
page
10 5
2
Describe
the
3
Describe
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4
Why
Each
your
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it
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certainly
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same
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perminute.
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of
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1
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15
2
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graph
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4
5
coordinates.
for
these
values
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x
and
y:
0 ≤ x ≤ 10, 0 ≤ y ≤ 200 (Please use a ruler when connecting the points.)
4
Represent
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water
wasted
5
Show
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6
Set
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7
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Name
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10 6
3
one
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verbal
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use
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time
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axes
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each
description.
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question.
y
y
b
10
10
9
9
8
8
7
7
6
6
5
5
4
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6
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2
3
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6
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10
9
9
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5
4
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8
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8
9
10
Continued
on
next
page
10 7
e
y
10
9
8
7
6
5
4
3
2
1
1
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After
fell
by
When
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To
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new
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that.
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the
following
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represent
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represent
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describe
tables
of
of
values:
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with
axes
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for
situation
that
every
habits.
of
a
The
could
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question
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1
of
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pattern.
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y
x
y
y
x
0
9
−6
−3
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7
1
11
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2
13
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6
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15
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−8
Continued
10 8
3
Linear relationships
on
next
page
A LG E B R A
3
The
increase
reduce
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our
cost
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demand
emission
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cost
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or
cars
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of
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lithium-ion
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lithium-ion
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batteries,
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used
decreasing
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by
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hour
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year.
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this
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the
pattern
as
a
set
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the
pattern
as
a
graph.
pattern
with
a
table
of
of
values
from
2010
until
the
current
year.
coordinates.
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your
domain
and
justify
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your
domain
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the
boundaries.
set
d
begun
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pattern
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verbal
description
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of
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relationship.
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4
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of
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think
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set
c
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give
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b
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to
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the
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on
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this
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expressed
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represent
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if
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million
million
to
be
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to
coordinates
your
people
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country
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of
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low-income
total
amount
do
with
wasted
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that
year
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in
high-
countries.
of
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many
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happen
Explain
a
do
ATL1
you
pattern
will
5
do
the
about
obstacles
the
and
question
or
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your
answers
preventing
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people
surprises
from
not
you?
Explain.
wasting
food?
10 9
Characteristics
Linear
them
relationships
relates
Rate
An
is
of
to
its
Constant
have
linear
very
steepness
relationships
specic
of
the
characteristics
graph
while
which
others
are
help
dene
specic
them.
points
on
One
the
of
graph.
change
important
that
the
of
rate
rate
property
of
of
of
change
change
a
is
line
is
its
rate
of
change.
The
denition
of
a
linear
relationship
constant.
(linear
relationship)
Time
Ozone
(years)
(DU)
0
100
1990
120
10
1100
1
104
1995
111
11
1500
2
108
2000
102
13
2300
3
112
2005
93
15
3100
4
116
2010
84
19
4700
24
6700
Air
quality
Population
Area
of
forest
Year
3
NOT
constant
rate
Time
of
(μg/m
change
(non-linear
2
)
(millions)
cut(km
)
relationship)
Year
e-waste
Population
Population
Population
(years)
(millions
(billions)
(billions)
of
(millions)
tonnes)
0
7
1
65
5.2
70
6.1
75
6.5
80
7.2
8
2
10
3
13
4
17
85
Reect
and
discuss
1804
1
1927
2
1959
3
1974
4
1987
5
1999
6
2011
7
7.4
3
The
●
Based
on
the
above
examples,
explain
what
it
means
to
have
rate
change
constant
●
In
which
you
able
rate
of
is
tell
was
that
the
the
rate
rate
of
change
was
less
obvious?
How
were
constant?
referred
to
slope
gradient
of
the
or
line.
measure
●
How
does
having
a
constant
Explain
with
an
rate
of
change
allow
you
to
3
Linear relationships
It
the
is
a
the
make
example.
line.
110
as
of
steepness
predictions?
often
change.
example
you
of
a
of
the
A LG E B R A
linear
relationships
calculate
it?
What
have
does
1
constant
rate
of
change,
how
do
you
mean?
–
Gradient
of
a
line
i
t
er
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r
Investigation
it
a
c
Incandescent
bulb
very
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light
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will
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to
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and
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by
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three
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table
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sense?
in
on
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in
of
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used
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the
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a
is
triangle
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the
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and
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of
energy
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shown
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incandescent
as
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of
You
Represent
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set
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decreased,
Choose
the
using
points
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third
changing
usage
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vertical
electricity
planet.
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Does
Choose
on
divided
change
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is
B
are
one-sixth
money,
joules
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notice?
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10
that
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.
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uses
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light
they
bulb.
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last
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to
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choose
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n
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on
to
graph
the
and
LED
verify
bulbs,
your
conclusion
Maryam’s
in
step
3.
electricity
below.
line
and
calculate
the
gradient.
y
3000
2750
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2500
2250
x
0
1
2
3
4
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5
of
replaced
6
7
8
9
10
bulbs
Continued
on
next
page
111
6
How
was
one?
7
Does
Based
a
the
line
on
this
your
when
Verify
your
of
make
work,
you
coordinates
8
slope
of
this
sense?
write
know
the
rule
line
dierent
the
slope
of
the
previous
Explain.
down
two
than
a
points
formula
on
it.
to
Use
calculate
variables
the
to
gradient
represent
of
the
points.
for
a
pair
of
points
on
each
graph
in
this
investigation.
9
Justify
why
Reect
●
What
your
formula
and
do
you
works.
discuss
think
is
the
4
gradient
change
●
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is
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slope
of
a
diculties
denition
a
would
horizontal
rather
you
in
line?
y
line
change
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of
in
if
in
x
change
in
y
than
x
encounter
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change
you
?
used
the
second
instead?
rise
●
The
gradient
is
sometimes
said
to
be
.
Does
this
make
sense?
run
Explain.
●
Which
line
gradient
Parallel
Two
a
or
of
However,
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In
a
line
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specic
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Explain.
linear
linear
to
This
cases
each
other
together
will
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lines
be
are
and
the
are
referred
focus
important
lines
that
parallel
lines
of
Unit
here
are
to
are
as
7.
lines
perpendicular
to
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steeper,
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and
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of
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perpendicular
discuss
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5
meaning
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and
the
meaning
of
lines.
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●
How
do
you
think
the
gradients
of
parallel
●
How
do
you
think
the
gradients
of
perpendicular
Explain.
112
3
Linear relationships
lines
compare?
lines
Explain.
compare?
A LG E B R A
–
Parallel
and
perpendicular
lines
i
t
er
i
o
2
r
Investigation
following
diagram
shows
a
n
c
The
parallelogram.
B
y
12
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10)
(8,
10)
10
8
6
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(6,
6)
6)
4
2
x
0
2
1
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2
Prove
of
3
its
are
that
properties
the
diagram
of
is
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6
8
10
parallelogram?
actually
a
parallelogram
by
calculating
the
slopes
sides.
Write
The
the
4
a
generalization
following
diagram
about
shows
the
a
gradients
of
parallel
lines.
square.
y
15
(2,
12)
(8,
10)
10
(0,
6)
5
(6,
4)
x
0
5
4
What
5
You
you
know
Write
7
Verify
8
the
properties
that
determine
6
one
are
a
angles
how
the
generalization
your
more
Justify
the
a
your
of
15
square?
in
a
square
slopes
about
generalizations
example
why
of
10
of
the
are
all
gradients
about
right
perpendicular
parallel
of
angles.
lines
Use
perpendicular
and
this
to
help
compare.
lines.
perpendicular
lines
using
each.
generalizations
make
sense.
1 13
Reect
●
Were
and
your
discuss
initial
perpendicular
6
conjectures
lines
about
the
slopes
of
parallel
and
correct?
Pairs
●
Given
why
that
what
the
you
gradient
found
of
out
a
line
about
is
a
the
measure
of
gradients
its
of
steepness,
parallel
lines
explain
makes
sense.
●
Danika
is
says:
always
“The
−1.”
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product
she
of
correct?
the
gradients
of
two
perpendicular
lines
Explain.
Intercepts
Other
important
These
are
the
(x-intercept)
not
the
that
points
and
only
passes
characteristics
the
types
y-axis
of
through
where
the
a
line
linear
passes
(y-intercept).
graphs
or
of
that
touches
have
the
x-
y
relationship
through
Graphs
x-
and
and
of
or
are
linear
has
intercepts.
touches
the
x-axis
relationships
y-intercepts.
y-axes
its
Any
these
are
graph
intercepts.
y
y-intercept
y-intercept
x-intercept
x-intercepts
x
x
Linear
Non -l ine a r
3
–
Finding
intercepts
i
t
er
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r
Investigation
For
each
of
x-intercept
the
and
graphs
the
y
a
below,
write
y-intercept.
b
down
the
Summarize
y
coordinates
your
c
results
y
B
table.
y
d
8
10
4
6
6
5
2
4
4
x
0
2
2
x
0
–2
a
the
8
–5
–4
in
of
2
x
0
–6
–4
–2
2
–5
–2
–10
–4
–15
–6
x
0
–2
5
n
c
1
2
Continued
114
3
Linear relationships
on
next
page
A LG E B R A
2
characteristic do all
3
x-intercepts have in common? What
What characteristic do all
How
can
you
use
y-intercepts have in common?
your
result
from
step
2
to
nd
the
x-
and
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2x
3y
+
=
6
2
4
Show
(0,
2)
that
and
a
relationship
(0,
dened
A
relationship
your
6
answer
Generalize
a
relationship
7
Generalize
8
Verify
your
is
by
dened
using
method
given
a
relationship
+
y
=
4
x-intercepts
has
at
−2).
2
5
2
x
by
its
method
given
its
method
by
an
for
x
2
y
+
online
=
4.
Find
graphing
nding
its
tool
y-intercepts.
like
Verify
Desmos.
the
x-
and
y-intercepts
of
a
the
x-
and
y-intercepts
of
a
graph.
for
nding
equation.
by
nding
the
x-
and
y-intercepts
for
the
3
relationship
9
In
Justify
linear
why
For
condition
when
of
in
watering
as
a
3
–
small
per
gardens
is
load
uses
often
activity
has
been
electricity
zero
is
bulbs
(see
named
page
wasted
usage
(see
with
LED
106),
when
pages
the
no
initial
the
initial
time
has
111–112),
bulbs,
she
is
using
like
water
near
a
house
Australia
from
these
even
toilet
a
where
tanks
some
and
is
popular
water
can
be
household
washing
the
shortages
used
for
applications
laundry
if
installed.
rainwater
many
−16.
day.
and
the
=
y-intercept
water
tanks
water
2y
faucet
Saving
countries
The
how
no
replaced
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as
−
works.
the
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energy
Installing
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method
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Activity
decision
the
was
For
she
3000 J
your
4x
by
relationships,
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passed.
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loads
50 L
of
of
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and
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if
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water?
x (number of loads)
y (liters remaining in tank)
0
2000
Continued
on
next
page
115
1
2
Represent
this
Represent
the
Plot
3
the
the
(y-intercept)
Write
5
How
6
a
last
7
in
tank
in
For
one
graph:
extend
change
can
of
0
≤
the
and
condition
be
values.
x
≤
line
state
using
washed
number
many
of
weeks
obstacle
and
home
school.
50,
on
the
0
≤
your
initial
y
≤
2000.
graph
using
a
ruler.
condition
coordinates.
before
the
tank
is
empty?
loads
of
would
laundry
the
your
rainwater
household
from
a
full
does
tank
or
one
challenge
to
installing
a
rainwater
2
whether
those
table
household?
your
Indicate
a
a
scenario.
initial
the
How
Practice
1
the
as
then
of
this
with
working.
your
Identify
rate
loads
your
week.
and
in
Approximate
in
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down
many
Show
pattern
points
Calculate
4
pattern
that
the
are
relationship
linear,
nd
the
a
rate
of
in
each
table
is
linear
x
32
9
78
14
37
10
72
16
42
12
60
18
47
16
36
22
0
52
c
d
y
x
non-linear.
y
12
20
or
change.
b
y
x
represented
x
y
2
33
−2
0
5
36
1
9
8
24
4
18
11
48
8
30
14
60
13
45
↔
2
Find
the
slope
of
each
line
a
A(−3,
−8)
and
B (12,
10)
b
A(10,
24)
and
B (−5,
18)
c
A(1,
−20)
and
B (
d
A(14,
1)
and
B (−7,
e
A(−7,
5)
and
B (3,
,
AB
−2)
11)
5)
Continued
116
3
Linear relationships
on
next
page
A LG E B R A
3
Are
the
lines
shown
on
the
grid
below
perpendicular?
Justify
youranswer.
y
8
6
4
2
x
0
–2
2
4
6
8
10
–2
4
You
you
want
are
to
given
a
Draw
b
Where
Verify
the
these
Draw
that
a
square
on
coordinates
three
should
square?
c
draw
points
the
the
you
point
have
Cartesian
A (−5,1),
on
fourth
the
a
and
be
label
chosen
B (−4,4),
Cartesian
point
the
C (−1,
plane
and
3).
grid.
placed
it
coordinate
in
order
to
make
the
D
correct
point
using
two
dierent
methods.
5
Find
the
x-
and
(There
may
a
5x
6y
d
y
be
y-intercepts
more
than
of
the
following
relationships.
one.)
2
−
=
30
b
y
=
e
3x
−3x
+
11
c
=
f
2x
−
4y
=
18
–5
2
=
x
+
10
2
+
2y
24
7
6
The
countries
rising
of
the
greenhouse
carbon
dioxide
European
gas
(CO
levels
),
one
by
of
Union
made
setting
the
most
the
targets
decision
for
prevalent
the
to
tackle
emission
greenhouse
of
gases.
2
The
target
CO
emission
levels
for
vehicles
in
2015
depended
on
2
the
weight
of
the
vehicle
and
followed
a
linear
2015
Weight of vehicle
pattern.
2020
CO
emissions
Weight of vehicle
CO
2
The
(g/km)
(kg)
1300
125
1500
1800
150
1800
targets
parallel
to
for
that
2020
of
are
the
emissions
2
(kg)
even
2015
more
ambitious,
(g/km)
100
though
the
line
is
targets.
Continued
on
next
page
1 17
a
Represent
the
b
Determine
2015
the
targets
rate
of
on
change
a
graph.
of
the
linear
relationship.
Explain
its
meaning
in
this
context.
c
Using
the
targets,
fact
represent
emissions
d
ATL1
7
What
In
a
a
are
target
the
previous
exactly
the
Show
that
the
for
2020
2020
2020
challenges
unit,
same
that
the
you
size
targets
for
of
trying
A,
their
B
and
relationship
on
the
vehicles
determined
and
triangles
linear
to
with
set
that
same
a
two
are
parallel
graph
mass
targets
on
of
a
triangles
corresponding
C
is
similar
sides
to
the
that
of
2015
the
2015
targets
to
nd
the
1800 kg.
continent-wide
are
are
each
as
to
similar
if
scale?
their
angles
are
proportional.
other.
y
6
4
A
2
B
–4
x
0
C
–6
–2
2
4
6
–2
–4
b
Explain
how
constant
rate
Algebraic
linear
You
have
verbal
118
3
is
change
triangles
(that
the
can
slope
representations
seen
how
with
a
you
the
is
focus
can
graph,
description.
representations
which
of
similar
help
One
the
of
a
of
represent
of
table,
the
algebraic
this
Linear relationships
a
linear
coordinates
most
and
important
representation,
section.
you
between
relationships
relationship
a
these
to
any
demonstrate
two
points
is
that
a
line
always
has
the
a
same).
A LG E B R A
Recognizing
What
does
recognize
the
a
linear
equation
linear
of
relationships
a
line
relationship
4
like?
without
– What
is
Is
it
possible
graphing
to
it?
linear?
i
t
er
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r
Investigation
look
Make
a
table
with
two
labelled“Non-linear”.
columns,
You
will
one
need
labeled
about
15
“Linear”
and
the
other
B
rows.
Linear
2
Using
a
graphing
following
the
display
equations
“Linear”
column.
a
Non-linear
calculator
at
Write
a
time.
the
(GDC)
Every
f
or
graphing
time
equations
b
e
of
you
the
nd
tool
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non-linear
like
Desmos,
that
is
graphs
c
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r
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Generalize
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straight
4
Verify
5
Justify
your
pairs,
results
with
your
and
discuss
relationships.
and
write
a
rule
for
how
you
can
tell
if
an
a
line,
in
the
draw
write
the
its
other
equation
graphs
of
equation
the
in
column.
represents
ornot.
rule
why
Reect
In
your
line
n
c
1
rule
two
more
works.
discuss
whether
Justify
examples.
or
each
7
not
the
following
equations
represent
linear
answer.
Pairs
●
4x
●
2x
−
2y
=
2
2
+
2x
●
10
3y
=
8
5 y
+
3
=
−4
2
3
●
y
=
−
4x
+
1
1 19
Gradient–intercept
While
can
be
all
linear
relationships
represented
intercept
form
have
in
a
variety
of
5
– What
a
constant
algebraic
rate
of
forms,
change,
such
as
they
gradient-
form.
does
this
number
do?
i
t
er
i
r
Investigation
n
c
B
l
i
n
k
b
e
Online
applications,
willallow
You
1
Using
can
sliders
you
use
to
such
perform
these
online
as
the
this
ones
on
the
investigation
tools,
a
graphic
or
a
graphing
tool,
change
the
or
a
graphing
tool,
select
value
Math
using
display
value
m
of
is
Fun
sliders
website
to
calculator
in
alter
or
on
the
(GDC)
desmos.com,
parameters
or
graphing
m
and
c.
software.
y = mx. Note the changes in the
graph.
2
Using
sliders
a
for
m
and
then
change
the
value
of
c
in
y = mx + c. Note the changes in the graph.
3
Using
sliders
changes
4
in
Generalize
include
or
the
a
do
results
following
m
tool,
change
the
values
and
c
on
the
eects
of
m
and
c
and
What
●
Which
parameter,
m
or
c,
aects
the
direction
●
Which
parameter,
m
or
c,
aects
the
steepness
●
What
the
m
both
and
c
in
y = mx + c. Note the
refer
eect
summarize
them
in
a
table.
of
to
in
terms
changing
c
of
on
the
the
components
graph
of
the
of
of
a
linear
graph
the
a
of
Verify
your
results
for
of
Justify
why
your
Representing
two
advantages
slope
the
12 0
and
graph
3
results
lines
when
the
in
it
more
make
linear
relationships:
graph
linear
of
y = mx + c and in what way(s)?
relationship?
y = −
x + 4 and y = 3x − 5.
sense.
gradient–intercept
comes
y-intercept
without
to
y = mx + c and in what way(s)?
2
6
sure
relationship?
1
5
Be
details.
●
is
of
graph.
your
the
graphing
can
having
Linear relationships
to
to
graphing
become
use
a
form,
these
tools
table
of
y
=
mx
+
c,
has
relationships.
that
allow
values
or
you
The
to
draw
coordinates.
A LG E B R A
Activity
Use
the
4
–
Graphing
following
equations
a
in
line
this
activity.
1
Equation
y = 3x
y = 2x − 1
y
=
−x
+
4
y = −2x
−
y
4
x
=
y = −4x
+
6
2
1
Without
using
of
graphs
these
its
steepness
technology,
crosses
(how
make
the
many
a
prediction
about
where
each
y-axis, the direction of the graph and
units
vertically
and
how
many
units
horizontally).
2
Using
set
of
the
information
axis
ranging
3
Verify
your
graphs
4
Summarize
the
from
from
−10
using
steps
step
to
1,
graph
each
of
these
equations
on
a
+10.
technology.
you
can
follow
in
order
to
graph
a
linear
relationship.
Example
Q
A
For
the
line
a
State
b
Graph
a
1
its
given
slope
the
by
and
linear
The
slope
is
The
y-intercept
y = 4x − 2
y-intercept.
its
relationship.
given
by
m,
so
this
is
4.
The
equation
form,
is
(0,
of
the
line
is
in
gradient-intercept
y = mx + c
−2).
y
b
When
6
you
plot
5
the
star t
that
point(0,
equation
with
the
point
on
is
in
value
the
slope –intercept
of
the
form,
y-intercept and
y-axis In this case, plot the
−2).
4
3
2
1
0
–3
x
–2
–1
–2
–3
–4
–5
–6
Continued
on
next
page
1 21
Gradient
=
4
rise
so
=
slope
you
conver t
the
( m)
slope
into
a
fraction.
run
4
=
Any
whole
number
is
represented
with
1
as
the
denomi-
1
nator.
the
If
the
slope
is
negative,
put
the
negative
sign
with
numerator.
y
From
y-intercept you go up for the rise (or down for
the
6
a
negative
number)
and
to
the
right
for
5
the
run.
In
the
right.
this
case,
you
will
go
four
units
up
and
one
to
4
3
2
Continue
1
also
plot
this
a
process
point
in
to
the
keep
plotting
opposite
points. You
direc tion
from
can
the
y -intercept (in this case by going four units down and
0
–3
x
–2
one
unit
to
the
left).
a
ruler
–1
Then
use
to
draw
the
straight
line
connecting
all
–2
the
plotted
points.
–3
Practice
3
1
following
2
Graph
the
a
y
=
x
b
y
=
−3x
c
y
=
5x
Which
linear
relationships
using
the
gradient
d
+
−
of
e
7
3
f
the
following
lines
are
y
=
− 4x
parallel?
Which
are
and
y-intercept.
g
y
=
2x
h
y
=
3
i
y
=
+
−
perpendicular?
x
Justify
your
answers.
a
y
=
5x
−
b
y
=
−5x
4
−
4
c
d
g
e
h
f
i
y
=
y
=
−5x
+
11
+
10
Continued
12 2
3
Linear relationships
on
next
page
A LG E B R A
3
As
a
way
of
companies
70%.
a
A
trying
are
Time
and
Calculate
c
Represent
d
Use
your
e
Use
the
the
equation.
the
4
is
versatility
0.82
the
in
tonne
of
of
how
and
own
is
climate
an
average
1
a
which
speed
of
train
reduce
80
km
emissions
per
by
hour.
2
3
4
condition
of
this
linear
relationship.
graph.
the
to
initial
locomotive
condition
verify
what
the
your
rate
most
widely
The
released.
used
to
travels
answer
of
in
represent
in
change
material
drawback
Increased
is
that,
CO
2
and
locomotives
sector,
3.5
the
step
and
hours.
linear
relationship
as
an
d.
initial
condition
mean
in
situation.
world’s
CO
as
the
initial
words
transportation
(km)
far
equation
the
values.
and
values
construction.
of
of
has
0
change
change
this
table
in
hybrid
locomotive
( x)
nd
your
emissions
eco-friendly
kilometres
of
to
your
context
Concrete
of
Use
in
in
table
graph
to
this
hours
rate
the
rate
Explain
in
carbon
electric
complete
travelled
b
f
diesel
travelled
Distance
limit
changing
hybrid
Copy
to
(apart
for
levels
from
every
have
water)
tonne
been
of
due
to
concrete
linked
to
its
produced,
global
warming
2
change.
Concrete
accounts
for
almost
10%
of
the
world’s
CO
emissions.
2
a
Represent
the
relationship
between
tonnes
of
concrete
produced
and
tonnes
of
CO
2
released
b
Find
c
Write
d
Use
for
the
up
rate
down
your
205 000
to
of
100
change
the
and
equation
equation
tonnes
tonnes
of
to
of
of
the
concrete.
initial
this
calculate
condition.
linear
how
relationship.
much
concrete
has
to
be
made
to
release
CO
2
e
More
than
4
billion
tonnes
of
concrete
are
produced
each
year.
How
much
CO
2
does
5
The
this
release
technology
annually?
exists
to
cut
CO
emissions
by
up
to
90%
by
using
dierent
concrete
2
recipes.
One
developer
claims
it
has
created
a
concrete
that
can
absorb
CO
from
the
2
air.
When
water
is
added,
every
tonne
of
concrete
absorbs
0.6
tonnes
of
CO
.
2
a
Write
b
If
the
an
equation
whole
to
world
represent
switched
to
this
scenario.
this
type
of
concrete,
how
much
CO
would
be
2
absorbed
c
Some
not
of
d
of
been
funds.
What
by
the
these
4
tonnes
innovative
popular,
Why
billion
do
obstacles
concrete
and
some
of
you
think
this
and
of
challenges
the
concrete
varieties
used
have
technology
is?
What
stand
in
been
has
action
the
each
way
year?
available
been
needs
of
this
for
abandoned
to
take
new
years
but
because
have
of
lack
place?
technology
being
used
ATL1
for
all
concrete?
Continued
on
next
page
12 3
6
The
bottled
cost
of
a
a
bottle
of
industry
water
in
Determine
the
equation
What
does
the
gradient
b
What
is
c
A
d
the
total
high-quality,
water
would
What
other
carry
7
water
The
their
and
you
own
buying
table
water
to
from
contains
operating
cost
or
the
for
States
a
in
one
in
capital
coal
Coal
Hydroelectricity
a
Determine
the
equation
that
(y)
of
water
cost
power
a
day
this
option
each
and
b
type
y-intercept
What
is
the
electricity
c
What
is
Coal
plants
plants
and
e
do
other
making
124
3
need
not.
bottles
of
inence
of
(cost
plant
a
year
$40.
be
(not
How
average
water.
more
someone’s
leap
cost
year)?
bottles
of
eective?
decision
whether
to
water?
to
build
and
a
the
plant
before
hydroelectric
cost
creating
power
Operating
cost
$250 000 000
$0.01
the
a
many
$0.20
the
these
producing
any
plant.
(per
kWh)
cost
total
cost
of
electricity
is
the
gradient
cost
to
build
cost
to
produce
the
facility
plus
the
equations?
5000 kWh
can
of
10 000 kWh
of
find
the
electricity.
cost
electricity
by
operating
cost
number
plant?
fuel,
What
if
in
do
x
$80 000 000
represents
producing
of
You
producing
multiplying
the
per
kWh
kWh
the
by
the
facility
hydroelectric
happen
price
apart
considering
Linear relationships
of
generates.
whereas
will
the
factors,
when
What
The
plant?
of
each
y-intercept)
What
of
each
cost
at
plant.
represent
cost
at
the
electricity
d
of
for
to
The
for
annually.
buying
approximately
bottles
Capital
billion
cents.
cost
for
might
buy
50
of
costs
order
to
is
$200
equation?
bottle
cost,
over
the
this
bottle
drink
bottle
worth
represents
represent
apart
water
United
that
of
have
currently
the
rellable
factors,
following
power)
cost
is
of
to
fuel
from
the
cost
functions
of
both
plants
(gradient
increases?
cost,
what
linear
type
might
of
inuence
power
plant
a
to
government’s
build
in
its
decision-
country?
A LG E B R A
Other
of
linear
The
of
algebraic
equations
equation
which
slope
m,
is
representations
of
a
line
can
be
given
gradient–intercept
y-intercept
c
and
in
form.
passing
a
variety
The
of
forms,
equation
through
point
of
(x
,
a
y
1
represented
in
the
following
one
line
)
with
can
be
1
ways:
y
Slope–intercept:
y
=
mx
+
c
6
(x
,
y
1
Point–slope:
y
–
y
=
m(x
–
x
1
)
1
)
1
4
slope
Standard
where
A,
form:
B
and
Ax
C
+
are
By
=
=
m
C,
integers.
2
(c,
0)
0
–6
x
–4
–2
–4
Activity
1
Copy
5
and
Moving
complete
represent
Point
–
its
this
equation
and
between
table.
in
the
Given
three
Point–slope
y – y
gradient
=
a
dierent
point
dierent
form
m(x – x
1
and
the
3)
and
m = 2
gradient
of
a
linear
relationship,
forms.
Gradient-intercept
form
y = mx + c
)
Standard
Ax +
form
By = C
1
y −
(1,
forms
3
=
2x
–
y = 2x + 1
2
y − 3 = 2(x − 1)
y = 2x + 1
−2x
+
y = 1
1
(−4,
−7)
and
m =
2
(2,
−1)
(0,
5)
and
and
m = −4
m = −1
2
(−6,
4)
m =
and
3
(−9,
2)
and
2
Which
3
What
The
the
form
are
form
of
easiest
doing
m = −7
do
some
the
way
you
of
prefer?
the
advantages
equation
to
Explain.
graph
of
it
a
and
linear
since
disadvantages
relationship
there
are
also
of
may
several
each
form?
determine
options
for
this.
12 5
Example
Q
Graph
2y
2
the
+
following
18
=
line.
−3x
If
you
the
are
intercept
A
2y
+
18
=
−3x
2y
=
−3x
−
using
equation
graphing
often
form.
software
needs
to
be
in
or
a
GDC,
slope–

18
Rearrange
the
equation
to
isolate
the
y
−3
y
x
=
− 9
2
−3
m
From
=
and
c
=
this,
extract
m and c
−9
2
y
5
Plot
0
the
y-intercept, which is (0, −9).
x
–10
–5
–10
–15
y
5
3
A
gradient
of
means
–
that
ever y
ver tical
change
2
0
–10
–5
x
of
5
3
units
change
go
down
and
down
to
the
3
and
extend
corresponds
right
the
of
move
line
2
right
in
–15
Are
12 6
there
3
other
ways
to
draw
Linear relationships
a
line,
given
the
same
2.
both
–10
information?
to
units.
a
horizontal
From
Join
the
these
directions.
y-intercept,
twopoints
A LG E B R A
Example
Graph
Q
2y
3
the
+
following
18
line.
−3x
=
Two
points
points,
x-
A
and
but
dene
the
18
=
−3x
To
2(0)
+
18
=
−3x
Solve
for
18
=
−3x
−6
=
x
Write
down
2y
+
18
=
2y
+
18
=
0
2y
=
−18
−3(0)
To
easiest
You
ones
may
tend
nd
nd
the
x-intercept, substitute
+
0)
line.
nd
to
be
any
two
the
y-intercepts.
2(0)
(−6,
a
y = 0.
x
the
the
ordered
pair.
y-intercept, substitute
x = 0.
y
Solve
for
Write
down
y = −9
(0,
−9)
the
ordered
pair.
y
5
x
0
–10
–5
5
10
Plot
the
connect
two
points
and
draw
a
straight
line
to
them.
–10
–15
Reect
●
Is
there
linear
best
●
In
such
a
discuss
thing
relationship?
method
which
most
●
and
a
so,
be?
situation(s)
useful?
Which
might
as
If
is
8
best
can
method
you
tell
of
graphing
from
the
a
particular
equation
what
the
Explain.
each
of
the
three
forms
of
linear
equation
Explain.
method
for
graphing
a
line
do
you
prefer?
Explain.
12 7
Practice
1
4
Represent
a
2y
=
each
−4x
+
of
the
following
8
b
d
x
−
y
=
2
Represent
3
Find
5
the
by
Which
of
a
x
–
7y
d
6x
g
28y
j
7x
+
=
3y
=
in
question
y-intercepts
to
see
following
for
10y
=
4x
lines
–
4x
on
a
each
=
in
the
form
20
1
of
the
these
2x
–
=
graph
parallel?
b
y
using
the
equations
intercepts
Which
=
12
following
lines
by
h
8x
+
k
4x
+
plotting
24
b
5x
−
2y
=
=
the
in
10y
are
are
on
the
following
lines
4x
−
2y
=
1.
c
+
9
−2
form.
Validate
your
lines.
+
Justify
6y
f
8
–
7x
i
y
=
−2x
0
=
2x
0
=
−3x
=
=
your
answers.
−2
−y
+
9
l
and
30
=
y-intercepts.
0
c
−
4y
+
12
f
using
a
method
of
your
choosing.
3x
−
12
b
3y
=
5x
+
15
c
d
y
=
−x
+
4
e
4y
−
8x
=
12
f
g
2y
10
3y
3x
=
−
8
+
−
c
y
2x
=
mx
x
a
=
2x
perpendicular?
e
Graph
f
the
1
−
=
question
−3
x-
0
=
gradient–intercept
8
4y
c
y
i
0
e
y
=
1
are
−5
–
−
whether
10
the
+
and
the
18
Graph
lines
d
6
−
h
checking
4y
=
0
the
x-
graphs
a
5x
relationships
e
g
4
linear
+
14
−
7y
h
i
7
Using
asphalt
because
the
in
Angeles
create
asphalt
atmosphere.
areas,
to
roads
absorbs
Temperatures
some
cases
elected
to
the
in
causing
paint
and
the
parking
Sun’s
these
severe
asphalt
lots
rays
zones
and
are
health
with
has
a
created
radiates
much
issues.
the
hotter
To
special
“urban
heat
than
combat
material
back
in
this,
that
heat
islands”
into
the
suburban
the
city
reects
of
the
and
rural
Los
Sun’s
rays.
2
a
Crews
and
b
an
What
can
paint
equation
do
the
4 m
each
written
gradient
in
and
minute.
all
three
Represent
forms.
y-intercept
this
Show
mean
in
relationship
your
the
using
a
table,
3
Linear relationships
graph
working.
context
of
this
situation?
Continued
12 8
a
on
next
page
A LG E B R A
c
Because
of
without
0.5 °C
What
this
in
do
on
World
the
One
in
new
material.
hour.
all
the
technology,
Suppose
Represent
three
forms.
gradient
temperatures
this
Show
and
the
y-intercept
o
temperature
relationship
your
cool
using
at
a
much
6 pm
table,
faster
is
a
at
night
28 °C
and
it
graph
and
an
than
drops
by
equation
working.
of
this
new
equation
mean
in
the
context
of
6
–
Wide
Turn
Fund
it
for
“OFF”
Nature
issues
tips
on
how
to
lessen
your
impact
environment.
of
use
new
situation?
Activity
The
the
every
written
d
this
these
and
tips
to
is
to
unplug
completely
chargers,
turn
even
if
o
no
equipment
device
is
when
not
attached.
Approximately
one-quarter
of
industrialized
power
mode.
“standby
1
all
residential
countries
These
power”
or
is
energy
used
devices
by
and
“phantom
consumption
electrical
chargers
in
devices
use
what
in
is
“idle”
called
power”.
Find out the cost per kilowatt hour that your electricity provider
charges.
2
Look
at
wasted
3
Create
your
on
a
current
devices
linear
phantom
electricity
in
“idle”
equation
power
in
and
assume
that
25%
is
mode.
(or
your
bill
linear model)
home.
What
is
to
represent
x
your
the
variable?
cost
What
of
is
your
y
variable?
4
Describe
this
the
5
Represent
6
Use
in
7
a
Go
rate
of
change
and
the
initial
condition
in
the
context
of
question.
the
your
model
year
if
you
through
model
to
as
predict
followed
your
a
graph.
how
the
house
much
given
and
energy
tip.
make
a
and
Show
list
of
all
all
money
your
you
could
save
calculations.
devices
and
cords
that
ATL2
draw
phantom
sources
your
8
and
power.
create
an
Research
action
ideas
plan
for
from
easy
a
variety
ways
to
of
save
reputable
energy
in
home.
What
are
the
challenges
involved
in
making
the
choice
to
turn
o
or
ATL1
unplug
using
your
computer/phone
Determining
In
order
point
on
charger/television
when
you
are
not
them?
to
the
determine
the
line
(x,
the
y)
equation
equation
and
the
of
of
the
gradient
a
line
line,
of
the
you
line
simply
need
a
(m).
12 9
Example
Q
Find
the
4
equation,
through
(2,
7)
and
in
slope–intercept
has
a
slope
of
form,
of
a
line
that
passes
5.
A
y
= mx
+ c
Star t
with
values
7 = (5)(2) + c
the
you
equation
have
been
and
substitute
in
the
given.
7 = 10 + c
Solve
for
the
Write
down
y-intercept.
3 = c
y = 5x −
Example
Q
Find
(9,
the
−2)
3
the
equation
in
the
required
form.
5
equation,
and
(−15,
in
standard
form,
of
a
line
that
passes
through
6).
A
Calculate
y
−
the
slope
by
using
the
formula.
y
2
m
1
=
x
−
x
2
1
6 − ( −2 )
m
=
−15 − 9
8
m
=
−24
1
m
=
−
3
Substitute
y
=
2
=
mx
1


−

2
=
1 =
+ c
−3 +
point
the
slope
into
the
and
the
coordinates
of
equation.


3
one
(9 ) + c

c
c
Write
down
the
equation
in
gradient–
1
y = −
x +1
intercept
form.
3
Multiply
3y
=
−x
+
that
x + 3y = 3
13 0
3
both
sides
of
the
equation
by
3
so
3
Linear relationships
the
all
coecients
equation
in
the
are
integers.
required
Then
form.
write
in
A LG E B R A
Example
Q
Find
the
the
line
6
equation
of
the
line
through
(6,
−8)
that
is
perpendicular
The ∴ symbol
to
means “therefore".
y = 3x + 5. Write your answer in slope–intercept form.
A
m
= 3
Determine
1
∴
m
=
it
is
If
it
the
parallel
to
slope
the
of
the
given
linear
line,
use
relationship.
the
same
If
slope.
−
perp
is
perpendicular,
use
the
negative
reciprocal.
3
y
=
mx
+ c
Substitute
1

−8 =

−



3
−8 =
−2 +
−6
=
c
=
−
into
the
the
slope
and
coordinates
of
the
point
equation.
(6) + c

c
1
y
x
−
Write
6
down
the
equation
in
the
required
form.
3
Practice
1
Represent
three
5
the
forms
a
following
linear
relationships
(gradient–intercept,
point–slope
algebraically.
and
answers
in
all
form).
8
8
6
6
4
4
2
2
0
your
y
b
y
standard
Write
x
0
x
–4
–4
–2
–4
–6
–8
–2
–4
–6
–8
Continued
on
next
page
13 1
y
c
8
8
6
6
4
4
2
2
0
–6
–4
y
d
0
x
–2
–16
2
–12
x
–8
–2
–2
–4
–4
–6
–8
2
Find
the
equation
form
without
following
lines.
line
with
a
slope
of
b
The
line
with
a
slope
of
c
The
line
with
a
slope
of
d
The
line
passing
e
The
line
with
a
f
The
line
with
an
g
The
line
that
passes
through
the
h
The
line
that
passes
through
(−1,
The
line
the
According
the
number
and
a
to
between
and
and
−2
(−6,
y-intercept
of
x-intercept
and
(−3,
−4)
of
the
answers
in
gradient–intercept
to
and
y-intercept
passing
an
−2)
4
of
through
x-intercept
and
and
7
of
(4,
an
that
is
origin
1)
5
(2,
of
3)
5
3)
x-intercept
of
−3
perpendicular
that
and
is
is
to
the
line
perpendicular
perpendicular
to
to
3x
the
the
–
6y
line
line
=
2x
that
8
+
3
=
12
passes
(x,
1)
has
a
slope
of
−5.
Determine
the
equation
of
Union
population
decrease.
ivory,
of
are
Conservation
decreased
Despite
which
number
for
from
warnings
the
main
elephants
and
died
Nature,
550 000
stier
causes
that
of
of
470 000,
penalties,
this
each
to
between
the
decline,
2006
and
illegal
and
the
poaching
continue.
year
2013?
Continued
13 2
7y
x
average
2006
your
1).
and
elephant
elephant
a
and
theInternational
continues
is
−4
through
(2,
value
African
selling
What
−2)
through
and
2013
(4,
Write
decimals.
The
line
4
using
the
a
through
3
of
Linear relationships
on
next
page
A LG E B R A
b
Assuming
the
population
continues
to
decrease
at
this
average
Set
rate
every
year,
represent
the
African
elephant
0
since
2013
with
a
linear
model.
Use
your
equation
to
and
c
African
Assuming
when
5
As
the
colder
sea
a
up
much
b
i
a
has
n
If
then,
linear
the
the
known
the
level
risen
trend
has
model
approximately
current
thermal
level
to
to
at
the
warmer
expansion.
the
risen
an
million
same
water
equation.
rate,
oceans.
In
average
started
people
when
will
takes
Melting
phenomenon.
NASA
continues,
year
extinct?
into
this
since
100
decrease
because
water
be
year.
become
rise
as
fresh
sea
equation
sea
levels
more
this
continues
elephant
sea
process
Since
a
up,
dumping
Worldwide,
level.
l
by
African
heats
water,
population
population
the
Earth
levels.
Set
the
will
problem
elephant
to
determine
predict
the
the
2006
population
of
1993,
3.2
all
of
these
started
millimetres
the
to
room
than
compound
NASA
According
within
more
glaciers
recording
live
up
this
per
the
measuring
year.
data,
how
data?
three
feet
people’s
(94 cm)
homes
of
be
sea
ooded?
k
b
e
Search for the video entitled “How Earth Would Look
If
All
The
Ice
continents
you
6
This
Melted”.
to
get
a
You
more
can
click
detailed
on
specic
look
at
where
live.
graph
represents
global
carbon
dioxide
levels
in
the
atmosphere
since
1980.
y
420
400
)noillim
380
rep
st rap(
360
slevel
340
OC
2
320
300
x
1980
1985
1990
1995
2000
2005
2010
2015
2020
Year
Continued
on
next
page
13 3
a
According
to
the
graph,
what
was
the
CO
level
in
1990?
What
was
it
in
2017?
2
b
According
to
the
graph,
what
is
the
change
per
year
of
CO
levels
in
the
atmosphere?
2
c
Use
the
graph
condition
d
Verify
you
e
determine
the
equation
of
the
line,
using
1980
as
your
initial
(y-intercept).
your
may
to
equation
using
dierent
points
on
the
graph.
Discuss
any
discrepancies
have.
Researchers
vulnerable
have
to
discovered
melting
once
that
CO
Antarctica’s
levels
exceed
ice
will
be
signicantly
more
600
2
parts
per
trend
million.
When
will
this
occur
if
the
current
Be
of
continues?
sure
to
Reference
f
Research
your
country’s
carbon
dioxide
emissions
ATL2
of
7
the
How
does
world?
your
Why
Chlorouorocarbons
aerosol
CFCs
in
our
propellants,
contribute
to
atmosphere
catastrophic
do
country
you
(CFCs)
cleaning
the
until
compare
think
were
global
of
the
variety
these
rest
know
and
in
the
state
your
source
how
is
you
reliable.
is?
created
solvents
destruction
a
this
with
a
sources.
per
answer
person.
check
reputable
and
the
decision
in
1928
and
refrigerants.
ozone
was
layer.
made
were
It
has
They
to
used
in
since
products
been
continued
address
this
such
proved
to
as
that
accumulate
potentially
issue.
y
Chlorouorocarbon
concentration
in
Ear th’s
atmosphere
600
550
500
)noillim
450
rep
st rap(
400
CFC
350
300
250
200
x
1978
1983
1988
1993
1998
2003
2008
2013
2 018
Year
A
graph
that
contains
two
sets
of
trends,
like
this
one,
can
be
referred
to
as
a
piecewise
relationship.
Continued
13 4
3
Linear relationships
on
next
page
A LG E B R A
a
Using
a
variety
of
sources,
research
the
global
eort
to
ban
CFCs
that
ocially
ATL2
know
b
c
Use
in
the
you
the
can
According
to
the
on
the
to
Discuss
the
after
all
concentration
to
the
Write
has
CFC
of
two
Formative
the
been
was
was
eort
it
was
global
rate
ban?
the
successful?
agreement
accomplished,
will
in
that
remain
what
you
you
Explain
how
you
to
ban
CFCs
was
CFCs
year
have
the
will
is
atmosphere
learned
CFC
from
dierent
the
points
concentrations
answer
decompose
be
using
in
using
in
dierent
the
points
have.
(which
there
concentrations
answer
in
your
may
CFC
have.
change
Verify
you
in
your
may
of
stopped
in
change
Verify
started?
has
of
rate
discrepancies
production
facts
say
presented.
the
the
global
what
any
trend,
view
discrepancies
global
CFCs
decreasing
down
the
of
whether
what
to
graph,
sources
answer.
any
Discuss
much
when
graph,
up
the
points
your
leading
graph.
Although
even
the
graph.
According
the
Do
determine
Explain
atmosphere
f
to
successful.
on
e
1980s.
trust
graph
atmosphere
d
late
not
yet
for
zero
this
very
the
over
CFCs
case
slowly,
case),
100
left
study
so
some
years.
in
the
and
According
atmosphere?
its
graph.
i
assessment
t
e
r
r
i
c
a
started
Reducing
the
emission
of
greenhouse
gases
has
been
a
major
focus
C, D
of
governments
countries
across
signed
and
their
eect
The
following
the
on
data
the
globe
Kyoto
global
since
Protocol,
the
mid-1990s.
agreeing
to
In
reduce
1997,
most
greenhouse
gases
temperatures.
represent
the
-equivalent
CO
concentration
of
all
greenhouse
2
gases
a
in
Plot
the
atmosphere
these
points
benchmark
b
According
gases
in
year
to
on
a
in
graph,
agreed
your
parts
linear
in
per
million
span
of
427
2000
441
2005
455
2010
469
2015
483
Kyoto
model,
a
1995
making
the
over
sure
you
start
Protocol).
what
was
the
Plot
your
the
20
years.
x-axis
years
concentration
at
1990
from
of
the
(as
1990
mix
that
to
of
was
the
2030.
greenhouse
1990?
Continued
on
next
page
13 5
c
According
to
greenhouse
d
Use
your
Write
graph
the
Verify
f
According
g
If
h
your
year
Draw
this
as
point
your
Reect
●
What
is
in
to
to
your
in
to
the
at
which
and
the
the
three
step
d
it
the
your
equation
using
threshold
on
is
dierent
what
change
per
year
in
the
concentration
is
is
the
line,
using
1990
as
Protocol,
the
500
concentration
ppm
reach
the
as
for
that
a
the
threshold.
greenhouse
concentration
amount
dotted
of
How
●
Given
many
this
emissions
to
At
●
countries
agreement,
data
continue
what
What
agreed
to
that
to
was
would
you
increase
the
you
graphed
in
the
it
if
current
horizontal
Verify
adopted
Kyoto
expect
in
short
that
line.
your
of
and
when
graphical
was
it
the
Protocol?
the
greenhouse
Formative
term?
In
the
gas
assessment
long
some
●
Whose
be
Kyoto
obstacles
to
reaching
and
enforcing
term?
an
agreement
Protocol?
responsibility
held
accountable
is
–
reducing
industry,
greenhouse
government
gases?
or
Who
should
individuals?
Explain.
Vertical
So
far,
the
sloping.
Can
with
13 6
its
and
lines
What
horizontal
you
have
happens
steepness
be
when
the
measured?
equations?
3
studied
Linear relationships
lines
have
line
How
all
is
been
vertical
can
you
diagonal
or
or
horizontal?
represent
these
this
year?
lines
gases,
in
continue?
Continue
rate?
are
the
condition.
greenhouse
trends
ATL1
like
initial
gases
9
when
the
forms.
enforced?
●
of
algebra.
graph
intersects
of
results.
discuss
Kyoto
the
concentration
threshold
algebraic
what
atmosphere?
model,
Protocol
will
model,
the
determine
answer
Kyoto
what
the
linear
equation
e
the
your
gases
your
line
results
to
are
show
the
same
A LG E B R A
Investigation
–
Equations
of
ver tical
and
i
lines
t
er
i
r
horizontal
6
Graph
the
following
lines
by
rst
creating
a
table
of
values
for
each
line.
n
c
1
Note
B
that
on
y = 2 means that the value of
one
set
of
y is 2 for any value of
axes.
y = 2
x
2
How
3
Pick
4
How
are
any
are
the
lines
two
the
x. Graph all three lines
y
related?
points
slopes
y = 3
on
x
What
each
related?
type
line.
How
y
y
of
lines
Calculate
does
−5
x
are
the
the
=
they?
slope
slope
y
of
relate
each
to
line.
the
steepness
of
the
graph?
5
Repeat
step
1
for
the
x is 2 for any value of
following
lines.
Note
x
How
7
Pick
8
How
the
are
any
are
these
two
the
lines
points
slopes
x = 2 means that the value of
y.
x = 2
6
that
x = 3
y
x
related?
What
on
line.
each
related?
How
y
type
of
x
lines
Calculate
does
x = −5
the
the
are
they?
slope
slope
y
of
relate
each
to
line.
the
steepness
of
graph?
9
Write a generalization about lines that are of the form
x = a, where a is a number.
10
Write a generalization about lines that are of the form
y = b, where b is a number.
11
Verify
12
Justify
your
why
generalizations
your
for
one
generalizations
more
are
case
of
each
type.
true.
13 7
Practice
1
Find
the
6
equations
of
the
lines
shown
on
the
set
of
axes
below.
y
6
4
3
2
1
x
0
–5
–4
–3
–2
–1
–2
–4
2
Find
the
equation
of
the
line
through
(6,
1)
that
is
parallel
3
Find
the
equation
of
the
line
through
(0,
1)
that
is
perpendicular
4
Find
the
equation
of
the
line
through
(5,
5)
and
5
Find
the
equation
of
the
line
through
(−3,
6
Find
the
equation
of
the
line
through
(4,
7
Find
the
equation
of
the
line
through
(−7,
13 8
3
Linear relationships
−2)
2)
and
that
8)
(5,
is
that
the
line
to
y
the
=
34.
line
x
=
−2.
−2).
(1,
−2).
parallel
is
to
to
the
line
perpendicular
to
x
=
the
−8.
line
y
=
−4.
Unit
Linear
summary
relationships
●
graph
●
table
●
coordinates
●
words
●
algebraic
Some
(or
of
The
its
change
point
as
x-intercept
or
the
(run).
(x
,
y
1
)
slope
ratio
The
and
positive
A
negative
of
of
(x
a
its
,
y
2
of
and
)
a
its
line
for
gradient
slopes
gradient
slopes
parallel,
If
two
lines
are
perpendicular,
of
state
the
This
also
application
nd
=
y-intercept
is
the
is
(for
equals
the
0
and
is
the
a
measure
change
the
of
its
(rise)
gradient
the
of
is
ways:
are
its
gradient
steepness.
to
its
the
It
is
horizontal
line
joining
the
to
by
as
Given
a
are
to
the
and
same.
are
).
product
right.
can
negative
Another
of
the
slopes
of
.
crosses
nd
the
the
x-axis.
x-intercept
by
x
a
initial
the
left
gradients
graph
you
where
the
right.
example
where
for
the
to
from
their
that
For
left
gradients
then
graph,
point
from
example,
solving
questions.
y-intercept
1.
point
of
referred
is
their
relationship
equation
y
then
other
lines
x-intercept
substituting
The
each
this
perpendicular
Given
of
relationship
downwards
are
The
variety
y-intercept.
upwards
lines
to
a
is
two
way
in
2
If
reciprocals
linear
vertical
formula
1
A
represented
characteristics
gradient
dened
be
equation.
the
slope),
can
graph
condition
equation
substituting
crosses
x
=
0
of
and
in
the
the
the
y-axis.
context
graph,
solving
for
you
of
can
y
13 9
To
graph
1
Plot
2
Starting
3
a
line:
the
y-intercept
from
the
coordinates
of
this
for
process
Draw
a
line
on
the
coordinate
y-intercept,
another
a
few
point
more
between
the
use
on
the
the
axes.
gradient
to
coordinate
determine
axes.
the
Continue
points.
y-intercept
and
the
other
points.
y
(1,
y
=
2x
+
5)
3
(0,
(–1,
3)
1)
x
(–2,
–1)
(–3,
A
linear
relationship
Gradient–intercept
Point–gradient
or
can
or
be
–3)
represented
in
(slope–intercept)
(point–slope)
form:
y
one
of
three
form:
y
–
m(x
y
=
=
mx
–
form:
Given
the
line
two
●
in
Select
use
graph
+
of
a
dierent
two
the
Ax
=
line,
C,
where
you
can
A,
B
and
determine
c
)
1
C
are
the
integers.
equation
of
the
ways:
points
gradient
By
+
x
1
Standard
forms:
on
and
the
a
graph,
point
to
nd
the
write
gradient
the
and
equation
in
then
point
gradient–form.
●
Use
the
gradient
y-intercept
and
algebraically,
gradient–intercept
Horizontal
horizontal
Vertical
vertical
14 0
3
lines
line
lines
line
where
have
is
is
a
and
a
gradient
of
the
gradient
always
cuts
then
the
y-axis
write
the
to
nd
the
equation
in
form.
always
have
it
of
the
Linear relationships
of
zero.
form
that
form
is
x
y
=
The
equation
a.
a
a.
undened.
=
of
The
equation
of
a
er
i
o
t
r
i
Key
review
to
Unit
review
Level 1–2
1
State
a
y
the
=
6x
n
c
Unit
A
question
levels:
Level 3–4
slope
−
and
the
Level 5–6
y-intercept
of
Level 7–8
each
of
the
following
lines.
12
2
b
y
=
−
x
+
23
7
2
Determine
the
x-
and
y-intercepts
of
the
following
lines.
y
a
10
8
6
4
2
0
x
–2
–2
–4
3
b
5x
−
4y
=
20
c
7x
+
3y
=
42
Plot
your
the
graph
of
each
of
the
following
lines
using
a
method
of
choosing.
a
y
=
−2x
b
y
=
x
−
+
8
7
141
4
Copy
and
complete
the
table
for
lines
A,
B,
C
and
D.
y
8
A
6
4
B
2
C
0
–4
x
–2
4
6
8
10
–2
–4
D
–6
Line
Slope
y-intercept
Equation
x-intercept
A
B
C
D
5
Represent
intercept
each
of
the
following
equations
using
gradient–
form.
1
a
y
=
x
− 12
b
− 2
e
5x
=
10y
−
20
c
x
=
8y
2
1
2
d
x
= 3 y
2
6
y
− 8x
=
−6
3
Find
the
gradient
of
the
line
L
that
passes
through
each
pair
of
points.
7
a
A(1,
5)
and
B(1,
–2)
b
A(–1,
c
A(0,
4)
and
B(2,
8)
d
A(3,
A
tank
has
and
falls
a
Is
this
b
Write
day,
c
142
3
slow
0.5 cm
a
a
leak
and
how
in
it.
The
2)
and
and
water
B(2,
B(5,
level
9)
4)
starts
at
100 cm
day.
constant
down
d,
After
a
6)
an
increase
equation
water
many
level,
days
Linear relationships
or
constant
showing
the
decrease
relationship
L.
will
the
tank
be
situation?
empty?
between
8
Graph
each
of
the
following
lines
using
a
method
of
your
choosing.
a
6y
+
c
2 y
3x
=
12
b
10y
−
x
=
x
1
+
−50
x
= 10
d
2 y
+ 3 =
4
3
10 + 2 y
e
0
=
24
+
8x
+
3y
4 x
f
=
3
9
Match
each
a
5y
of
the
following
equations
to
its
1
3x
−
=
15
b
y
=
4
x
c
2
1
x
3
y
+
y
+
4
=
0
2
y
ii
i
graph.
2
2
1
0
–4
–2
x
2
–2
0
x
–4
–4
–1
–2
–8
–3
–10
–4
–5
y
iii
3
2
1
0
x
–4
–1
–2
–3
14 3
10
The
line
through
Determine
11
Write
the
passes
12
Find
13
Write
that
14
the
Write
the
HIV
the
there
living
a
Write
by
and
a
number
newly
b
If
the
this
new
c
Is
3
of
(−1,
with
in
1)
has
a
slope
of
in
parallel
that
line
point
the
linear
to
3x
+
goes
through
perpendicular
(2,
y
−
4
=
0
that
line
(−3,
to
9)
3x
+
and
y
−
(9,
4
=
1).
0
−5).
parallel
to
x
=
of
the
a
total
HIV
,
that
local
−3
passes
perpendicular
of
and
year.
36.63
there
The
million
were
rate
300 000
technology
equation
in
with
of
1.8
realistic
Why
million
new
people
and
the
in
that
models
the
millions
HIV
infections
trend,
to
y
=
4
that
people
who
since
per
year
to
will
been
follow
there
model
or
Linear relationships
why
to
use
not?
for
such
year
support
2016.
continues
what
have
be
an
new
infections
governments.
people
of
line
infections?
a
−5.
6).
drug
infected
rate
this
of
−5).
approximately
linear
epidemic?
14 4
(2,
the
the
were
advancements
agencies
line
line
equation
infections
decreasing
a
(c,
−7).
through
already
the
of
through
(6,
2016,
of
and
c
point
equation
Determine
In
the
of
of
equation
passes
passes
16
equation
equation
the
−4)
value
through
through
15
the
(2,
no
of
is
due
to
global
17
According
States
in
of
2016.
to
The
14 000
done
in
is
the
What
is
there
number
in
or
main
the
Academy
were
had
The
when
of
average
farms
this
of
Sciences
cheetahs
the
decision
livestock
cause
7100
decreased
1975,
Africa.
agricultural
that
a
theNational
America,
estimated
was
to
left
in
signicantly
last
to
of
United
the
from
wild
an
comprehensive
convert
has
the
count
wilderness
caused
the
loss
of
areas
habitat
decline.
decrease
in
the
number
of
cheetahs
Set
each
1975
to
year?
be
year
0
and
determine
b
Assuming
this
is
still
the
average
rate
of
decline,
what
is
the
the
equation.
cheetah
c
Assuming
the
18
19
21
this
cheetah
this
average
become
the
equation
4x
+
2y
−
7
=
that
2x
+
3y
−
12
Find
−
the
6y
0
=
1
=
0
of
parallel
b
perpendicular.
ABC
C(−1,
5).
Fully
justify
of
decline
has
k
if
has
of
the
the
the
line
same
lines
vertices
Determine
3x
your
A(3,
whether
the
18.6%
of
the
world
In
2015,
10.9%
of
the
population
was
b
Assuming
the
this
population
c
Determine
d
Assuming
average
rate
that
an
e
What
this
is
−
2y
−
5
B(−3,
triangle
population
was
continues,
=
as
0
−5)
is
a
to
will
to
the
the
line
line
and
and
right
triangle.
people
in
the
when
undernourished.
undernourished.
is
the
this
represent
continues,
was
change
what
undernourished
rate
year?
percentage
of
the
year?
this
scenario.
will
world?
per
Do
there
you
be
no
think
this
is
possible?
decisions
driving
x-intercept
percentage
equation
undernourished
actually
when
answer.
1991,
What
perpendicular
−1),
In
a
continues,
are:
a
Triangle
rate
0.
value
+
year?
extinct?
Determine
kx
20
population
this
at
local,
country
and
world
level
could
be
change?
14 5
22
The
decision
color
and
of
style;
which
car
it
also
environmentally
necessity
States,
mile
and
a
when
the
to
costs
Determine
between
and
b
What
you
is
the
costs)
c
What
you
is
total
were
d
Which
or
i
planning
ii
the
US
When
a
f
is
What
On
of
average,
of
conversion
14 6
kg
of
go
that
3
cost
is
the
meat
of
car
to
at
8
is
is
the
sells
now
United
cents
for
per
$21 600
it
both
What
do
the
equations?
miles?
If
reached
60 000
it
buying
option?
miles?
If
reached
cheapest
option?
relationships
(gradient
if
increases?
car
rebate
electric
exactly
mileage
using
your
cost
(a
partial
cars?
the
of
same?
the
cars
original
might
go
Write
to
when
equations.
into
your
buy?
of
feed
consume
this
this
information
are
(this
between
must
have
linear
animal
high,
kg
at
of
on
make
Linear relationships
the
sure
a
feed
least
model,
meat
in
6
with
is
needed
called
table
and
for
the
kg
of
showing
kg
rows
y-axis.
your
a
relationship
cheapest
the
new
from
kilograms
we
the
be
the
the
these
larger
each
apart
Focus
once
US
a
In
costs
30 000
linear
of
relating
in
be
car
the
oers
and
once
aected
in
cars
car.
(including
at
car
these
cheapest,
which
car
the
purchase
total
3.85
table
Sketch
the
factors
relationship
b
gasoline
new
traveled.
car
would
be
a
of
that
every
feed
ratio).
Represent
Your
would
statement
other
kilogram
a
on
model
decision
23
of
the
general
each
car
of
miles
each
one
about
run.
car
would
sell
government
refund)
e
which
for
represents
the
just
comparison
to
each
of
to
choosing
represent
car
characteristic
price
the
longer
electric
each
sell
cost
no
$19 000
that
of
to
which
y-intercept)
the
cost
total
60 000miles,
for
mile
of
and
planning
the
per
y-intercept
30 000miles,
cost
around
sells
cost
is
involve
A
equation
total
and
were
cents
the
the
buy
zero-emission
3
running
gradient
Focus
The
only
friendly.
shopping
Ford
run.
may
to
of
animal
meat.
data.
feed
Even
x-axis
of
the
if
goes
on
the
your
to
x-axis
data
130 kg.
and
doesn’t
c
Research
person)
your
d
the
data
graph
How
most
to
many
livestock
for
recent
your
country.
represent
kilograms
raised
for
meat
Draw
a
per
capita
horizontal
line
(per
on
this.
of
one
consumption
feed
are
person’s
needed
meat
to
sustain
consumption
the
in
your
country?
e
Research
f
weight
annual
meat
corn
crops
annually
need
to
eaten
To
or
you
Edible
viable
we
insect
at
that
with
is
fed
corn
to
meet
in
4445 kg
to
your
of
livestock,
order
perspective,
you
live.
needed
sustainable,
and
ratio
for
of
to
is
the
country’s
grain
how
feed
research
How
to
feed
the
per
acre
many
acres
livestock
does
the
it
the
area
compare
of
the
with
city
the
livestock?
example,
1
to
1
so
meat
eat.
The
the
range
lean
from
ecologically
actually
some
of
the
have
they
than
a
are
any
average
approximately
with
which
we
could
for
producing
also
compared
meats
a
source
animals
is
of
in
which
replacement
conversion
typical
needed
What
country?
area
are
Crickets,
ecient
country.
know...?
meat
eat.
60%
farmland
insects
feasible
in
feed
your
approximately
planted
this
of
the
yield
your
town
of
of
demands?
and
be
in
put
area
Did
population
overall
If
g
the
50%
cuts
of
28%
a
feed
more
of
the
edible
protein,
the
to
be
meat
big
four
34%.
14 7
r
assessment
i
t er
i
C, D
Challenges
By
2030,
8.5
it
billion
is
of
feeding
estimated
people.
How
that
can
we
we
will
do
a
growing
need
that
to
in
a
feed
a
way
planet…
world
that
is
population
of
environmentally
sustainable?
You
have
been
and
report
and
how
sections
your
on
it
below,
answers
of
both
for
how
a
to
and
1
There
–
been
pork,
has
Nations,
a
will
the
Food
person).
a
world
linear
Use
your
c
Predict
d
Discuss
whether
e
Do
think
you
model
the
meat
to
per
this
14 8
3
Linear relationships
on
completed
to
(2-minute
at
least
in
two
that
video
includes
the
pros
arguments
rainforest
your
the
summarize
max.)
summarizing
the
farmers
types
and
each
deforestation.
report
and
indicate
capita
since
meat
to
in
is
that
the
meat
humans
eat),
meat
per
the
was
24.2 kg
capita
equation
for
United
in
per
2015.
worldwide
1965.
capita
a
of
1965
41.3 kg
consumption
per
this
produce
Organization
consumption
meat
in
of
to
environment.
determine
increase
Explain.
them
other
on
think
have
consumption
consumption
you
as
analyse
consumption
presentation
Brazilian
increased
predict
you
to
reputable.
meat
had
experts
Prezi/Powerpoint
include
of
of
meat
short
Agriculture
model,
consumption
b
and
a
a
well
pressure
eects
This
as
to
are
with
Once
using
meat
and
group
create
reference
chicken
a
either
results
sources
annual
Assuming
sure
increasing
the
to
below,
Be
undesirable
the
(per
meat
the
of
associated
create
sources,
lamb,
to
part
presentation
against
of
a
need
Worldwide
According
capita
will
questions
know
has
which
you
a
be
deforestation.
deforestation.
you
(beef,
the
variety
Part
to
You
or
to
problems
relates
presentation
Use
the
ndings.
cons
asked
in
realistic
per
capita
this
year.
2030.
model.
consumption
is
sustainable?
a
Summative
c
ATL2
To
put
which
this
in
perspective,
compares
meat
take
a
look
consumption
at
in
the
following
developed
and
table,
developing
countries.
1997
meat
consumption
Average
(million
in
tonnes)
Developed
countries
Developing countries
f
Graph
each
set
of
data
meat
1997
annual
(million
98
0.8
111
4.6
on
the
same
axes.
increase
consumption
Assume
a
since
tonnes)
linear
You
model
for
each.
to
g
Determine
the
equation
h
What
the
slope
of
each
think
the
line.
of
each
line
represent?
Which
slope
i
Will
concerning?
there
ever
developing
Explain.
If
be
Explain.
a
time
countries
it
is
approximately
is
equal
possible,
when
use
this
the
to
that
your
will
meat
of
consumption
developed
equations
to
try
their
growth
developed
and
when
about
is
in
more
want
populations
and
does
may
developing
countries.
of
countries?
to
nd
out
happen.
14 9
Part
2
–
The
Approximately
trees
in
cattle
an
total
graph
in
and
rainforest
clearcutting
Brazilian
farming.
This
(cutting
rainforest
accounts
deforestation,
down
is
to
for
making
it
all
or
provide
almost
the
most
land
15%
largest
for
of
the
cause
of
worldwide.
below
Rainforest
of
the
annual
deforestation
The
70%
area)
ranches
world’s
Brazilian
since
represents
the
forest
cover
in
the
Brazilian
1970.
y
mk(
2
)
4 000 000
nozamA
3 800 000
naillizarB
3 600 000
3 400 000
ni
revoc
3 200 000
tseroF
3 000 000
x
1970
1980
1990
2000
2010
2020
2030
2040
Year
a
Determine
between
and
the
b
the
a
the
linear
amount
year.
gradient
Assuming
To
put
town
that
with
d
of
15 0
the
Given
does
this
you
it
3
area
live
forest
does
the
represents
cover
in
the
y-intercept
of
of
in
in
deforestation
forest
to
town
in
the
square
rate
city?
Linear relationships
an
rate
is
is
the
relationship
Brazilian
represent?
of
research
that
area
area
constant,
cleared
kilometres.
rainforest
clearcut
or
that
perspective,
calculated
take
your
the
area
area
the
What
of
that
Amazon
What
does
represent?
approximate
c
equation
of
is
the
the
annually.
the
How
area
of
does
it
cleared
clearcut
calculate
each
each
rainforest
the
city
or
compare
year?
year,
that
how
is
the
long
size
e
Some
size
scientists
to
less
than
irreversible
f
Use
we
a
in
your
a
Write
done
b
Write
a
c
what
the
report
Action
of
continues
variety
need
75%
that
year
of
and
at
will
sources
Brazilian
the
its
consequences
clearcutting
graph,
warn
rainforest
1970
to
the
the
size
rainforest
deplete
catastrophic
ecosystem.
rate
in
If
represented
in
the
occur?
and
research
rainforest.
indicate
without
constant
that
cannot
how
three
Reference
you
know
major
your
they
are
reasons
sources
why
in
reputable.
plan
an
to
action
plan
address
down
3
dierence
Summarize
challenges
personal
and
your
presentation
the
containing
using
a
of
decisions
describe
action
at
their
plan
ve
feeding
that
you
eects
and
program
least
on
as
that
growing
can
the
decisions
such
a
things
in
make
could
be
planet.
that
will
make
environment.
an
interactive
Prezi.
1 51
4
In
this
unit,
complex
3D
you
will
problems
see
with
versatile
shapes
personal
applications
Personal
Expressing
You
are
used
can
and
beliefs
shapes
structures
on
a
be
to
and
shapes
range
of
ancient
truly
in
have
helped
fascinating
other
us
to
solve
products.
contexts
as
some
But
well,
these
from
history.
expression
values
surrounded
studied
the
3D
instrumental
cultural
constantly
the
how
shapes
planet.
in
by
this
Many
3D
unit
shapes.
to
create
structures
Architecture,
some
also
of
have
a
the
be
it
most
strong
ancient
or
inspiring
cultural
modern,
and
and
has
memorable
personal
signicance.
The
the
pyramids
Seven
World.
tombs
believed
allow
and
live
were
dead
that
the
Egypt
Wonders
They
of
of
of
built
are
the
to
pharaohs,
the
shape
pharaoh
forever.
to
one
of
Ancient
house
and
that
climb
it
the
was
would
skyward
Pisa
was
ruled
port
once
over
and
the
city,
deserved
a
intricate
was
the
of
cathedral.
today
it
is
to
soft
lean
the
3.99°
–
triangles
15 2
its
tower
to
famous
–
it
house
on
was
in
its
built
began
its
Today
at
could
your
too!
for
and
leans
this
in
of
However,
early
interest
it
nearby
ground,
tower
Italian
prosperity.
bell
construction.
the
and
and
imperfection
on
four
Mediterranean.
constructed
bells
of
symbol
importance
An
one
about
be
of
study
of
It
republics
was
an
that
important
Orientation
Civilizations
Three
on
were
soccer
The
at
or
same
the
of
shadows
time
have
celebrate
in
an
to
and
see
ancient
been
an
interact
stone
game
integral
with
one
goalposts
played
part
with
of
how
another.
Modern
surrounding
a
some
spherical
a
of
the
greatest
visitors
huge
rubber
to
Chichen
ballcourt.
ball,
not
Itza
These
dissimilar
from
basketball.
people
center
people
shapes
surprised
used
and
interaction
Earth
sometimes
goals
a
and
space
dimensional
peoples
are
in
of
who
the
Chichen
seemed
Chichen
Itza
to
played
would
bring
that
Itza
game
site.
come
the
also
Every
spring
together
serpent
on
constructed
for
the
a
side
and
the
autumnal
dazzling
of
famous
the
light
Castillo
equinox,
show
temple
El
to
as
the
pyramid
Mayan
sunlight
and
life.
15 3
4
3D
shapes
Products,
KEY
CONCEPT:
Related
Global
While
in
as
unit,
at
you
both
surprising
the
the
will
global
natural
interesting
of
and
some
improve
of
their
the
own.
dene
results
of
of
scientic
ingenious
man-made
that
Statement
Generalizing
to
and
outcomes
analyze
Generalization,
context
examine
designed
measurements
more
quality
Your
them
human
of
study
will
and
and
Measurement
technical
solutions
processes,
human
which
of
take
to
on
life.
You
have
a
innovation
problems,
will
as
tour
some
a
some
shapes
of
well
take
produced
mathematical
you
natural
and
of
the
forces.
Inquiry
relationships
generate
between
products,
Objectives
●
solutions
context:
products
look
and
RELA
TIONSHIPS
concepts:
exploring
this
processes
measurements
processes
and
Inquiry
Calculating the surface area and volume of
3-dimensional shapes involving cylinders,
can
help
solutions.
questions
What
is
volume?
F
What
is
surface
area?
cones, pyramids and spheres
●
Applying
mathematical
problems
involving
3D
strategies
shapes
to
solve
How
C
are
surface
area
and
volume
related?
How
do
we
between
What
generalize
relationships
measurements?
makes
for
an
ingenious
solution?
D
How
can
a
product
solve
a
problem?
G E O M E T R Y
A
TL1
Thinking:
thinking
Make
and
Y
ou
1
guesses,
generate
should
ask
“what
testable
already
Find
the
area
of
a
Find
the
area
of
the
following
A
TL2
Creative-
Thinking:
skills
A N D
T R I G O N O M E T R Y
Transfer
skills
if”
questions
hypotheses
know
how
skills
4
with
to
knowledge,
create
understanding
products
or
and
solutions
to:
circle
circle
Combine
Find
the
volume
of
prisms
Find
the
volume
of
the
the
following
shapes.
dimensions.
a
radius
=
12 cm
b
diameter
c
circumference
16 cm
=
10 cm
=
25 cm
8 cm
2
Simplify
algebraic
Simplify
the
3
a
8r
b
3x
following
2
+
2
9r
2
−
2r
a
expressions
20 cm
expressions.
3
−
11r
3
(x
)
c
10 cm
3
Solve
equations
Solve
the
your
following
answers
where
to
6 cm
b
equations.
the
nearest
Round
tenth
necessary.
2
a
x
=
14
2
b
3x
=
7 cm
25
2
25 cm
c
c
15 5
Introducing
The
products
around
to
you.
solve
easier.
that
human
Some
problems
Tools
we
of
can
amazing
3D
are
are
understand
structures
on
phenomena
not
example,
cylindrical
middle
or
been
this
into
you
see
been
see
to
create
●
you
and
What
life
●
Name
4
by
and
products
of
of
of
the
natural
human
straightforward
little
bit
phenomena
Some
result
a
some
as
eort
you
so
most
forces,
and
and
natural
might
think.
the
it
of
your
like
a
and
can
inspire
own.
discuss
the
white
or
your
product
1
cylinder
in
man-made
the
photo
forces?
on
this
Describe
page
what
is
you
the
think
it
reasoning.
do
without
you
use
most
often
each
day?
What
would
your
them?
that
you
problem
3D shapes
will
these
natural
the
some
You
maybe,
explain
and
have
new
product(s)
be
better.
the
life
natural
all
which
now
think
describe
15 6
of
and
product
is
are
make
are
health,
delve
processes
analyze
and,
Reect
Do
planet
as
might
will
versions.
to
products
you
in
mathematics
you
study
our
natural
How
day
may
replaced
how
help
●
photo
some
every
improved
simply
to
ingenuity
improve
the
you
products,
which
and
to
formed?
unit,
solutions,
of
this
the
always
formation
man-made?
have
In
of
is
our
Earth
between
is
others
created
distinguishing
For
creativity
designed
while
also
shapes
use
that
it
that
solves
solves.
a
particular
problem
and
G E O M E T R Y
A N D
T R I G O N O M E T R Y
Cylinders
Cylinders
form
processes.
of
a
Surface
on
as
tall
from
as
84 m
trees
you
red
problem
the
of
analyze
largest
sequoia
has
surface
these
naturally
in
an
humans
totem
can
by
in
area
and
and
greater
occuring
California,
average
have
poles
to
British
out
canoes
of
many
of
leaky
many
trees,
be
of
canoe
having
all
incredible
created
that
Like
their
Some
attempts
Other
the
trunk
people
entire
tree.
Not
task.
of
its
the
to
the
products
volume
depth.
cylinder
and
as
calculate
you
giant
problem
carved
water.
some
Gwaii
an
cedar
they
a
to
interesting
USA,
diameter
produced
created
by
can
of
some
many
“cylinders”
grow
8 m.
It
is
incredible
native
North
know...?
the
carving
of
many
groups.
Haida
solved
allow
this
of
how
The
like
such
American
The
are
planet.
products
Did
will
area
trunks
the
basis
Knowing
cylinder
Tree
the
of
First
in
carry
the
a
Haida
in
canoes
single
forest
heavy
were
from
North
people,
to
tree
to
by
Western
well
Gwaii’s
forests
products
native
a
in
Nations
the
however,
found
Columbia
seams
solve
to
the
the
suited
to
abandoned
this
trees
day.
include
American
the
totem
poles
groups.
157
Investigation
1
1
–
Sur face
area
of
a
cylinder
Using three pieces of paper and some tape, create a model of a cylinder. Use one
piece of paper as the “body” or “tube” of the cylinder. Cut the other piece of paper
to form the base shapes so that they t perfectly on the top and bottom of the tube.
2
Draw
the
net
of
the
cylinder,
the
surface
indicating
important
measurements
on
your
diagram.
3
Calculate
about
how
4
Draw
a
5
Find
6
Generalize
to
cylinder
the
total
cylinder
if
area
calculate
and
label
surface
your
you
work
know
of
the
some
its
area
to
the
of
cylinder.
the
height
of
this
Refer
to
your
model
if
you
are
and
radius
with
measurements
of
your
●
Explain
area
●
of
How
the
is
and
determine
a
formula
measurement
how
a
of
its
for
nding
height
h
the
and
surface
and
the
area
radius
The
the
nding
surface
Kake
who
tell
the
a
us
totem
in
of
surface
a
about
area
prism?
a
in
of
a
of
a
Which
Alaska
It
was
honor
While
the
in
world.
tree
poles
relate
pole
the
died.
totem
trees
the
area
single
had
circumference
circle
is
related
to
the
surface
cylinder
is
easier?
dierent
than
nding
Explain.
1
tallest
from
the
2
cylinder.
Example
Q
discuss
the
tree’s
that
human
of
a
natural
are
is
one
of
carved
rings
chief
of
a
tree
history,
carved
from
story.
2
Find the surface area (in m
) of the
Kake totem pole that was available for
carving. Assume that it has a height of
40.2 m and a diameter of 2.6 m. Round
your answer to the nearest hundredth.
(Assume that no carving was to be done
on the bottom surface of the pole.)
Continued
15 8
4
3D shapes
choice.
cylinder.
base.
Reect
unsure
areas.
on
next
page
of
r
any
of
the
G E O M E T R Y
A
The
available
surface
area
of
the
totem
The
formula
for
the
A N D
sur face
T R I G O N O M E T R Y
area
of
a
cylinder
is
2
pole
is
given
SA
by
=
πr
2πr
+
2πrh
However, the totem pole will not be carved on the
2
SA
=
2πr h
+
bottom surface, so remove the area of one circle.
r
=
2.6
r
=
1.3 m
÷
With r =
the
2
The radius of the totem pole is half of its diameter.
1.3 m
surface
and
area
h =
40.2 m,
Substitute
the
measurements
into
the
formula.
is
2
SA
=
π(1.3)
SA
=
333.67 m
+
2π(1.3)(40.2)
Calculate
the
answer
and
round
to
the
nearest
2
hundredth.
Practice
1
Find
the
1
surface
area
of
the
following
cylinders.
Round
your
answers
to
the
nearest
tenth.
15 cm
3 cm
24 cm
5 cm
4 cm
b
a
10
cm
c
14 cm
11.7 cm
23 cm
31.4 cm
10 cm
d
e
4.3 cm
f
Continued
on
next
page
15 9
2
Copy
and
Round
complete
answers
to
the
the
following
nearest
table
for
the
surface
area
of
a
range
of
cylinders.
tenth.
2
Radius
3
In
the
tool,
(cm)
12
18
10
11
985
3
100
piece
Marvin
of
of
takes
21.5 cm
The
in
glass
you
What
c
and
its
the
julep
not
was
rst
using
like
to
a
there
the
wrap
drinking
include
reasons,
to
make,
diameter
is
It
an
was
of
which
is
the
are
of
walls
a
is
a
what
strips
part
move
was
residue
straw.
exible
20 cm
0.65 cm?
architectural
designed
Dante’s
the
work
presents
consists
outer
in
the
calculate
walk
meters
area
4
did
solution
creating
completes
by
artwork
height
walls
16 0
he
of
or
away
then
caused
paper
Most
a
from
the
by
standard
the
around
straws
even
and
straw
Show
work
with
your
of
created
a
radius
grass
a
pencil
nowadays
spoon
plastic
on
one
and
are
end.
back
to
art
on
by
the
top
the
Comedy.
visitors
with
0.65 cm
The
amazing
of
the
ARoS
or
art
Danish–Icelandic
representation
Divine
of
a
working.
building’s
to
of
a
3 m
diameter
middle
the
the
is
of
the
oor
wide
52
you
circle.
inside
of
and
its
made
glass
of
the
the
3D shapes
panorama
inside
of
and
glass.
required
structure.
to
is
outside
Find
build
the
the
circular
meters.
the
distance
complete
area
of
walkway?
The
3
a
and
walked
would
the
mint
)
of
Hell,
journey
views
museum
artist
Purgatory
to
paradise
of
the
city
all
the
colors
and
ends
bathed
in
colors.
walkway,
b
His
modications
Denmark.
panorama,
spectrum
If
liquid.
Panorama
inspired
dierent
a
his
However,
material
with
Eliasson
Paradise
The
and
drank
environmental
straw
Aarhus,
the
the
more
Rainbow
Olafur
at
grass.
together,
plastic
for
Stone
(cm
straws.
Which
4
rye
in
them
However,
paper
area
8
glue
made
Surface
611.4
disintegrating
and
(cm)
7
1880s,
a
Height
walkway
in
of
the
G E O M E T R Y
5
One
of
the
sequoia
tallest
tree
California,
average
by
a
a
used
USA.
the
world
“General
With
diameter
the
to
of
The
surface
carve
bottom
next
a
8 m,
has
surface
a
giant
Sherman”
height
the
is
of
tree
84 m
can
in
and
be
an
modeled
of
pole.
not
giant
height
be
tree
Find
that
(Assume
could
that
be
the
carved.)
sequoia
of
between
the
(“General
81.7 m
the
and
an
dierence
General
Grant
average
in
and
Sherman.
the
height
surface
average
will
7.6 m.
area
General
same
of
a
area
totem
largest
diameter
Find
a
surface
Grant”)
c
named
in
of
area
radius
of
a
giant
as
sequoia
General
with
the
Sherman,
but
an
9 m.
V
Volume
As
Its
T R I G O N O M E T R Y
cylinder.
Find
b
trees
A N D
with
of
all
volume
a
3D
objects,
can
=
A
base
×
h
cylinder
be
a
cylinder
calculated
in
a
takes
up
similar
space.
way
to
h
the
volume
of
Activity
1
Draw
a
1
a
–
prism.
Volume
diagram
hexagonal
of
a
rectangular
Write
down
the
formula
3
Write
down
the
general
4
Draw
a
diagram
the
a
cylinder
prism,
a
triangular
prism
and
a
prism.
2
down
of
of
formula
a
for
the
formula
cylinder.
for
volume
the
for
Using
volume
of
the
of
of
volume
your
a
each
result
of
the
prisms.
any
from
prism.
step
3,
write
cylinder.
1 61
Reect
●
Explain
and
how
discuss
the
volume
of
3
a
cylinder
is
related
to
the
area
of
a
circle.
●
If
the
cylinder
Example
Q
When
no
with
the
designed
apparatus
called
the
of
today’s
Cousteau
to
explore
seen
Blue
Belize
years
The
ocean
an
precursor
Great
does
problem
underwater,
Cousteau
been
top,
that
aect
its
volume?
Explain.
2
faced
breathe
has
explorer
Aqualung.
scuba
places
he
cylindrical
forming
to
Jacques
breathing
was
and
that
Hole,
a
1972,
It
gear
In
began
trying
underwater
before.
which
of
had
the
allowed
never
explored
the
formation
over
70
in
million
ago.
Great
Blue
Hole
has
a
depth
of
roughly
3
125 m.
nd
its
If
there
radius.
is
9 132 700 m
Round
your
of
water
answer
to
in
the
the
Great
nearest
Blue
Hole,
tenth.
The
formula
for
the
volume
of
a
cylinder
is
2
A
V
=
πr
h
2
V
=
the
πr
h.
In
depth
this
of
case,
the
the
Great
height
Blue
of
the
cylinder
is
Hole.
2
9 132 700
=
πr
(125)
Substitute
the
the
information
that
you
know
into
formula.
2
9 132 700
=
(392.70)
=
r
=
r
r
Solve
the
equation
calculations
to
the
for
r.
Round
nearest
intermediate
hundredth.
9 132 700
2
392.70
2
23 256.18
To
isolate
r, you need to take the square root of
2
23 256.18
152.5
The
16 2
4
radius
each
r
=
= r
of
the
3D shapes
Find
Great
Blue
Hole
is
approximately
side.
the
answer
152.5 m.
and
round
to
the
nearest
tenth.
G E O M E T R Y
Practice
1
A N D
T R I G O N O M E T R Y
2
Find
the
volume
your
answers
to
and
the
surface
nearest
area
of
each
shape.
Show
all
of
your
working
and
round
tenth.
8 cm
10 cm
24 cm
2 cm
15 cm
7 cm
a
b
c
10 cm
30 cm
8 cm
40 cm
6 cm
13 cm
d
e
20 cm
f
9 cm
32 cm
14 cm
g
2
Find
the
missing
hundredth
dimension
where
Radius
of
of
each
cylinder.
Round
your
answers
to
the
nearest
necessary.
base
Height
of
cylinder
Volume
3
4.5 cm
19.4 cm
3
15 cm
3
Designing
one
a
of
tools
the
makes
back
is
its
explore
problems
solution,
product
to
build
the
a
NASA
prototype
Deep
own
that
planets
Phreatic
decisions
on
550 cm
and
moons
continuously
product
and
Thermal
where
to
then
that
has
go,
to
test
Explorer
which
are
it
very
solve.
here
dierent
Often,
on
(DEPTHX),
samples
to
NASA
Earth
an
from
rst.
our
will
One
underwater
collect
and
even
own
is
propose
such
robot
how
that
to
get
home.
Continued
on
next
page
16 3
Created
the
El
to
Zacatón
Sinkholes
and
they
The
are
can
depth
319 m.
4
explore
The
sinkhole
the
be
of
Europa,
result
found
the
El
Find
the
volume
b
Find
the
percentage
c
A
of
3 m.
water
from
to
The
of
our
the
the
is
water-lled
involving
335 m,
by
a
the
robot
sinkhole
acidic
swimming
of
although
cylinder
sinkhole.
sinkhole
number
deepest
moons,
was
in
tested
the
rainwater
and
in
world.
limestone
planet.
sinkhole
in
Jupiter’s
processes
modeled
water
the
trac
in
which
friction
initial
the
tunnel
skates
reduced
be
the
of
that
pool
whose
Show
is
lled
has
a
your
with
length
Olympic-size
the
of
diameter
the
is
water
is
only
110 m.
working.
water.
of
pools
depth
Show
50 m,
that
a
your
width
could
be
working.
of
25 m
lled
and
with
a
the
Zacatón.
solve
underground
electric
of
Find
El
natural
over
Olympic-size
depth
order
all
can
a
In
of
smallest
Mexico,
Zacatón
sinkhole
typical
in
the
and
tunnel
problem
the
will
air
shape
enable
of
in
a
Los
Angeles,
cylinder.
them
to
He
travel
Elon
Musk
proposes
much
faster
that
than
wants
cars
on
to
will
build
be
an
placed
roadways
due
on
to
resistance.
beneath
Los
Angeles
has
a
diameter
of
4.11 m
and
a
length
of
2.6 km.
a
b
Find
the
your
working
If
the
volume
5
Dating
new
processes
alunite,
cave
by
to
cave
as
is
a
acid.
material
new
tool
the
our
a
riddle
of
not
to
just
is
dating
could
climate
was
easy,
to
the
in
order
nearest
by
how
to
create
the
tunnel.
tenth.
much
would
the
volume
that.
the
eroded
the
One
the
that
from
over
on
below
of
ability
hoping
the
of
method
age
The
these
forms
formations
scientists
change
of
amount
the
cave.
However,
been
promising
nd
may
in
to
to
them
solve
history
of
planet.
Continued
16 4
Show
especially
that
have
mineral
to
and
for
removed
doubled,
create.
processes
Another
caves
be
is
years
chalky
date
be
answer
formations
inside
to
working.
measuring
accurately
had
your
tunnel
limestone
radiocarbon
organic
of
and
involves
walls
the
accomplish
which
that
round
your
millions
sulphuric
uses
as
technologies
developed
dirt
formations
such
taken
of
Show
natural
examples
have
and
diameter
increase?
of
4
3D shapes
on
next
page
G E O M E T R Y
The
Watchtower
is
a
The
Watchtower
is
15.25 m
a
Find
the
holes
b
The
in
100
can
added
of
damage
it
fabric
the
If
(in
a
Natural
and
of
has
a
Bridge
Cavern
circumference
material
your
cave
portion
it
of
the
to
Show
took
very
in
the
to
is
the
the
of
near
San
Antonio,
Texas.
6.71 m.
Watchtower,
of
Watchtower
assuming
there
are
no
that
they
of
will
your
cubic
the
a
is
The
to
of
an
form.
need
oils
Bridge
most
height
they
size
centimeters
ice
of
cube
Show
all
a
every
of
your
year.
Natural
that
to
16
Watchtower
sensitive.
the
up
with
about
nearest
very
fabric
slow,
(That’s
for
owners
use
all
is
years.
answer
The
meters)
Watchtower.
100
formation
decide
square
column
long
forever.
they
a
every
how
round
surface
visitors.
tall
the
amount
forming
Find
and
covering
the
of
being
years!)
The
in
T R I G O N O M E T R Y
it.
working
c
approximate
process
column
column
A N D
of
so
at
3
on
a
Cavern
risk
of
meters,
that
person’s
it
are
being
nd
goes
all
hand
thinking
touched
the
the
of
by
amount
way
of
around
working.
Cones
A
cone
has
a
is
at,
When
the
a
three-dimensional
circular
vertex
base
(or
and
apex)
a
is
object
single
ver tex
that
vertex.
directly
height
slant
above
right
right
the
cone.
center
In
cones,
simply
as
of
this
the
unit,
which
base,
you
will
be
it
is
will
called
only
referred
a
height
study
to
cones.
radius
Reect
●
How
●
Why
●
Write
is
a
and
cone
does
a
down
discuss
dierent
cone
an
have
than
two
equation
4
a
prism
or
“heights”?
relating
the
cylinder?
Explain.
Explain.
cone’s
radius,
height
and
slant
height.
16 5
of
Because
cone
the
a
whole
as
relationship
you
cone
does
gure,
procedure
as
a
will
for
have
nding
a
prism
between
see
not
in
the
or
the
volume
shape
of
that
cannot
cylinder.
volume
next
2
base
follow
However,
a
cone
runs
the
there
and
through
that
same
is
of
a
a
cylinder,
investigation.
– Volume
of
a
cone
i
t
e ri
r
Investigation
its
a
c
Your
teacher
may
give
you
manipulatives
and
sand
in
order
to
nd
the
B
relationship
l
i
n
between
the
volume
of
a
cone
and
that
of
a
cylinder.
k
b
e
If
you
don’t
have
demonstration:
“volume
the
1
Look
at
notice?
ATL1
of
to
the
materials
needed,
you
can
watch
the
dimensions
Discuss
your
and
water”
on
YouTube.
It
uses
water
instead
of
sand,
of
the
cone
observations
discuss
and
with
the
a
cylinder.
What
do
you
peer.
5
Before doing the activity or watching the video, guess how many cones
full of sand it will take to completely ll the cylinder. Compare your guess
with a peer. Your teacher may collect the guesses of the entire class.
2
Use
3
Write
4
the
Use
formula
to
the
If
a
the
of
and
you
cone
4
the
have
and
determine
how
relationship
cylinder
generalize
cone
16 6
down
to
volume
3
5
sand
for
a
with
the
the
for
measurement
other
3D shapes
many
between
same
volume
formula
of
the
of
with
the
same
a
cones
the
it
height
and
cylinder
height
verify
takes
volume
h
a
ll
your
cylinder.
and
result
its
base
the
the
from
step
volume
radius
relationship
dimensions.
the
cone
between
and
your
of
to
radius.
and
relationship
its
manipulatives,
cylinder
following
https://www.youtube.com/watch?v=9fUmkZuaRQM
cone
same.
the
Reect
a
access
for
of
r
another
a
but
or
video
search
the
o
n
Volume
for
results
are
G E O M E T R Y
Activity
The
Orion
deeper
2
The
Spacecraft
into
designed
–
our
to
up
T R I G O N O M E T R Y
frustum
is
solar
carry
A N D
currently
system
to
four
being
than
crew
we
built
have
with
ever
the
gone
goal
of
traveling
before.
It
is
members.
The height of the crew section of the spacecraft is 3.3 m, with a large
base diameter of 5 m and a smaller top diameter of 2 m. The shape of the
crew module is called a frustum, which is a truncated (chopped o) cone.
1
Using
the
2
Use
the
spacecraft
similar
been
Calculate
4
Only
27%
the
the
by
the
How
What
would
of
Research
living
on
of
the
nd
the
be
it
like
the
guide,
of
of
total
volume
draw
a
diagram
of
the
smaller
height
of
the
cone
that
has
cone.
spacecraft.
is
habitable
volume
the
habitable
volume.
that
can
6
your
compare
the
height
Calculate
discuss
in
a
frustum-shaped
spacecraft’s
crew.
as
dimensions.
nd
Hence
the
does
it
diagram
its
dimensions
astronauts
period
●
of
to
volume
and
Estimate
volume.
●
the
o.
and
label
triangles
accessed
Reect
●
and
chopped
3
be
photograph
classroom
with
the
and
calculate
habitable
space
the
available
to
spacecraft?
being
in
that
size
space
for
an
extended
time?
what
the
NASA
hopes
spaceship
in
to
gain
such
a
by
these
conned
explorations.
space
be
worth
Would
it?
167
Practice
1
Find
the
3
volume
of
the
following
shapes.
Round
your
answers
to
the
nearest
tenth.
7 cm
8 cm
250 cm
8 cm
90 cm
6 cm
a
b
c
21 cm
7.5 m
78 cm
9 cm
0.5 m
d
2
e
Italo
Marchiony
New
York
glass
dishes
he
ice
a
cream
If
If
4
an
volume
Marchiony
rst
cream
maker
1900s.
broke
Marchiony
cone
wanted
had
ice
to
would
a
was
were
dish
went
base
the
have
to
who
his
taken
on
sold
to
by
use?
he
a
ices
served
folded
on
his
To
Wall
product
solve
wae.
Street
this
This
in
in
small
problem,
was
the
rst
patent.
ll
volume
lemon
customers.
from
diameter
to
his
pushcart,
made
needed
double
he
From
or
cone-shaped
lemon
radius
3D shapes
early
which
of
ice
often
edible
cone,
base
an
the
they
Marchiony’s
what
16 8
in
but
designed
what
b
City
was
f
of
the
of
Show
50 mm
and
a
height
of
120 mm,
cone?
the
cone,
your
but
keep
working.
the
same
height,
G E O M E T R Y
3
A
piping
bag
cone-shaped
depending
of
canvas
easier
to
is
a
product
with
on
the
with
a
a
small
style
of
plastic
used
to
hole
the
at
decorate
the
icing
interior
or
cakes
apex.
Dierent
decoration
even
eciently
tips
required.
food-grade
and
are
A N D
precisely.
placed
Piping
silicone,
T R I G O N O M E T R Y
over
bags
which
can
It
is
the
be
makes
hole
made
them
clean.
a
9 cm
b
10 cm
c
8 cm
31.5 cm
34 cm
39 cm
Given
the
Round
4
your
Popcorn
is
“popcorn
for
a
answers
typically
cones”
to
of
to
each
the
served
serve
piping
nearest
in
bags
bag,
nd
the
volume
of
icing
that
it
can
contain.
tenth.
or
popcorn,
boxes.
However,
especially
at
events
some
companies
where
popcorn
are
is
using
given
away
free.
Each
cone
volume
b
dimensions
Why
of
has
a
base
popcorn
would
a
diameter
that
will
company
use
t
a
of
11 cm
in
the
cone
to
and
a
slant
height
of
20 cm.
Find
the
cone.
serve
free
popcorn?
What
problem
might
that
solve?
5
It
has
day,
been
with
estimated
American
Scandinavians
year!
Making
paper.
In
an
introduced
The
of
on
that
attempt
reusable
diagram
shows
over
coee
drink,
all
that
to
billion
drinkers
average,
coee
reduce
coee
a
1.4
drinking
about
often
the
cups
10
need
for
over
or
requires
of
11
coee
three
are
cups
kilograms
lters,
paper
drunk
which
a
of
day
on
coee
are
products,
worldwide
some
average!
per
usually
every
person
made
out
companies
every
of
have
lters.
lter
with
a
radius
of
12.8 cm
which
can
3
hold
a
total
volume
a
Find
the
height
b
Find
the
slant
of
of
200 cm
the
height
of
coee.
lter.
of
the
lter.
16 9
Surface
Finding
area
the
of
surface
a
cone
area
of
a
cone
involves
opening
it
out
into
its
net.
r
r
l
h
r
=
radius
l
h
l
=
=
height
slant
height
2
Because
It
can
the
be
lateral
shown
area,
Hence,
base
can
the
is
a
that
be
total
circle,
the
area
found
surface
its
of
using
area
area
the
A
of
can
=
a
be
side
of
found
the
using
cone,
A
=
called
πr
the
πr l
cone
(including
the
base)
is
given
2
by
A
=
πr
+
Example
Q
The
but
If
3
original
there
when
and
πr l
ice
was
a
vendors
the
diameter
nearest
cones
demand
were
packaging
minimum
cream
ice
of
not
creams
each
amount
for
in
of
were
the
the
for
cone
frozen
area.
sale
is
sold
person
treat
to
by
be
Companies
in
5 cm
packaging
in
street
vendors,
available
began
even
producing
stores.
and
the
required.
height
Round
is
14 cm,
your
nd
answer
the
to
the
hundredth.
5 cm
A
The
formula
for
the
sur face
area
of
a
cone
is
2
SA
the
=
πr
+
πr l,
where
l
is
the
slant
height
of
cone.
14 cm
2
2
+
2.5
14
2
=
Find
l
the
slant
height
of
the
cone
using
Pythagoras’ theorem.
2
202.25
14.22
=
=
l
l
Substitute
2
SA
=
=
The
π( 2.5)
17 0
minimum
amount
131.32 cm
4
area
formula.
measurements
into
the
sur face
Find
the
131.32
2
is
the
+ π( 2.5)(14.22)
3D shapes
of
answer
packaging
hundredth.
and
round
to
the
nearest
G E O M E T R Y
Practice
1
A N D
T R I G O N O M E T R Y
4
Find
the
surface
your
answers
to
area
the
and
volume
nearest
of
each
gure.
Show
all
of
your
working
and
round
tenth.
10 cm
18 cm
4 cm
6 cm
3 cm
a
b
12 cm
c
22 cm
4 cm
3 cm
d
26 cm
e
30 cm
16π cm
50 cm
f
40 cm
5.7 cm
g
4.8 m
12.2 m
2.6 m
8.6 m
h
14 m
i
3 m
Continued
on
next
page
171
®
2
The
to
Watercone
help
water.
solve
The
is
the
a
product
problem
circular
tray
of
at
that
was
invented
poor-quality
the
bottom
is
lled
®
with
the
salty
sun.
water
As
free
water
and
trickles
the
and
the
water
condenses
into
the
Watercone
evaporates
on
the
rim
of
side
the
it
is
left
rises.
of
the
cone,
in
Salt-
cone
where
it
is
collected.
Find
the
drops
of
3
at
curved
any
80 cm
You
make
time,
a
height
buy
a
set
needed
the
a
one
and
can
material
surface
hole).
of
to
area
inside
knowing
of
that
cone
the
that
cone
could
has
a
be
base
covered
diameter
make
piping
each
your
bags
bag
dierent
(before
answers
b
10 cm
of
to
the
the
end
dimensions.
of
nearest
each
cone
of
In
do
the
you
desert
make
with
no
approximately
yakhch āls
keep
top
the
conical
underground
structures
sand,
The
and
were
lime,
inside
ice
Find
in
the
ash,
of
a
ice
the
BC,
in
cavern
of
cavern
is
cold.
172
4
and
base
3D shapes
area
is
and
surface
out
the
of
with
built
and
the
inside
conical
mixture
even
empty,
710 m
months
rose
a
middle
Persians
of
goat
the
clay,
hair!
water
it.
area
of
a
yakhchāl
is
31.5 cm
refrigerator?
These
sarooj,
2
18 m
the
keeping
underneath
approximate
or
Heat
whites
cone
in
colder
structure,
egg
ice
ancient
the
year.
made
the
keep
electricity
400
make
throughout
of
and
to
and
shape
8 cm
31.5 cm
How
the
tenth.
c
8 cm
Find
39 cm
4
in
30 cm.
three
Round
the
whose
height
is
amount
cut
o
of
to
G E O M E T R Y
A N D
T R I G O N O M E T R Y
A
The
Elizabethan
shame”,
injured
a
is
one
areas
truncated
collar,
sometimes
solution
after
cone
a
to
the
medical
because
it
is
referred
problem
of
procedure.
a
cone
to
as
pets
The
whose
the
“cone
trying
shape
apex
to
is
has
of
lick
called
been
x
truncated
plastic
If
the
(removed).
and
is
A
attached
dimensions
of
typical
to
the
Elizabethan
pet’s
collar
is
made
of
collar.
an
D
Elizabethan
in
the
collar
diagram,
are
nd
as
the
area
E
7 cm
shown
of
20 cm
plastic
required
to
make
it.
B
C
12 cm
assessment
–
Solar
energy
1
i
t
e ri
r
Formative
c
Solar
for
energy
the
sun
into
in
a
areas
In
this
improve
throughout
Cylindrical
the
traditional
it
is
energy,
shape
about
panel
a
every
overhead.)
painted
even
that
wind
and
the
is
of
the
When
on
as
is
of
of
a
of
sunlight.
wide
potential
the
cells,
power.
of
energy
which
Mounting
range
products
solar
one
transforming
photovoltaic
analyze
underside
o
larger
it
of
rectangle.
as
of
passes
these
solutions
that
have
Please
of
cells
be
has
been
created
your
D
the
must
have
been
show
solution
to
working
to
on
a
is
The
makes
will
these
cylinder
from
(A
that
just
traditional
has
have
absorb
shape
solar
directly
businesses
panel
roof.
The
rays
roof
claim
collecting
overhead.
Sun
However,
panel
of
Sun’s
the
many
the
the
solar
way
buildings.
the
when
which
expensive
a
ecient
mounted
(such
resistance,
less
lot
and
cylindrical
more
ecient
reected
will
panel
absorption
white
a
own,
you
use
eectiveness
solar
angle
most
the
as
cells
especially
is
However,
receive
its
important
task.
much
allows
needs.
on
task,
solar
manufacturers
that
that
the
increasingly
requires
problem
proposed.
The
energy
electricity
become
to
becoming
planet’s
mounted
try
is
o
n
5
also
solar
been
today),
sunlight
reduces
cells
easier
install.
17 3
The
height
of
each
cylinder
that
is
covered
by
photovoltaic
cells
is
The
100 cm.
The
diameter
of
each
cylinder
is
is
a
Find
the
total
surface
area
that
is
area
that
22 mm.
exposed
to
the
Sun
of
200
covered
in
of
photovoltaic
these
solar
cells
are
installed
on
the
roof
of
a
does
b
Find
Spin
the
cell
amount
of
space
(volume)
that
the
200
cells
take
the
up.
cones
not
top
of
the
so
do
these
Spin
cell
cones
use
some
of
the
solar
energy
that
they
generate
include
and
but
to
collect
of
the
their
the
cone
is
manufacturer
Sun’s
rays
covered
from
in
explains
every
that
angle
photovoltaic
the
rotation
without
cells,
with
allows
not
in
include
your
to
overheating.
the
exception
them
Most
of
its
base.
c
d
ATL2
If
the
radius
area
of
Find
the
Making
the
have
e
cone
typical
f
of
a
section
the
of
roof.
solar
Calculate
the
cones
the
area
roof?
exposed
to
4
flat
by
an
its
height
is
0.82 m,
find
the
cells.
array
are
factor(s)
3D shapes
the
of
50
cover
surface
will
area
the
the
area
with
of
these
spin
dimensions
measures
that
installed
Using
will
roof
panel
panels
them.
can
of
of
the
white
solar
be
installed
spin
the
cell
roof.
Calculate
Explain.
1 74
solar
and
cell
cones.
1.65 m
cover
that
side
they
by
the
will
side
dimensions
maximum
of
these
of
by
10 m
1 m.
by
6 m.
Sketch
maximum
the
possible
cover.
and
need
given
possible
cylindrical
at
above,
area
least
sketch
of
panels
the
that
the
roof.
will
be
Sun.
configuration
What
0.55 m
in
Calculate
that
Spin
on
a
these
total
to
possible
h
the
exposed
cell
of
cells
between
configuration
a
is
occupied
rectangular
Cylindrical
20 mm
g
cone
covered
volume
configuration
area
one
choices
You
A
of
How
the
so
cones
that
that
many
total
they
will
cones
surface
touch
cover
will
area
of
each
the
you
the
other.
Sketch
maximum
be
able
cones
to
that
place
will
be
Sun.
would
influence
your
choice
of
solar
panels
to
install?
base
cylinder,
calculation.
rotate,
cells
building.
G E O M E T R Y
A N D
T R I G O N O M E T R Y
Pyramids
A
a
pyramid
cone
All
is
of
with
the
its
after
triangular
between
to
shape
circular
of
a
the
shape
various
that
base,
pyramid
pyramids,
the
you
another
sides
named
help
is
of
a
has
surface
There
a
base
are
and
However,
that
each
is
type
square
etc.
(height,
area
apex.
and
pyramids,
measurements
the
an
triangular
base.
hexagonal
determine
at
pyramid
are
its
meets
a
unlike
polygon.
of
pyramid
pyramids,
Relationships
length,
volume
apothem)
of
any
can
pyramid.
apothem
base-side
Square
Did
A
length
T
riangular
pyramid
you
pyramid
Hexagonal
pyramid
know...?
triangular
pyramid
is
often
referred
to
as
a
tetrahedron.
Some
molecules
H
in
chemistry,
central
atom
like
is
the
methane
located
at
the
molecule
center
of
shown
the
here,
pyramid
are
tetrahedrons.
(carbon,
in
this
The
case),
C
while
the
four
surrounding
atoms
(hydrogen)
occupy
the
H
vertices.
H
H
In
the
of
same
volume
of
a
relationship
a
pyramid
way
as
you
pyramid
to
the
by
did
of
a
cone,
out
an
prism
– Volume
you
a
nd
investigation
with
of
can
similar
out
the
into
its
dimensions.
pyramid
i
t
e ri
r
3
the
carrying
volume
Investigation
for
c
Your
teacher
between
height
l
i
n
the
and
may
give
volume
base
you
of
a
manipulatives
rectangular
and
pyramid
sand
and
in
order
that
of
a
to
nd
prism
the
with
o
n
Volume
relationship
the
B
same
area.
k
b
e
If
you
don’t
have
demonstration:
“volume
of
a
access
to
the
materials
needed,
you
can
watch
the
following
https://www.youtube.com/watch?v=O2wenAIf0H8
pyramid
water”
on
or
video
search
for
YouTube.
Continued
on
next
page
175
1
Look
at
What
ATL1
the
do
Reect
●
Before
doing
your
Will
your
Repeat
Write
a
Use
If
answer
sand
depend
to
step
2
shapes
down
of
the
watching
take
a
the
pyramid
with
a
with
the
same
base.
peer.
7
to
peer.
on
the
Your
type
triangular
how
the
video,
completely
of
prism
many
in
prism?
square
guess
ll
teacher
prism
rectangular
determine
with
are
the
a
the
may
and
the
how
many
prism.
collect
the
pyramid?
same
way
Will
as
a
Explain.
pyramids
it
takes
to
ll
the
other
pyramids
and
their
corresponding
prisms
if
available.
relationship
prism
formula
down
base
same
How
the
with
with
the
between
same
base
the
volume
and
of
height.
a
pyramid
Write
this
and
the
relationship
and
your
Investigation
volumes
of
Generalize
4
the
volume
for
the
measurements
verify
does
base
for
formula
your
l
of
a
prism
volume
and
relationship
of
and
a
your
square
result
from
pyramid
step
with
4
for
another
2
8
result
investigation
(the
cones
the
shape.
3D shapes
this
investigation
and
cylinders)?
formula
for
the
h
w
discuss
in
to
height
pyramid
and
cylinder
with
dimensions.
Reect
its
a
and
ndings
prism.
possible,
the
17 6
class.
prism
your
formula.
and
●
with
entire
and
the
or
guess
ll
Use
the
will
pyramid
write
●
it
pyramid
as
6
the
activity
sand
triangular
volume
5
of
rectangular
those
4
of
of
Discuss
discuss
the
Compare
square
3
and
full
a
2
notice?
pyramids
guesses
●
dimensions
you
into
the
relate
to
your
relationship
result
in
between
the
Explain.
volume
of
any
pyramid,
regardless
of
G E O M E T R Y
Practice
1
Find
the
A N D
T R I G O N O M E T R Y
5
volume
of
the
following
gures.
Round
answers
h
=
to
3
s.f.
10 cm
9 cm
6 cm
6 cm
8 cm
7 cm
10 cm
a
8 cm
b
8 cm
c
18 cm
h
25 mm
20 mm
20 mm
d
36
23
14 mm
e
f
7.5 cm
13 cm
mm
mm
13 cm
27
mm
g
2
10 cm
h
Find
the
missing
measurements
in
each
of
the
following
square
If
pyramids.
you
are
nding
calculating
Height
Base
side
length
Apothem
Volume
dicult,
this
try
3
(cm)
(cm)
(cm)
(cm
)
adding
an
column
4
extra
for
the
6
base
16
12
area.
10
13
20
1000
8
314
3.25
702.15
Continued
on
next
page
17 7
3
In
the
Strip
to
1990s,
began
attract
casinos
to
rides
is
themed
Great
resort
With
Why
step
beam
a
inside
would
a?
a
The
these
of
At
in
the
night,
the
actual
the
world
and
hotel
replaces
created
situate
the
with
Egyptian-
107 m
the
the
a
for
emanates
square
base
bedrooms,
space
available
from
with
a
the
sides
top
of
convention
be
less
than
of
the
197 m,
center,
the
the
to
be
not
only
of
are
a
viewed
replace
volume
114 996 cm
b
of
What
the
to
simple
from
allow
anywhere
air
in
by
base
of
inside
opening
two
in
a
wall.
or
even
three
viewing
from
all
in
the
a
More
Cairo,
many,
not
Using
would
178
4
a
to
to
sides.
sides,
The
you
space
and
was
theatres?
calculated
but
pyramid
replace
a
in
63 cm.
a
it
also
to
is
Find
the
length
of
the
the
was
model
lit
3D shapes
while
in
1989,
much
your
if
the
sides
of
working.
also
meant
preserving
Paris
the
is
Great
a
giant
structure
was
more
accepted
and
to
help
since
of
visitors
that
the
solve
the
the
the
the
across
Giza
criticized
problem
entrance
museum.
underground
view
of
even
original
to
pyramid
Pyramid
the
designed
volume
in
of
collections
large
pyramid
replace
Show
museum
now
Louvre’s
the
pyramid
scale
is
of
length?
Completed
handle
well
is
the
designs
pyramid
room.
square
volume
Louvre
pyramid
the
height
the
of
is
although
glass
be
the
that
The
its
doubled
area
Egypt.
accessing
could
much
restaurants
modern
base.
are
glass
admired.
of
the
entrance
made
and
happens
Surface
The
how
amount
3
sides
pyramid.
in
Explain.
replace
was
of
so
build
of
shape
light
The
enough
to
One
Giza.
of
not
complete
an
the
of
height
Traditional
the
in
Pyramid
available
4
Luxor,
try
children.
was
hotels,
Vegas
to
clientele,
attractions.
The
brightest
b
new
Las
project
were
devised
themed
hotels
a
this
the
with
shows
they
and
on
ten-year
families
in
solution
large,
a
and
bring
hotels
lobby
courtyard
often
replace
make
it
allow
design
possible
to
G E O M E T R Y
from
any
homage
within
How
need
a
to
the
the
the
to
the
extensive
glass
do
Louvre
nd
pyramid.
Egyptian
Creating
atten
3D
a
the
Copy
out
you
think
also
paid
collection
Pyramid?
how
to
was
To
needed
answer
calculate
the
3
net
–
is
Surface
an
shape
and
to
ecient
see
complete
all
way
of
this
area
its
to
of
to
this,
you
surface
area
Dimensions
pairs,
surface
add
the
area
areas
of
of
a
3D
the
shape,
since
individual
you
can
sections.
to
and
shapes
determine
of
sections
surface
area
SA
formula
(SA)
pyramid
pyramid
Hexagonal
In
the
then
pyramid
Rectangular
Square
and
table.
solid
Triangular
pyramids
determine
faces
needed
2
It
pyramid.
Activity
1
outside
T R I G O N O M E T R Y
museum.
much
make
of
point
A N D
pyramid
follow
these
of
steps
●
Draw
the
net
each
●
Label
the
dimensions
●
Find
●
Total
●
Use
to
3D
determine
shape
in
the
the
formula
for
each
pyramid
listed
in
the
table.
table.
Pairs
the
the
this
Your
teacher
their
nets
l
i
n
area
or,
of
areas
to
of
write
may
each
all
the
of
each
section.
of
the
sections.
formula
provide
alternatively,
section.
you
for
with
you
can
the
surface
area
manipulatives
look
at
virtual
of
of
3D
the
prism.
pyramids
that
can
be
broken
down
into
manipulatives.
k
b
e
On
the
section
how
Annenberg
and
the
net
search
Learner
for
unfolds
–
website
“pyramid”.
use
this
as
(www.learner.org)
Select
a
“Explore
visual
to
help
&
Play
you
go
to
with
the
“Interactives”
Pyramids”
determine
the
to
see
formulas.
17 9
Example
Q
The
4
Louvre
measuring
the
Pyramid
35.4 m
amount
square
of
and
glass
meters.
is
a
square
whose
needed
Round
your
pyramid
apex
to
is
cover
answer
whose
21.6 m
the
to
above
sides
the
base
of
has
the
the
nearest
sides
ground.
pyramid
square
Find
in
meter.
A
Draw
the
net
of
the
3D
gure
and
indicate
the
21.6 m
known
measurements.
To
the
35.4 m
35.4 m
height
slant
height
nd
sur face
area
Pythagoras’ theorem
2
s
2
(s).
Round
calculations
of
each
triangle,
calculate
the
use
slant
2
=
h
+
=
21.6
=
779.85
2
s
height
to
1
2
2
+
17.7
2
s
the
s
=
middle
of
a
to
the
nearest
hundredth
in
solution.
27.93
1
A
=
(35.4)(27.93)
Find
the
area
of
one
triangular
face
and
multiply
2
by
A
=
four.
494.36
You
The
area
of
all
4
triangles
=
do
since
=
not
need
to
nd
the
area
there
is
no
glass
on
the
Round
=
the
answer
to
the
nearest
1977 m
(as
18 0
4
3D shapes
the
base
base.
1977.44
2
SA
of
4(494.36)
specied
in
the
question).
square
meter
G E O M E T R Y
Practice
1
Find
the
Round
A N D
T R I G O N O M E T R Y
6
volume
answers
and
to
surface
the
area
nearest
of
the
following
gures.
tenth.
12 cm
15 cm
10 cm
17 cm
10 cm
10 cm
2
area
16 cm
a
43.3 cm
base
b
24 cm
8 cm
8 cm
5 cm
c
16 cm
5 cm
8 cm
d
4 m
10 cm
17 cm
12 cm
3.5 m
e
2
Dice
play
games.
it
dierent
A
4,
important
Because
options,
with
an
was
not
6,
8,
12
four-sided
pyramid,
all
of
and
die
of
all
part
of
to
sides.
even
You
20
faces
have
make
dice
can
six
with
now
get
dice
sides!
(abbreviated
whose
role-playing
scenarios
necessary
numbers
30 cm
f
d4)
are
is
a
triangular
equilateral
triangles.
a
If
the
apothem
is
18 mm
triangles
b
Is
a
d4
and
is
of
the
any
length
10.4 mm,
fair?
side
of
nd
of
the
the
the
pyramid
sides
of
surface
the
area
of
the
dice.
Explain.
Continued
on
next
page
181
3
Many
of
the
corridors.
possible
three
The
to
Pyramid,
the
designer
illuminate
there
main
(with
collections
are
exception
the
of
the
these
also
hallways
at
of
the
Louvre
can
smaller
These,
be
accessed
Pyramid
subterranean
three
meet.
Louvre
wanted
to
passageways.
ones
too,
underground
are
that
serve
square
as
use
In
as
much
addition
skylights
pyramids
and
via
to
that
are
hallways
natural
the
let
light
main
in
made
as
Louvre
light
of
and
where
glass
base).
2
If
each
of
apothem
base
4
of
Using
of
three
each
each
small
One
shaped
Users
using
The
the
is
interact
their
is
a
as
to
interaction
covered
sides
with
The
a
Find
the
surface
b
Find
the
percentage
an
in
a
measures
total
of
6.75 m,
79 m
what
of
is
glass,
the
side
and
when
that
objects
or
using
sits
on
a
tabletop.
the
pyramid
tablet.
height
of
of
problem
pyramid-
inside
pyramid
area
a
this
of
the
the
with
the
entire
total
sides
pyramid
is
23 cm.
device.
surface
area
that
is
available
for
viewing.
Spheres
A
In
in
sphere
is
a
perfectly
mathematics,
three
it
is
dimensions
round
often
that
three-dimensional
dened
are
the
as
“the
same
set
object.
of
distance
all
points
from
a
xed
point”.
Reect
●
Explain
and
how
dimensions
●
Is
●
What
18 2
the
4
a
the
sphere
that
sphere
is
discuss
the
are
3D shapes
represents
the
entire
“xed
9
same
solid
point”
the
set
distance
“ball”
called?
or
of
points
away
just
from
the
in
a
shell?
three
xed
point.
Explain.
the
length
individual
solve
developed
display
square
45 cm.
seen
hopes
smartphone
device
measuring
often
have
holographic
can
is
pyramid?
group
They
pyramids
triangular
company
promoting
technology.
small
of
technology
activity.
by
the
r
of
the
G E O M E T R Y
Unlike
down
area
prisms,
into
of
a
accurately.
sphere
cylinders
individual
How
4
can
–
pyramids,
nor
you
can
you
a
sphere
draw
determine
its
area
will
orange,
net
of
surface
of
a
a
be
broken
sphere
area?
sphere
i
t
e ri
o
c
Sur face
a
cannot
r
Investigation
and
faces
then
T R I G O N O M E T R Y
n
Surface
A N D
For
this
one
l
i
investigation,
whose
n
shape
is
you
as
need
spherical
as
an
a
tangerine
or
a
clementine.
Choose
B
possible!
k
b
e
If
you
don’t
have
an
orange
available,
you
can
watch
the
https://www.youtube.com/watch?v=jaL8Kuv6YHo
sphere
1
Place
the
around
2
Repeat
3
In
its
4
the
ATL1
1
middle
the
is
in
of
to
following
search
video
for
demonstration:
“surface
area
of
YouTube.
piece
make
times
of
a
paper.
shape
so
that
Holding
that
you
is
your
roughly
have
pen
or
pencil
upright,
trace
circular.
between
4
and
6
circles
that
are
size.
each
circle,
write
the
formula
for
the
area
of
the
circle,
assuming
r
orange
Reect
on
a
several
equal
ngernail
on
orange
step
radius
Peel
orange
the
roughly
orange”
or
of
in
your
and
small
pieces.
Try
to
make
the
pieces
no
larger
than
the
thumb.
discuss
10
Before doing the activity or watching the video, guess how many circles
you will completely ll with orange peel. Compare your guess with a
peer. Your teacher may collect the guesses of the entire class.
5
6
Completely
ll
leaving
space
How
many
terms
7
8
any
of
circles
circle
before
between
did
you
moving
pieces.
ll?
You
What
is
on
may
the
to
the
not
total
ll
next.
Try
every
area
of
to
avoid
circle.
those
circles
in
r?
Collect
data
person
ll?
Based
each
on
from
your
several
result,
of
write
your
down
peers.
the
How
formula
many
for
circles
the
did
surface
each
area
of
a
sphere.
18 3
ATL1
Reect
●
Why
does
sphere?
●
and
What
the
discuss
amount
of
11
orange
peel
represent
the
surface
area
of
a
Explain.
other
objects
could
you
use
to
perform
this
investigation?
Explain.
●
What
factors
more
or
less
Example
Q
have
the
contributed
correct
to
number
students
of
in
circles?
your
class
lling
Explain.
5
Throbber
be
may
than
heating
controlled
aluminium
balls
using
and
use
a
are
designed
smartphone
induction
to
to
app.
heat
heat
The
up
food,
balls
any
and
are
can
made
of
liquid.
Suppose six heating balls are placed in a pan of soup. If the balls have a
diameter of 5 cm, nd the total area of the heating balls that is in direct
contact with the soup. Round your answer to three signicant gures.
5
A
r
Find
=
the
radius
of
each
ball
by
dividing
the
2
diameter
=
by
2.
2.5 cm
Find
the
sur face
area
of
one
ball
using
the
2
SA
2
4 πr
=
formula
SA
=
4πr
2
=
Keep your working in terms of
4 π( 2.5)
π
at this stage, so
you don’t have to round your answer until the
2
very end of the question.
= 25 π cm
2
Total
surface
area
=
6
Since
× 25 π cm
there
sur face
are
area
of
6
heating
one
ball
balls,
by
multiply
the
6.
2
=
150π cm
=
471 cm
2
Calculate the value and round your answer to 3 s.f.
2
The
18 4
4
total
surface
3D shapes
area
of
the
6
heating
balls
is
471 cm
G E O M E T R Y
Practice
1
Find
the
A N D
T R I G O N O M E T R Y
7
surface
area
of
the
following
shapes.
Round
answers
to
the
10 cm
9 cm
a
nearest
tenth.
14 cm
b
c
5 cm
Circumference
=
47 mm
20π m
5 cm
d
2
e
With
the
recent
technology,
someone
cameras
when
all
was
are
its
that
only
invented
into
in
a
portable
matter
spherical
to
the
surface,
amateur
interesting
a
designed
thrown
over
tool
it
advances
f
take
air.
the
time
camera.
before
These
panoramic
With
ball
of
camera
photos
cameras
camera
photographers
is
can
a
ball
mounted
new
use
to
take
shots.
2
If
3
there
is
available
for
diameter
of
The
mirrored
ball
was
around
early
onto
as
the
it.
The
of
disco
small
clubs
balls
and
and
as
the
order
the
to
often
the
sphere
light
turns.
wide
is
reect
while
reect
ball
ball
what
The
rotates
gained
are
on
clubs.
mirrors
club
the
in
night
mirrors
the
1920s,
of
area
cameras,
created
inside
small
parts
the
1970s
“disco
the
with
dierent
the
small
camera?
shines
area
mounting
ball
covered
If
surface
a
light
in
of
1520 cm
a
is
light
into
Used
as
popularity
referred
to
as
balls”.
typical
can
be
nearest
disco
ball
covered
has
with
a
radius
mirrors?
of
42 cm,
Round
what
your
surface
answer
to
hundredth.
Continued
on
next
page
18 5
4
Which
each
5
would
with
a
radius
and
In
Aquilino
1963,
plastics,
resistant
infants
used
6 cm
a
newborns,
a
regular
balls
in
a
wide
a
By
to
what
factor
doubled?
b
By
what
tripled?
c
Volume
Since
Like
there
the
How
Can
be
a
the
you
is
4
chair.
of
were
have
exercise
the
each
10
with
a
Throbber
radius
of
used
sphere
shape,
does
using
to
even
Cosani’s
from
to
area
justify
surface
area
justify
as
of
used
not
a
do
sphere
an
in
of
an
your
of
an
exercise
ball
increase
when
its
radius
is
ball
increase
when
its
radius
is
answer.
an
exercise
answer.
increase
in
radius
and
you
is
nd
the
related
to
area
that
of
of
a
a
sphere?
cylinder.
12
of
a
to
nd
have
a
dierent
cylinders.
3D shapes
your
exercise
22 cm
your
between
how
volume
a
Show
been
regular
prism
or
a
cylinder?
What
on?
be
12 cm?
balls,
with
sphere.
the
heating
puncture-
since
and
sizes,
surface
the
discuss
principle
and
liquid,
manufacturer
produce
they
relationship
based
calculated
prisms
balls
a
diameter.
does
a
to
they
desk
volume
nd
it
but
range
does
base
and
do
a
of
heating
up
sphere
no
cone,
this
Because
the
heating
Italian
rst,
mathematics
area
is
principle
for
Use
in
an
mathematics
factor
of
Reect
●
Use
Generalize
surface
●
85 cm
at
answer.
therapy,
to
5
process
At
alternative
come
or
Cosani,
balls.
physical
eective
your
perfected
and
for
of
justify
plastic
diameter
18 6
more
working
of
to
be
the
volume
base
of
shape,
method
than
a
its
sphere?
Explain.
volume
the
will
standard
have
one
its
eect
on
the
G E O M E T R Y
5
– Volume
of
a
T R I G O N O M E T R Y
i
sphere
t
er
i
r
Investigation
A N D
this
investigation,
your
teacher
may
provide
you
with
manipulatives
and
n
c
For
sand.
B
l
i
n
k
b
e
If
manipulatives
are
not
available,
you
can
watch
the
following
https://www.youtube.com/watch?v=8jygxFuLoCk
and
1
ATL1
If
you
sphere
have
Reect
water”
on
manipulatives,
and
or
video
search
for
demonstration:
“volume
of
a
cylinder
YouTube.
ll
discuss
the
sphere
with
sand.
13
Before doing the activity or watching the video, guess the volume of
the sphere as a fraction of the volume of the cylinder. (Assume that the
sphere and cylinder have equal radii and that the height of the cylinder
is equal to the diameter of the sphere.) Compare your guess with a peer.
Your teacher may collect the guesses of the entire class.
2
Determine
pouring
3
Write
step
down
2,
equal
4
What
what
sand
the
adapt
height
is
fraction
from
the
formula
this
and
of
the
sphere
for
formula
cylinder’s
into
the
to
the
volume
write
volume
is
the
volume
of
the
sphere
by
cylinder.
of
down
a
a
cylinder.
formula
Based
for
the
on
your
volume
work
of
a
in
sphere
of
radius.
relationship
between
the
height
of
the
cylinder
and
the
radius
of
the
sphere?
5
Write
down
Substitute
6
the
this
height
into
of
your
the
cylinder
formula
from
in
terms
step
of
the
radius
of
the
sphere.
3.
Simplify your formula, so that you end up with the formula for the volume of a
sphere.
Reect
●
How
close
volume
●
How
have
and
of
does
discuss
was
the
this
studied
your
initial
sphere
result
in
this
and
14
guess
the
relate
to
of
the
relationship
between
the
cylinder?
the
volume
of
other
shapes
that
you
unit?
18 7
Example
Q
The
Super
1970s.
and
it
6
it
It
was
was
could
Ball
was
a
one
of
bouncing
said
that
bounce
if
it
higher
the
ball
was
many
made
thrown
than
a
toy
of
a
fads
of
the
material
hard
called
enough
three-story
1960s
onto
building!
and
Zectron
the
One
ground
SuperBall
3
contains
your
of
451.76 cm
answer
to
the
Zectron.
nearest
Find
the
radius
of
the
ball.
Round
hundredth.
4
4
A
V
3
The formula for the volume of a sphere is
3
πr
V =
πr
=
3
3
4
451.76
3
πr
=
Substitute
the
given
value
into
the
formula.
3
3
3
(451.76)
=
4
4

4
3
πr


3


Solve
for
r by
rst
multiplying
each
side
by
the

4
reciprocal
of
and
simplifying.
3
338.82
=
3
πr
3
πr
338.82
Divide
=
π
each
side
by
π
and
simplify.
π
3
107.84756
=
r
Take
3
3
the
cube
root
of
each
side.
3
r
107.84756
=
r
=
Round your answer to the nearest hundredth.
4.76 cm
Only the nal answer should be rounded to the
nearest hundredth.
Intermediate calculations
should use more signicant gures in order for
The
18 8
4
radius
of
the
3D shapes
SuperBall
is
4.76 cm.
the answer to be as exact as possible.
G E O M E T R Y
Practice
1
Find
the
nearest
A N D
T R I G O N O M E T R Y
8
volume
and
surface
area
of
the
following
shapes.
Round
your
answers
to
the
tenth.
32 cm
15 mm
a
67 mm
b
c
Circumference
=
40π
41 mm
r
d
2
mm
e
Copy
and
sphere.
complete
Round
all
this
table,
values
to
assuming
the
nearest
Radius
Diameter
(cm)
(cm)
that
each
set
of
measurements
are
for
a
tenth.
Surface
area
Volume
2
(cm
3
)
(cm
)
15
50
1000
1000
3
The
water
ball)
was
“walk”
and
the
user
a
If
walking
invented
on
water.
ball
is
ball
to
It
(or
allow
is
zipped
aqua
zorbing
people
made
closed
of
to
plastic
with
the
inside.
the
surface
area
of
a
water
walking
2
ball
is
inside
b
In
16π m
the
order
,
nd
the
volume
of
air
ball.
for
an
adolescent
to
be
Continued
on
next
page
18 9
b
In
order
for
an
adolescent
to
be
able
to
breathe
enough
oxygen
for
30
minutes,
they
3
need
an
c
ATL2
2 m
What
During
small
are
a
The
of
a
water
per
5
What
your
do
Give
three
the
96
these
in
try
to
Los
reduced
for
30
could
problems
reduce
black
be
a
water
be
associated
amount
85
the
of
each
Reservoir
by
ball
that
would
with
the
water
walking
in
to
city
water
of
Los
that
measuring
an
90
attempt
percent,
Angeles
evaporated
10 cm
to
in
block
saving
from
the
over
1
sunlight.
trillion
enough,
volume
the
of
balls
water
were
was
lled
in
all
with
of
the
water
balls
so
in
that
the
they
Los
would
Angeles
not
you
think
black
spheres
were
used
instead
of
some
other
shape
or
reasons.
of
the
following
shapes.
8 cm
20 cm
3 cm
22
cm.
ball
7
Find
b
is
the
of
the
radius
volume.
19 0
4
packaged
Assuming
percentage
blow
working.
volume
soccer
liters
Reservoir?
16 cm
A
its
diameter,
5 cm
6
ball?
used
15 cm
a
allow
overcome?
spheres,
evaporation
walking
uninterrupted.
California,
the
Angeles
of
minutes
that
southern
million
the
it
diameter
problems
drought
to
in
the
year.
Why
Find
Find
the
could
into
Interestingly
Show
of
About
balls
air.
remain
spheres
released
away.
b
How
black
shade
to
some
severe
reservoirs.
were
fresh
adolescent
Explain.
4
of
3D shapes
that
the
volume
of
a
c
in
a
ball
of
sphere
cubic
box
touches
the
that
box
has
is
an
with
all
side
faces
wasted
equal
of
lengths
the
of
box,
what
space?
surface
area
and
color?
G E O M E T R Y
–
Solar
energy
2
i
t
e ri
c
You
have
and
cones,
Solar
The
the
solar
is
the
how
solar
designs
made
tools
of
cells
have
have
also
been
been
made
in
the
form
of
cylinders
D
attempted.
for
at
designed
solar
cells
much
a
cell,
smaller
such
as
on
help
solve
each
This
applications
camping
construction
25 cm
to
overheating.
site.
side
The
and
or
square
has
a
35 cm.
Assuming
solar
was
the
measures
height
If
of
cylindrical
charging
base
other
pyramid
problem
than
b
but
seen
pyramid
device
a
already
o
n
assessment
T R I G O N O M E T R Y
r
Formative
A N D
cells.
each
the
pyramid
(Assume
pyramid
is
sits
that
lled
on
they
with
the
can
air,
ground,
be
nd
nd
placed
the
all
the
the
volume
of
surface
way
air
to
area
the
inside
available
for
top.)
each
pyramid.
Rawlemon
Inspired
by
architect
combines
potential
70%
the
of
more
is
Sun
or
daughter’s
Brossel
benets
solar
works
capable
Moon
of
of
than
with
toy
a
a
order
typical
both
to
the
its
to
glass
produce
and
up
the
to
panel.
moonlight
movement
motorized
It
with
photovoltaic
sunlight
the
German
Rawlemon.
magnifying
in
following
thanks
marbles,
created
energy
energy
Rawlemon
and
his
Andre
base.
of
the
The
large
crystal ball is lled with water to help magnify the light
rays on the solar cells.
The
c
Rawlemon
comes
in
a
variety
of
sizes.
Find the area of the sphere that is exposed to sunlight if it has a diameter of 1.8 m.
(Assume that the whole of the sphere’s surface area is exposed to sunlight.)
d
ATL2
Find
the
Making
e
Try
volume
of
water
in
a
sphere
with
a
diameter
of
1.8 m.
choices
out
several
product
the
on
dierent
which
surface
solar
area
shapes,
cells
then
would
available
for
choose
be
one
to
design
your
own
placed.
f
Find
solar
cells,
rounded
to
the
nearest
tenth.
g
Find the volume of space that your design occupies, rounded to the nearest tenth.
h
Explain the usefulness of your product. What is the most appropriate use for
your design (e.g on a building, for charging a small device)? Who would want
to use it?
19 1
Unit
The
of
volume
the
the
summary
base
gure
of
any
gure
that
3D
and
has
shape
can
multiplying
been
repeated
be
by
in
calculated
the
height.
order
Shape
to
by
taking
The
base
produce
Surface
the
area
the
gure
3D
shape.
2
2π r
r
is
Volume
2
Cylinder
area
+ 2π rh
r
h
h
2
Cone
r
1
+ π rs
2
π r
h
3
s
h
r
2
Square
b
+
2bs
1
2
b
pyramid
h
3
s
h
b
b
Rectangular
lw
+
2bs
1
lwh
pyramid
s
3
w
l
2
Sphere
4
4
3
r
π r
3
r
The
surface
area
of
any
pyramid
can
be
calculated
using
its
net.
1
The
volume
of
a
pyramid
the
is
volume
3
same
19 2
base
4
shape
3D shapes
and
same
height.
of
the
prism
with
the
t
e ri
o
c
n
r
i
Unit
Key
to
review
Unit
review
Level 1–2
1
Calculate
Round
A
question
levels:
Level 3–4
the
volume
answers
to
Level 5–6
and
the
surface
nearest
area
Level 7–8
of
each
of
the
following
shapes.
tenth.
6.2 mm
12 mm
5.6 cm
22 cm
15 cm
a
b
c
Slant
l
=
height
26 cm
8 cm
24 cm
6 cm
7 cm
7 cm
20 cm
d
2
In
a
7 cm
e
order
place
built
the
to
to
three
help
get
Seattle
and
away
giant
spheres,
providing
reduce
a
the
tropical
from
spheres
climate
nice
weather.
employees’
break
The
gardens.
their
in
is
oce
very
similar
the
spheres
levels
cubicle,
downtown
from
The
stress
that
sometimes
also
largest
contain
sphere
give
Amazon
Seattle,
to
and
USA.
of
cold
has
Inside
Costa
and
waterfalls,
measures
them
Rica,
rainy
a
river
40 m
in
diameter.
19 3
a
Find
the
Round
b
If
the
your
Airbags
1980s
them
and
a
impacted
are
the
fully
so
A
car
is
in
cylinder
has
a
the
volume
b
the
amount
order
inventor
design.
to
James
turned
the
material
problems
Dyson
could
center
of
At
the
around
motor
same
a
time,
with
particular
its
airbags,
passenger.
wheel
by
25 cm
of
a
and
to
the
car,
a
radius
fully
ball
moved
and
the
to
make
into
and
t
when
28 cm,
nd:
airbag.
and
stability,
vacuum
inside
allows
of
the
his
increased
ball
driver’s
inated
manoeuvrability
a
The
cylinder.
when
required
be
glass
will
and
airbag
of
of
between
specic
gravity
corners
for
they
car
inserted
motor
the
machine.
the
area
injuries
Airbags
of
the
the
car.
steering
in
in
passengers
a
sphere.
had
multiple
height
nd
and
the
largest
making
approximated
air
of
solve
The
lowered
of
a
cars
of
barrier
have
the
be
in
crash,
and
the
tenth.
glass,
feature,
protect
in
can
a
in
sphere.
belts
the
soft
can
located
seat
nearest
in
method
of
a
to
used
driver
that,
create
the
largest
safety
inside
inated
the
If
a
the
positioned
airbag
In
the
driver/passenger
each
4
as
the
and
windows.
If
standard
installed
deploy
as
to
available
covered
Although
persisted
space
widely
1990s
safer.
are
cover
became
become
still
to
of
answer
spheres
needed
3
volume
the
the
the
cleaner
ball,
stability
machine
which
of
the
to
be
ease.
vacuum
cleaner
has
a
volume
3
of
4200 cm
used
19 4
4
to
,
hold
nd
it.
3D shapes
the
radius
of
the
smallest
ball
that
could
be
5
Three-dimensional
adolescents,
The
especially
wooden
pieces
but
comes
cube,
is
with
and
puzzles
puzzle
very
the
the
since
shown
dicult
pieces
aim
are
is
popular
the
Rubik’s
here
to
has
solve.
arranged
to
toys
in
rearrange
cube.
only
The
the
for
10
puzzle
shape
them
to
of
a
make
3
a
square
of
6
pyramid.
wood,
The
nd
LifeStraw
waterborne
water
into
inches
that
7
choose
sphere
for
If
a
within
the
a
is
Find
The
sphere,
the
zorb
with
is
a
option”,
ride
sphere
and
of
with
down
has
2 m,
outer
a
liters
length
of
of
the
the
of
contaminated
straw
volume
is
of
9
water
and
on
0.8 mm
your
surface
even
makes
a
of
hot
3 m
the
the
each
air
inner
layer
material
which
answer
of
day.
and
of
plastic
thick,
area
in
can
volume
(The
a
inner
This
diameter
spheres.
hill
air
can
the
hill
the
of
Zorbs
zorbers
the
a
actually
water.
Round
of
down
where
nd
approximately
volume
1000
99.9999%
its
layer
and
lled
for
rolling
people
partly
to
calculate
home
three
calculations.)
the
is
shocks.
diameter
is
ball.
“aqua
removes
the
inch,
absorb
the
inner
these
3D
to
up
If
known
involves
a
1
help
outer
sphere
8
to
is
activities,
This
turn
cm
pyramid.
lter
water.
is
the
9
hold.
which
refreshing
the
has
can
inatable
up
can
diameter
straw
between
It
contains
of
water
the
“zorbing”.
hold
personal
drinking
adventurous
sphere
puzzle
dimensions
bacteria.
Zealand,
an
the
safe
and
the
New
in
the
If
to
of
3
can
for
be
sphere
between
each
ignored
in
s.f.
the
following
shapes.
8 cm
8 m
1.8 m
5 m
10 cm
7 m
1.8 m
8 cm
a
3 m
b
c
19 5
6 mm
c
m
14 mm
r
20 cm
26.5 mm
d
9
e
Specialized
to
protect
the
and
silo
to
is
the
one
of
height
how
needed
The
of
Morning
in
Australia.
known
from
a
the
west
study
use
a
such
tool
attached
(which
These
coast
are
shown
here.
of
are
mostly
not
in
Carpentaria
sea
breezes
breezes
of
phenomena,
a
a
from
scientists
radiosonde,
weather
balloon
spherical!).
cylindrical
of
cloud
150 m.
cloud
Find
the
surface
3D shapes
formations
Moving
moving
formation.
4
when
The
very
causes
of
the
formation
occurs
Gulf
of
Find
silos
cloud
forms
meet
called
to
is
radius
a
it
part
primer
ve
its
with
and
15 m.
of
paint.
20 m.
north
Although
the
is
layer
then
17 m
and
the
the
process
a
life.
east.
To
lot
paint
photo)
on
over
is
is
Glory
the
and
silo
silo
precisely,
October
during
silo
protect
predictably
a
the
with
cylindrical
the
much
to
(shown
the
of
their
specialized
the
agricultural
diameter
out
of
height
total
19 6
painted
two
designed
against
prolong
primers
layers
been
silos
rust-resistant
The
10
has
storage
corrosion
First
paint
at
can
speeds
up
be
to
1000 km
long
60 km/h,
with
there
is
a
overhead.
area
Round
of
your
the
Morning
answer
to
Glory
the
cloud
nearest
hundredth.
b
If
the
cloud
factor
would
answer
c
If
the
restaurants
grinder.
pepper,
used.
of
a
radius
contain
12
Access
many
as
to
was
a
ball
The
Q
volume
simply
of
by
only
long,
area?
by
what
Justify
your
half
surface
the
radius,
area?
by
Justify
what
your
as
a
own
possible
is
a
in
of
a
a
until
wide
over
rolling
the
5 cm.
Find
it
has
as
the
to
cylinder.
in
especially
located
need
the
transporter
problem.
a
large
distances
drum.
the
if
pepper
often
this
long
of
(sphere).
transport
water
use
radius
who
water
to
cylinder
problem
are
people
a
grinder
countries,
solve
of
ball
pepper
it
their
come
and
of
Drum
to
pepper
grind
volume
the
can
to
many
the
shape
water
allows
anyone
as
sizes.
sources
developed
is
can
14 cm
same
from
half
surface
the
in
fresh
the
shape
clean
design
that
in
of
water
away
as
and
developing
water.
Its
the
clean
far
in
of
users
it
shapes
is
had
found
grinders
height
the
change
kitchens
allows
is
only
calculations.
keeping
Another
the
and
product
with
that
product
Pepper
range
One
It
change
formation
with
common
were
calculations.
would
answer
A
that
with
cloud
factor
11
formation
This
device,
means
including
3
children.
The
of
Its
water.
50 cm.
the
13
Find
Q
Drum
height
the
is
holds
36 cm
radius
of
50 000 cm
and
the
the
inner
diameter
hole
of
through
the
the
base
is
center
of
drum.
The
traditional
approximately
traditionally
and
the
made
poor
of
by
its
conductor
of
inside
snow,
heat,
the
dome,
Canadian
Greenland.
compacted
generated
with
hemispherical
built
people
of
igloo,
so
Inuit
It
is
which
any
structure
is
is
a
heat
will
stay
inside.
19 7
a
If
the
radius
inside
an
igloo
is
1.8 m,
nd
the
volume
of
air
Disregard
contained
the
inside.
entranceway
b
If
the
and
thickness
the
surface
14
The
both
the
to
ensure
medicine
gelatin
11 mm
and
Find
that
the
a
hold.
your
Find
to
19 8
4
in
to
a
volume
Include
of
a
of
of
one
such
as
but
of
can
with
showing
the
capsule.
of
gelatin
capsule.
1847,
having
manufacturer
5 mm.
size
0.8 m,
is
nd
was
medication
without
length
diagram
amount
a
in
medicine
this
the
of
is
snow
0.5 m
your
this
the
answers
question.
nd
the
and
surface
needed
to
lls
designed
and
taste
the
antibiotics.
to
make
the
hollow
area
of
to
Just
volume
igloo.
patented
sizes,
of
compacted
the
dosage
total
calculations,
3D shapes
of
medicines
many
of
entrance
swallow,
diameter
produce
the
outside
correct
has
capsule
the
of
capsule,
with
dimensions
b
walls
pharmaceutical
one
a
the
the
come
common
of
easy
The
the
radius
hard
capsule
Capsules
a
area
two-piece
medicine.
a
inner
of
in
the
hemisphere.
i
t
e
r
Summative
i
c
ATL2
a
r
assessment
C, D
Designing
Heating
pads
muscles
and
sprains,
joint
a
are
for
injuries.
Your
task
is
pain,
can
you
to
get
design
use
also
into
a
used
such
long-term
They
when
bag
commonly
relieve
and
example
heat
as
to
be
a
worldwide
arthritis,
alleviate
used
cold
to
help
to
headaches,
chronic
simply
to
strains,
back
warm
relax
pain
you
and
up,
for
bed.
microwavable
heat
bag
with
a
volume
of
3
1500 cm
wheat,
bag
.
The
ax
or
The
several
bag
cannot
Part
a
to
so
1
The
Design
using
it
ve
each
that
the
the
a
not
of
the
●
Square-based
●
Cylinder
●
Cone
●
Square-based
●
Sphere
of
a
be
so
ller,
lled
you
or
touch
the
of
bags
following
a
ller
roof
that
want
a
oven,
dish
of
so
The
the
The
to
as
longer
the
experiment
it
in
most
maximum
microwave
basic
have
such
combination.
found
30 cm.
possible
heat
may
even
glass
diameter
with
microwaved.
microwave
circular
product:
possible
can
better,
inside
has
will
completely
kinds
than
which
bag
–
heat
t
be
grain
dierent
larger
microwaves
the
its
needs
be
will
other
maintains
with
of
bag
the
is
height
20 cm.
shapes
given
volume
shapes.
rectangular
prism
pyramid
19 9
b
Draw
a
diagram
material
Show
you
all
of
accuracy
round
Part
a
2
–
your
design
need
to
working.
will
use
and
make
Think
(how
calculate
each
(the
about
many
the
surface
what
decimal
amount
area).
degree
places
of
of
you
will
The
a
product:
heat
bag
shape
that
a
creative
has
made
up
the
of
given
two
shape
volume
or
more
but
of
is
the
a
above
shapes
together
Draw
a
diagram
material
Think
c
would
you
compound
b
each
to).
Design
put
of
Show
you
about
that
of
will
need
what
your
your
design
to
make
degree
chosen
of
and
it.
calculate
Show
accuracy
dimensions
all
you
the
of
will
produce
a
amount
your
of
working.
use.
3D
shape
with
a
3
d
volume
of
Choose
a
senior
use
A
specic
citizens).
to
common
diagram
bag
on
of
fabric.
Part
You
3
–
will
the
that
●
and
●
an
from
the
initial
analysis
your
nal
of
4
your
heat
design
of
You
of
for
bag
this
scents,
fabric
will
110 cm
fabric
your
(e.g.
children,
market
etc.
Add
–
you
these
can
design
wide.
is
110 cm.
position
possible
net
is
to
the
You
in
Construct
net
of
should
order
to
your
use
a
heat
the
minimize
waste
scale.
materials
materials
beginning
will
to
end
create
a
for
as
your
well
folder
product
as
how
(digital
to
you
or
show
plan
hard
its
to
copy)
following:
designs,
of
complete
(dimensions,
each
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20 9
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The followng data represents the lfe expectancy at brth of a chld n
several countres as compared wth that country’s gross domestc product
(GDP). The GDP s a measure of a country’s economy, and ncludes the value
of everythng produced by all of the ctzens and companes n a country.
Gross
domestic
product
Life
expectancy
Country
(PPP
Australa
per
capita)
(years)
43 000
82
7 000
68
43 100
82
Egypt
6 600
73
Japan
37 100
84
Mozambque
1 200
52
Nepal
1 500
75
Oman
29 800
67
Peru
11 100
73
102 100
78
37 300
80
58 000
79
Bhutan
Canada
Qatar
Unted
Kngdom
Unted
States
Reect
and
discuss
3
In small groups, dscuss the followng questons.
●
What varable wll you graph on the horzontal axs? Justfy your choce.
●
What
wll
●
s
set
What
the
up
range
each
scale
wll
of
of
each
the
you
varable?
axes
use
for
of
your
each
How
does
scatter
varable?
ths
aect
how
you
plot?
Explan
your
choce.
Contnued
210
5
Bivariate data
on
next
page
S TAT I S T I C S
1
Based
a
on
scatter
2
Do
the
3
Would
4
What
you
5
your
data
t
ponts
make
pattern
Descrbe
Does
the
there
Explan
the
8
of
on
a
must
questons,
represent
the
data
on
the
n
Explan.
data
the
ponts?
data?
Explan.
Descrbe
any
general
trends
that
Is
the
●
Does
be
is
data
does
pars,
of
the
relatonshp
of
ts
between
ctzens?
to
mply
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about
a
country’s
your
what
GDP
and
the
answer.
aects
the
length
of
the
the
2
to
–
nd
suggest
on
list
valid.
you
about
to
the
Is
The
know
What
there
the
quality
is
of
the
that
the
their
that
of
a
data
major
statistics
web
the
for
you
source
quoted
information,
site.
motivation
current
sources
that
importance
webstes
the
is
the
for
nding
bias?
website
it
that
internet
surng
a
websites
worldwde
ensure
trustworthiness
where
other
need
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data?
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ve
collectng
you
do
the
data?
website
data
and
accurate?
about
links
the
Explan.
how
information
Activity
drecton
relatonshp
seems
reliable
but
are
think
the
a
statistics,
collected
the
to
and
plot.
experence.
sharing
●
for
see
scatter
experence?
website
In
jon
expectancy
collected
Who
1
prevous
perfectly?
strength
the
anythng,
contain
ATL1
lfe
analyzing
and
up
to
you
form,
information,
you
●
can
n
what
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human
have
lne
sense
seem
human
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the
see.
average
7
to
P R O B A B I L I T Y
plot.
represented
6
answers
A N D
you
its
you
of
thnk
and
up
to
date?
information?
can
it
check?
reliable
would
Does
be
data
trustworthy
sources
data.
Pairs
2
Check
each
ste
for
valdty
and
relablty
and
share
your
ndngs
To
wth
the
nvestgate
queston,
3
As
a
class,
access
come
up
wth
a
lst
of
approved
webstes
you
can
use
to
nformaton.
to
nd
Use
the
webstes
you
have
found
to
try
to
answer
the
you
the
statstcs
(or
4
ths
class.
cty)
wll
need
poverty
of
a
and
country
the
crme
followng
rate
n
that
same
queston:
country
“Is
crme
related
to
(or
cty).
poverty?”
Contnued
on
next
page
21 1
Thnk
about
these
●
How
many
●
How
wll
●
Whch
countres/ctes
you
wll
Represent
scatter
the
on
wll
whch
you
need
to
look
countres/ctes
ndependent
varable
and
at?
to
nclude?
whch
wll
be
the
varable?
the
plot
decde
be
dependent
5
questions.
data
seem
wth
to
a
scatter
ndcate
n
plot
drawn
response
to
by
the
hand.
What
queston
does
you
your
were
nvestgatng?
Practice
1
Describe
1
the
form,
strength
and
direction
y
of
these
scatter
plots.
y
y
b
a
c
500
100
110
100
90
90
400
80
80
70
70
300
60
60
50
50
200
40
40
30
30
100
20
20
10
10
x
x
0
2
1
3
0
5
10
15
20
25
100
200
300
400
500
y
y
y
d
x
0
4
e
f
1
0
20
25
9
15
8
20
10
7
5
6
15
5
0
4
10
–5
3
–10
2
5
–15
1
–20
x
x
x
0
0
2
1
2
Humans
are
enjoyment.
use
some
People
For
of
are
one
in
of
4
the
Whereas
their
the
the
–30 –20 –10
few
most
time
prepared
example,
shown
3
for
to
cost
animals
leisure
spend
of
following
species
a
tickets
–
lot
that
seem
focus
for
a
to
their
simply
of
0
10
do
on
on
for
that
leisure
basketball
100
5
Bivariate data
200
humans
interest
them.
activities.
game
are
table.
Contnued
212
150
pure
survival,
things
their
professional
50
30
activities
eorts
enjoying
money
20
on
next
page
S TAT I S T I C S
Number
of
tickets
Total
cost
1
143
2
286
3
429
4
572
5
715
6
858
7
1001
8
1144
9
1287
10
1430
11
1573
12
1716
a
Is
each
data
b
Which
is
the
dependent
c
Which
is
the
independent
d
Represent
e
Do
f
Describe
the
set
the
data
discrete
data
on
points
the
or
line
continuous?
up
relationship
your
Justify
answer.
your
answer.
plot.
perfectly?
that
Explain.
Justify
variable?
scatter
P R O B A B I L I T Y
($)
variable?
a
A N D
is
Would
represented
it
in
make
the
sense
scatter
to
join
plot
them?
(form,
Explain.
strength,
direction).
g
3
Represent
While
all
birth
living
celebrate
the
Día
way
of
died.
using
death
only
death.
With
los
all
Muertos
over
the
are
The
end
Día
los
cultural
(Day
world
October
November.
to
the
Attendance
period
is
of
have
given
the
like
their
who
own
have
in
Los
days
beginning
in
working.
Dead),
festival
the
your
to
events
several
at
Show
actually
the
those
lasts
equation.
common
Muertos
California,
of
ve-year
de
an
humans
commemorating
Angeles,
the
and
data
things,
de
humans
the
from
of
festival
over
following
a
table.
Contnued
on
next
page
213
Attendance
Year
(number
of
people)
2013
24
882
2014
30
941
2015
33
127
2016
35
081
2017
39
572
a
Represent
b
Describe
the
the
data
using
a
relationship
scatter
between
plot.
the
two
variables
(form,
strength,
direction).
4
c
Predict
d
Suggest
One
of
are
most
and
also
amount
festival
reasons
the
habitats
we
the
of
why
the
notable
for
festival
human
2018.
traits
of
transportation.
capable
of
destroying
is
remaining
represented
in
in
is
the
natural
table
Population
your
keeps
ability
However,
Indonesia
the
Justify
attendance
modes
forest
population
attendance
a
increasing.
to
create
unlike
resources
as
answer.
on
function
products,
most
a
other
large
of
the
species,
scale.
The
country’s
below.
Area
of
forest
Year
2
(millions)
233
964
870
2008
236.2
958
020
2009
239.3
951
170
2010
242.5
944
320
2011
245.7
937
476
2012
248.9
930
632
2013
252
923
788
2014
255.1
916
944
2015
258.2
910
100
Represent
b
What
c
How
and
e
)
2007
a
d
(km
the
pattern
do
you
2017?
data
is
for
suggested
think
the
the
strength,
direction).
is
by
area
the
of
and
area
using
a
scatter
plot.
data?
forest
in
Indonesia
changed
during
2016
Explain.
Describe
Suggest
population
relationship
reasons
unrelated
to
why
the
the
that
area
increasing
is
of
represented
forest
is
in
the
scatter
decreasing.
population?
Is
it
plot
(form,
possible
that
it
Explain.
Contnued
21 4
5
Bivariate data
on
next
page
S TAT I S T I C S
5
As
humans,
some
of
our
component
The
the
Index
June
score
in
in
in
What
If
a
The
table
years,
The
below
along
shows
be
of
for
part
to
of
the
is
a
key
community.
percentage
Myanmar
average
give
others
a
Myanmar’s
the
to
is
often
World
of
at
or
Giving
temperature
in
years.
June
World
Giving
Index
29
25.92
58
25.85
64
25.95
66
26.12
70
26.37
65
data
to
country
with
willing
empathy
according
25.47
the
are
P R O B A B I L I T Y
using
a
scatter
plot.
Be
score
sure
to
choose
suitable
scales
axes.
relationship
think
the
nature
that
average
of
is
represented
temperature
citizens
in
in
in
June
Myanmar?
Is
the
has
this
scatter
any
plot
(form,
impact
represented
on
in
the
the
Explain.
might
of
could
plot
are
necessarily
contrast,
of
part
the
but
has
the
may
of
relationship
a
a
“average
with
seem
the
temperature
World
Giving
in
June”
Index
human
Does
does
that
that
mean
mean
that
that
one
the
two
variable
Explain.
to
queen,
also
pattern,
related?
other?
animals
of
instead
Explain.
necessarily
beehive
money
a
plot
demonstrates
aects
species
a
you
demonstrate
Myanmar?
variables
earns
charity.
and
22
variable
example,
human
countries
and
This
25.83
scatter
Many
be
need.
(%)
direction).
you
in
citizens
(°C)
strength,
Do
to
those
the
score
In
in
to
ranks
consecutive
Describe
that
6
list.
fellow
those
means
give
temperature
your
graph?
e
it
our
help
Index
that
Myanmar
charitable
d
about
to
who
of
Represent
for
c
Giving
top
Average
b
what
population
the
care
wealth
of
World
near
a
we
A N D
have
clearly
worker
bees,
experience
full
the
need
is
to
dened
“roles”.
For
drones,
foragers
and
choosing
feel
a
useful
job
and
that
guards.
not
only
valued.
Contnued
on
next
page
21 5
Data,
like
person
those
changes
in
the
jobs
table
by
the
below,
age
Average
of
suggest
30
number
is
of
that
the
changing
number
with
each
of
times
a
generation.
job
Generation
changes
Matures
Baby
X’ers
Dene
3.7
6.4
each
continuous.
b
Can
c
What
d
Do
so
you
that
f
g
you
one
this
categorical,
ordinal,
discrete
or
answers.
data
be
using
evident
make
represent
your
scatter
correct
the
strength,
direction).
a
plot
answer?
the
5
to
and
Describe
Make
either
a
scatter
in
this
changes
data
to
using
plot?
data?
the
a
Explain.
Explain.
independent
scatter
plot.
variable
Draw
the
plot.
relationship
prediction
generation.
216
can
as
your
seems
research
Compare
than
Justify
pattern
scatter
e
variable
represent
some
30
2.4
Mllennals
a
age
2.1
Boomers
Gen
by
Justify
Bivariate data
about
your
with
three
others
in
your
class.
Is
represented
in
your
scatter
there
more
Explain.
that
the
is
average
prediction.
number
of
job
plot
changes
for
(form,
your
S TAT I S T I C S
Line
In
of
best
general,
do
not
need
data
to
line
a
curve
through
should
follow
so
it
If
that
the
you
have
If
to
you
of
as
and
had
would
t
to
look
trend
lke?
a
be
a
as
of
well
plot
best
line)
as
t
in
the
curve
of
is
plot
usually
points,
pattern
will
but
the
it
data
possible.
a
line,
(often
line).
discuss
draw
will
called
data
scatter
line
scatter
You
may
them
the
the
the
a
relationship
general
your
drawn
Reect
●
on
the
(which
all
the
on
something
represents
curve
referred
points
perfectly.
curve,
This
go
data
up
P R O B A B I L I T Y
t
approximate
with
tting.
not
all
the
A N D
sngle
Justfy
4
lne
your
or
curve
through
each
of
the
followng
scatter
plots,
what
answer.
y
y
b
a
$700
$700
$600
$600
selaS
selaS
$500
$400
$500
$400
$300
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$200
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$0
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x
x
10 12 14 16 18 20 22 24
T
emperature
10
26
15
20
25
30
35
T
emperature
(ºC)
40
45
(ºC)
y
y
c
d
10000
15
9000
14
8000
s ruoH
)srallod(
16
13
12
11
ecirP
10
9
8
7000
6000
5000
4000
3000
2000
7
1000
x
0
4
8
12
16
20
24
x
0
1
Age
2
3
4
5
Age
●
Whch
●
For
of
the
those
above
6
7
8
9
10
(years)
scatter
relatonshps
plots
that
are
suggest
lnear,
a
how
lnear
could
(years)
relatonshp?
you
nd
the
equaton
of
the
trend
lne?
21 7
Activity
Perform
3
ths
–
The
actvty
line
wth
a
of
best
t
partner.
Pairs
1
Look
lne
at
of
a
the
best
followng
t
s
well
scatter
drawn.
plots
and
Justfy
lnes
each
of
of
t.
Dscuss
whether
or
not
you
thnk
the
decsons.
y
b
y
best
your
1.5
16
14
12
1
10
8
6
0.5
4
2
x
x
0
0
1
2
3
4
5
6
2
7
4
6
8
10
y
y
c
d
2000
85
1800
75
1600
1400
65
1200
1000
55
800
45
600
400
35
200
0
25
0
e
x
x
1840
.01 .02 .03 .04 .05 .06 .07
f
y
1860
1880
2000
2020
2040
y
10
9
7
6
5
4
3
2
1
x
x
0
1
2
Usng
of
a
best
2
3
4
5
straght
t
for
6
7
8
edge
each
of
91011
(preferably
the
followng
y
a
a
transparent
scatter
decde
where
you
would
draw
the
lne
plots.
y
b
$70
ruler),
100
90
$60
80
$50
70
$40
60
50
$30
40
$20
30
$10
20
$0
x
0
2
4
6
8
10
10 12 14 16
x
0
2
4
6
8
10 12 14 16 18 20
Contnued
218
5
Bivariate data
on
next
page
S TAT I S T I C S
y
A N D
P R O B A B I L I T Y
y
c
d
25
23
21
20
19
15
17
10
15
5
13
x
30
20
3
Do
all
lnes
4
Descrbe
5
Once
Drawing
a
of
the
you
40
50
best
t
aspects
have
line
have
of
drawn
x
60
to
pass
drawng
the
of
best
t
If
you
nd
1
0
lne
can
the
of
be
through
lne
best
of
t,
made
the
orgn
best
how
easier
2
t
can
if
(0,
that
you
you
3
0)?
are
nd
4
Explan.
dcult.
ts
know
equaton?
at
least
The
one
point
on
it.
the
mean
of
each
variable,
use
that
the
create
an
ordered
pair,
(x,
y),
and
then
plot
that
point.
This
symbol
mean
one
point
that
must
be
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21 9
Age
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220
5
Bivariate data
on
next
page
S TAT I S T I C S
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172
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2 23
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2 24
5
Bivariate data
on
next
page
S TAT I S T I C S
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on
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2 25
7
Abraham
Maslow,
psychologist,
human
must
the
need
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share
in
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that
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data
to
life).
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Contnued
226
5
Bivariate data
on
next
page
S TAT I S T I C S
Analyzing
As
you
saw
direction
focusing
of
on
determine
be
bivariate
previously,
the
relationship
data
the
measured
variables.
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correlation.
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be
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strength
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here,
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relationship?
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relationships
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Perform
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data
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follow
strength
Describing
Correlation
you
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a
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2 27
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Explan.
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your
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assocated
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your
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as
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Explan
the
discuss
−1,
how
that
are
are
+1
Negative
data
perfectly
on
and
previously,
called
well
depending
between
learned
5
−1
the
can
linear
can
correlation
be
linear
have
direction
be
depending
of
an
the
described
on
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strength
of
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–1.0
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coefcient
Bivariate data
relationship
gnortS
5
linear
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of
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S TAT I S T I C S
Practice
1
Describe
correlation.
a
graph
For
or
as
those
weak.
having
with
Justify
positive
correlation,
your
f
your
3
would
Would
Justify
4
g
strong,
y
d
y
x
y
y
h
x
x
y
x
the
correlation
you
you
your
expect
expect
a
the
coecient
(r)
to
each
graph
in
question
1.
Justify
b
Number
c
Probability
a
d
Number
of
calories
e
Number
of
connections
f
Number
of
pets
either
awarded
of
data
between
correlation,
a
the
following
positive
pairs
correlation
of
or
data
no
sets
to
correlation?
answer.
Level
the
correlation
negative
a
For
no
answers.
What
be?
for
is
or
x
y
j
value
correlation
correlation
x
y
a
the
x
y
Assign
negative
whether
c
x
2
state
a
choices.
x
i
correlation,
y
b
y
a
a
x
e
P R O B A B I L I T Y
3
each
moderate
A N D
cases
an
of
assignment
the
person
sets
strong,
on
has
versus
had
a
consumed
versus
in
u
in
moderate
the
Grades
question
or
versus
3,
Number
romantic
versus
brain
in
of
of
hours
people
versus
of
Age
versus
siblings
of
worked
immunized
relationship
Number
you
an
in
Age
on
the
against
of
a
assignment
the
u
person
family
adolescent
school
classify
weak.
Number
the
Justify
relationships
your
that
do
have
a
correlation
as
choices.
Contnued
on
next
page
2 2 9
5
All
animals
that
have
an
education.
its
own
are
learn,
system,
designed
quality
sorts
of
Data
of
it
organized
While
productive
but
to
each
all
by
only
system
education
and
humans
of
country
produce
citizens
life
is
adopts
systems
informed,
to
opening
improve
doors
to
their
all
careers.
on
the
represented
unemployment
in
the
scatter
rate
plot
in
the
United
States
and
educational
level
are
below.
y
7.5
7
6.5
6
5.5
)%(
5
etar
tnemyolpmenU
4.5
4
3.5
3
2.5
2
1.5
x
0
10
12
14
Y
ears
of
16
education
18
20
(years)
Contnued
23 0
5
Bivariate data
on
next
page
S TAT I S T I C S
a
Describe
the
correlation
between
the
two
variables
as
A N D
P R O B A B I L I T Y
positive/negative
and
weak/moderate/strong.
b
What
your
c
do
you
think
What
For
of
relationship
each
two
your
of
the
seems
rate?
variables
able
to
that
actually
you
are
that
can
calculate
not
be
the
expected
represented
by
a
you
although
the
hs
work
Pearson
hs
there
at
between
years
a
coecients,
correlation
of
for
education
write
with
exact
1
an
that
down
this
and
an
value.
−0.9
data?
Justify
the
example
Explain
is
the
guess
Pearson
r
and
it
0.5
appropriate
important
a
−1
coecient
with
associated
make
coecent
based
study
skill,
with
a
every
but
do
correlation?
time!
correlation
measures
how
One
well
Surely,
value
coecient.
how
you
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is
data
can
be
contnued
coecent.
also
wrote
Asde
from
also
you
ways
the
named
work
correlaton
that
work
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about
tme
Pearson’s
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can
to
use
coecient
mathematics
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a
of
wth
19th
travel,
work
Karl
Francs
and
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the
after
Galton.
regresson
the
the
on
and
calculaton
century
of
Pearson,
Englsh
fourth
the
dmenson
study
Albert
of
Ensten
who
work.
are
value
on
of
approximate
your
www.desmos.com
correlation
the
to
was
the
deas
Pearson’s
coecient,
those
is
symbol
who
correlaton
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While
is
line.
poneered
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read
have
relationship
value
correlaton
mathematcan,
and
coecient
know...?
Pearson
Galton
a
calculated
the
was
correlation
correlation
vocabulary
by
of
exist
correlation
might
the
represented
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to
−0.2
describe
mathematical
The
the
choices.
Calculating
Did
of
Explain.
following
0.8
Being
value
answer.
unemployment
6
the
of
r
is
GDC
or
value
online
of
The
that
is
formula
studied
the
tools,
www.alcula.com,
eciently.
content
or
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in
to
for
correlation
such
as
calculate
the
calculating
higher-level
courses.
2 31
Activity
One
6
–
Calculating
characterstc
communcate
that
wth
denes
one
“r ”
humans
another
usng
s
our
ablty
language.
to
Whle
other
anmal
ATL2
speces
use
language
can
n
speak
the
gestures
lke
that
of
and
sounds
humans.
to
The
(expressive vocabulary)
table
communcate,
number
changes
of
as
none
words
she
have
that
gets
a
a
chld
older,
as
seen
below.
Age
Expressive
(years)
vocabulary
(words)
1
20
2
240
3
950
4
1500
5
2200
6
2600
7
3500
1
Enter
the
data
2
Draw
the
scatter
3
Estmate
the
nto
your
GDC
or
onlne
tool.
plot.
value
of
the
correlaton
coecent.
Justfy
your
answer.
4
Use
your
GDC
correlaton
5
How
close
of
Explan.
r ?
Finding
the
include
pattern
232
5
of
was
onlne
your
value
relationship
data
or
tool
to
calculate
the
value
of
r,
the
coecent.
of
between
an
most
estmate?
r
helps
two
outlier
of
the
Bivariate data
(a
Are
you
to
you
interpret
variables.
value
data)?
surprsed
that
the
However,
simply
by
the
exact
value
potential
what
does
not
happens
follow
if
the
the
S TAT I S T I C S
Investigation
2
–
The
eect
of
an
outlier
on
A N D
P R O B A B I L I T Y
“r ”
t
er
i
r
i
Humans
are
the
speces
extracts
speces
n
the
perl.
world’s
only
speces
more
The
to
use
resources
populaton
resources
are
vast
from
of
amounts
the
planet,
varous
represented
n
of
resources.
sometmes
countres
the
natural
table
and
puttng
ther
No
other
B
other
percentage
n
c
1
use
of
below.
Percentage
of
the
Population
Country
world’s
natural
(millions)
resources
Thaland
69
2.1
128
7.3
82
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261
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208
2.9
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128
2.2
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96
1.4
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1403
20.2
65
3.5
66
4.1
Japan
Germany
France
Unted
a
Usng
Copy
technology,
the
scatter
Are
c
Recalculate
the
2
Kngdom
b
there
ths
data
two
Despte
natural
usng
any
our
pont
of
have
area,
others
to
y
to
the
and
nd
the
correlaton
coecent.
notebook.
data
set?
Explan
coecent
strength
resources,
example
ts
decreasng
low
mangrove
relates
the
plot
(%)
how
wthout
and
you
the
drecton
know.
outler
of
the
ncluded.
How
relatonshp
has
between
Explan.
natural
for
scatter
the
correlaton
ncrease
been
the
your
n
aected
resources,
drones
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outlers
the
use
draw
plot
varables?
trees
Snce
used
over
trees
amount
plantng
ablty
due
the
grow
of
by
sol
n
salt
to
to
the
also
trees.
replant
human
and
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trees
seed
scentsts
water
make
n
actvty.
depost
water,
n
humans
an
country
areas
Whle
pods
have
eort
n
of
the
s
mangrove
drones
desred
studed
replensh
Myanmar
where
some
to
map
the
locatons.
growth
rate
as
t
(salnty).
Salinity of water
Growth rate of mangrove
(grams of salt per kg of water)
(grams/day)
0
0.004
5
0.007
10
0.0036
15
0.0032
20
0.003
23
0.0024
Contnued
on
next
page
23 3
a
Usng
Copy
scatter
b
Are
there
c
Recalculate
ths
the
3
technology,
the
data
two
Descrbe
make
the
pont
a
Activity
7
Competton
breedng
–
s
a
rghts.
n
aected
the
plot
and
nd
the
correlaton
coecent.
notebook.
data
set?
Explan
coecent
the
outler
strength
can
regression
on
Making
of
lfe
However,
have
and
process
based
fact
scatter
how
wthout
and
you
the
know.
outler
drecton
of
the
ncluded.
How
relatonshp
has
between
Explan.
an
useful
predictions
the
your
correlaton
eect
linear
is
n
outlers
varables?
the
Performing
coecient
any
draw
plot
in
the
on
the
results
calculating
statistics
data
that
as
the
it
you
of
lnear
regresson
analyss.
correlation
allows
you
to
have.
predictions
for
every
humans
speces
are
on
unque
the
n
planet,
whether
organzng
t
be
for
food
competton
on
from
the
a
or
habtat
worldwde
or
scale.
ATL2
The
nd
100
Olympcs,
out
m
who
race
for
s
example,
the
snce
world’s
1980
are
brng
together
fastest
gven
human
the
best
beng.
Usan
Justn
Maurce
Carl
for
the
fastest
Time
in
100
(seconds)
9.63
Gatln
2004
1.85
9.85
2000
1.76
9.87
1996
1.85
9.84
1992
1.88
9.96
1988
1.88
9.92
1980
1.83
10.25
Baley
Chrste
Wells
m
(cm)
1.95
Greene
world
male
n
order
sprnters
5
Bivariate data
n
the
on
next
race
Contnued
234
to
Year
2012
Lews
Allan
tmes
Bolt
Donovan
Lnford
The
around
below.
Height
Runner
athletes
page
S TAT I S T I C S
1
a
A N D
P R O B A B I L I T Y
Usng a GDC or onlne tool, calculate the correlaton coecent and nd the equaton of the
lne of best t for the relatonshp between a runner’s heght and hs tme n the 100 m race.
b
Descrbe
the
c
Use
equaton
your
1.87 cm.
2
Show
your
predct
the
moderate
wnnng
or
tme
weak
of
and
Hasely
as
postve
Crawford,
or
negatve.
whose
heght
s
workng.
a
Usng
a
GDC
lne
of
or
best
onlne
t
for
b
Descrbe
the
c
Use
equaton
d
Are
Based
the
your
your
you
on
Find
the
or
less
analyss,
the
as
strong,
predct
the
condent
whch
heght
the
correlaton
between
moderate
wnnng
the
or
tme
coecent
year
weak
of
and
and
Hasely
and
the
as
nd
the
wnnng
postve
Crawford,
or
who
equaton
tme
n
the
of
100 m.
negatve.
won
n
1976.
of
s
the
a
n
your
better
runner
latest
predcton
predctor
or
the
year
of
n
the
of
Crawford’s
wnnng
whch
the
tme
race
wnnng
n
was
the
run?
tme?
men’s
Explan.
100 m
at
Explan.
4
correlation
relationship
to
calculate
relatonshp
workng.
your
Olympcs:
tool,
the
relatonshp
more
Practice
between
coecient
the
x
y
4
for
variables
a
ATL2
strong,
How does the correlaton coecent aect the condence you have n your predcton? Explan.
Show
1
to
as
d
the
3
relatonshp
each
as
of
the
positive
b
or
following
negative
sets
and
of
as
strong,
x
y
19
32
77
7
12
25
88
12
24
41
72
15
22
18
100
11
25
27
89
8
11
21
95
x
y
c
data.
Describe
moderate
the
or
weak.
d
x
y
2.3
2.5
12
39
3.5
7.8
18
33
11.7
12.4
13
38
9.8
11.3
22
29
10.2
12.1
32
19
12.9
13.7
19
32
15.4
13.9
25
26
6.6
5.6
Contnued
on
next
page
23 5
2
a
Interpret
b
Find
c
Verify
d
Can
e
Find
the
For
it.
3
An
or
line
out
marry.
part
close
to
marriage
happen
of
the
of
human
are
event
is
line
best
in
t
data
any
of
for
for
set
of
best
data
d
the
t
set
data
is
the
in
set
on
other
for
d
question
1.
d.
the
data
other
line
you
sets
data
in
found
in
question
sets
in
step
1?
b
Explain.
question
1
using
a
in
step
e,
verify
whether
or
not
any
of
the
points
are
on
results.
in
with
others.
couple
and
in
existence
friendly,
the
celebrate
period
of
in
calculated
that
rate
line
points
relationships
important
10-year
the
Ceremonies
worldwide
The
you
relationships
so
the
coecient
tool.
your
important
this
of
thing
online
each
become
of
equation
Explain
many
to
each
same
the
seeking
correlation
equation
that
the
GDC
f
the
the
New
may
decide
are
used
commemorate
life
of
a
Zealand
represented
While
others
parties
and
is
in
the
human.
over
a
table
below.
Marriage
rate
Year
(number
a
per
100
inhabitants)
2006
14.17
2007
13.98
2008
14.11
2009
13.7
2010
13.04
2011
12.46
2012
12.51
2013
11.44
2014
11.57
2015
11.13
2016
10.95
Which
is
the
dependent
variable
and
which
is
the
independent
variable?
Justify
your
answer.
b
Use
technology
nearest
to
calculate
the
correlation
coecient.
Round
your
answer
5
the
hundredth.
Contnued
23 6
to
Bivariate data
on
next
page
S TAT I S T I C S
c
If
you
swap
correlation
d
Verify
e
Using
t
for
the
dependent
coecient
your
answer
technology,
the
original
in
and
will
be?
step
c
graph
data
calculating
scatter
for
variables,
what
do
you
P R O B A B I L I T Y
think
the
Explain.
by
the
and
independent
A N D
the
plot
r
with
the
variables
nd
the
equation
and
swapped
data.
Describe
reversed.
of
any
the
line
of
similarities
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23 8
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next
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23 9
3
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university
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S TAT I S T I C S
assessment
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money
buy
P R O B A B I L I T Y
happiness?
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buy these possessons – so does that mean that money can buy happness?
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241
Unit
summary
Bivariate
data
Bivariate
scatter
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data
diagram
scatter
plot
is
the
can
or
name
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represented
scatter
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related
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24 3
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Unit
Key
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Unit
review
Level 1–2
1
For
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question
levels:
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describe
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represented
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24 4
might
plot.
y
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v
Level 7–8
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y
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Copy
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complete
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table.
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variable
variable
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2
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2
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of
tmes
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take
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indicate
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4
values
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only
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following
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scatter
plot.
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x
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countries
females
are
with
given
the
in
highest
the
table
life
expectancy
for
males
and
below.
Female life expectancy
Male life expectancy
(years)
(years)
Japan
86.8
80.5
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85.3
81.3
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86.1
80.0
Australa
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shapes
can
to
be
make
in
a
turned
the
shapes
that
has
no
tessellation
are
(rotated)
ipped
pattern.
or
These
are
also
all
tessellations.
and
how
discuss
each
Tessellations
is
the
examples?
Is
pattern
of
the
2
examples
shown
here
ts
the
denition
of
tessellation.
What
●
a
no
as
(reected)
●
is
there
a
that
single
has
shape
been
being
repeated
tessellated
in
each
over
of
and
the
over.
above
Explain.
more
example?
involve
shape
than
one
possibility
for
the
tessellating
shape
in
each
Explain.
25 9
Activity
1
Look
to
at
tile
your
a
1
the
–
Which
shapes
oor
with
shapes
below.
no
(Your
gaps
or
tessellate?
teacher
overlaps?
may
You
give
may
b
c
e
f
g
i
n
cut-outs.)
or
ip
the
Which
shape
shapes
if
could
necessary.
you
use
Justify
choices.
a
l
you
turn
d
k
b
e
Search
it
to
for
“Tessellation
investigate
drag
your
Experiment
The
Alhambra
tessellations.
and
palace,
The
northern
tiles
Africa)
the
choice
with
of
kinds
the
tools
Granada,
in
the
by
of
shapes
in
and
Creator”
shapes
onto
for
Christian
the
that
the
contains
were
laid
artisans
With
a
a
partner,
What
discuss
original
the
following
shape
has
been
create
the
pattern
questions.
Pairs
tessellated
shown
b
How
so
the
photograph?
was
the
original
that
it
ipped,
3
to
in
Research
would
shape
tessellate?
altered
(e.g.
turned)
the
Alhambra
and
two
other
ATL2
examples
and
paste
these
and
260
6
Islamic
examples
buildings
add
each
of
a
into
architecture.
of
a
the
Word
commentary,
piece
represents
art
a
Copy
found
in
document
explaining
why
tessellation.
Geometric transformations
and
and
by
of
and
the
both
were
illuminations.nctm.org.
those
position
rotating
some
out
who
website:
tessellate
screen
14thcentury.
2
NCTM
duplicating,
Spain,
Alhambra
on
inspired
to
coloring
most
the
that
them
by
the
famous
Moors
the
do
not.
make
You
can
patterns.
shapes.
examples
(Muslims
Moors’
Use
of
from
style
in
Spain
the
G E O M E T R Y
Reect
●
Why
●
Where
do
and
you
else
discuss
think
have
tessellations?
Tessellations
have
to
be
changing
are
the
was
used
geometric
specic
so
much
shapes
in
this
being
style
used
to
of
art?
create
examples.
transformations
so
or
shape.
its
of
that
form
include
changing
Congruence
seen
patterns
shape
transformations
without
you
modied
a
enlarging
geometry
Give
Congruence
3
shapes.
they
are
can
called
moving
When
size,
a
it
Sometimes
tessellate.
shape
is
are
the
to
moved
undergoes
transformations
shapes
The
ways
transformations.
the
shape
the
a
a
a
of
this
location
new
congruence
focus
Examples
new
to
of
of
or
location
transformation.
rst
section.
Translations
A
geometric
down.
The
original,
been
translated
so
the
shifted
in
Activity
1
translation
2
shapes
shapes
one
–
moves
or
are
a
shape
remain
congruent
more
left
or
exactly
to
each
right
the
and/or
same
other.
size
They
up
as
have
or
the
just
directions.
Maze
Describe the path you need to take to get
from the left side of the maze to the right side
of the maze. (e.g. “Move 2 cm up, then move
3 cm right,…”) Use a ruler to measure the
length of each movement (translation). You
can only move up, down, left and right.
2
Now design a maze of your own, no bigger
than 20 cm by 20 cm and list the translations
needed to get through the maze on a
separate piece of paper. Remember, you can
only move up, down, left and right.
3
Your
the
class.
your
4
teacher
your
You
peers,
When
will
you
path
collect
will
listing
have
with
try
all
to
the
nd
the
nd
answer
the
they
and
path
translations
nished,
the
mazes
redistribute
through
the
(movements)
creator
wrote
of
the
as
them
maze
you
maze
among
of
one
of
go.
and
compare
down.
Pairs
2 61
When
shape
using
is
describing
is
called
the
uppercase
referred
uppercase
to
as
a
shape
preimage
letters
the
letters
pronounced
how
with
and
(e.g.
image
a
“A-prime,
has
been
the
vertices
ABCD).
and
the
“prime”
B-prime,
transformed,
The
to
often
translated
vertices
next
are
are
each
C-prime,
the
labeled
shape
labelled
(e.g.
original
using
A′ B ′C ′D ′,
D-prime”).
B
B′
Preimage
Image
A
C
A′
C′
ΔABC
Example
Q
Specify
ΔA′B ′C ′
1
the
preimage
→
translation
is
the
blue
of
this
gure
in
shape
the
that
upper
has
occurred.
left-hand
The
corner.
Select any point on the preimage – in this case, the rectangle in the
A
7
units
right
top left– and count how many units you have to go right or left.
5
units
down
Count
Translation:
Q
Translate
3
units
7
the
right
and
shape
on
5
down
the
grid
how
Record
below
8
the
many
units
translation
units
to
the
you
have
using
right
to
go
proper
up
or
down.
notation.
and
up.
Continued
262
6
Geometric transformations
on
next
page
G E O M E T R Y
A
Select
any
number
of
direction
to
the
It
is
also
possible
structure
to
be
of
the
plotted
translation.
in
to
translate
Cartesian
space
Activity
3
–
1
table.
plane
and
Specialized
shapes
then
using
allows
its
notation
Translations
each
image
is
used
on
the
for
the
point
on
after
describe
the
given.
all
on
the
preimage
required
In
the
exact
Cartesian
plotted
to
units
this
for
case,
the
go
3
and
count
translation
units
up
the
in
and
the
8
units
right.
Repeat
Draw
point
the
image
plane.
the
a
other
points.
of
the
original
shape.
The
preimage
given
translation.
Cartesian
plane
ATL2
Copy
this
indicated
and
Translate
enter
the
each
new
of
the
following
coordinates
in
the
preimage
Image
Preimage
Image
3
units
left
7
units
(0,
(0,
unit
down
your
own)
7)
−4)
0)
10)
(−12,
What
(Create
−11)
(−5,
2
Image
right,
9)
(−2,
(8,
units
up
1
(3,
as
Image
2
point
points
table.
0)
happened
under
the
rst
to
the
value
translation?
of
the
x-coordinate
of
each
point
Explain.
Continued
on
next
page
2 63
3
What
happened
under
4
a
b
the
What
point
How
would
The
correct
plane
( x ,
is
Describe
to
under
to
you
+ 6,
what
the
the
of
for
this
the
y
coordinate
of
each
point
Explain.
value
third
what
notation
in
value
represent
show
shown
y) → ( x
the
translation?
happened
each
notation
5
to
second
of
x
the
and
translation?
this
the
translation?
happened
to
representing
y
coordinate
of
Explain.
each
Create
your
own
point.
translations
on
the
Cartesian
example:
y − 4)
this
translation
does
to
all
of
the
points
in
a
preimage.
6
Describe
the
three
translations
in
the
table
using
the
correct
notation.
7
Create
a
translation
notation
to
translate
location
each
of
Practice
1
For
each
image
is
for
describe
of
each
the
the
your
nal
preimage
image
column
translation.
points
of
Use
in
your
your
the
table.
Use
translation
table;
record
correct
to
the
point.
1
shape,
specify
the
translation
that
has
occurred.
The
preimage
is
blue
the
green.
a
b
Continued
264
and
6
Geometric transformations
on
next
page
G E O M E T R Y
2
a
On
squared
paper,
copy
each
shape
and
translate
it
by
the
specied
units.
i
5
units
down,
2
units
left
ii
6
units
up,
3
units
right
iii
1
b
Could
the
instructions
completed
c
3
Write
the
The
logo
The
rings
Union
of
companies’
ring
on
fewer
three
the
company
Audi,
the
four
DKW,
to
right.
produce
Use
7
units
iii
left,
be
4
units
down,
simplied
so
9
that
units
the
right
translation
can
be
Explain.
using
Auto
correct
Union
original
Wanderer
contributions,
how
the
steps?
up,
translation
translations
represent
AG:
Describe
in
for
unit
they
each
correct
are
of
car
made
Horch.
for
three
each
up
of
that,
Out
represented
other
notation
is
companies
and
all
the
AG
notation.
in
of
if
rings,
together,
respect
the
rings
four
for
as
shown
make
each
up
of
here.
Auto
the
original
symbol.
they
are
all
translations
of
the
translation.
Continued
on
next
page
2 6 5
4
Plot
the
Then
5
shape
plot
Given
the
the
with
vertices
image
of
translation
this
(x,
(1,
8),
shape
y)
(x
(−3,
−5),
after
+
7,
(−4,
each
y
−
of
3),
7)
and
these
nd
the
(−6,
−2).
translations.
image
of
each
of
these
preimagepoints.
a
6
(5,
9)
Given
b
the
(−2,0)
c
translation
,
nd
the
(0,−8)
preimage
of
the
following
imagepoints.
a
7
(−6,
15)
Pac-Man
1980s.
It
creator
more
b
was
was
to
innovative
become
a
a
even
at
still
be
Pac-Man
down.
has
to
make
8
This
is
to
Count
the
a
time.
It
video
attempt
games,
which
later
animated
song.
many
only
the
time
game
left,
translations
eat
all
each
Gherkin
four
dot
to
was
one
building
0)
the
its
to
own
series
legacy
right,
can
games.
up
and
Pac-Man
large
as
(0,
very
on
video
that
c
attract
television
today’s
move
in
with
Pac-Man’s
of
−11)
game
went
phenomenon,
an
in
can
List
ghosts).
the
pop
seen
video
deliberate
social
a
rst
video
merchandise,
and
popular
the
made
girls
a
(13,
dots
using
the
shortest
path
possible
(avoiding
the
unit.
in
London,
UK.
In
order
to
ATL2
be
energy
much
(such
ecient,
energy
as
creating
its
a
If
the
most
curved
natural
architecture
a
as
can
vertical
b
Is
the
the
building
large
shape
both
height
20 m,
was
and
the
system)
each
Many
gaps
functional
of
designed
buildings.
ventilation
be
approximately
on
the
in
of
its
the
the
half
as
features
oors
concept
that
creative.
diamond-shaped
describe
use
between
express
and
to
translation
section
of
this
is
shape
building.
pattern
on
the
building
an
example
of
a
tessellation?
Explain.
Continued
266
6
Geometric transformations
on
next
page
G E O M E T R Y
9
a
This
wall
bricks,
What
have
b
Is
has
been
230 mm
wide
translations
taken
this
an
built
using
and
of
the
standard
76 mm
brick
high.
shape
place?
example
of
a
tessellation?
Explain.
10
This
is
rugs
just
a
Peruvian-style
and
other
artistic
used
and
to
gods
many
and
identity
Identify
the
describe
world
can
from
Peru,
more
were
histories
people
found
the
than
often
around
be
original
the
In
personal
between
the
textiles
were
They
communicate
Peruvian
a
products.
relationships
their
in
textiles
rug.
and
them.
expressed
region.
shape(s)
transformations
and
that
have
occurred.
Indicate
b
Is
this
an
example
of
a
tessellation?
Explain.
right,
up
translations
and
left,
down.
Rotations
Geometric
called
to
the
dene
wish
to
occurs.
you
rotations
center
the
wish.
negative
two
the
can
shapes
When
and
an
go
in
is
a
the
go
an
you
the
object
rotate
number
direction
object
rotations
image
are
turning
rotation,
rotate
rotations
the
of
shape
Positive
translation,
the
rotation.
center
rotate
You
of
involve
by
in
a
the
a
of
in
a
rotation
degrees
direction
as
need
you
the
direction.
size
point,
you
degrees
many
clockwise
same
shape,
which
however
counterclockwise
exactly
around
the
and
As
with
a
preimage,
so
congruent.
2 67
Rotations
are
most
clockwise
often
described
(CW)
or
as:
counterclockwise
(CCW).
1
11
2
10
3
9
4
8
7
7
5
5
6
6
Drawing
on
a
are
image
coordinate
the
This
the
same
will
after
grid,
that
from
the
1
–
sure
the
can
each
center
image
is
be
tricky.
vertex
point
and
as
congruent
Rotations
on
side
on
to
After
the
the
the
rotating
of
the
image
preimage.
preimage.
Car tesian
plane
i
t
er
i
r
Investigation
rotation
make
distance
ensure
a
rotated
center
1
can
around
of
x-
the
and
be
any
rotation.
Consider
the
also
done
point,
You
y-axes?
at
will
Cartesian
on
a
Cartesian
this
focus
plane.
Justify
your
stage
on
plane.
you
will
rotations
What
is
the
Although
always
of
90,
use
180
measure
of
a
shape
the
and
the
can
origin
270
as
n
c
Rotations
be
the
B
degrees.
angle
formed
by
answer.
y
6
5
4
3
2
1
0
–6
–5
–4
–3
x
–1
–1
–2
–3
–4
–5
–6
2
268
Moving
in
a
the
of
the
size
a
East
b
South
c
West
6
(the
clockwise
angle
positive
(the
(the
direction
turned
x
true
to
north
reach
(the
each
positive
of
the
y-axis),
nd
following.
axis)
negative
negative
from
through
y-axis)
x-axis)
Geometric transformations
Continued
on
next
page
G E O M E T R Y
l
i
n
k
b
e
Go
to
use
ct.excelwa.org,
this
shape
line
app
or
or
a
3
Copy
90°,
and
around
point.
with
180°
origin
To
plot
input
270°
this
in
a
the
of
or
lines.
the
APPS
the
single
point,
You
a
menu,
geometric
more
either
table.
through
all
two
straight
or
complete
the
select
perform
shape,
together
by
to
enter
ordered
can
then
clockwise
Rotate
each
indicated
of
and
choose
an
ordered
pairs
which
rotate
(CW)
the
number
Polygon
transformations
the
or
a
point
and
then
or
Transformer.
step
3
click
using
shape
and
around
(CCW)
You
any
“Draw”.
plotted
counterclockwise
following
of
pair
are
in
To
can
given
draw
a
connected
the
origin
direction.
points
If you are doing
degrees.
the rotations on
Point
90°
CW
90° CCW
180° CW
180° CCW
270°
CW
270° CCW
paper, a good
procedure to help
(3,
9)
you to visualize
(−2,
7)
the rotation is to
graph the original
(8,
−11)
point, then
(−5,
−4)
physically “rotate”
(0,
your graph paper
0)
through the given
(0,
10)
rotation around
(−12,
the origin. The
0)
new location
4
Copy
this
following
results
table.
Describe
clockwise
using
the
in
words
rotations
correct
the
about
eect
the
on
origin.
a
point
Then,
of
each
of
generalize
the
the
of your point
represents the
coordinates of the
notation.
image.
Degree
of
90°
5
Verify
6
Justify
7
While
rotation
Result
of
rotation
Notation
CW
( x,
y)
→
(
,
)
180°
CW
( x,
y)
→
(
,
)
270°
CW
( x,
y)
→
(
,
)
your
why
results
each
for
rule
technology
two
points
of
your
own
choosing.
works.
makes
drawing
easier,
are
there
any
advantages
ATL1
to
drawing
eective
rotations
by
hand?
How
could
this
skill
make
you
a
more
learner?
2 6 9
Reect
●
In
a
the
●
rotation,
same
Name
why
●
and
When
someone
this
to
depicting
artists
God
life.
a
the
is
wall
Spain,
a
move?
Do
they
all
move
that
produce
the
same
image.
Explain
little
changes
gone
sense.
a
their
opinion
complete
What
would
be
360
a
or
actions,
degrees.”
more
it
is
Explain
logical
way
of
change?
living
as
Muslim
not
picture
the
its
This
or
of
to
helps
to
copy
images
shows
of
that
attempt
life
use
opposed
idea
Alcazar
palace
for
things.
creating
in
rotations
famous
should
This
preimage
Explain.
“They’ve
patterns
the
by
the
2
art
geometric
express
on
completely
say,
makes
Practice
Islamic
points
dierent
describing
1
all
4
happens.
common
why
do
distance?
two
this
discuss
tiles
of
on
Seville,
constructed
by
the
Moors.
a
Does
this
represent
tessellation?
has
been
If
so,
a
what
shape
tessellated?
b
Identify
and
describe
two
examples
of
translations
c
Identify
and
describe
two
examples
of
rotations
of
rotation
and
angles
of
in
in
this
this
pattern.
pattern.
Indicate
the
rotation.
Continued
270
6
centers
Geometric transformations
on
next
page
G E O M E T R Y
2
Use
a
squared
Rotation
paper
of
180°
to
graph
about
the
the
image
origin
of
each
b
gure
Rotation
y
using
of
the
180°
transformation
about
the
given.
origin
y
A
D
A
x
B
x
D
C
B
C
c
Rotation
about
of
the
90°
counterclockwise
d
Rotation
of
90°
clockwise
about
the
origin
origin
y
y
A
B
C
x
x
D
3
The
Olympic
rings
express
y
the
ATL2
ideals
Each
behind
of
the
continent,
each.
The
represent
unite
over
a
ve
with
how
from
represents
dierent
are
the
and
Games.
color
interlocked
Olympic
people
a
for
to
Games
from
x
all
world.
Describe
circles
Olympic
rings
a
rings
athletes
the
the
how
could
the
each
have
blue
of
the
been
circle.
other
translated
Use
correct
notation.
Continued
on
next
page
2 71
b
Describe
circles.
c
4
the
describe
5
(5,
(−6,
The
the
circle
to
could
indicate
in
be
the
the
in
same
traditional
windows
in
the
time,
while
Describe
a
because
summer,
the
royal
maintaining
any
translated
to
produce
another
pair
produced
angle
nd
of
the
from
the
rotation
of
another
circle
about
rotation.
image
of
the
following
preimage
points
and
(−2,
0)
c
nd
the
image
of
the
(0,
−8)
following
preimage
points
and
(13,
marble
they
the
their
c
window
from
incorporated
open
ladies
tessellations
−11)
windows
could
observe
Jaipur
two
(0,
in
0)
India.
important
allowed
daily
values.
breezes
events
Architects
to
cool
through
With
the
the
included
soaring
building.
spaces
in
the
privacy.
that
you
see.
What
is
the
largest
shape
that
has
been
tessellated?
b
Describe
c
Explain
any
how
rotations
a
shape
that
has
are
been
represented
in
this
window.
transformed
to
create
the
tessellation.
Continued
272
6
of
words.
b
shows
been
notation.
,
rotation
have
words.
15)
pattern
a
mathematical
rotation
ornate
the
could
,
rotation
temperatures
At
sure
circles
rotation
the
photo
these
Be
one
of
b
describe
6
pair
9)
Given
a
one
correct
how
origin.
Given
a
Use
Describe
the
how
Geometric transformations
on
next
page
G E O M E T R Y
7
This
and
building
design.
design
and
disciplines
b
Identify
in
the
directions
graphics.
through
“logos”
Logo
Well,
l
i
the
building,
–
a
Logo
also
was
for
an
designed
is
and
to
was
created
encourage
you
can
explain
icon
see
how
to
or
computer
which
in
the
the
–
a
to
university
express
for
digital
current
collaboration
tile
ideas
among
the
media
in
dierent
design.
shapes
“reason”.
The
It
new
again
and
the
the
predict
name
was
programming
language
followed
understand,
commands.
other
now
that
educational
turtle
students
“word”
from
tiling,
UK
have
been
tessellated.
Academy
computer
means
old
shape(s)
onscreen
was
of
dierent
was
was
its
London,
Ravensbourne.
Turtle
aim
which
at
University
including
transformations
small
use
It
original
1980s,
The
what
n
any
4
to
the
was
The
Ravensbourne
studied
Describe
Back
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creation.
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Activity
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Logo
chosen
to
languages
Turtle
is
In
top
that
students
instructions
and
explain
was
derived
express
which
the
were
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the
used
turtle’s
from
idea
to
produce
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that
number
give
line
movements
Greek
word
logic-based
based.
back!
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b
e
Head
how
to
over
to
is
turtleacademy.com.
control
make
Here
to
the
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turtle
basic
turtle.
turn,
example
Go
go
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through
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as
move
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forward,
square
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forward
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right
90
forward
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right
90
forward
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right
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forward
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right
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is
select
the
etc.
the
“Lessons”
tutorials
to
50units
make
long
a
as
you
tab
to
need
learn
in
order
shape.
on
each
side.
Continued
on
next
page
273
1
Once
you
have
then
make
Here
is
an
a
gone
shape
through
that
example
to
you
get
the
can
your
lessons
and
translate.
creative
feel
Feel
juices
comfortable
free
to
be
with
the
commands,
you
can
creative.
owing.
60
90°
90°
30
90°
30
60
90°
30
90°
30
Star t
2
When
3
Translate
4
Can
you
rotate
your
shape
180
5
Can
you
rotate
your
shape
90
6
Now
this
your
shape
your
that
completed,
shape
you’ve
software
is
had
and
250
an
these
units
translate
shape
100
units
to
the
right.
down.
degrees
degrees
opportunity
skills
your
can
be
clockwise?
Explain.
counterclockwise?
to
play
applied
around
in
Explain.
Turtle
World,
can
today?
Reections
Reections
see
your
Take
and
2 74
a
at
6
are
own
look
the
all
around
you;
when
you
look
in
the
mirror
you
reection.
at
this
photo
reection
of
a
of
a
human
natural
Geometric transformations
landscape
landscape
on
reected
page
258.
in
water,
you
think
of
where
G E O M E T R Y
Reect
●
In
the
and
photos,
discuss
which
is
the
5
image
and
which
is
the
preimage?
Explain.
●
How
does
image?
●
A
rotation
rotation.
A
reection
reection,
the
the
is
is
Example
A
Draw
the
is
the
preimage
a
a
point
known
the
point
necessary
when
corresponding
Q
of
requires
What
also
image
size
compare
with
the
size
of
the
Explain.
same
point
as
a
of
rotation,
to
or
shape
mirror
distance
in
dene
the
a
is
line.
from
original
an
angle
and
a
direction
of
reection?
ipped
In
the
a
across
reection,
line
of
a
line
each
reection
of
point
as
in
the
preimage.
2
line
of
reection.
Look
at
halfway
the
two
gures.
between
corresponding
ever y
point
Continued
in
on
Determine
point
the
next
in
a
the
image.
line
that
preimage
Draw
in
is
exactly
and
the
its
line.
page
2 75
Q
Draw
the
image
after
being
reected
in
the
given
mirror
line.
A
First,
measure
preimage
Then,
side
to
mark
of
image
the
will
the
the
the
line
be
your
of
Reect
●
Why
●
Where
is
and
a
discuss
reection
a
ver tices
of
from
if
of
the
reection,
you
so
reection
the
is
in
transformation?
turn
line
vertical.
Explain.
ATL2
is
the
preimage?
●
Where
do
school?
276
6
line
of
reection
in
relation
to
both
the
image
and
Explain.
you
What
see
do
reections
you
think
is
in
the
the
Geometric transformations
art
and
purpose
architecture
of
these
of
your
reections?
ver tices
the
image
the
size
6
congruence
the
of
the
reection.
reections
easier
paper
of
identical
Sometimes
seem
distance
line
on
same
and
the
opposite
distance
shape,
away. The
simply
ipped.
G E O M E T R Y
Practice
1
Copy
3
these
reections
on
squared
paper
a
2
image
logo.
It
the
shows
has
simply
of
no
represents
way:
underneath.
a
Explain
using
the
b
a
of
A
coat
The
of
the
of
of
all
the
a
coat
line
of
reection
on
each.
eye-
the
can
can
symbol
are
arms
where
the
be
created
the
of
center
rotation.
that
represents
often
is
symbols
in
it,
the
middle,
to
to
a
placed
shown
throughout
ability
created
is.
angle
refer
A
logo.
be
Indicate
Indicate
in
as
letters
stylized
trace
arms
people
their
an
a
logo
of
cross
cross
initial
logo
the
of
main
the
work
is
Coats
of
or
the
and
arms
Serbian
people
with
rotation.
Serbian
around
U
the
how
Serbian
Two
on
the
shields
the
The
motto
that
this
photo.
eagle
nation
and
identity
stylized
reminds
and
or
ags.
national
four
heritage
family
or
in
double-headed
history.
their
person,
shield
represent
popular
save
on
a
Cs
Serbian
if
they
together.
Describe
Indicate
b
the
Armour
signicance,
in
reection
rotation
country.
a
how
Explain
a
a
name
Copy
of
Under
the
reection.
line
using
the
cultural
company
catching
3
draw
b
This
it
and
the
the
Describe
reection
location
two
that
of
examples
the
of
was
line
used
of
to
create
the
coat
of
arms.
reection.
translations
that
were
used
to
create
the
coat
of
arms.
Continued
on
next
page
277
4
Copy
the
shapes
and
mirror
lines
onto
squared
paper.
Draw
the
reected
image
of
eachshape.
a
b
d
e
c
f
I
B
G
A
C
H
D
K
J
i
Investigation 2 – Reections on the Cartesian plane
t
e
r
i
c
a
r
The
easiest
way
to
perform
a
reection
is
simply
to
draw
in
the
reection
line.
B, C
Then
draw
opposite
l
i
n
each
side,
point
so
that
exactly
the
the
mirror
same
image
distance
is
away
identical
to
from
the
the
line,
original
but
on
the
preimage.
k
b
e
You
can
perform
ordered
“Draw”
point
use
pair
to
in
the
most
of
to
plot
the
plot
x-axis,
won’t
two
reections
to
the
your
You
Go
Polygon
be
able
on
Transformer
geometric
a
point,
point
the
to
y
or
or
several
draw
axis
reect
or
in
app
that
you
transformations
any
the
ordered
your
shape.
other
lines
y
in
pairs
You
vertical
=
x
or
used
this
to
can
or
in
Investigation
investigation.
plot
then
a
shape.
reect
horizontal
y = −x
using
1
Input
Then
the
to
one
click
shape
or
line.
the
app,
so
do
these
paper.
ct.excelwa.org,
select
the
APPS
menu,
and
choose
Polygon
Transformer.
Continued
278
6
Geometric transformations
on
next
page
G E O M E T R Y
1
Reect
each
of
the
given
points
in
the
following
reection
lines.
If
Point
y
x-axis
-axis
x
=
y
1
=
3
y
=
x
y
=
you
the
−x
on
(3,
are
reections
paper,
a
good
9)
method
(−2,
7)
you
the
(8,
doing
to
to
help
visualize
reection
−11)
is
(−5,
−4)
to
graph
original
the
point,
thenphysically
(0,
0)
fold
(0,
your
paper
(−12,
Copy
and
these
alongthe
reection
0)
The
2
graph
10)
complete
reections
on
this
a
table.
point.
Describe
Then
in
words
generalize
the
the
eect
result
of
using
each
of
correct
of
new
your
line.
location
point
represents
the
coordinates
notation.
of
the
image.
Reection
4
Justify
of
reection
Notation
( x,
y)
→
(
,
)
y
-axis
( x,
y)
→
(
,
)
x
(x,
y)
→
(
,
)
−x
(x,
y)
→
(
,
)
y
Verify
Result
x-axis
y
3
line
=
=
each
rule
your
Reect
for
two
more
points
of
your
choice.
rule.
and
discuss
7
Make a prediction about the following combined reections, and then
try an example to test whether your predictions are correct:
●
What
single
consecutive
●
Would
the
parallel
●
What
lines?
single
Would
reections
same
consecutive
●
transformation
the
result
produces
over
be
two
the
parallel
achieved
over
same
result
as
lines?
any
even
number
of
Explain.
transformation
reections
same
result
produces
over
be
three
the
same
parallel
achieved
over
result
as
lines?
any
odd
number
of
parallel
lines?
●
What
single
consecutive
transformation
reections
produces
over
two
the
same
intersecting
result
as
lines?
279
Practice
1
Use
a
4
squared
Reection
paper
in
y
=
to
graph
the
image
of
each
−2
shape
b
using
the
Reection
transformation
in
the
given.
x-axis
y
y
A
B
D
C
x
x
A
D
B
C
c
Reection
in
y
=
−x
d
Reection
in
y
=
−1
y
y
A
A
D
x
x
D
B
B
C
C
2
Write
a
rule
to
describe
each
transformation.
y
a
y
b
C ′
A
B
D
D′
A′
D′
′
A
B′
A′
x
x
D
C
C ′
C
Continued
280
6
Geometric transformations
on
next
page
G E O M E T R Y
y
y
c
d
A′
B′
A
A′
D′
B
B′
C ′
x
x
C
D
C
C ′
B
A
3
Give
has
a
4
(5,
has
5
It
a
reected
(−6,
For
coordinates
reected
that
even
of
create
it.
angle
is
While
a
the
the
logos
design
are
that
from
many
the
mountains
the
have
the
that
logo
are
occurred
the
following
preimage
points
c
after
each
of
the
(0,
preimage
points
−8)
following
on
tell
transformations
describe
draw
in
the
any
(0,
to
0)
create
transformations
reection
transformations.
to
look
Native
interpretations
will
use
c
lines
their
that
and
logos.
have
indicate
been
any
used
center
appropriate.
Navajo
you
of
the
the
homeland,
one
is
it
about
transformations
that
rug.
pattern,
small
with
of
What
at?
American
that
representing
the
most
diamond
each
four
sacred
it.
shape(s),
and
of
−11)
to
here,
pleasing
surrounded
original
(13,
and
where
Navajo
diamond
each
0)
point
love
shown
based
a
grandmothers
of
Identify
are
after
y-axis.
each
rotation
symbolizes
corner
logos
(−2,
image
superheroes
Copy
logos
there
Navajo
the
of
above
produces
shape
in
of
point
x-axis.
b
each
The
that
the
image
15)
seems
This
the
b
the
and
6
in
of
9)
been
to
b
coordinates
been
Give
a
the
describe
explain
how
the
the
transformations
shapes
are
tessellated.
Continued
on
next
page
2 81
This
carved
Uzbekistan,
with
depiction
living
faith.
It
avoid
a
of
also
using
often
made
Identify
its
screen
costly
of
things,
the
are
shapes
the
clay
original
that
have
the
c
Describe
how
portions
or
single
of
of
the
Muslim
requirement
artistic
pieces
describe
and
could
the
to
were
the
explain
have
how
as
reection
in
the
x-axis
b
a
reection
in
the
line
c
a
reection
in
the
line
followed
as
that
by
a
followed
the
been
angle
by
Y
our
a
a
translations.
created
and
the
180-degree
by
with
using
direction
same
result
of
rotations.
rotation.
as:
rotation
90-degree
rotation
270-degree
personal
rotation
clockwise
clockwise.
brand
i
t
e ri
r
–
have
achieves
followed
assessment
created
could
well
transformation
a
been
pattern
rotation
a
Formative
so
by
the
brass.
pattern
center
a
religious
avoids
tessellated.
how
down
required
occurred
Describe
Write
from
patterns,
shape(s),
b
the
as
materials,
wood,
reections
window
geometric
expresses
the
Indicate
8
wooden
c
As
ATL2
you
have
person,
the
a
you
beliefs,
some
have
Copy
the
c
each
Design
at
least
grid
own
of
the
d
Use
notation
e
Below
f
Create
and
6
a
of
arms
small
your
own
to
you
poster,
of
coats
and
an
of
arms
coats
of
of
that
the
be
used
arms
individual
arms
each
can
or
are
to
identify
areused
to
a
C
express
group.
dierent
congruence
than
the
ones
transformations:
a
document
and
annotate
the
image
to
show
shows.
of
arms
that
beliefs,
of
expresses
values
congruence
features,
describe
give
such
each
include
details
using
personal
logos
of
as
or
something
aspirations.
It
transformation.
reection
lines,
important
must
Draw
this
centers
to
incorporate
of
logo
on
an
Geometric transformations
of
app
logo/coat
transformation.
an
explanation
the
like
of
of
the
signicance
transformation(s)
Canva,
arms.
with
the
of
your
involved.
examples
you
a
rotation,
etc.
must
and
it
types
translation,
image,
into
coat
important
of
coat
exemplify
identity,
three
a
rotation.
arms
or
your
description
the
of
Many
and/or
to
and
or
aspirations
logos
logo
of
indicate
correct
logo
transformation
part
two
and
logo/coat
282
of
and
already
logo/coat
a
nation.
reection,
your
some
or
unit,
values
studied
type(s)
you:
this
company
translation,
b
in
organization
ideals,
Find
seen
studied
o
n
7
G E O M E T R Y
Reect
Your
teacher
these
●
A
will
questions
logo/coat
arms
●
and
Can
you
of
of
gure
any
out
Activity
5
1
Download
2
Select
the
or
device.
3
Select
Then
and
–
of
person
it?
you?
who
the
classroom.
everyone’s
personal
to
Explain
why
You
that
Cultural
app
no
if
to
the
is
at
called
“Symmetry”
image.
Answer
work.
Which
logos/coats
of
Explain.
created
with
an
each
logo/coat
example
you
can
a
each
of
than
from
of
your
arms,
class.
just
now
learned
analyze
about
are
at
tessellations
the
to
is.
tessellations
and
to
you
create
four,
identify
the
will
a
be
mobile
each
help
and
person
you,
grid
will
in
a
number
using
that
on
basic
turn
help
a
Work
turn
draw
templates
background
device.
oered
transformations
To
(rectangles)
dierent
to
tessellations.
with
tessellation.
the
The
will
menu
template
altered.
you
“KaleidoPaint”
designs
more
create
rst
select
shape
very
transformations
the
See
template
a
around
seen
personal
the
template
groups
work
have
tessellation.
how
dierent
is
8
tessellations
congruence
heart
arms
looking
your
you
most
identify
by
Analyzing
The
display
after
seemed
simply
discuss
to
you
in
allow
own
each
“Grids”.
shape/design.
see
to
pairs,
their
the
of
how
your
identify
the
transformations.
4
Once
of
you
your
gallery
are
own
walk
developed
a
class,
explain
have
that
of
by
with
how
is
all
nished
based
the
it
the
devices
dierent
students
exploring
on
and
people
taking
expresses
the
theme
in
see
your
turns
to
app,
of
how
the
group.
hold
create
“culture”.
up
theme
Come
their
a
tessellation
Conduct
has
been
together
design
a
as
and
“culture”.
2 83
Activity
You
briey
construct
6
–
Islamic
looked
two
of
at
the
this
art
tiling
shapes
in
pattern
it
and
from
the
investigate
Alcazar
how
of
they
Seville
on
can
used
be
page
to
270.
You
create
will
this
now
tessellation.
The larger star
1
On
a
piece
centered
2
Using
3
Draw
the
the
in
left
x-axis
of
at
grid
the
same
a
paper,
origin.
center
diameter
to
the
and
right
the
as
from
of
x-
draw
(Using
in
and
y-axes
compass
step
the
the
a
top
large
1,
draw
of
the
circle.
in
will
a
the
smaller
large
Make
middle
make
circle
sure
this
circle
to
the
of
the
paper.
Draw
a
giant
circle
easier.)
inside
the
your
bottom
lines
are
large
and
circle.
another
perpendicular.
diameter
Label
from
these
the
y-axis.
4
Cut out the large circle, then fold the paper in half, and then again into quarters, along the axes.
5
Fold
the
spaced
6
From
line
paper
radii
the
to
from
point
the
further
the
an
center
where
point
into
one
where
even
of
number
your
circle
fold/radius
the
next
of
sections
or
draw
an
even
number
to
the
circumference
of
the
larger
meets
the
circumference
of
the
larger
fold/radius
intersects
the
smaller
circle.
See
of
evenly
circle.
circle,
the
draw
a
diagram
below.
7
Then
draw
8
Continue
a
line
from
around
the
this
point
circle
to
to
the
produce
point
a
where
the
next
fold/radius
meets
the
larger
circle.
star.
y
x
Keep
in
mind
that
this
is
an
example.
Your
star
may
have
more
or
fewer
points.
Continued
284
6
Geometric transformations
on
next
page
G E O M E T R Y
9
Color
in
the
star,
making
sure
you
can
still
a
Describe
any
reections
in
the
x-axis.
b
Describe
any
reections
in
the
y-axis.
c
Could
and
d
How
the
e
this
the
shape
angle
could
photo
Paste
how
can
have
star
be
start
onto
be
would
been
created
using
the
a
folds
or
rotation?
lines.
If
so,
dene
the
center
of
rotation
rotation.
the
star
that
you
this
at
your
shape
of
see
modied
of
the
it
activity?
another
piece
tessellated.
do
so
could
be
tessellated
to
create
a
mosaic
like
the
one
in
Explain.
of
paper
Explain
why
and
it
is
ll
in
around
possible
to
your
star
tessellate
to
this
create
base
a
base
shape
and
it.
The smaller star (8-pointed)
1
Create
2
How
3
Color
an
8-pointed
could
in
you
the
star.
describe
star,
making
The
visuals
these
sure
steps
you
below
to
can
still
Describe
any
reections
in
the
x-axis.
b
Describe
any
reections
in
the
y-axis.
c
Could
d
and
the
Can
this
shape
angle
star
of
be
have
been
each
somebody
a
this
show
created
see
using
the
a
who
folds
step
did
or
rotation?
of
the
not
lines.
If
so,
process.
have
the
Indicate
dene
the
visuals?
the
x-
and
center
of
y-axes.
rotation
rotation.
tessellated
to
create
a
mosaic
like
the
one
at
the
start
of
the
activity?
Explain.
Reect
●
Do
you
and
nd
it
discuss
easier
to
9
follow
written
instructions
or
visual
ATL1
instructions?
to
●
become
What
of
a
makes
art?
Why?
more
this
How
could
ecient
style
of
art
you
and
so
use
this
eective
unique
information
to
help
you
learner?
and
distinct
from
other
styles
Explain.
2 8 5
Investigation 3 – Shapes that tessellate
t
er
i
r
i
will
be
tessellate.
virtual
l
i
n
looking
You
can
at
regular
do
this
polygons
using
and
determining
manipulatives
(a
set
of
whether
regular
they
n
c
You
will
polygons)
B
or
manipulatives.
k
b
e
If
you
the
not
have
Tessellation
page
1
do
Determine
set
Creator
of
regular
on
the
polygons
NCTM
to
website
use
for
this
investigation,
illuminations.nctm.org
you
(see
can
use
Activity
1
on
260).
which
themselves
complete
a
of
the
(without
this
following
needing
another
polygons
shape
to
ll
tessellate
gaps).
by
Copy
and
table.
Does
Regular
regular
this
tessellate?
polygon
(yes
Equilateral
or
no)
triangle
Square
Regular
pentagon
Regular
Regular
hexagon
heptagon
Regular
2
Of
the
3
What
4
How
5
Looking
regular
are
the
many
interior
to
octagon
at
polygons
sizes
of
degrees
your
angles
the
the
you
interior
make
results
of
that
in
up
a
4
which
angles
full
steps
regular
tried,
of
ones
the
tessellated?
shapes
that
tessellated?
rotation?
and
5,
polygons
what
that
pattern
can
tessellate?
you
What
nd
in
allows
the
them
tessellate?
Continued
286
6
Geometric transformations
on
next
page
G E O M E T R Y
6
Can
you
Copy
predict
and
which
complete
regular
this
Degree
Number
of
polygons
will
tessellate
by
themselves?
table.
of
interior
Number
of
polygons
needed
to
sides
angle
ll
a
full
rotation
3
4
5
6
7
8
9
10
11
12
n
7
Given
the
20-sided
8
Verify
9
Justify
You
formula
gure
your
why
have
you
would
formula
your
found
found
for
two
formula
the
in
the
last
row
of
the
table,
test
to
see
if
a
tessellate.
more
polygons.
works.
regular
polygons
that
tessellate
by
themselves,
In a semi-regular
but
what
if
you
the
rotation?
combine
two
or
more
shapes
together
to
complete
tessellation, two
Tessellations
of
two
or
more
dierent
regular
or more shapes are
polygons
such
that
the
same
polygon
arrangement
exists
at
every
used to complete
vertex
are
called
When
you
semi-regular
tessellations.
a tessellation with
no gaps, hence
name
these
types
of
tessellations,
you
name
them
by
the vertices of the
the
number
of
sides,
starting
with
moving
in
the
polygon
with
the
smallest
shapes connect
number
of
sides
and
a
clockwise
direction.
and the shapes
t together to
complete a full
360-degree
rotation.
2 87
ATL2
Activity
There
1
are
Your
7
eight
task
steps
–
is
semi-regular
to
determine
●
test
●
sketch
a
nd
all
tessellations
tessellations.
eight
dierent
semi-regular
tessellations.
For
a–c:
●
There
Semi-regular
them
a
the
combinations
using
full
manipulatives
rotation
of
each
(real
and
or
virtual)
explain
why
it
tessellates.
are:
three combinations of regular polygons that will tessellate with
exactly three shapes put together (two of which may be the same
shape)
b
two
combinations
four
c
three
ve
2
shapes
Why
of
of
combinations
shapes
do
(two
we
(up
only
to
regular
which
of
four
consider
polygons
may
regular
of
be
the
that
polygons
which
regular
may
will
same
be
that
the
polygons
tessellate
with
shape)
will
tessellate
same
and
not
with
shape).
irregular
polygons?
You
the
have
Alhambra.
such
as
the
famous
works
of
for
of
space
things,
6
his
and
the
of
M.C.
the
wanted
logic
a
same
his
bird,
of
in
appear
Escher
“impossible”
Escher
as
tessellations
Tessellations
shapes,
such
applied
at
works
art.
geometric
288
looked
to
space
car,
underlying
in
a
a
his
range
and
both
look
or
principles
a
as
as
of
who
his
the
artwork.
tiles
dancer
Geometric transformations
such
(1898–1972),
express
in
art,
wide
constructions
tessellating
a
Islamic
sh.
with
art
is
in
forms,
most
tessellated
mathematics
Instead
like
those
of
being
recognizable
However,
regular
he
polygons.
G E O M E T R Y
Activity
Below
is
a
a
an
Research
polygon
●
Do
●
Why
●
Do
as
the
Escher’s
that
can
you
created
closely
count
triangle
was
style
think
think
What
that
would
if
produce
at
the
of
you
use
and
point
is
No.44”,
where
which
three
completing
tessellated
select
and
three
works
describe
how
completed
beaks
the
using
he
of
the
rotation.
a
same
This
1941
color
makes
in
ink
meet,
the
and
you
can
tessellating
rotation.
of
it
in
art
was
to
analyze.
Identify
the
base
tessellated.
10
Why
saw
to
“Bird
shapes
which
art?
Escher
was
a
or
why
used
piece
identify
not?
so
by
much
Escher
in
this
you
style
would
of
art?
recognize
it?
transformations
preimage.
produce
Sometimes,
image
images
called
six
geometry
you
M.C.
tessellated
transformations
an
of
discuss
this
you
art
tessellations
that
and
like
do
produces
These
look
and
shape
you
all
Escher
you
Similarity
Not
The
equilateral
Reect
it?
If
rotation
shape
–
piece
watercolor.
see
8
are
that
is
similar
similar
when
larger
rather
gures
images
are
or
a
that
shape
smaller
than
are
is
same
size
transformed,
than
congruent.
called
the
similarity
the
it
preimage.
Transformations
transformations.
2 8 9
Dilations
A
dilation
changes
the
(also
the
or
point
type
all
of
the
is
of
an
smaller
pre-image.
the
size
dilation
larger
known
A
enlargement)
enlargement
the
image
which
also
the
transformation
other
an
theshape.
dilation
from
as
types
of
Thescale
or
a
will
a
dilation
that
a
factor
reduction
be
needs
is
transformationthat
determines
and
compared
center
is
changes
the
transformation
to
many
the
be
of
the
simply
times
original
dened:
Dilations
size
are
with
point
made.
how
whether
are
this
the
original
is
only
shape;
movements
of
the
preimage.
i
Investigation 4 – Dilations
t
e
r
i
c
a
r
You
will
be
using
dynamic
geometry
software,
such
as
the
Polygon
B, C
ATL2
Transformer
1
Create
2
From
the
point
(0,
3
Copy
a
app
on
triangle
the
A
ct.excelwa.org
(1,
2),
B
transformations
0)
and
and
a
scale
complete
(3,
C
menu,
factor
this
3),
of
website,
(5,
to
complete
this
1).
construct
a
dilation
image
with
table.
Relationship
Image
Vertex
A
(1,
2)
Vertex
B
(3,
3)
Vertex
C
(5,
1)
Horizontal
distance
A
to
B
Horizontal
distance
A
to
C
Horizontal
distance
B
to
C
A
to
B
Vertical
distance
A
to
C
Vertical
distance
B
to
C
Area
4
Create
5
What
a
of
preimage
and
image
triangle
new
kind
between
triangle
triangle
distance
center
3.
Preimage
Vertical
investigation.
of
table
scale
and
repeat
factors
steps
produce
1–3
using
a
scale
enlargements
and
factor
what
of
kind
.
produce
reductions?
Continued
290
6
Geometric transformations
on
next
page
G E O M E T R Y
6
Generalize
using
the
Verify
8
Justify
9
Construct
10
the
How
of
your
rule
why
How
could
12
How
would
Reect
●
What
if
center
●
Use
the
A
the
Q
of
is
→
the
to
×
If
of
you
to
a
scale
the
the
factor
origin
notice?
determine
origin
using
examples
not
are
created
of
your
choosing.
as
you
the
the
arrow
and
Explain
each
why
of
this
location
of
the
vertices
makes
other
sense.
dilations
center?
notation?
have
studied
in
this
investigation
11
the
center
happens
you
on
found
how
center
What
is
the
point
A
(16,
5
point
the
coordinates
→
to
to
point
the
in
the
create
point
36)
A
and
of
the
original
dilation?
triangle
investigation
a
the
of
of
A’
the
A’
of
scale
Move
and
the
the
dilation
scale
to
image
factor
of
write
step-
regardless
the
of
dilation.
shape
is
at
will
factor
(−2,
5
9),
The
be
of
becomes
factor
with
what
the
will
center
the
be?
45)
(−10,
the
A’
45).
dilation
(4,
dilation
image
point
factor.
In
(with
is
by
this
center
at
centered
at
the
multiplying
case,
the
the
the
scale
origin)
origin,
so
ordered
factor
is
you
pair
can
by
nd
the
the
scale
5.
if
9)?
1
The
4
factor
dilation
scale
1
scale
dilation
point
(−10,
scale
a
=
=
16
the
using
what
undergo
The
so
dilations
3
origin.
16
lines
the
was
instructions
9)
÷
and
through
do
dilations
write
see
(−2,
4
pass
discuss
to
4
A
how
dilation.
points
What
these
write
origin
coordinates
A
explain
notation?
location
shape
at
that
relationships
Example
Q
to
image.
by-step
the
lines
you
point
dilated
new
of
1–4
works.
triangle
and
the
rule
steps
center
two
use
you
arrow
from
triangle.
you
original
the
for
three
original
11
using
as
your
could
the
results
origin
7
of
your
the
factor
is
centered
by
dividing
x-coordinate
of
the
at
the
the
origin,
so
you
x-coordinate
of
can
the
nd
the
image
by
preimage.
is
4
Continued
on
next
page
2 91
Check:
Verify
1
36
=
×
=
Q
Draw
2,
scale
factor
by
the
by
multiplying
scale
factor.
If
the
your
original
answer
is
correct,
9
you
4
the
y-coordinate
36
should
get
the
y-coordinate
of
the
image.
4
the
image
centered
at
of
the
triangle
ABC
after
a
dilation
with
a
scale
factor
of
origin.
y
3
B
2
1
C
0
x
1
2
3
4
5
6
–1
–2
A
–3
A
(1, −2)
→
×
2
→
The
(2, −4)
dilation
image
(4, 2)
→
×
2
→
→
×
2
→
centered
multiplying
at
the
the
origin,
ordered
so
pair
you
of
can
each
nd
the
ver tex
by
(8, 4)
the
(5, 1)
by
is
scale
factor.
In
this
case
the
scale
factor
is
2.
(10, 2)
y
Plot
the
points
of
the
ver tices
of
the
image
triangle
5
label
B′
them
correctly.
Join
them
4
3
B
C ′
2
1
C
0
x
1
2
3
6
8
9
10
–1
–2
A
–3
–4
A′
–5
Just
like
the
Cartesian
explore
292
6
other
plane
this
transformations,
can
be
notation
represented
in
the
next
Geometric transformations
dilations
by
that
specic
activity.
are
on
notation.
the
You
will
with
straight
lines.
and
G E O M E T R Y
Activity
1
9
Complete
dilation
Create
paper.
at
3
the
this
with
(x,
2
y)
two
3
4
Describe
a
image
A′
Using
a
image
B′
Using
a
image
C′
D′
5
a
6
F′
7
of
2,
Cartesian
notation
centered
at
in
order
the
plane
to
show
a
origin.
)
dilations
of
of
your
own
reduction
your
of
dilation
of
of
and
the
of
of
the
the
the
the
scale
Find
What
dilated
of
do
of
a
four-sided
one
using
shape
enlargement,
on
both
squared
centered
correct
notation
and
nd
the
images.
scale
B
point
C
of
factor
(12,
factor
(3,
4
factor
(20,
of
the
center
at
the
origin,
nd
the
coordinates
of
the
with
the
center
at
the
origin,
nd
the
coordinates
of
the
27).
factor
the
with
−7).
(−12,
scale
factor
you
A
scale
point
scale
E′
scale
point
dilation
the
dilations
the
of
dilation
39).
(0,
correct
factor
,
one
of
the
5
(27,
Find
(
each
becomes
b
scale
→
each
of
Using
Find
example
on
origin.
Practice
2
Dilations
a
Include
location
1
–
with
the
center
at
the
origin,
nd
the
coordinates
of
the
0).
dilation
the
(with
dilation
center
(with
at
center
the
at
origin)
the
if
origin)
point
if
D
point
(9,
E
13)
(48,
becomes
−20)
−5).
think
factor
a
of
negative
the
scale
dilation
factor
(with
indicates?
center
at
the
Explain.
origin)
if
point
F
(0,
−70)
becomes
21).
Rewrite
questions
4–6
using
correct
notation.
Continued
on
next
page
2 9 3
8
Graph
the
centered
image
at
the
of
the
quadrilateral
ABCD
after
a
dilation
after
dilation
with
a
scale
factor
of
3,
origin.
y
A
B
x
D
C
9
Graph
at
the
the
image
of
the
triangle
ABC
a
with
a
scale
factor
of
centered
origin.
y
A
B
x
C
10
This
is
the
company.
name
as
for
the
logo
the
constellation
merged
294
the
to
(FHI),
the
So
the
star
on
words
a
united
6
cluster
in
is
of
in
idea
form
logo
and
that
stars
known
(part
of
“Subaru”
ve
FHI
–
the
also
this
companies
Heavy
represents
of
car
Japanese
company
how
cluster
the
Japanese
Fuji
parent
Subaru
English
Taurus).
“united”
expresses
the
“Subaru”
Pleiades
means
for
Industries
of
Subaru.
both
a
considers
companies.
Geometric transformations
play
itself
Continued
on
next
page
G E O M E T R Y
11
a
Describe
any
congruence
b
Describe
any
similarity
This
a
pattern
Identify
is
the
based
on
original
transformations
transformations
traditional
shapes
and
that
that
Aztec
describe
are
print
all
are
visible
visible
in
in
this
this
logo.
logo.
designs.
the
transformations
you
can
see
in
the
design.
b
12
Would
This
is
an
you
consider
example
never-ending
a
Describe
b
Could
pattern
the
this
of
a
this
tessellation?
fractal
repeated
dilations
shape
a
be
that
design.
at
In
Justify
math,
increasingly
you
see
tessellated?
in
the
a
your
response.
fractal
small
fractal
is
an
abstract
object
formed
by
a
scales.
design.
Explain.
2 9 5
l
i
n
b
e
Go
to
two
to
also
need
to
this
On
a
a
Below
–
you
to
sure
3
to
carry
Creating
On
a
of
Round
1
5
your
person
order,
and
a
the
to
“Transformation
download
for
the
a
for
each
and
print
player
to
the
use
in
game
as
a
You
board
counter.
game.
transformation
be
Game”.
groups
of
puzzle
three.
and
x
and
label
plane,
all
of
the
list
the
y-axes
longer
than
−10
to
10,
vertices.
written
a
no
instructions
reection,
information
a
for
rotation
that
four
and
someone
dierent
a
dilation.
would
need
transformation.
representation
of
paper,
list
the
four
ordered
pairs
of
the
quadrilateral.
each
your
applying
drawing
play
translation,
sheet
visual
on
with
algebraic
separate
Represent
Pass
each
to
triangle
to
need
plane
include
out
need
a
search
representation
Cartesian
the
vertices
4
will
visual
transformations:
Be
out
and
Creating
quadrilateral
the
will
cut
instructions
Cartesian
draw
2
the
You
to
10
activity
Creating
1
four.
play”
Activity
For
nrich.maths.org
transformation
representation
right.
each
each
They
will
using
and
written
carry
transformation
resulting
image.
arrow
out
to
instructions
the
the
They
notation.
result
will
to
the
transformations
of
then
the
color
one
the
in
before
nal
image.
Round
2
6
your
Pass
They
will
algebraic
draw
the
transformation,
each
7
29 6
using
only
shape
to
and
algebra
to
the
the
person
image
determine
image.
Compare
6
representation
original
your
answers
and
check
Geometric transformations
for
accuracy.
on
your
after
the
left.
each
coordinates
of
can
and
play
in
cards.
Then
groups
You
of
will
followthe
“How
G E O M E T R Y
ATL1
Reect
Reect
and
and
discuss
discuss
12
12
Discuss these questions with your group.
●
●
Compare
the
visual
representation.
Which
Do
know
you
just
●
using
need
one?
How
process
how
to
did
do
you
both
and
nd
the
algebraic
easier?
methods
or
Explain.
could
you
rely
on
Explain.
does
problem
to
representation
knowing
make
you
how
a
to
more
use
multiple
eective
methods
learner?
to
solve
a
Explain.
2 97
Unit
A
tessellation
no
A
summary
overlaps
is
the
and
transformation
and
produces
Translation,
a
tiling
no
of
a
changes
shape
rotation
an
and
size
create
is
a
and
similarity
images
that
are
proportional
in
Translation
moves
change
must
to
be
–
geometric
shapes
with
shape,
called
to
are
all
congruence
create
the
preimage.
transformation.
Similarity
identical
preimage,
image.
transformations
shape
the
in
shape
to
the
images
that
transformations
preimage
and
are
size.
size.
If
moved
Translations
the
reection
are
Dilation
original
called
Congruence
in
using
gaps.
transformations.
identical
surface
a
in
can
a
point
shape
the
be
is
same
or
shape
in
translated,
direction
represented
any
every
and
using
direction
the
by
vertex
the
with
of
same
the
no
shape
distance.
notation
,
where
h
and
Reection
reection
–
or
number
–
of
Rotations
298
6
are
ips
the
Reections
Rotation
k
a
real
point
mirror
can
be
turns
a
or
shape
shape
The
the
over
a
given
line
called
the
line
of
line
represented
degrees.
about
numbers.
using
notation:
a
reection
in
the
x-axis
a
reection
in
the
y-axis
a
reection
in
the
line
a
reection
in
the
line
about
a
center
origin
this
can
xed
is
point
called
be
the
through
point
represented
of
determined
rotation
using
this
notation:
a
90-degree
a
180-degree
rotation
clockwise
a
270-degree
rotation
clockwise
a
360-degree
rotation
clockwise
Geometric transformations
rotation
a
clockwise
Dilation
–
image).
The
of
the
of
the
Dilations
can
scale
dilation
center
a
creates
be
a
bigger
factor
must
be
dilation
with
the
smaller
image
than
indicates
the
size
given.
the
examples
is
In
always
origin
represented
or
as
using
the
this
the
change
the
original
and
in
the
this
(pre-
center
book,
the
origin.
center
of
dilation
and
scale
factor
notation:
,
where
a
is
any
real
number.
Scale
factors
larger
Scale
factors
between
than
the
images
upside
original
that
are
than
0
1
produce
and
1
(reductions).
on
the
other
enlargements
produce
images
Negative
side
of
the
scale
of
that
the
are
factors
centre
of
preimage.
smaller
produce
dilation
and
are
down.
2 9 9
t
er
i
r
i
Key
review
to
Unit
review
Level 1–2
1
n
c
Unit
Copy
and
A
question
levels:
Level 3–4
complete
the
Level 5–6
table
Translation
Level 7–8
below.
Translation
Translation
down
2
units
Dilation
by
a
scale
factor
Point
up
(0,
0)
(6,
1)
(3,
(−8,
Plot
a
units
left
5
units
and
right
7
units
of
4
about
the
origin
−2)
(−4,
2
3
7)
−5)
the
image
Translation
and
4
units
of
4
each
units
shape
on
your
own
right
b
copy
of
the
Translation
down
and
3
units
y
Cartesian
2
units
grid.
right
up
y
B
C
A
x
x
C
B
D
A
3
The
in
photo
the
Parc
region
park’s
of
Gaudí,
his
Spain.
tiles.
who
It
300
this
6
was
a
of
the
Barcelona,
and
bench
is
created
his
work,
Serpent
in
winds
covered
designs.
the
symbols,
design
in
expressed
commissioned
Is
detail
The
fantastical
Greek
a
Güell,
perimeter
ceramic
in
shows
by
love
a
the
of
wanted
Geometric transformations
artist
Güell,
and
of
Antoni
forms
who
Gaudí
Explain
the
mosaic
natural
symbols
tessellation?
Catalonia
around
in
Eusebi
Christian
the
Bench
to
represent
symbols
why
or
why
of
the
connection
Catalonia.
not.
between
ancient
4
Describe
the
transformations
that
have
occurred
y
a
in
each
diagram.
y
b
C ′
C ′
C
B
B′
B′
A′
C
A
x
A′
x
B
A
y
c
y
d
B′
B
B′
B
x
A
A′
A′
A
C′
x
C
C
C ′
y
e
x
B
B′
A
A′
C
5
Using
the
6
a
dilation
image
What
T
C′
(6,
is
L′
the
−18)
is
of
of
the
scale
T′
scale
point
factor
(2,
factor
L
of
(12,
the
5,
with
the
center
at
the
origin,
nd
the
coordinates
of
−9)?
dilation
(with
center
at
the
origin)
if
the
image
of
point
−6)?
3 01
7
Plot
(−3,
8
the
0)
image
using
of
the
the
following
a
(x, y)
→
(x
+
b
(x, y)
→
(x
− 3, y
Copy
and
shape
4, y
vertices
(−5,
8),
(3,
−6)
and
translations.
− 9)
+
complete
Reection
with
6)
the
table
Reection
below.
Reection
Reection
Reection
Reection
Point
in
(0,
0)
(8,
2)
(7,
(0,
in
y
=
x
in
y
=
−x
in
y
=
in
−3)
0)
the
image
Cartesian
of
each
shape
on
your
own
copy
of
the
grid.
Rotation
about
the
90°
clockwise
b
origin
Rotation
180°
about
the
origin
y
y
x
x
D
A
C
C
A
B
302
3
1)
Plot
a
y-axis
−10)
(9,
9
in
−6)
(−4,
(−5,
x-axis
6
Geometric transformations
B
x
=
−4
10
This
is
a
folk
Romanian
every
color
of
where
the
it
age
and
make
c
11
and
0)
(8,
2)
(7,
(0,
the
the
region
indicate
the
woman
shape(s)
used
to
pattern.
the
transformations
that
have
occurred
to
the
how
the
shapes
complete
90
this
of
degrees
are
table.
tessellated.
All
Rotation
180
rotations
of
are
Rotation
degrees
270
clockwise
about
of
the
point
Rotation
degrees
360
clockwise
(0,
of
degrees
clockwise
0).
Rotation
−90
of
degrees
clockwise
1)
−3)
0)
Plot
a
Even
−10)
(9,
12
of
the
−6)
(−4,
(−5,
can
status
clockwise
(0,
of
expresses
and
that
woman’s
family.
original
Rotation
Point
said
shape(s).
Explain
Copy
traditional
blouse.
this
original
is
aspect
and
made
the
Describe
an
social
the
It
in
Romanian
pattern
was
Identify
b
a
history
the
wearing
a
on
represents
woman’s
used
embroidery.
form
blouse
pattern
the
image
Reection
of
in
x
each
=
shape
on
your
−3
own
b
copy
of
Reection
y
the
in
y
Cartesian
=
grid.
x
y
D
A
D
B
C
x
x
A
C
B
3 0 3
13
Describe
diagram
the
and
transformation
express
a
it
using
that
has
correct
been
arrow
performed
notation.
y
D
C
B
E
A
x
D′
C ′
B′
E′
A′
y
b
A
C
A′
C ′
x
B ′
B
304
6
Geometric transformations
in
each
y
c
A
B
C
E
D ′
x
D
E′
C′
B′
A′
y
d
B
A
C ′
D′
x
D
C
A′
B′
14
Given
the
image
of
a
(7,
b
(−12,
c
(0,
rotation
the
given
(x, y)
→
points
( y , − x ),
and
nd
describe
the
the
coordinates
rotation
in
of
the
words.
14)
−5)
3)
3 0 5
15
Given
of
the
the
reection
image
of
the
(x, y)
given
→
( − y , − x ),
points
and
nd
the
describe
coordinates
the
reection
in
words.
16
a
(0,
b
(14,
c
(3,
This
in
−9)
−4)
5)
door
is
Beijing,
expressed
the
ideas
in
Identify
b
Describe
c
Explain
Is
a
Explain
In
ancient
small
to
oors
and
geometric
of
oors
people
The
symbols
own
is
(the
to
two
of
306
6
that
in
symbolizing
in
this
that
design.
have
tessellated.
− x
the
same
as
a
rotation
of 180°?
were
used
covered
is
the
a
ancient
Mosaic
were
used
since
such
mosaics
a
important
to
most
luxury.
had
mosaics
symbolism,
or
=
mosaic
aord
intertwined
immortality
y
are
power,
Some
depict
full
one
within
express
shape(s).
artists
found
and
not
meaning.
meant
one
wealth
that
Turkey.
this
the
notation.
This
mosaic
could
line
of
wealth.
shape(s)
pictures
walls.
as
and
red
are
throughout
colors
color
“tesserae”
tiles
Ephesus,
express
the
Rome,
such
the
palace
religion
power
Even
shapes
arrow
bigger
the
original
the
in
squarish
make
city
the
using
is
luck
a
architecture
family.
with
and
transformations
how
reection
as
original
the
to
and
City,
the
City,
Philosophy
happiness,
a
Forbidden
art
beliefs,
occurred
18
the
imperial
and
power,
the
China.
Forbidden
Chinese
17
in
their
are
events
such
bands)
eternity.
Geometric transformations
as
or
the
which
people.
A
mosaic
Solomon’s
often
knot
expressed
a
like
this
gure
belief
in
a
Identify
preimage
shapes
in
the
mosaic
that
have
been
transformed.
b
Describe
the
preimage
c
19
Explain
What
the
a
transformations
that
have
occurred
to
these
shapes.
how
single
the
shapes
are
transformation
tessellated.
gives
the
same
result
as
each
of
following?
A
reection
in
the
x-axis
in
the
line
followed
by
a
reection
in
the
y-axis
b
A
reection
clockwise
20
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cubic
− 5
=
9
7
3
12
−
2
4w
+
2w
−
w
=
2
2
3y
6z
−
+
11
19
=
=
−7y
2(3z
3
5p
−
4)
+
p
2
−
2p
+
4
=
−2p
3
−
8p
2
7h
2
Based
−
9h
on
=
−12(3h
your
●
linear
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quadratic
●
cubic
+
2)
results,
dene
the
following
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equation
equation
equation
3
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do
you
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think
choice
of
a
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equation”
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or
m,
is?
etc.)
Give
aect
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the
example.
classication
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In
this
unit,
noticed
that
variable.
sucient
You
to
will
each
When
have
nd
You
complex
be
solving
equation
there
the
already
equations.
more
you
is
only
value
learned
will
in
apply
of
linear
the
one
that
investigation
variable,
variable
strategies
those
equations.
to
same
a
(or
solve
You
may
contained
single
only
equation
“solve
the
dierent
strategies
have
here
is
equation”).
types
to
one
of
solve
equations.
319
Example
Q
Solve
the
1
following
x
equations.
2x
2( x
+ 7)
2
a
=
15
−
6
b
x
+
75
=
100
3
If
=
15
−
you
have
a
fractional
denominator
6
and
equation,
multiply
ever y
nd
a
term
common
in
the
3
equation
by
that
common
denominator.
2 6 (2 x )
6 x
=
6(15) −
Simplify.
6
3
x
=
90
−
4x
4x
=
90
−
4x
Rearrange
+
6
2x
a
x
+
5
x
A
2x
=
c
+
operation
5x
the
equation
by
doing
the
opposite
4x
to
isolate
the
variable.
90
=
5
5
x
=
You
have
nished
variable
on
one
Simplify
and
when
you
have
isolated
the
side.
18
2
b
x
+
75
=
100
rearrange
the
equation
by
doing
2
x
+
75
−
75
=
100
=
25
−
the
75
opposite
operations
to
isolate
the
variable.
2
x
You
2
x
=
have
variable
25
square
x
2( x
=
±
5
When
= 2x
the
of
and
you
a
when
side.
have
you
Note
number,
negative
distributive
inside
× 2( x
one
have
that,
you
must
answers
brackets
in
isolated
when
into
the
the
nding
take
both
the
the
account.
equation,
apply
+ 6
5
5
root
positive
+ 7)
c
nished
on
of
outside
+ 7)
= 5( 2 x
the
of
law,
so
brackets
the
that
is
ever y
term
multiplied
by
on
the
ever y
term
brackets.
+ 6)
5
Simplify
2x
−
2x
+
14
=
10x
14
−
30
=
8x
−
2x
+
the
−16
+
30
−
and
rearrange
the
equation
by
doing
30
opposite
operations
to
isolate
the
variable.
30
8x
=
8
8
You
x
=
have
variable
320
7
nished
when
−2
Linear systems
on
one
side.
you
have
isolated
the
A LG E B R A
Reect
●
Classify
●
and
each
●
equation
your
answers.
How
is
the
equation?
With
a
discuss
number
of
in
2
Example
solutions
1
(linear
related
to
or
the
quadratic).
degree
of
Justify
the
Explain.
peer,
review
each
of
the
examples
and
help
each
other
to
ATL2
understand
Practice
1
each
a
+
4)
strategies
that
were
used.
1
Solve
3(x
the
equation.
=
−3x
b
c
d
e
f
g
h
i
j
k
3
2
l
4x
2
−
11
=
2x
9
Create
a
linear
3
Create
a
fractional
4
Create
7
and
is
a
3
−
12x
+
5
=
4x
3
+
9x
−
11
+
3x
2
+
2
answer
7x
−
3x
equation
that
linear
requires
equation
multiple
that
steps
requires
to
solve
multiple
where
steps
to
the
solve
answer
where
is
−4.
the
2.
quadratic
equation
that
requires
multiple
steps
to
solve
where
the
answers
are
−7.
3 21
Did
you
Most
companies
which
prot
is
to
know...?
typically
and
are
revenue
increase
the
or
goal
of
many
and
people
implement
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bank;
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if
know
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will
need
97%
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are
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the
is
of
32 2
each
If
least
these
of
of
of
this
on
of
of
created
small
a
variables
using
a
million
and
families
people
have
in
his
in
What
group
in
it.
if
of
to
both
there
of
two
nd
a
but
are
that
can
variables,
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values
you
linear
what
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equations
linear,
simply
variety
variable,
With
order
are
or
one
have
system.
methods;
these
the
will
be
you
of
both
what
You
is
can
methods
graphing
used
apply
coordinate
one.
Linear systems
can
by
the
to
solve
skills
plane,
a
you
but
linear
have
now
system
learned
there
will
native
borrowed
equations
with
unknown?
equations
9
women.
equations
is
individuals
Nearly
linear
equations
that
you
just
a
unit.
methods
lines
the
help
is
been
systems
Here,
to
Yunus
equations
system
of
impoverished
have
the
linear
the
social
reach
provides
equations
two
a
help
can
business.
one
two
linear
the
to
small
of
with
to
credit
bank
solve
for
instead
7
to
goal
change.
that
them
system
graphing.
lines
a
A
focus
graphing
of
the
they
support
for
His
than
linear
Solving
One
how
system
this
for
systems
at
variables.
called
start
business
solution
With
using
company.
Muhammed
Bank.
more
variables?
a
a
microloans)
can
Solving
You
right,
Grameen
loans
ask
ideas
enterprises,
ultimate
the
media,
believing
fundamental
the
and
their
example,
run
issue.
social
The
of
nding
environmental
internet
prot”
performance
data.
value
Social entrepreneurs
primary
“for
measure
in
be
two
Bangladesh
money
from
so
the
A LG E B R A
Activity
1
On
the
1
–
same
y
2
Why
do
3
Find
the
=
Graphing
coordinate
2x
−
these
4
and
lines
y
linear
grid,
=
−3x
intersect?
coordinates
of
draw
the
+
systems
the
following
lines.
1
Justify
point
your
where
answer
the
two
mathematically.
lines
cross
(the
point of intersection).
4
Verify
the
that
5
Explain
6
Repeat
why
point
a
y
=
−5x
b
y
=
3
−
+
of
the
steps
Reect
Reect
ATL1
the
coordinates
1
8
2x
point
intersection
point
of
through
2x
and
and
and
and
of
this
4y
−
7
is
the
intersection
5
y
+
into
using
=
=
the
on
both
original
solves
lines
the
following
by
substituting
equations.
linear
system.
equations.
2
−6x
discuss
discuss
3
3
Answer the following questions in pairs.
●
What
are
the
dierent
ways
of
graphing
a
line
such
as
3x
–
6y
=
12?
Pairs
Which
●
●
method
Describe
by
Why
good
is
it
l
i
Is
n
it
a
you
prefer?
disadvantages
equations
system?
●
any
do
you
see
to
solving
a
system
of
linear
graphing.
idea
to
verify
your
answer
when
solving
a
linear
Explain.
possible
for
a
linear
system
to
have
no
solution?
Explain.
k
b
e
If
you
do
graphing
this
not
have
tool
graphing
access
such
as
utility.
to
a
graphic
Desmos
Be
sure
to
to
display
graph
write
each
your
calculator
of
the
(GDC),
lines.
equations
in
Visit
you
can
use
an
online
www.desmos.com
gradient–intercept
to
use
form.
323
Activity
An
artist
for
the
in
2
Colombia
is
similar
to
selling
price
1
y
be
the
from
Solve
4
The
To
the
why
the
point
in
amount
the
make
he
a
cost
of
system
prot,
to
the
hammock
you
think
there
is
money
by
a
Linear systems
a
do
artist's
on
a
start
be
a
can
is
microloan
a
business
the
like
$500
from
the
and
a
which
the
both
think
he
North
dierence
in
that
the
lines
one
the
pictured
cost
community
he
hopes
this
needs
of
here.
materials
The
for
development
will
support
his
cost
each
bank
family.
The
in
from
sales
equation
can
on
be
the
y
given
same
a
=
sold.
of
a
product.
Explain
why
the
y
25x
+
500.
revenue
is
called
70x.
by
the
equation
coordinate
company’s
costs
=
grid.
and
is?
to
to
American
Is
hammocks
represents
begins
price?
of
takes
by
hammocks
revenue
before
number
given
system
you
the
company
be
graphing
of
Why
hammocks
out
x
and
making
produce
same
to
hammocks
$70.
hammocks
The
7
of
cotton
the
take
order
is
intersection
have
make
to
dollars
break-even point.
does
to
in
loss?
macramé
decides
Bank
the
linear
of
makes
hammock
amount
is
3
324
a
revenue
Explain
6
of
or
needed
He
Grameen
2
5
$25.
Revenue
the
Prot
equipment
hammock
Let
–
be
more
make
market
this
unfair
a
than
prot?
would
and,
if
his
costs.
Explain
cost
so,
the
to
How
your
many
customer
whom?
hammocks
answer.
$120?
Why
do
A LG E B R A
Practice
1
Rewrite
a
2y
=
2
each
– 4x
+
equation
in
gradient–intercept
8
b
d
g
5x
=
10y
–
20
10y
=
20
c
0
=
2x
–
3y
+
9
=
2x
–
4y
+
12
f
h
x
=
8y
j
Find
the
a
+
3y
x-
4x
and
=
y-intercepts
of
20
the
b
d
3
–
e
i
2
5x
form.
following
5x
–
10y
–
linear
30
=
equations.
0
c
e
Solve
each
a
y
=
2x
–
b
y
=
–x
+
linear
6
and
2
and
c
system
y
=
y
x
=
+
5x
by
0
f
graphing.
Verify
your
answers.
3
–
4
and
You
d
y
=
–2x
–
10
and
your
by
e
y
–
7
=
6x
and
can
3y
+
12x
–
6
=
0
coordinates
back
3x
+
2y
=
16
g
5y
–
35
=
–
A
video
and
2x
+
3y
=
answers
substituting
the
f
verify
into
the
14
equations.
4
do
cost
is
many
the
some
not
and
streaming
considering
where
5x
have
$6
a
the
the
movies
loyalty
couple
of
per
company
new
proceeds
nancial
movie.
as
of
you
program
The
putting
together
promotions.
will
go
resources
loyalty
want
and
is
for
then
a
to
to
of
providing
aord
program
$35
the
Two
their
the
rental
them
internet.
cost
fee
is
are:
or
just
The
and
loyalty
to
local
current
can
can
per
a
access
you
you
$4
strategy
involve
internet
options
monthly
pricing
either
pay
$10
is
program
students
average
rental
download
a
month
who
to
as
join
movie.
Continued
on
next
page
325
a
Create
a
options
5
that
Graph
c
The
on
values
shows
these
results
the
for
for
each
cost
Write
of
more
customer
rentals
How
Which
of
general
is
the
marketing
number
a
option
a
on
customers
month.
this
options
of
average,
each
e
of
of
the
renting
three
from
1
to
movies.
b
d
table
same
survey
download
promotion
rentals?
axes.
showed
5
is
about
economically
on
of
that,
movie
the
rentals
better
deal
Explain.
statement
depending
set
the
when
each
attractive
number
to
of
the
movie
downloaded.
could
company’s
such
promotions
ability
to
raise
increase
money
for
a
their
cause?
Explain.
Number
The
of
systems
produced
solutions
of
one
linear
Could
ecient
way
there
of
a
equations
solution.
answer?
to
Is
it
ever
linear
that
you
possible
be
investigating
no
for
system
have
there
solution?
the
answer
solved
to
be
these
far
more
Graphing
to
so
all
than
systems
is
one
an
questions.
Investigation 2 – Classifying systems of equations I
t
er
i
r
i
are
three
linear
n
c
Here
systems.
B
A
B
y = 2x + 1
2y
y = –x + 7
1
Graph
4y
each
pair
your
teacher,
2
How
many
3
Research
of
you
–
8
2x
may
names
=
=
lines
solutions
the
+
C
x
6x
use
does
used
a
separate
set
technology
each
to
20
–
2y
y + 3x – 5 = 0
–16
on
=
linear
classify
if
of
axes.
you
the
permission
of
wish.
system
the
With
have?
three
types
Explain
of
your
linear
answers.
systems.
Continued
326
7
Linear systems
on
next
page
A LG E B R A
4
Is
it
possible
nd
y
=
the
mx
to
determine
solution
c
+
and
by
how
graphing?
copy
and
the
lines
Rewrite
complete
the
will
all
intersect
the
table
above
y
=
system
mx
Slope
+
of
c
actually
in
the
trying
to
form
below.
A
Linear
before
equations
B
C
in
form
each
line
y-intercept of each
line
Number
of
solutions
5
Write
a
brief
linear
systems.
6
Verify
your
7
Justify
results
why
Practice
ATL2
summary
Answer
giving
1
your
for
of
at
results
the
results
least
two
make
of
your
more
investigation
linear
into
the
three
types
of
systems.
sense.
3
question
1
in
pairs.
If
someone
needs
help,
ask
leading
questions
instead
of
answers.
Without
Justify
graphing,
your
classify
answers
each
linear
system
based
on
the
number
of
solutions.
mathematically.
Pairs
a
y
=
c
4x
3x
=
−
6y
5,
+
2x
1,
,
e
g
4x
–
2y
i
y
–
3
k
y
+
3x
=
=
=
9y
6y
14,
4x,
5,
6
−
=
–
12x
–
5x
6y
2y
+
1
=
6x
2x
8
b
−5
d
12
f
x
h
– x
j
y
=
2x
l
x
+
y
+
–
6y
=
8x
=
5
3y
=
=
0
40
,
=
–
2x
=
−1,
+
y
2y
–
+
4y,
=
=
=
+
5x
4y
+
=
10y
−2
–
30
=
0
3
5
2,
4
3x
–
6x
0,
y,
–
−x
3y
4x
+
+
=
3y
=
5
2
4y
+
8
=
0
Continued
on
next
page
327
2
Given
y
=
4x
the
2kx
+
y
+
=
linear
system
6
2,
a
Find
the
b
What
value
values
of
of
k
k
that
will
makes
result
the
in
one
linear
system
solution?
have
no
solution.
Explain.
3
The
YMCA
countries.
in
in
a
mentorship
A
local
fee
and
(which
b
Graph
c
If
you
If
the
you
a
Can
not
has
$5
a
for
each
of
sign
country
needs.
charitable
people
who
membership
every
values
you
ended
you
of
day)
will
membership
up
from
day
can
to
to
to
the
bought
vastly
YMCAs
raise
money
including
both
social
charges
facilities
to
facilities
oer
become
that
operating
and
activities,
oer
represent
country
Local
want
that
be
organization
use
the
119
dierent
so
providing
that
programs
they
classes
and
used.
a
$90
one-time
Alternatively,
a
joining
day
7
pass
facilities.
options.
up
for
a
think
up
for
use
fee
only
the
and
using
for
the
membership
of
a
another
facilities
this
to
6
times
be
facilities
be
a
reason
good
why
the
you
month,
better
once
idea?
a
a
how
long
will
it
take
Linear systems
to
pay
option?
week
(4
times
a
month),
would
Explain.
would
only
use
the
day
pass
option
and
membership?
Continued
328
can
entrepreneurs.
youths
are
in
system.
think
signing
e
variety
$0.75
worldwide
community
young
table
the
dier
local
for
costs
up
non-prot,
wide
then
Set
d
to
YMCA
a
o
a
YMCAs
response
engage
is
on
next
page
A LG E B R A
f
Sometimes
if
the
up
g
4
You
joining
for
How
the
would
are
the
given
the
Classify
b
Verify
c
Change
one
number
of
Show
fee
run
this
your
system
system.
to
of
the
your
be
your
by
−6x
why
correct
by
Create
a
linear
system
where
there
is
6
Create
a
linear
system
where
there
are
While
linear
dicult
to
to
the
solve
Solving
used
1
pairs,
–
linear
3
–
the
good
are
+
going
answer
2y
in
=
2
by
solving
order
and
once
for
−6x
lower
a
week?
the
there
+
4y
joining
=
Would
system
to
be
fee.
of
no
What
you
sign
equations.
solution?
−4.
system.
4y
=
−4
so
that
the
system
has
an
innite
works.
graphing
no
visual
the
solution
innitely
system
but
this
many
it
is
and
by
is
with
not
your
new
equation.
immediately
solutions
but
this
is
obvious.
not
fractions
and
There
may
more
are
of
two
the
be
ecient
methods
elimination.
substitution
substitution
following
or
easier
algebraically.
systems
representation
decimals
Sometimes
substitution
the
linear
a
that
system
The
complete
Given
you
answers
linear
Activity
In
gives
determine.
commonly
signicantly
obvious.
graphing
system,
+
the
this
5
immediately
on
modied
3x
a
answer.
equation
is
oer
plan
your
graphing
Explain
answer
to
and
you
equations
Justify
in
and
Justify
have
classication
term
$1
then?
problem
solutions.
that
promotions
dropped
membership
a
d
ATL1
YMCAs
method
activity.
system
Pairs
3x
y
a
–
=
2y
−2x
=
+
2
3
substitute
equation
b
1
8
solve
this
the
expression
for
y
from
equation
2
into
1
linear
equation
for
x
Continued
on
next
page
32 9
2
c
substitute
the
d
write
answer
Given
4x
a
+
your
the
y
=
linear
−2,
Which
Isolate
c
Substitute
this
equation,
3
Solve
e
Find
–
variable
b
the
the
as
an
into
equation
ordered
pair
(x,
2
to
nd
the
value
of
y
y).
2y
=
4,
would
be
easiest
to
isolate
in
one
of
these
Explain.
variable
the
as
seen
linear
how
in
one
expression
value
Summarize
x
for
system
−2x
equations?
d
value
of
the
for
two
this
equations.
variable
into
the
other
previously.
equation.
of
to
the
other
solve
a
variable.
linear
system
using
the
substitution
method.
Reect
and
discuss
4
Pairs
In pairs, answer the following questions.
●
In
question
equation
●
What
form
Can
ATL2
and
these
when
Solve
4x
A
–
−5x
to
the
3y
+
+
what
=
2y
+5x
made
you
By
+
have
C
=
any
2
easy
to
substitute
into
with
use
a
to
peer
the
do
if
equation
2
were
written
in
standard
0)?
disadvantages
and
help
substitution
of
the
each
substitution
other
to
fully
method?
understand
how
method.
2
following
−11,
=
−5x
linear
+
=
2y
5x
2
=
using
the
substitution
method.
12
one
of
the
variables
equations.
+
5x
=
2y
system
Isolate
12
+5x
2y
equation
?
identify
Example
Q
1
would
(Ax
you
Discuss
1,
12
12
+
2
2
5
y
x
=
+
6
2
Continued
330
7
Linear systems
on
next
page
in
one
of
the
A LG E B R A

5

Substitute
4 x
− 3
x


+ 6
the
expression
into
the
other
=  −11


2
equation
to
solve
for
the
other
variable.
15
Use
4x
x
−
−
18
=
the
distributive
proper t y
to
expand
the
−11
2
brackets.
15
2(4x
x
−
−
18
=
−11)
2
8x
−
Multiply
each
eliminate
15x
−
36
=
−22
−7x
−
36
=
−22
S olve
+ 36
for
all
side
of
of
the
the
equation
by
2
to
denominators.
x
+36
−7x
=
14
x
=
−2
y
=
5
( −2 ) + 6
2
4(−2)
–
y
=
1
3(1)
=
−11
−11
=
−11
2(1)
=
12
12
=
12
Substitute
the
to
value
nd
Verify
−5(−2)
The
+
solution
is
(−2,
1).
the
your
value
of
answer
of
the
in
x
into
other
each
of
the
equation
for
y
variable.
the
original
equations.
Write
the
solution
as
an
ordered
pair.
3 31
Activity
According
women
related
cause
is
a
to
and
to
of
per
15%
the
1
If
Health
die
every
cooking
in
by
Organization,
year
smoke.
children
under
from
5
years
approximately
lower
Globally,
organization
supplying
in
a
use
buy
their
stove
Copy
World
Project
this
of
age.
that
clean-burning
respiratory
is
is
the
The
to
million
diseases
number
one
Paradigm
trying
cooking
4
Project
address
stoves
to
this
people
in
countries.
They
they
of
Paradigm
entrepreneurial
living
day.
day.
the
children
issue
family
The
indoor
developing
A
–
death
social
serious
4
a
poor
30%
pay
and
of
area
their
for
on
complete
with
this
fuel.
the
of
to
pay
for
costing
How
money
has
many
income
cooking
$35,
they
an
they
days
save
fuel
will
will
on
it
of
$5
each
only
take
cooking
use
for
fuel?
table.
Total
Number
Guatemala
stove
cooking
itself
in
income
clean-burning
income
to
rural
cost
of
fuel
Total
cost
of
fuel
days
(no
stove)
0
(including
0
new
stove)
$35
1
2
3
4
5
2
Graph
3
Determine
4
relationship
the
equations.
Verify
your
answer
What
are
the
eciency
Once
you
the
other
How
7
it
will
quality
332
of
solve
each
solving
the
line,
the
linear
and
system
apart
of
the
from
the
fuel
savings?
the
the
life
is
paid
family
saves
of
benets
stove,
stove
think
money
7
by
and
substitution.
clean-burning
6
equation
linear
by
5
each
on
stove
for
o,
will
what
buy
else
with
do
the
fuel?
improve
the
Linear systems
family?
the
overall
system.
hence
create
a
system
of
A LG E B R A
Practice
1
2
3
Solve
the
following
a
4x
+
y
=
1,
c
5x
–
y
=
−1,
e
5x
−
2y
Solve
a
y
=
c
5x
e
y
6x
–
=
3y
−2,
phone
=
to
break
even.
d
When
Which
Solving
how
much
will
4
4
by
graphing
7y
=
9
business
to
that
start-up
y
=
60x
–
d
2x
f
x
7y
–
=
=
5y
−5,
by
0,
=
3x
6,
4x
+
2y
–
y
d
3x
+
7y
–
1
f
6x
+
4y
=
14,
cell
=
−8,
2x
=
=
phone
+
0,
5x
You
sell
the
phone
=
30x
+
1300
−6
Verify
3y
=
=
+
=
answer.
3x
9
with
are
each
10
8
2y
covers
business
=
−18
–
the
answer.
17
4y
2y
for
y
=
+
substitution.
4x
and
each
4y
−3x
b
sells
–
costs
manufacture.
equations
x
and
−2
a
b
Verify
the
aim
of
$1300
and
each
covers
for
represent
$60
the
donating
cell
each.
revenue
and
you
substitution
revenue
start
do
linear
seen,
ecient,
of
that
by
to
you
you
make
prefer,
to
will
a
nd
make
prot?
graphing
how
at
many
the
covers
break-even
you
must
sell
in
order
to
point.
Explain.
or
substitution,
or
does
it
depend
on
the
Explain.
have
not
+
$30
system
method
one
method
–
=
2x
=
the
substitution.
functions.
Find
does
why
3y
charity.Your
costs
using
−7
starting
local
systems
systems
–
6y
are
the
isolate
3
=
+
3y
−3x
+
Solve
very
3y
=
=
−2,
a
Explain
you
x
y
−5x
question?
be
5,
you
c
As
+
−10x
6,
cover
cost
b
2x
linear
following
–
Imagine
a
=
the
money
4
4
rely
on
solving
a
especially
the
can
systems
variables.
be
used
to
isolating
a
by
elimination
linear
when
The
system
it
is
fairly
method
solve
any
by
of
substitution
can
straightforward
elimination
linear
system.
is
This
to
another
method
variable.
3 3 3
Activity
1
Given
5
the
–
equation
by
2
b
multiply
the
equation
by
−3
c
divide
d
Explain
the
equation
why
Given
the
to
multiply
the
multiply
multiply
the
the
By
what
The
−4x
is
to
solve
linear
to
manipulate
is
y
x
5y
=
c
are
−7,
an
integer
so
that
the
by
an
integer
so
that
the
by
an
integer
so
that
the
you
uses
multiply
to
the
the
the
equation
original?
concept
Creating
more
following
through
by
could
system.
Perform
a
4.
method
6
in
equation.
equivalent
Activity
created
−25
is
not
you
12
is
them
–
+
equation
of
elimination
a
x
number
equation
original
equation
of
coecient
d
the
equation
of
−2.
equations
equation
coecient
c
the
by
equations
8,
the
b
&
=
multiply
coecient
1
2y
–
a
a
ATL
4x
equation
equivalent
2
Equivalent
of
equivalent
so
that
the
new
Explain.
equivalence
equations
in
order
allows
you
easily.
Elimination
method
2
the
activity
in
pairs.
Your school is putting on an event to support an upcoming fundraising
walk. Visitors to the event can buy the 2/1 combo (2 slices of pizza and
Pairs
adrink) for $5.20. They can also buy the 6/2 combo (6 slices of pizza and
2 drinks) for $14.40. How much does one slice of pizza cost? How about
onedrink?
1
Starting
subtract
many
2
the
drinks
Repeat
result
with
the
in
the
rst
are
expensive
combo
left
same
step
more
and
from
what
process,
it?
is
combo,
How
their
what
many
cost?
subtracting
the
happens
slices
of
Explain
2/1
if
pizza
your
combo
you
and
from
your
1.
Continued
334
7
how
answer.
Linear systems
on
next
page
A LG E B R A
3
4
What
does
is
price
the
Represent
the
Multiply
How
7
does
think
this
Refer
back
rst
8
are
step
8?
Find
the
Consider
you?
original
to
1
equation
the
the
by
is
of
How
How
to
the
what
linear
−3
of
linear
could
x?
you
y?
the
you
Solve
13
Summarize
can
and
●
by
Which
of
pizza?
What
system
of
of
slices
equations.
of
pizza
Use
and
d
to
by
2
and
combo
you
did
2
then
subtract
this
equation.
in
steps
1
and
2?
Why
do
you
elimination?
then
and
d.
Verify
you
add
created
it
to
the
dierences
your
2x
combine
combine
the
and
either
by
solution
− 5 y
+ 20 y
in
step
second
between
4.
Multiply
the
equation.
step
5
and
in
both
original
equations.
= 12
= −18
the
two
equations
to
eliminate
the
the
two
equations
to
eliminate
the
elimination.
process
of
add
or
which
you
operation
result
linear
equations
does
multiply
will
a
your
answer.
system
by
elimination.
5
subtract
How
Verify
solving
discuss
method.
subtracting?
●
a
number
system
system
system
elimination
value
slice
Explain.
12
You
a
Explain.
could
variable
●
the
called
similarities
value
the
Reect
of
Explain.
variable
11
price
with
equation
2x
10
the
drinks.
from
relate
method
to
is
combos
represent
combo
this
What
drink?
number
equation
What
9
p
the
the
tell
a
the
equivalent
6
of
variable
represent
5
this
your
the
in
when
choice
of
equation(s)?
making
fewer
you
use
the
operation
aect
the
Explain.
mistakes,
adding
or
Why?
Once
you
have
the
value
value
of
the
other
value
of
the
second
one.
of
the
What
rst
variable,
options
do
you
you
need
have
for
to
nd
nding
the
the
variable?
3 3 5
Example
Q
Solve
3
the
following
x
−
y
= 1
x
+
y
=
linear
system
using
the
elimination
method.
3
2
(2 x
A
− 3 y
= 1
)
×
3
In
3
3x
+ 5 y
=
×
− 2
top
−6 x
− 10 y
–
a
multiplied
variable,
by
a
both
value.
equations
Multiply
that
by
add
the
3
and
the
the
coecients
bottom
of
x
are
equation
6
and
−6,
equations.
3(0)
= 0
=
1
to
2x
=
solve
the
for
solution
the
other
into
one
of
the
equations
variable.
1
Another
option
is
to
use
elimination
again,
1
this
x
time
eliminate
the
y
=
2

1




2
 −  3 ( 0 ) = 1
Verify
your
answer
in
each
of
the
original

equations.
1
3

1



2
−
0
=
5(0)
=
1
3
+

2
3
3
=
2
2

The
solution
7
1

,
is

336
by
and
= 3
Substitute
2
the
= −3
− 19y
2x
eliminate
equation
so
then
− 9 y
to
be

−2
6x
to


2
order
need
0
2
Linear systems

Write
the
solution
as
an
ordered
pair.
but
A LG E B R A
Activity
A
benet
use
their
concerts
(2003),
Live
Hand
and/or
8
in
help
do
the
the
to
a
for
Write
of
school
for
and
money
local
One
Hurricane
to
see
and
awareness
Some
Conspiracy
The
Love
Relief
what
musicians
abroad.
(2001),
(2007),
for
or
(1985),
City
which
they
SARS
of
for
large
Hope
Benet
Manchester
a
and
Tour
Concert
(2017),
(2017).
were
raising
money
has
like
for
the
great
20%
crew
an
to
children
local
of
is
ongoing
organize
the
a
in
bands
price
$4000.
of
You
connection
concert
the
to
with
raise
orphanage.
play
each
are
to
and
ticket
going
to
You
they
sold.
sell
an
money
have
have
The
agreed
rental
tickets
to
of
the
$20.
down
represent
concert
Earth
in
for.
some
concert
Aid
York
concerts
school
auditorium
raise
either
Live
Benet
would
get
help
New
Live
A
big
your
You
build
concert
2
these
to
performance
crisis,
been
for
(2005),
that
managed
to
have
Hand:
concerts
musical
status
awareness
orphanage.
to
a
Concert
Research
Imagine
is
humanitarian
The
1
Benet
celebrity
(1986),
and
–
concert
particular
famous
7
in
tickets
an
the
equation
revenue
terms
sold.
of
the
Justify
to
from
the
number
your
equation.
3
Write
down
represent
on
the
4
in
to
putting
terms
tickets
the
sold.
of
the
Justify
revenue–cost
graphing,
break-even
5
Solve
6
The
the
and
a
system
tickets
total
will
nd
system
the
point.
auditorium
raise
of
equation.
Solve
by
of
equation
cost
concert
number
your
an
the
of
by
has
elimination.
a
capacity
$10 000.
you
need
to
Is
this
sell?
of
1000
going
Show
to
people
be
your
and
your
possible?
If
goal
so,
is
how
to
many
working.
3 37
3
–
Classifying
systems
of
equations
i
II
t
er
i
r
Investigation
are
three
linear
n
c
Here
systems.
B
A
y
=
B
2x
+
2y
1
4y
y = –x + 7
1
You
have
already
classifying
based
on
–
8
2x
=
=
x
–
6x
these
linear
(Investigation
2).
Solve
each
system
by
substitution.
3
Solve
each
system
by
elimination.
4
Write
a
summary
linear
system
5
Verify
your
6
Justify
Practice
1
Solve
when
results
your
each
20
2y
–
systems
Write
in
down
the
the
rst
investigation
classication
of
on
each
system
results.
2
brief
=
y + 3x – 5 = 0
16
graphed
systems
those
+
C
of
using
with
results.
the
at
Why
an
results
your
algebraic
least
do
of
two
they
investigation
into
how
to
classify
a
method.
more
make
linear
systems.
sense?
5
system
a
of
linear
equations
using
b
the
method
c
x
+
−3x
d
e
g
h
of
5y
–
6
elimination.
=
−2
=
15y
f
4x
−10x
+
−
8y
20y
=
=
−3
i
12
j
Continued
338
7
Linear systems
on
next
page
A LG E B R A
2
a
Verify
b
How
three
did
method
3
a
Verify
of
you
of
the
linear
decide
which
graphing?
three
of
the
systems
in
linear
question
systems
1
to
by
graphing.
verify
using
the
Explain.
systems
in
question
1
using
the
method
of
substitution.
b
How
did
method
ATL
1 &
Reect
you
of
decide
which
substitution?
and
discuss
linear
systems
to
verify
using
the
Explain.
6
2
Discuss the following with a peer.
●
Explain
when
you
would
use
the
graphing
method
to
solve
a
linear
Pairs
system.
●
Explain
when
you
would
use
substitution
●
Explain
when
you
would
use
elimination
●
Make
a
ow
depending
graphing
and
can
how
to
often
solve
nd
of
the
to
use
x-
and
a
need
be
dierent
information
solving
decisions
they
the
method
gradient,
Problem
When
chart
on
to
when
be
modeled
linear
Be
graphing
a
nd
using
can
based
linear
help
solve
solve
you
sure
linear
made
system
to
strategies
given.
y-intercepts,
with
to
to
a
linear
linear
would
system.
system.
use
include
linear
two
a
system
which
( y-intercept
points).
systems
on
a
variety
of
relationships.
you
to
make
factors,
Knowing
an
informed
choice.
3 3 9
Activity
8
–
Creating
a
system
of
equations
ATL1
Translating
problem.
information
With
a
into
partner,
a
copy
system
and
ll
of
in
equations
this
is
an
important
rst
step
in
solving
a
table.
Pairs
Solving
the
system.
Creating
Information
Dening
variables
Which
method?
equations
Justify
The
sum
their
of
two
dierence
numbers
is
is
1211
and
283.
Let
S
represent
smaller
Let
L
represent
larger
Four
is
times
16 g
the
less
the
The
sum
basketball.
is
of
a
of
represent
L
Let
b
____
Let
B
Let
__
represent
____
Let
__
represent
____
Let
__
represent
____
Let
__
represent
____
Let
__
represent
____
Let
__
represent
____
Let
__
represent
____
Let
__
represent
____
a
their
masses
represent
hamburgers
$5.00.
drinks
Three
cost
a
The
with
length
triple
its
in
scored
drink
cost
and
____
two
a
perimeter
which
is
2 m
of
less
28 m
than
width.
Tritons
shots
a
$7.90.
rectangle
has
and
hamburgers
a
a
made
a
total
basketball
total
combination
of
of
96
of
game.
points
2-point
42
They
through
and
a
3-point
shots.
Seven
times
numbers
number
the
the
340
is
larger
eleven
smaller
nine
178.
is
the
of
times
When
number
times
result
7
the
plus
is
two
the
ten
larger
times
increased
smaller
230.
Linear systems
by
number,
+
L
=
1211
the
number.
baseball
mass
of
S
756 g.
Two
A
mass
than
the
number.
–
S
=
283
your
choice.
A LG E B R A
Organizing
with
the
use
Example
Q
You
A
of
If
information
a
in
problems
is
sometimes
made
table.
you
$325
in
counted
donations,
46
bills
in
some
in
all,
how
bills.
$5
bills
many
of
and
others
each
type
in
of
$10
bill
do
have?
Let
x
represent
the
number
of
$5
Let
y
represent
the
number
of
$10
Write
a
statement
Number
$10
5x
y
10y
46
325
bills
TOTAL
x
y
=
46
10y
=
325
y
=
46
y
=
46
+
Value
x
bills
($)
Setting
+
x
+
−
up
a
equation.
set
up
up
an
t wo
Select
the
solve
10y
=
325
5x
+
10(46
−
x)
=
325
5x
+
460
−
10x
=
325
−5x
+
460
=
325
−5x
=
−135
x
=
27
each
in
the
most
the
each
of
the
to
represent
appropriate
this
organize
contains
the
information
the
question.
question
In
help
column
equations
information
x
might
to
elimination.
+
char t
–
to
5x
what
represents.
information
S et
5x
explaining
bills.
variables
$5
easier
4
receive
bills.
you
the
–
and
either
case
use
ecient
method
substitution
or
substitution.
S olve.
x
+
y
=
46
27
+
y
=
46
y
=
19
19
=
46
10(19)
=
325
Substitute
27
5(27)
+
+
equations
Verify
Write
You
have
19
$10
bills
and
27
$5
your
a
the
to
solution
solve
answer
concluding
for
in
into
the
the
one
other
of
original
sentence
the
original
variable.
equations.
with
the
answers.
bills.
341
Example
Q
You
invested
interest
has
A
each
more
you
5
$12 000
year.
risk.
invest
in
If
you
each
into
The
two
other
earned
types
bond
$1160
of
bonds.
yields
12%
interest
in
One
bond
interest
a
year,
how
Write
a
x
be
the
amount
invested
at
8%.
Let
y
be
the
amount
invested
at
12%.
risk
is
not
much
statement
variables
at
8%
year
but
did
bond?
Let
Note:
yields
each
factored
into
the
explaining
what
each
of
the
represents.
question
all.
Amount
Interest
Interest
invested
rate
earned
8%
x
0.08
0.08x
12%
y
0.12
0.12y
Total
12 000
Setting
up
a
char t
information
set
up
an
–
might
each
help
column
organize
contains
the
information
to
equation.
$1160
Set
x
+
y
=
12 000
0.08x
+
0.12y
=
1160
−0.08x
−
0.08y
=
−960
+0.08x
+
0.12y
=
1160
0.04y
=
200
y
=
5000
in
up
the
Select
to
two
equations
to
represent
the
information
question.
the
solve
most
the
appropriate
question
substitution
or
Multiply
rst
the
–
in
and
this
elimination.
equation
Use
by
ecient
case
method
either
elimination.
−0.08
Solve.
x
y
=
12 000
5000
=
12 000
+
Substitute
x
+
solve
7000
+
x
=
7000
5000
=
12 000
Verify
0.08(7000)
+
0.12(5000)
560
You
invested
bond
(12%)
bond
(8%).
+
600
$5000
and
in
=
1160
=
1160
the
$7000
in
higher
the
7
Linear systems
the
the
your
solution
other
into
one
of
the
equations
variable.
answer
in
the
original
equations.
risk
lower
risk
Write
342
for
a
concluding
sentence
with
the
answers.
to
A LG E B R A
Reect
and
discuss
7
Can you think of a
real-life situation
that could be
modeled using a
linear system in
which there is no
solution? Can you
think of one in
which there are an
innite number of
solutions?
Activity
A
coee
9
–
farmer
kilogram
to
Go
in
grow
Costa
and
land
management
for
guaranteed
a
1
Create
2
Solve
a
linear
the
Rica
currently
harvest
per
$1.50
year.
per
system
system
substitution,
big?
the
She
coee,
can
sell
costs
plus
her
of
80
$1000
green,
cents
of
per
xed
costs
unroasted
for
beans
kilogram.
model
using
has
two
elimination)
in
for
of
this
the
order
scenario.
three
to
methods
calculate
(graphing,
the
break-even
point.
The farmer knows that her coee is being sold to consumers for $8 per
kilogram and is considering getting into the rest of the supply side of
the business. A social entrepreneur said she would help her start up a
farmers’ cooperative where they will grow, roast, package and export
the coee directly to the customers. The costs associated with the entire
process are $5 per kilogram and the xed costs for growing, processing
and exporting the coee would be substantially more at $8000 per year.
3
Create
4
Solve
to
a
linear
the
system
system
calculate
your
the
model
using
the
break-even
answer
using
for
this
method
new
you
scenario.
did
not
use
before
in
order
point.
5
Verify
one
of
the
other
two
methods.
6
Determine the prot function for both scenarios. The farmer produces
6000 kilograms of coee per year. What is her prot in both scenarios?
7
Based
Justify
on
your
your
analysis,
what
would
you
recommend
that
she
do?
answer.
Continued
on
next
page
34 3
8
Why
two
9
is
there
such
a
large
dierence
between
the
prot
made
in
the
scenarios?
What
are
the
advantages
and
disadvantages
of
starting
the
cooperative?
Practice
Answer
the
6
rst
three
questions
in
pairs.
When
someone
encounters
diculty,
ask
ATL2
leading
1
Ann
bills
Pairs
2
A
questions
has
$300
does
she
parking
instead
made
of
up
giving
of
$5
the
and
answers.
$10
bills.
If
there
are
39
bills
machine
quarters.
dimes
3
Saira
one
at
4
were
each
Plan
contained
will
5
A
be
total
girl
in
to
a
20
part
$3.05
coins
interest
at
8%
from
plays
she
in.
many
oers
gifts
developing
school.
equipped
manager,
How
many
made
up
of
dimes
in
all.
How
many
dime
quarter
is
10
is
25
cents
and
7
a
cents.
per
annum
these
and
the
investments
remainder
was
$84.
at
How
9%
per
much
annum.
did
she
and
soccer
asked
The
If
how
for
buy
is
oered
she
20
options
It
costs
gifts
girls
at
will
negotiating
$800 000
would
hope
country.
many
player
team
games
you
of
for
the
$6000
need
to
a
€500
total
be
her
play
for
to
to
After
invest
of
the
a
a
gift
to
support
schoolroom
€6515,
how
and
many
Linear systems
a
€295
to
classrooms
school?
an
game
select
equip
contract.
plus
every
you
cost
sent
year,
for
where
Using
the
additional
played
team’s
and
oer
to
advice
$1500
for
$600 000
be
the
of
her
every
for
better
game
the
year.
option?
Continued
34 4
$5
rate?
professional
she
were
$1000,
International
a
how
there?
her
community
send
There
invested
year,
all,
have?
A
and
in
on
next
page
A LG E B R A
6
Supplying
were
24
benet
7
Jen
Two
well
gifts
of
and
22
social
adults,
event,
Many
human
is
of
being
United
get
with
in
a
10
a
that
the
year
basic
Costco
before
system
the
to
show
$160
per
hour.
What
Over
1
billion
service
is
Indian
800 000
If
each
that
how
the
solar
per
people
many
range
lanterns
Rs
350
sells
the
for
who
lanterns
company
is
and
per
3
she
same
the
at
of
least
the
the
for
lantern
after
its
A
If
there
would
one
be
year
the
was
paid
and
live
the
cost
the
each
more
social
use
one
is
have
it
to
day
and
adult
prohibits
physical
of
conduct
those
and
per
in
you
to
the
year.
at
the
You
back
Costco
buy?
Set
up
problem.
them
teenagers
a
for
teenager?
paid
in
nothing
spend
paid
are
this
Costco
code
solve
and
the
attend
which
$120
for
5
to
membership
to
and
teenagers
the
the
Costco
and
a
events?
especially
option
adults
the
where
billion
A
3
to
all
and
pay
for
children
tracking,
ensure
you
30
chains.
membership
better
for
were
suppliers
human
funds
attended
both
membership
would
the
for
global
To
raise
adult
supply
basic
premium
is
each
same
to
there
suppliers,
options
rate
and
What
adults
night
example
of
20
ethical
for
for
group
did
the
more
employed
grid.
which
were
money
rate
1
much
teenagers
world
interest
and
second
premium
much
hourly
the
the
violations.
membership
day
scout
children
facilities
the
and
AU$35.
bought?
The
environments.
with
costs
the
total
a
of
total
of
(Assume
same
rate.)
electrical
people
entrepreneur
of
xed
materials,
costs
are
in
Rs
month.
lantern
the
What
is
while
local
How
membership
the
paid
On
such
How
around
of
of
adults
was
are
out
the
5
unreliable
rupees,
year,
following
people
completely
manufactures
risk
25
chain,
the
were
10%.
conduct
working
hive
family?
a
prices
of
bee
each
at
toward
supply
purchases
employed
adults
code
premium
The
the
at
by
£170.00.
audit
membership.
hour.
all
may
per
per
live
b
$60
a
a
a
rate?
night,
ticket
of
to
each
moving
their
more
of
and
rest
£150.00.
the
unsafe
they
$224
grid
a
is
rst
was
has
in
and
are
on
contractor
that
11
abuses
the
organized
the
are
at
many
hive
and
invest
that
AU$50
how
bee
9%
revenues
today
enforced,
back
linear
A
with
costs
or
been
Corporation
States
2%
have
evening
workers
countries
she
assuming
rights
at
During
companies
Wholesale
abuse
that
sheep
part
did
Tanzania.
family
AU$990,
a
$8000,
events
for
a
totaling
much
social
in
to
supplying
How
revenue
9
sheep
invested
$740.
8
a
to
Rs
really
must
break
revenue
475
(costs
need
be
are
them
made
kept
can
and
low
buy
sold
to
ensure
them),
each
nd
month
for
even.
at
the
break-even
point?
Convert
this
amount
into
your
local
currency.
Continued
on
next
page
34 5
12
Find
the
point
13
A
of
equation
of
intersection
rectangle
algebraic
has
the
of
4x
vertices
solution
to
line,
–
at
in
y
=
standard
3
A (2,
and
2),
determine
2x
+
B (2,
the
form,
y
6),
=
that
passes
through
(–7,
3)
and
the
9.
C (4,
coordinates
6)
of
and
the
D (4,
point
2).
of
Develop
an
intersection
of
the
diagonals.
Thailand
is
producers.
the
xed
one
For
costs
of
a
(in
per
the
costs
hectare.
per
a
Rice
Thai
rice
is
are
sold
are
season
and
12 000 THB
for
rice
farm,
baht)
growing
top
per
19 500 THB
hectare.
Set
up
many
break
a
the
c
Convert
average
live
this
on
Suppose
the
prices,
the
the
to
nd
would
of
a
how
need
be
can
the
rice
to
to
of
be
new
is
the
local
currency.
season?
If
break-even
–
what
Do
farmer
the
you
The
has
selling
point?
How
no
price
will
social
does
think
this
selling
control
drops
this
Sustainable
corporate
prot
the
is
farmer
enough
make?
money
Explain.
downward.
small
signicant.
starting
hectares,
price
over
by
aect
of
rice
is
uctuations
15%,
Thai
as
it
rice
has
by
in
recently,
farmers?
denim
responsibility
set
r
i
t er
i
c
are
local
6
trending
assessment
companies
is
growing
rice
and
farm
your
whole
price
which
will
size
market
Formative
More
you
amount
for
global
what
model
even.
If
to
linear
hectares
b
d
world’s
typical
16 000 THB
variable
the
a
14
(CSR)
C, D
reporting,
“people,
the
sometimes
planet
initiative
to
and
communities
Unfortunately,
order
work
and
to
in
cut
in
an
in
which
prot
costs,
can
that
is
still
be
as
triple bottom line
means
that
supply
chains
and
they
the
that
they
driving
force
especially
extensive
the
companies
are
have
(triple
ethically
positive
for
take
and
interactions
with
operate.
unhealthy
unsustainable
the
This
employees,
and
to
their
responsible
dangerous
shipping
used
prot”).
ensure
environmentally
the
referred
of
those
conditions.
and
the
Earth’s
many
in
companies
developing
Pollution
natural
in
and
in
countries,
may
manufacturing
resources
can
be
way.
Continued
346
7
Linear systems
on
next
page
A LG E B R A
Case
Two
one
study:
premium
is
a
denim
socially
common
in
jean
factories
responsible
factory
in
Asia
and
have
the
the
second
following
is
a
costs
traditional
per
month.
factory
of
a
The
type
rst
still
Asia.
Socially
responsible
Cost
Traditional
factory
factory
Materials
per
Wash/dye
pair
of
process
jeans
$12*
$7
per
$0.25
$2**
pair
of
jeans
Labour
cost
per
pair
of
$1.50***
$0.50
$10 000****
$5800
jeans
Rental
of
utilities,
building,
admin
programs,
*
All
raw
costs,
etc.
materials
environmental
**The
factory
uses
environmental
uses
non-toxic
rather
than
that
go
into
making
the
denim
jeans
meet
the
highest
ethical
and
standards.
organic
impact
and
organic
the
cotton
not
a
water
contaminate
vegetable
harmful
and
chemical
dye
to
dyes
recycling
local
ensure
used
in
system
community
that
no
harm
traditional
to
reduce
water
is
jean
the
supplies.
done
to
the
It
also
workers,
manufacturing.
*** Workers earn a living wage, which is 3 times the average of workers in the industry,
allowing them to better support themselves and their families and even to save for the
future.
****
in
Fixed
the
costs
factory.
production
are
require
every
component
Build
a
linear
more
of
jean
systems
to
time
better
courses
and
ventilation,
and
every
light,
training
worker
is
space
courses
rotated
to
in
and
all
general
facilities
components
ensure
they
are
of
jean
skilled
in
making.
model
Derive
b
If
c
Graph the cost functions and the revenue functions of both factories on one set of axes.
d
What
the
cost
due
language
a
the
a
higher
English
jeans
is
function
are
the
to
both
the
break-even
factories.
wholesaler
point
for
for
both
$30
a
pair,
factories?
determine
Comment
on
a
revenue
the
function.
dierence
between
two.
e
Verify
f
At
your
what
wanted
answer
price
the
How
much
by
would
same
appropriately
g
sold
for
the
explain
is
this
the
socially
break-even
and
more
solving
as
degree
pair
algebraically.
responsible
point
the
per
system
as
a
the
of
factory
Show
have
traditional
to
your
sell
factory?
a
working.
pair
Round
of
jeans
your
if
they
answer
accuracy.
percentage
of
the
original
price?
Continued
on
next
page
34 7
Reect
●
and
Describe
in
●
the
Given
end
the
the
nal
Do
Discuss
of
think
needed
in
be
for.
to
5
What
pairs
place,
following
to
get
makes
these
shipping,
stores),
up
both
in
solution
sense
problem.
companies
chains
the
still
8
your
(including
them
be
not
the
could
sells
price
supply
or
placement
jeans
you
of
steps
customer
factory
●
whether
context
product
of
discuss
the
nal
times
would
of
the
put
though
questions
selling
the
to
the
and
price
amount
the
approximate
jeans?
should
even
jeans
marketing
in
sustainable
this
groups
decreases
of
their
prot?
Explain.
3.
ATL1
●
Whose
responsibility
is
it
to
ensure
sustainable
growth?
Companies?
Governments?
Consumers?
●
Each
help
●
person
generate
How
could
current
●
If
you
i
n
a
and
were
discussion,
l
in
your
group
change.
factory
future
to
Who
with
do
the
you
ethical
role
of
think
is
practices
one
in
of
the
such
these
best
as
the
key
players
position
one
to
and
lists
generate
described
in
the
how
they
could
change?
case
study
aect
generations?
write
what
takes
a
headline
would
that
that
captured
headline
be?
the
Share
most
your
important
headlines
aspect
with
the
of
this
task
and
your
class.
k
b
e
Research
in
the
these
Activity
A
10-acre
of
a
entrepreneur
industry
and
premium
broader
denim”
textile
to
see
industry.
what
Find
out
some
the
companies
motivation
are
for
doing
creating
companies.
10
plot
nearby
“social
denim
–
of
city.
Urban
forest
One
has
planning
been
proposal
designated
comes
from
a
for
a
new
housing
development
company
that
aims
to
due
to
create
the
a
expansion
CO
neutral
2
community.
left
on
the
In
other
land
to
words,
convert
they
the
want
of
CO
to
the
develop
people
the
housing
living
in
the
estate
estate.
so
The
that
enough
problem
trees
they
are
must
2
solve
is
determining
Factors
An
to
the
number
of
houses
they
can
build.
consider:
average
human
exhales
around
2.3
pounds
of
CO
in
a
day.
2
The
average
tree
can
convert
48
pounds
of
CO
per
year.
2
There
Each
are
currently
house
will
approximately
have
a
700
trees
3000-square-foot
per
acre
in
the
forested
region.
plot.
Continued
348
7
Linear systems
on
next
page
A LG E B R A
Process:
1
What
units
2
Given that
will
x
you
use?
Explain.
will be the number of houses and
y
will be CO
production per year, what does
2
the gradient of each line represent?
3
Assuming
an
represents
working
4
Is
the
5
What
6
the
to
people
living
the
the
y-intercept
Determine
the
linear
sure
the
8
What
9
Graph
the
both
of
If
them
10
What
11
Verify
your
12
Given
does
the
will
linear
When
they
using
into
the
What
What
Why
working
positive
on
can
the
be
an
in
make
that
sure
linear
you
equation
show
all
of
that
your
the
trees
calculate
the
negative?
this
in
the
housing
development.
gradient.
Explain.
scenario?
set
using
of
axes,
making
technology
that
will
living
you
need
take
to
sure
(graphic
screen
hand
that
you
clearly
calculator
shots
of
your
or
show
the
graphing
solutions
and
in.
represent?
can
plot
or
sure
you
algebraic
original
to
same
done
which
houses
do
happen
How
you
would
How
if
you
are
think
used
would
How
happen
How
originals.
be
developments
originals.
would
change?
Make
the
of
method.
be
built
land
Show
and
that
all
still
will
be
working.
remain
used
ecologically
for
housing
viable,
and
the
calculate
percentage
forested.
change?
the
development.
represents
intersection
using
of
determine
Explain.
represent
report,
the
house,
being
that
built,
are
these
percentages
maintained?
How
are
is?
model:
would
would
with
the
of
that
your
line
This
of
each
represent?
technology,
point
left
housing
of
this
your
solution
dierent?
Changing
all
in
housing
negative?
equations
number
be
or
y-intercept
percentage
that
of
the
equation
intersection.
software).
the
show
gradient
does
point
copy
you
in
living
gradient.
does
Will
15
people
calculate
4
positive
7
14
of
gradient
Make
13
average
it
would
if
there
change?
that
people
that
it
would
5
change?
were
aect
only
living
the
500
Determine
aect
the
in
Determine
point
point
trees
the
of
each
the
of
per
new
house
new
as
linear
the
average?
equation
Which
and
line
compare
it
intersection?
acre
of
linear
the
forest?
equation
Which
and
line
compare
would
it
with
the
intersection?
34 9
Unit
summary
A
linear
A
quadratic
A
cubic
An
equation
is
an
equation
equation
equation
is
equation
is
an
an
degree
equation
equation
generally
of
of
has
the
is
group
of
degree
degree
same
one.
two.
three.
number
of
solutions
as
its
degree.
A
system
for
at
If
each
least
the
of
two
Linear
of
equations
equations
linear
the
a
variables
systems
nding
Linear
equations
the
two
system
or
of
in
are
uses.
order
both
equations
can
point
systems
it
be
of
of
equations
With
to
nd
linear,
or
by
the
you
simply
solved
two
a
that
variables,
values
have
linear
graphing
can
of
be
you
both
what
is
solved
will
need
variables.
called
a
system.
each
of
the
lines
and
intersection.
can
also
be
solved
algebraically
using
substitution
elimination.
To
solve
a
equations
to
expression
To
solve
a
equations
then
you
add
can
Always
linear
isolate
into
the
linear
so
the
system
that
one
for
verify
of
other
system
the
the
your
substitution,
the
variables.
equation.
by
This
other
of
will
by
rewrite
Then,
Solve
elimination,
coecients
equations.
solve
by
x
or
this
one
are
eliminate
new
this
equation.
one
exact
the
the
substitute
multiply
y
of
or
both
opposites
variable
and
and
then
substitution.
answers
by
substituting
have
solution,
into
the
original
equations.
Linear
many
systems
no
one
solution
or
innitely
solutions.
Graphically,
a
parallel
to
you
given
are
variables
example
are
0
=
example
7
0
other.
the
There
same
there
0
or
2
−4
there
and
=
system
line
are
you
or
−4
=
is
Linear systems
are
no
solution
innitely
when
many
the
lines
solutions
are
when
twice.
and
you
many
end
solutions
up
with
a
when
true
all
of
the
equation,
for
−4.
no
end
=
has
innitely
eliminated
Algebraically,
eliminated
linear
each
Algebraically,
350
can
7.
solution
up
with
when
an
all
of
equation
the
that
variables
is
not
are
true,
for
er
i
o
t
r
i
Key
review
to
Unit
review
Level 1–2
1
Solve
a
2x
b
3(5
c
5(x
x
n
c
Unit
7
+
=
levels:
and
Level 5–6
verify
your
Level 7–8
answers.
11
4)
+ 3
d
equation
2x)
+
question
Level 3–4
each
+
A
=
=
x
–21
3(x
–
3)
+
5
− 1
=
4
2
1
e
2
y
− 3 =
y
2
+ 4
3
3
(
f
+ 2) =
y
2 y
− 11
4
2
Sketch
the
of
each
a
y
=
–3x
b
y
=
2x
following
point
+
of
10
graphs
and
write
down
the
coordinates
intersection.
and
y
=
2x
–
5
3
–
6
and
y
=
x
− 6
4
3
Find
the
systems
exact
using
intersection
point
of
the
following
two
linear
technology.
1
a
y
=
x
–
8
and
y
=
−
x
− 4
3
1
b
x
=
5y
–
75
and
x
+
y
− 7 =
0
2
4
Solve
=
the
a
y
6x
b
3a
c
14x
d
2y
–
4
e
6x
=
2y
+
+
2b
+
following
7
=
21
=
and
16
=
x
–
and
–21y
and
8
3y
+
2a
–
5y
systems
12x
+
and
3y
and
linear
3b
2x
6x
–
–
=
+
=
5x
6
=
algebraically.
0
14
3y
=
12
=
0
–3
+
10
3 51
5
Classify
each
of
the
following
linear
systems
and
justify
your
answers.
6
a
4x
−
3y
=
15
b
4x
−
3y
=
5
and
8x
–
6y
=
10
c
2x
=
3y
−
6
and
4x
–
6y
=
24
When
ve,
7
the
the
Two
result
the
larger
A
sum
four-sided
is
and
smaller
10
You
=
times
negative
integers
integer
15
a
2.
Find
are
is
number
equal
gure
with
a
is
less
than
this?
result
is
3.
number
three
is
two
Find
4
twice
dollars’
your
than
of
divided
by
less
20.
than
has
What
If
three
double
a
the
width
kind
of
answer.
is
divided
numbers
double
of
is
than
120 cm
length.
two
worth
ten
number.
eight
numbers
these
less
to
its
and
smaller
perimeter
Justify
between
the
both
integers.
30 cm
have
9y
both
dierence
the
–
nd
quadrilateral
The
is
8x
four
smaller
integer,
which
9
of
consecutive
times
8
and
the
coins
if
2
triple
larger
in
by
your
the
number.
piggy
bank
A
consisting
of
nickels,
dimes
and
quarters.
The
number
a
dimes
is
double
one-third
your
11
is
You
and
piggy
could
pay
$1.14
a
month
How
to
song
songs
a
download
the
you
of
downloaded
Linear systems
or
nickels.
pay
The
How
number
many
an
service
average
of
download.
using
technology.
month
to
cheaper
think
nickels.
streaming
you
scenario
pay-per-song
you
7
this
many
service
352
each
of
of
total?
$9.99
b
the
in
music
Graph
Can
bank
number
a
a
c
the
number
use
for
have
of
the
would
make
the
you
streaming
option?
why
you
option
20
would
instead,
songs
a
choose
even
month?
if
of
coins
nickel
is
5
cents,
of
quarters
are
in
dime
and
25
a
is
10
cents
quarter
cents.
is
12
Some
talk
teachers
on
student’s
an
A
ticket
going
microlender
ventures,
with
one
In
one
higher
year
invested
14
some
have
in
aord
them
per
the
can
to
least
of
Year
chains
at
which
42
tickets
and
to
each
nd
cost
3
students
some
$582
teacher’s
the
low
AU$20 000
risk
yielding
number
in
a
social
total.
ticket
of
to
If
costs
teachers
each
$25,
and
in
yielding
8%
per
a
combination
3%
per
annum.
calculate
If
annum
the
algebraically
of
and
interest
the
two
one
after
amount
venture.
in
out
the
get
it
a
operate.
month
number
30
to
of
sold
for
for
They
cover
to
the
for
free.
$8.
must
the
meals
days)
to
meals
meal
is
United
meals
delicious
and
(assume
group
talk.
AU$850,
have
each
month
with
giving
$2.50
month
$2000
the
communities
started
is
$12
invested
each
still
meal
speak.
method
to
a
supply
costs
risk
was
trucks
is
will
algebraic
students
13
taking
sustainable
entrepreneurs
use
are
States,
homeless.
sale,
The
The
cover
of
need
the
cost
a
to
to
free
make
cost
who
of
of
trucks
the
cannot
make
truck
prot
the
food
While
people
food
make
costs
they
local
each
costs
at
least
meals.
and
the
$3000
What
sell
free
in
a
meals?
3 53
15
In
some
you
could
natural
to
areas
start
don’t
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up
other
16
You
into
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(€60
and
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light
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in
baby
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years
a
drill
would
where
$5000
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then
good
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for
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it
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many
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with
many
packages.
354
gas
and
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nets
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blankets
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blankets
to
prevent
and
with
medicine
the
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mosquito
In
the
the
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number
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kits.
package
calculate
care
number
package
the
line
Would
second
are
drill
How
you
spread
(€20
the
medicine
If
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The
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gas.
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you
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grid.
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electricity.
year.
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The
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own
boxing
countries.
world,
worth
line
factors
are
bites
each
your
gas
the
for
$7500
use
the
drill
gas
produce
of
Linear systems
medicine
the
of
nets
each
kits
and
of
the
and
solar
four
blankets
kits
cost
items
cost
€3000
€3400,
there
are
in
the
assessment
r
i
t er
i
a
c
Summative
C, D
Running
Part
Your
set
1
–
the
to
event.
show
Part
You
a
by
to
2
–
will
or
a
a
raising
need
to
what
to
a
to
start
all
a
a
day
at
products
which
that
organization
and
yer
out
business
like
you
all
the
will
made.
would
your
like
prots
the
from
charity
event.
pricing
to
to
have
summarizing
during
and
they
that
donating
students
make
make
your
(for
product.
example,
baked
jewellery).
the
your
the
or
hand
would
needed
of
sell
costs
toy,
nd
up
fundraiser,
one-page
and
up
you
simple
utensils
itemize
set
a
awareness
create
for
event
to
charity
Business
a
business
hosting
market
nd
online
need
be
customers
Decide
Go
at
Then
goods,
b
will
and
support
own
fundraising
booths
Research
to
A
school
up
your
approximate
business
to
make
costs
so
cost
of
(including
your
that
goods).
you
can
everything
buying
Make
include
all
sure
you
the
will
tools
you
them
in
your
report.
c
Approximate
sold
(i.e.
Make
per
d
of
sure
item
the
you
and
Decide
how
It
be
must
sell
the
the
a
cost
making
materials
show
include
much
all
it
you
realistic
product
of
but
you
of
in
will
your
your
would
selling
also
use
individual
to
make
working
to
item
the
to
be
product).
calculate
the
cost
report.
like
price
raise
each
to
so
charge
that
money
for
for
you
your
will
your
be
product.
able
to
charity.
3 5 5
Part
a
3
–
Create
Determine
product.
the
a
linear
equation
What
do
the
model
that
represents
gradient
and
the
cost
y-intercept
of
making
represent
in
the
this
equation?
b
Determine
that
and
c
you
Graph
make
The
the
the
by
How
of
costs
many
as
this
Verify
your
selling
the
system
well
as
equal
in
this
the
you
product
of
the
do
What
have
point
so
revenue
(the
do
the
money
gradient
equation?
created.
the
break-even
prot
you
Remember
to
intersection.
represents
revenue,
your
represents
product).
that
intersection
of
(Solve
that
represent
linear
axes
point
where
e
equation
y-intercept
label
d
the
at
have
this
to
point,
point
sell
to
is
zero.
break
even?
graphically.)
point
of
intersection
by
solving
the
system
algebraically.
f
Write
a
general
product
sold
making
g
h
How
10
ii
1000
units
What
to
sell
money
of
units
your
of
happens
the
point
in
you
to
are
you
the
losing
lose
quantity
money
of
and
your
when
you
are
make
if
you
make
and
sell
product?
the
original
break
of
or
product?
intersection
to
parts
will
your
of
order
following
when
relating
prot.
much
i
and
a
to
statement
your
line
(gradient
(number
even)
linear
when
of
you
and
y-intercept)
products
alter
you
each
of
need
the
system?
This
so
i
Change
of
start-up
ii
Change
of
variable
iii
Change
of
selling
is
costs
(material
costs)
the
the
above
three
scenarios
in
a
sketch
separate
revenue
an
lines
increase
sets
in
and
of
one
a
calculate
points
axes
–
colour
decrease)
drawing
and
in
the
the
new
dierent
original
lines
cost
and
(showing
colours.
show
Label
both
the
lines
shifted
356
7
point
of
intersection
Linear systems
–
and
the
new
how
the
have
and
point
break-even
points.
how
of
intersection
changed.
original
break-
on
the
three
not
price
just
of
to
actual
even
each
do
costs
need
Show
conceptual
you
has
Part
Use
4
–
Y
our
publishing
eye-catching
report
software
yer
that
to
create
a
summarizes
professional-looking,
the
information
about
the
charity/organization.
In
your
a
●
list
you
a
●
report
of
costs,
derived
linear
verication
●
three
the
●
Describe
the
Given
point
costs
and
showing
and
you
calculations
determined
shows
cost
showing
the
and
selling
how
price
revenue
point
break-even
and
how
graphically
break-even
point
how
using
changes
break-even
algebra
in
costs
and
selling
price
points.
discuss
degree
whether
that
What
make
What
you
units
of
money
of
of
or
accuracy
not
could
such
the
of
your
answer
solution.
makes
sense
in
the
context
of
Given
only
your
you
have
product
that
you
estimated
do
to
one
do
ensure
day
you
to
sell
calculated,
selling
that
your
anticipate
will
product,
selling?
you
how
Using
make
or
the
lose
quantity?
you
make
money?
Can
you
even
assumptions?
other
school
will
your
intersection
using
factors
determining
how
do
you
many
of
need
your
to
take
into
product
you
consideration
will
make
for
when
the
fundraiser?
the
projected
product/model
it
explanation
problem.
many
●
an
which
and
Explain
●
the
graphs
●
●
of
sketches
Reect
●
all
and
●
the
with
model
functions
alter
include:
that
break-even
you
would
point,
is
change?
there
If
so,
anything
what
about
your
would
be?
●
How
●
What
can
is
you
our
make
a
dierence
responsibility
to
as
those
a
social
in
our
entrepreneur?
community
and
other
communities?
3 57
Answers
1
Number
Practice
1
1
a
Rational
b
Rational
c
Rational
d
Irrational
e
Rational
f
Rational
g
Irrational
h
Rational
i
Irrational
j
Rational
19
2
7
a
8
b
22
c
50
33
2587
g
5
2
3
3
a
d
b
Individual
response.
which
up
Practice
add
to
Individual
Possible
response
answers
are
of
74
99
99
form 0.abc , 0.xyz
1
are
c
three
digit
numbers
d
1
e
I
1
j
64
1000
1
g
64
1
7
l
m
27
o
4
1
p
r
0
b
0
, 5
1

1
, 3
,

5
c
0
4
“Zero
5
Correct.
5s”
6
“Half
7
Correct.
of
is
an
A

3
−2
is
,
of
not
multiplying,
exponent
times
1
is
taking
not
is
an
the

11

Exponential
form
7

10
an
10
kilo
10
the
of
times
the
Expanded
form
1 000 000 000
1 000 000
1000
1
1
10
10
1
2
centi
10
100
1
3
milli
10
1000
1
6
micro
10
1 000 000
1
9
nano
10
1 000 000 000
1
12
pico
10
1 000 000 000 000
1
15
femto
10
1 000 000 000 000 000
1
18
atto
10
1 000 000 000 000 000 000
3 5 8
Answers
2

base
base.
3
deci
5
10
e


5
2
−2
, 2
−3
, 1

1
d
0

,
1




2
1
, 3
0
,1

exponent
6
mega
,

9
giga

−1
c

4
8
Prex
d
2
exponent
number
by
1


taking
divided

,

8
4
1



positive
number
5

−1
, 3

multiplying,
8”
2
b


32
2
2
2
t
4
5
4
1
s
144
6
8
9
q
144
125
n
9
1
the
64
16
k
4
1
h
343
9
is
xyz
1
b
1
a
abc ,
2
f
3
where
999
a
a
225
c
36
2
j
55
25
1
1
2033
I
625
1
500
312
h
15
3
e
9
2
f
6437
d
5
value
is
multiplied
and
a
negative
exponent
(
7)
Practice
3
1
1
5
9
1
a
b
5
c
d
4
x
4
6
8
1
5
e
1
11
y
f
s
g
h
7
12
10
1
12
1
2
i
5
9
j
k
l
2
a
10
2
x
1
1
1
m
1
n
o
5q
w
p
11t
y
2t
6
m
a
3
30
3
q
r
6a
2
b
5
4
c
s
14 p
4
q
t
4
2
y
7
z
x
3
3
7qr
q
5
ef
u
v
w
p
x
6
2 p
5
a
3
3
b
1
15
2
a
18
10
b
1
12
0
c
10
d
e
24
5
10
2
3
a
3
e
2
4
10
1
b
c
10
d
equal
5
f
equal
b
2
1
4
1
a
c
2
d
12
1
5
a
Individual
Practice
response
b
m
C
4
81
1
9
a
b
3
c
4
9
25
f
g
1
h
7
i
j
9
21
a
e
16
3
2
10
d
25
1
9
5
b
c
5
a
d
a
h
648 y
1
2
y
4
5
2t
f
14
x
x
5
e
g
20
2
z
7
3
c
4
i
9x
3
5
y
j
k
6a
6
l
xy
4
2 ab
5
189 x
5
y
3
3
z
3
5c
6
m
n
6 g
2
h
o
p
7
a
2
16 a
ab
2
6
q
4c
8
d
10
e
r
11
s
6 cd
1
8
243 j
Practice
a
2
a
b
e
0.0
f
3
g
000
000
506
f
.8 × 10
b
.002
f
.00
807
000
−6
3
Smallest
to
4
2.99792 × 10
8
.658 × 10
6
c
2.1 × 10
d
c
9800
d
.65 × 10
9
5.69 × 10
1 450
4
2
5
2.35 × 10
5
e
t
2
k
5
4
1
5
b
largest: 302 × 10
−4
2
, 0.0025,14, 2.876 × 10
5
2
km/s;
9.80665 cm/s
;
000
000
002
, 9.83 × 10
5
700
4.5 × 10
3
, 7.8 × 10
8
teslas ;
2.25 × 10
4
, 1.42 × 10
9
m/s; 10
meters
100
100
5
a
10
10
b
Practice
Individual
a
c
100
10
d
1
6.5 × 10
b
e
6.916 × 10
i
3.36 × 10
2
.43 × 10
11
c
5
5.66 × 10
4
d
f
.724 × 10
j
.149 148 381… × 10
g
6.4 × 10
k
5.776 × 10
h
6
q
097 744 K
n
a
610 176 × 10
2 30
76 9
7
l
.6248 × 10
2
o
1.31 578 947 … × 10
2
p
9.6824 × 10
21
1.95 916
005 376 × 10
19
2
2.7
6.394
5
11
8.89
.312 × 10
2
1
m
× 10
6
6
1
response
1.84 × 10
7
b
.1536 × 10
11
c
5.85 × 10
(to
3
s.f.)
Answers
3 5 9
13
3
a
4.375
b
5.03 × 10
b
862
6
2
m
2.51 × 10
c
2
5
m
1.01 × 10
d
2
m
4
e
5.22 × 10
4
a
22
5
4.32 × 10
2
hours
or
about
36
days.
c
1.30 × 10
(to
3
s.f.)
4
Unit
1
km
Review
a
Rational
b
Irrational
c
Rational
d
Irrational
e
Rational
f
Rational
g
Rational
h
Rational
2
2
1157
a
b
217
10
9
8
, 10
, 10
7
, 10
6
, 10
6661
g
h
500
, 10
9
3013
f
11
5
990
4
, 10
, 10
1
4
10
431
90
10
d
99
e
22
c
9
3
31
3
, 10
2
, 10
1
, 10
0
, 10
−
1
, 10
900
−
2
−
3
, 10
, 10
−
4
, 10
−
5
, 10
−
6
, 10
1
a
9
b
c
125
3
d
27
5
16
5
e
49
f
g
2
4
h
2
36
7
a
5
5
a
b
1
c
28m
5
7
n
d
1
e
18 x
y
3
b
9
2x
2
g
2
h
y
a
2 c
7
a
39.2
b
7
2 × 10
a
647 × 10
days
f
3.038 × 10
b
288
7.75 × 10
c
−7.87 × 10
b
5.531 × 10
f
4.842 × 10
10
a
8.569 × 10
7 p
11
5.627 × 10
c
8.569 × 10
4
10
b
2.7158 × 10
b
Stars
11
5
greater,
by
6
factor 8.17 × 10
6
q
12
1
v
b
a
6
a
7
4
2
c
6m
2
a
9
b
5
b
x
19
5
n
m
7
d
10
64 u
e
y
c
6
r
4a
d
3
10
13
476 × 10
4.973 × 10
9
a
2.
190
9
4.491 × 10
9
3
d
days
2
5.8 × 10
13
e
c
1
8
8
2n
3
1.2 × 10
1.19 117
4
x
5
e
i
4
x
14
6
5m
105
24 y
f
2
6
10 j
f
g
7
7
4
3b
11
2 g
0.4
12
2
a
10
b
2
h
6
2.2
(≈2.51)
c
10
(≈158)
Triangles
Practice
1
10
6
1
a
9.2 mm
b
a
Yes,
b
No,
the
c
Yes,
3
a
40 cm
4
7.07 cm
5
a
6
90.0 m
to
3
s.f.
7
60.2 m
to
3
s.f.
8
No:
the
triangle
triangle
the
8.4 m
contains
does
triangle
not
c
a
right
a
right
d
24 cm
e
10 m
f
8.7 m
angle
contain
contains
77.2 cm
a
right
angle
angle
2
to
5.77 m
3
to
b
1200 cm
b
3.88 m
s.f.
3
s.f.
to
3
s.f.
2
the
inches
similar
than
maximum
=
39.9
argument
42
height
inches,
for
of
which
the
television
longer
than
comparative
inches).
3 6 0
the
is
Answers
can
the
heights,
be
13
width
or
inches
of
the
and
this
cabinet
alternatively
nd
corresponds
(note
the
that
it
is
diagonal
to
a
television
also
of
the
width
acceptable
cabinet
to
and
of
42
provide
show
2
13
a
this
is
less
9
a
5
b
10
Work
13
c
15
d
9
out:
2
2
π × 1.5
area
of
semicircle
with
diameter
3
(a)
+
area
of
semicircle
with
diameter
4
(b)
π × 2
25 π
+
=
=
2
2
8
2
π × 2.5
area
of
semicircle
with
diameter
5
(c)
25π
=
=
2
so
Pythagoras’
the
area
of
Practice
1
a
the
theorem
still
semicircle
applies
on
the
for
8
semicircles:
hypotenuse
is
equal
to
the
sum
of
the
areas
of
the
semicircles
on
the
other
two
sides
2
7.2
b
17.8
c
9.8
d
8.9
2
Original
Final
Location
location
(2,
(
6)
1,
3
1)
7)
i
b
Individual
No
e.g.
11
5
2
6
2
or
7310
miles
to
Not
similar
data
end
supposed
1)
5
6
N
5)
7.81
7
N
13
13
Y
5)
Hit
(Any
iii
pair
of
coordinates
such
Hit
that
iv
the
distance
between
the
up
where
to?
No
(Y/N)
Hit
points
is
less
than
2.
(0,1))
similar
possible
e
Not
f
Similar
a
ΔOKL
=
6
(corresponding
(AA
to
∼ ΔSTR ,
∼ ΔCDE
0.67
the
not
all
right
same
Not
triangles
proportion)
similar
are
similar
(AA
postulate)
b
ΔABE
∼ ΔACD
ΔQPS
∼ ΔSQR,
,
AA
postulate;
2.8 cm
AA
postulate;
17.5
cm
e
AA
postulate;
x
=
15 cm,
y
=
25 cm
ft
angle
oor
enclosed
and
angles
applied,
the
are
and
Measure
the
tell;
∼ ΔOMN
ΔABE
Two
in
m
ΔPQR
The
are
postulate)
c
=
sides
d
d
c
the
was
3
c
b
it
3s.f.
Not
to
Did
(d)
14
Similar
a
was
travel
Y
5,
and
data
to
10
ii
b
x
3
4,
Distance
supposed
10
response
a
x
(
distance
travelled
Hit
(0,0)
Practice
1
(2,
(
a
4
14)
3)
(1,
(0,
(8,
Actual
the
by
the
line
of
sight
of
the
man
and
the
oor,
and
the
angle
enclosed
by
the
line
from
the
the
the
same
triangles
horizontal
proportionality
of
in
each
are
triangle,
so
all
three
angles
are
the
same.
Therefore,
the
AA
postulate
can
a
ΔPQR
∼ ΔTRS ,
AA
distance
scaling.
between
Then,
the
measure
man
the
and
the
vertical
surface,
distance
and
of
the
the
distance
man’s
eye
between
from
the
the
surface
oor
and
to
P
x
T
176 cm
S
Q
160 cm
13 320 cm
Answers
nd
scale
postulate
R
be
similar
proportionally
4
light
oor
3 61
this
5
b
14652 cm
c
When
other
Practice
2
his
146.5 m
shadow
Hypotenuse:
Two
1
=
200
sides:
(to
is
1
the
d.p.)
same
length
height,
the
pyramid’s
shadow
would
be
the
same
length
cm
4
a
0.85
b
0.21
e
0.71
f
0.88
a
30.0°
b
48.6°
e
39.8°
f
8.6°
a
sin B
b
sin B
c
0.96
d
0.50
c
67.4°
d
33.1°
c
sin B
d
sin B
3
=
12
=
17
sin C
7
5
=
sin C
17
=
13
4
=
6
=
5
15
sin C
his
10 cm
8
3
as
3
=
5
sin C
=
13
7
1
sin B
e
=
f
sin B
=
0.69
2
3
sin C
15.9
=
sin C
=
2
4
5
22
a
29.2 cm
b
11.5 cm
e
38.7°
f
36.9°
a
5.80 m
to
3s.f.
b
4.66 m
to
Closed
Practice
1
2
a
0.469
b
0.306
e
0.875
f
0.695
a
33.69
b
33.56
e
65.80
f
73.70
a
i
d
18.6 cm
c
0.999
d
0.675
c
35.10
d
70.71
iv
1
3s.f.
length:
2.3 m
3
4
ii
iii
5
5
3
5
3
v
vi
5
b
28.1°
5
4
3
c
The
4
side
adjacent
Adjacent
∴ cos B
to
angle
B
=
a
2.12 m
5
a
18.5 m
6
34.4°
7
Horizontal
to
is
the
to
side
opposite
C,
and
the
side
adjacent
to
B
is
adjacent
to
C
C
=
Hypotenuse
4
B
Opposite
= sin C
and
similar
for
the
second
equality
Hypotenuse
3s.f.
b
2.12 m
b
Yes,
because
the
tree
is
over
18
metres
tall
3s.f
1
component:
19.2
ms
1
Vertical
Unit
1
2
3
component:
16.1
ms
Review
Individual
response
a
No,
b
Yes,
the
the
triangles
are
similar
(corresponding
angles
are
equal)
c
Yes,
the
triangles
are
similar
(corresponding
angles
are
equal)
d
Yes,
the
triangles
are
similar
(corresponding
sides
e
No,
a
1.19
the
3 6 2
triangles
triangles
b
are
are
not
not
0.62
Answers
similar
are
proportional)
similar
c
0.26
d
0.14
e
0.33
f
0.49
as
its
height
4
a
51.3°
b
53.1°
e
62.9°
f
44.8°
5
a
17
b
6
6
a
b
Postulate
Postulate
AAA
14
6
AAA
c
26.4°
c
13.3
c
Postulate
x
6
7
a
5.03 cm
8
73 m
9
a
10
a
to
3s.f.
b
73
58.0°
b
to
3s.f.
145
3
s.f.)
x
=
angle
of
d
12.3
d
57.3°
d
5
b
21.8°
(to
3s.f.)
AAA
=
=
7.5
30
c
74.8 cm
c
2
to
3s.f.
13
x
100 km
49.0°
= 15
y
z
(to
d
to
3s.f.
subduction
x
250 km
2
11
a
4 cm
12
a
12.0 m
to
13
a
31.3 m
to
14
a
b
12 cm
3s.f.
b
11.9 m
3s.f.
b
82.5°
to
to
3s.f.
3s.f.
C
700 000 km
B
1738 km
A
E
D
384 400 km
150 000 000 km
ΔABE
b
The
to
the
will
15
a
∼ ΔACD
angle
nearest
be
;
Individual
by
the
response
is
postulate;
by
the
hundredth.
blocked
rectangle
AA
enclosed
more
from
the
top
of
corresponding
the
angle
Sun
for
to
the
the
earth
Moon
is
and
the
0.52
°.
Earth
the
Therefore,
centre
the
of
the
majority
Sun
of
the
is
0.53
°
sunlight
Moon
(any
than
lines
The
suitable
double
the
rectangle
area
of
with
the
diagonals
smaller
measuring
80
and
55,
such
that
the
area
of
the
larger
rectangle)
b
Area
=
2735
sq in
80
Scale
factor
80 in
39.221
337
337
9
55
55 in
Scale
factor
26.96
337
47.94
So
the
80-inch
TV
has
an
area
that
is
more
than
double
the
area
of
the
55-inch
TV
Answers
3 63
16
a
Let
the
pair
of
points
be
(x
,
y
1
Then
the
(x
x
distance
between
2
−
1
3
b
i
c
386.7
390.0
Linear
a
( y
−
y
1
light
light
)
z
)
and
1
(x
,
2
points
2
+
2
Practice
1
)
,
1
the
y
2
,
z
)
2
i s
2
+
(z
2
−
z
1
years
ii
4.3
)
2
light
years
iii
8.6
c
iii
light
years
years
Relationships
1
ii
b
iv
d
v
y
e
i
y
16
1
14
x
0
5
4
3
2
1
1
12
10
2
8
3
6
4
2
x
0
1
c
Individual
2
3
response
y
8
6
4
2
x
0
–4
–2
2
–2
–4
–6
–8
3
a
Year
Cost,
$/kWh
2010
1000
2011
875
2012
750
2013
625
2014
500
2015
375
2016
250
2017
125
3 6 4
Answers
b,
y
c
1000
900
800
700
)$(
600
hWk
500
,tsoC
400
300
200
100
0
x
2010
2011
2012
201 3
201 4
201 5
2016
2017
2018
Year
d
Individual
response
e
Individual
response
4
Individual
5
a
response
Year
(all
parts)
Million
tonnes
wasted,
high-income
1
670
630
2019
1340
1260
2020
2010
1890
2021
2680
2520
2022
3350
3150
8710
c
Individual
response
d
Individual
response
e
Individual
response
a
million
tonnes
(high
Linear,
a
2.5
b
Linear,
8190
million
tonnes
(low
income),
from
baseline
6
c
Non-linear
d
2
b
5
3
income),
wasted,
zero
in
2017.
2
6
2
tonnes
low-income
2018
b
Practice
Million
3
10
c
5
Linear,
36
d
e
21
No
Answers
3 6 5
0
4
y
a
5
B
4
C
3
2
A
1
D
x
–6
b
(
a
x
–5
–4
–3
–2
2, 0 )
–1
0
1
c
Individual
b
x
response
11
5
=
6;
y
=
−5
=
9
;
y
= 11
c
x
= −3, x
= 3;
y
= −
3
d
x
= 14;
2
4
e
x
=
−
8,
x
=
8;
y
=
−
12 ,
y
=
12
f
x
= 16;
y
=
3
6
a
y
160
2015
150
)mk/g(
140
snoissime
130
2
120
OC
2020
110
100
x
1000
1100
1200
1300
140 0
Weight
b
0.05(g / km ) / kg
a
Triangle
A:
vehicle
ratio
of
sides
1600
1700
1800
(kg)
c
d
4
7
of
1500
2
2
=
=
2
= 2
;
triangle
B:
ratio
of
sides
= 2;
=
1
1
2
triangle
C:
ratio
of
sides
= 2
=
;
therefore
triangles
are
similar.
1
b
Gradient
Practice
1
a
of
hypotenuse
is
same
3
y
5
4
y
=
x
3
2
1
x
0
1
3 6 6
2
3
4
Answers
5
for
all
similar
triangles
with
two
sides
parallel
to
axes.
y
= 10
y
y
8
7
7
6
y
=
3x
+
7
5
6
4
5
3
4
2
3
1
2
y
=
5x
3
1
0
x
0
1
x
1
2
3
4
5
1
2
2
3
y
y
5
0
x
1
2
3
4
5
6
1
4
2
3
3
2
2
y
=
x
+
1
3
4
1
1
y
=
x
–
4
2
5
x
0
1
2
3
4
5
6
6
7
8
2
a,
3
a
f
perpendicular;
Time
in
b
e
f
4
5
6
7
y
=
80 x ;
The
=
km
b,
d
perpendicular;
d,
g
perpendicular;
c,
h
perpendicular
0
1
2
3
4
0
80
160
240
320
0
Distance
train
starts
a
Individual
0.82; x
a
parallel;
280 km
b
a
in
80 km / h; x
d
g
hours
Distance
c
b,
=
y
=
0.6 x
y
=
0.5 x
b
$182.50
a
Coal:
b
Coal:
y
=
80
from
response:
×
rest
3.5
table
0
or
c
y
b
The
cost
of
one
graph.
=
80 000 000
Hydroelectricity:
cost
×
=
80 km
This
every
should
hour
start
0.82 x
billion
bottle
of
tonnes
0.2
Hydroelectricity:
×
at
a
from
constant
(0,
0)
speed.
and
go
to
d
250 000
tonnes
c
Individual
response
d
Individual
response
(100,
e
d
82)
.28
billion
Individual
tonnes
response
water.
80
= 80 000 000 + 0.2 x
=
280 km
travels
2.4
c
cost
=
and
5000
250 000 000
=
×
y
=
250 000 000 + 0.01 x
$80 000 000 000;
0.01
×
5000
=
$250 000 050
Answers
3 67
c
Coal:
cost
=
80 000 000
Hydroelectricity:
cost
d
Individual
response:
e
Individual
response
Practice
×
=
0.2
×
10 000
250 000 000
slope
increases
×
=
$160 000 000 000;
0.01
for
×
coal,
10 000
not
for
=
$25 000 000 000
hydroelectricity; y-intercepts
4
1
1
a
y
=
−2 x
+ 4
y
b
=
2
x
− 2
c
y
=
2
e
y
=
unchanged
3x
x
+ 3
d
y
=
6x
+ 10
3
+ 9
y
y
5
5
4
4
3
3
y
=
2x
+
4
2
2
1
1
1
y
=
x
–
2
2
0
x
0
1
2
3
4
x
5
1
2
y
y
y
8
22
18
7
20
16
6
18
14
y
=
6x
+
10
y
=
3x
+
9
2
y
5
=
x
+
3
16
12
4
14
10
3
12
8
2
10
6
1
8
4
6
2
4
0
3
0
x
1
2
3
4
5
6
x
1
2
3
4
2
0
x
1
2
3
4
9
3
a
x
=
2,
y
=
4
b
x
=
4,
y
=
−2
c
x
= −
5
,
y
= 3
d
x
= −
2
e
x
=
−3,
y
=
,
y
= 10
3
9
1
4
a,
e,
g
parallel
(gradient
=
1
);
h,
i,
k
parallel
(gradient
=
–2);
j
perpendicular
to
a,
e,
g
(gradients
7
perpendicular
to
c
(gradients
are
2
and
Answers
3
);
2
3 6 8
–7
and
;
7
1
b
are
d
perpendicular
to
l
(gradients
are
2
and
2
).
3
y
y
8
1
6
0
x
5
3y
+
4x
=
24
4
1
2
2
5x
10y
30
=
0
3
x
0
2
4
6
8
y
y
12
4
10
0
=
2x
−
4y
+
12
2
8
y
–
3x
=
6
2
1
6
4
x
6
5
4
3
2
1
0
1
2
2
x
4
3
2
1
0
1
2
y
y
1
16
x
0
18
16
14
12
10
8
6
4
2
2
1
10
2
1
1
y
=
2x
+
4y
4
4
=
x
–
12
2
8
3
6
4
4
2
x
2
1
0
1
2
y
y
5
2
3y
=
5x
+
15
4
0
x
2
3
4
2
6
1
y
=
3x
−
12
8
x
3
2
1
0
1
2
10
12
Answers
3 6 9
y
y
4
4
3
3
2
2
0
=
−3x
+
14
−
7y
1
1
y
x
0
1
2
3
4
=
−x
+
4
x
0
5
1
2
3
4
y
y
4
3
3
2
y
4y
−
8x
=
–
2
=
12
−2x
3
1
1
x
0
1
2
x
2
1
0
1
2
3
y
y
5
0
x
1
2
3
1
4
2
3
3
2
1
2y
=
2x
−
10
−2
=
3x
y
2
4
1
5
x
1
7
a
y
=
4 x,
y
− 0
Individual
b
c
Rate
y
=
of
−0.5 x
Rate
of
Practice
1
4( x
painting;
+ 28,
Individual
d
=
y
cooling,
− 28
−
y
=
for
=
0
and
at
−0.5( x
table
=
5x
y
=
−3 x
+ 5,
+ 7,
y
y
− 5
− 7
=
c
y
=
−2 x
− 5,
y
+ 5
1
graph
start.
− 0 ),
and
y
=
+
y
=
28
initial
time x
=
0
5( x
− 0 ),
5x
=
−3( x
− 0 ),
=
−2 ( x
− 0 ),
−
y
3x
=
+
2x
y
=
x
+ 3,
=
y
=
y − 3 =
(x
− 0),
x
− 2 y
= −6
+ 5
b
y
=
7
x
1
+
c
y
=
−2 x
+ 5
y
d
=
3
+ 4
f
y
=
−2 x
x
+1
+ 14
g
y
=
x
+
7
x
y
h
=
2
+ 6;
x
2
7
x
−5 x
pm
−5
3
3
6
2
−4 x
=
at
7
4
y
taken
−5
y
+
3
e
is
temperature
1
a
0.5 x
graph;
1
2
2
1
5
y
=
4 x
table
painted
initial
a
y
for
area
response
b
d
− 0 ),
response
0
3
10
3
= 1
80 000
4
a
7
80 000
b
Individual
response
number
years
according
to
year;
following
the
Tip,
the
equation
is
y
= −
x + 550 000
7
c
of
since
2006
2054
37 0
Answers
and
y
is
number
of
elephants
where
x
is
5
6
7
a
y
=
3.2 x
where
b
2287
a
353 ppm;
e
2135
is
number
399 ppm
a
Individual
e
2271
Practice
x
b
response
of
x
= −4,
y
= 1
3
y
= 1
4
5
x
y
=
and
.7 ppm/year
y
c
response
b
Individual
response
f
Individual
response
is
y
sea
level
= 1.7 x
rise
in
mm
since
+ 336
d
17 ppm/ year
c
1993
Individual
d
response
2 ppm/ year
6
x
= −
5
,
x
= 2,
y
= −3,
y
=
,
y
= 5
2
5
=
−2
6
x
=
4
7
x
=
−7
Unit
1993
Individual
2
2
since
f
3
1
years
review
2
1
a
6,
12
, 23
b
7
2
a
x
=
4,
y
=
8
b
x
=
4,
y
=
−5
c
x
=
6,
y
= 14
y
y
8
0
x
5
7
1
6
2
y
=
−2x
+
8
5
3
4
4
3
5
2
6
1
7
y
=
x
−
7
x
0
1
2
3
4
5
4
Line
Slope
y-intercept
x-intercept
Equation
5
5
A
5
y
2
=
2
B
x − 5
2
0
3
None
2
10
y
= 3
y
= −
1
1
C
5
D
Undened
None
a
y
=
y
= −4
1
x
− 4
b
y
=
6
e
x
4
1
5
x + 2
5
= 12 x
1
x
+ 2
c
y
=
2
1
x
d
8
y
=
2
x
6
+
3
− 9
6
a
Undened
b
7
a
Constant
b
decrease
=
−0.5d
+ 100
c
2
c
200
d
Answers
3 71
y
y
3
0
x
6y
+
3x
=
10
12
2
1
1
2
20
30
40
3
x
0
1
2
3
4
5
10y
−
x
=
−50
4
5
y
y
5
0
x
10
4
1
x
2y
1
2y
+
x
=
+
3
=
4
10
3
3
2
2
3
1
x
0
10
20
30
40
y
y
1
x
0
4
3
2
1
1
0
x
1
2
1
3
2
4
3
5
4
10+2y
4x
=
3
6
0
=
24
+
8x
+
5
3y
6
8
9
a
ii
b
iii
c
i
b
2022
c
Individual
10 1
11
y
=
−3 x
12
y
= −
+ 1
2
x
+ 7
3
1
13
y
17
=
x
−
3
3
14
x
=
6
15
x
=
−1
16
a
17
a
y
=
−0.3 x
+ 1.8
response:
trend
not
likely
to
be
linear
6900
≈ 168
41
6900
b
Individual
response
according
to
year:
equation
is
y
= −
x
41
and
c
y
is
number
of
cheetahs
2058
372
Answers
+ 14000
where
x
is
years
since
1975
1
18
y
=
x
− 3
2
19
a
9
b
20
Yes.
4
2
Gradient
AB
=
3
;
gradient
BC
21
a
b
c
22
so
y
=
e
Individual
y
=
0.08 x
b
$21400;
c
$23 800;
Individual
e
52
000
according
where
000;
y
$23
is
to
year
years
since
1991
and
y
is
number
miles
of
If
the
miles
=
0.03 x
number
driven
response:
is
feed
in
is
miles
more
values
on
is
less
than
table
than
52 000
should
52 000
it
is
it
is
cheaper
cheaper
satisfy
to
who
are
undernourished.
to
drive
equation
y
=
a
drive
the
Ford
zero-emission
3.85 x
where
x
is
Focus,
electric
animal
but
if
the
Focus.
meat
in
kg
kg.
Individual
response
d
Individual
response
Individual
response
g
Individual
response
e
Individual
response
Shapes
1
2
2
150.8 cm
b
2
628.3 cm
c
2
e
people
+ 21 600
f
3D
of
400
Individual
y
x
c
a
= −1
2
response
response
Practice
1
3
response
Individual
and
4
+ 18.6
+ 19
d
a
3
× −
because
$22500
number
23
response
−0.321 x
2049
f
perpendicular
2
0.321% / year
Individual
d
a
2
=
3
703.7
2
cm
d
319.5 cm
2
432.3 cm
f
2535.2
cm
2
2
Radius
3
20 cm
4
a
(cm)
Height
(cm)
Surface
area
12
18
2261.9
10
11
1319.5
7
6.9
8
11.6
3
2.3
straw
with
a
radius
(cm
)
611.4
985
100
of
0.65 cm
2
101π
=
317.3
m
(1 d.p.)
b
03 π
=
951.9
2
5
a
672 π
Practice
=
2111.2
m
2
(1 d.p.)
c
606 π
= 1903.8
2
m
(1 d.p.)
b
160.5
m
a
(1 d.p.)
c
37.3
m
Surface
Area:
512π
Volume: 1536 π
= 1608.5
2
cm
b
=
4825.5
Surface
Area:
200π
=
628.3
cm
3
Volume:
cm
375 π
= 1178.1 cm
2
Surface
Area:
36 π
2
= 113.1 cm
d
Surface
Area:
112 π
=
351.9
3
Volume:
28 π
=
88.0
Surface
Area:
Volume:
693.2
cm
3
cm
Volume: 160 π
=
502.7
cm
2
e
(1 d.p.)
2
3
c
(1 d.p.)
2
2
1
m
2
cm
f
Surface
Area:
5170.8 cm
3
1102.7
Volume:
30 283.2 cm
2
g
Surface
Area:
7803.7 cm
3
Volume:
23 411.1 cm
2
Radius
of
base
Height
of
cylinder
Volume
3
4.5 cm
0.30 cm
19.4 cm
3.42 cm
15 cm
550 cm
3
Answers
373
3
3
a
3 031 558.4 m
c
808
a
34 494.3 m
to
the
to
the
nearest
nearest
whole
tenth
b
95.2%
b
The
to
the
nearest
tenth
number
3
4
volume
would
increase
by
3
5
a
54.6 m
Practice
to
the
nearest
tenth
b
factor
of
4
341 495 541
years
c
20.1 m
to
the
nearest
tenth
3
3
1
a
2
3
a
301.6 cm
e
4.6 m
a
78 539.8 mm
3
b
102.6 cm
f
1781.3 cm
3
c
2 120 575.0 cm
b
35.4 mm
b
2884.0 cm
3
d
67 858.4 cm
3
3
2
to
the
nearest
tenth
to
3
3
a
4084.1 cm
c
2111.2 cm
a
609.1 cm
b
Individual
a
4.7 cm
the
nearest
tenth
3
to
the
nearest
tenth
to
the
nearest
tenth
to
the
nearest
tenth
3
3
4
5
Practice
to
to
the
nearest
response
the
tenth
(e.g.
nearest
it
would
use
less
material
tenth
than
a
cylinder
b
7.9 cm
to
b
Surface
the
of
the
same
nearest
a
Surface
area:
Volume:
area:
to
produce)
452.4 cm
3
Surface
24.9 cm
area:
Volume:
639.8 cm
2
201.2 cm
d
Surface
area:
3
Volume:
75.4 cm
3
157.1 cm
Volume:
37.7 cm
2
Surface
area:
2
1255.8 cm
f
Surface
area:
1002.2 cm
3
Volume:
3
3893.5 cm
Volume:
1710.2 cm
2
g
cheaper
2
66.0 cm
2
e
be
4
3
c
so
tenth
2
1
height
Surface
area:
2
16336.3 cm
h
Surface
area:
705.3 m
3
Volume:
3
150796.4 cm
Volume:
1508.6 m
2
i
Surface
area:
54.2 m
3
Volume:
29.1 m
2
2
6283.2 cm
3
a
4
1107.6 m
5
1230.5 cm
to
the
nearest
tenth
2
2
b
1264.9 cm
2
994.4 cm
c
816.8 cm
c
80 cm
2
to
the
nearest
tenth
2
Practice
to
the
nearest
tenth
5
3
1
a
3
160 cm
b
e
3
168 cm
3
3
d
3
1633.3 mm
f
1332.9 mm
3
292.5 cm
g
3
22728.0 mm
h
363.6 cm
2
Height
Base
side
length
Apothem
Volume
3
(cm)
(cm)
(cm)
(cm
)
4
6
5
48
6
16
10
512
12
10
13
400
7.5
20
12.5
1000
8
10.9
9.7
314
3.25
25.5
13.1
702.15
3
3
a
1 384 190 m
b
Individual
4
a
74 m
3 74
response
(For
example,
b
Answers
it
wouldn't
Increases
by
a
be
scale
possible
factor
to
of
4
use
the
space
right
up
to
the
apex
or
in
the
corners.)
Practice
6
2
1
a
Surface
Area:
2
800 cm
b
Surface
Area:
223.3 cm
3
Volume:
3
1280 cm
Volume:
167.4 cm
2
c
Surface
Area:
2
1065.5 cm
d
Surface
Area:
70.9 c m
3
Volume:
3
2048 cm
Volume:
25.0 cm
2
e
Surface
Area:
3
84.5 cm
f
Surface
Area:
864 cm
3
Volume:
2
42.4 cm
Volume:
960 cm
2
2
a
374.4 mm
3
1.95 m
4
a
to
b
Yes
Each
face
has
the
same
area
so
there
is
an
equal
chance
of
landing
on
each
face
3s.f.
2
Surface
Practice
Area:
to
4920.8 cm
the
a
2
22.0 cm
tenth
b
58.8%
to
the
nearest
tenth
7
2
1
nearest
2
1017.9 cm
to
b
the
2
314.2 cm
nearest
c
2
2463.0 cm
d
2
235.6 cm
e
2
6939.8 mm
f
942.5 m
tenth
2
3
22 167.08 cm
4
5
5
heating
balls
each
with
a
radius
of
12 cm
10
a
The
surface
area
increases
by
a
factor
of
4
b
The
surface
area
increases
by
a
factor
or
9
c
If
the
radius
of
a
sphere
is
increased
by
a
is
1440π
factor
5
of k
is
2880π
then
the
surface
area
of
the
sphere
2
increases
Practice
by
a
factor
of
k
8
2
1
a
Surface
area:
2
2827.4 mm
b
Surface
area:
3217.0 cm
3
Volume:
3
14 137.2 mm
Volume:
17 157.3 cm
2
c
Surface
area:
2
56 410.4 mm
d
Surface
area:
15 843.1 mm
3
Volume:
3
1 259 833.1 mm
Volume:
144 347.8 mm
2
e
Surface
area:
5026.5 mm
3
Volume:
33 510.3 mm
2
Radius
Diameter
(cm)
(cm)
Surface
Area
15
30
2827.4
14 137.2
25
50
7854.0
65 449.8
8.9
17.8
1000
6.2
12.4
Volume
2
(cm
3
)
(cm
)
2973.5
483.6
1000
3
3
to
the
a
33.5 m
b
1.56 m
c
Individual
to
the
nearest
nearest
response
tenth
hundredth
e.g.
it
could
e.g.
black
sink
or
burst
3
4
a
50 265.5 m
b
Individual
response
surfaces
absorb
heat
3
5
a
6
47.6%
7
3
Unit
42.4 cm
to
radiation
a
the
nearest
b
2077.6 cm
shiny/light
colored
b
Surface
ones
c
5294.4 cm
c
Surface
tenth
Review
Surface
area:
2
294.1 mm
area:
394.1 cm
3
Volume:
Surface
362.3 mm
area:
Volume:
1961.6 cm
3
735.6 cm
Volume:
5183.6 cm
2
1440 cm
e
Surface
area:
3
Volume:
2
area:
3
2
d
than
3
2
1
better
3
280 cm
3
3200 cm
Volume:
3
280 cm
2
2
a
33 510.3 m
3
a
61 575.2 cm
b
5026.5 m
b
9324.2 cm
3
2
Answers
3 75
4
10.0 cm
5
Base:
3 cm
by
3 cm;
height:
3 cm
3
6
7.07
to
inches
the
nearest
hundredth
3
7
79.6 m
8
a
2
Surface
Area:
2
97.1 m
b
Surface
Area:
Volume:
2
1306.9 m
3
c
Surface
Area:
176.3 m
3
62.5 m
Volume:
3
4155.3 m
Volume:
107.3 m
2
d
Surface
Area:
1309.3
e
Surface
Area:
1680.5 cm
3
Volume:
3
3152.7 cm
Volume:
4593.2 cm
2
9
Enough
to
10
a
11
6.40 cm
12
13.5 cm
13
a
14
Volume:
cover
4957.1 m
2
942.62 km
to
the
nearest
b
Approximately
b
33.2 m
b
Form:
c
Approximately
c
Form:
2
hundredth
3
2
to
12.2 m
2
the
nearest
tenth
3
183.3 mm
3
Area:
5
172.8 mm
Bivariate
Practice
1
a
data
1
Form:
Linear
Strength:
Moderate
Direction:
d
Form:
a
Discrete:
b
Total
e
Form:
No
pattern
No
because
Strong
Direction:
f
correlation
Form:
Strength:
Strong
Direction:
only
certain
values
are
possible,
i.e.the
number
of
tickets
and
the
of
the
total
cost
is
dependent
on
the
number
of
total
tickets
tickets
Justication:
this
is
the
quantity
that
is
being
varied
d
1800
1600
1400
1200
)$(
tsoc
1000
l at o
T
800
600
400
200
0
2
4
Number
Answers
6
of
8
tickets
10
12
to
observe
a
change
in
the
total
Positive
Non-linear
cost
Number
37 6
Positive
Direction:
Linear
Strength:
Negative
Justication:
c
Strong
Direction:
Moderate
Direction:
2
Positive
Linear
Strength:
Non-linear
Strength:
cost
Negative
cost
are
integers
e
The
data
form
f
a
Form:
points
line.
perfectly;
there
is
a
linear
relationship
so
it
would
make
sense
to
join
the
points
response)
Positive
Denote
the
number
Denote
the
total
Then,
up
Strong
Direction:
3
line
Linear
Strength:
g
do
(Individual
t
of
cost
tickets
by n
by
= 143n
a
4
×
10
4.4
4.2
4
)elpoep
3.8
fo
3.6
rebmun(
3.4
ecnadnettA
3.2
3
2.8
2.6
2.4
2013
2013.5
2014
2014.5
2015
2 015.5
2016
2016.5
2017
2017.5
2018
Year
b
Form:
Linear
Strength:
Moderate
Direction:
Positive
c
Individual
response
(e.g.
between
d
Individual
response
(e.g.
greater
40 000
and
popularity
44 000)
through
publicity
or
word
of
mouth)
Answers
377
to
4
a
5
×
10
9.8
9.7
9.6
)mk
qs(
9.5
tserof
9.4
fo
aerA
9.3
9.2
9.1
9
225
230
235
240
245
Population
b
The
c
Individual
greater
d
Form:
of
(student
2 60
265
Indonesia,
answers
the
should
smaller
the
mention
area
that
of
the
remaining
area
of
forest
forest
Negative
Individual
response
population,
but
e.g.
this
the
scatter
does
not
plot
mean
suggests
that
it
is
that
the
there
is
increase
negative
in
in
Indonesia
likely
correlation
population
that
a
70
65
60
)%(
55
erocs
50
xednI
45
gniviG
40
d l r oW
35
30
25
20
25.4
25.5
25.6
25.7
25.8
Average
378
is
to
have
decreased
further)
Strong
Direction:
the
5
population
255
Linear
Strength:
e
the
response
250
(millions)
Answers
25.9
temperatu r e
26
in
2 6.1
Ju n e
(°C)
26.2
26.3
26.4
is
between
causing
the
the
area
of
forest
decrease
in
and
area
b
Form:
linear
Strength:
weak
Direction:
c
the
6
positive
Individual
response
citizens
d
Individual
e
No;
a
Generation
can
of
response
correlation
only
No
c
The
d
Individual
f
Form:
more
e.g.
does
plot
the
not
suggests
recent
jobs
the
an
number
imply
categorical;
change
b
that
the
average
Averages
whole
of
people
who
number
number
generation,
the
of
y
=
a
weak
aect
on
greater
e
of
job
the
are
Buddhists
changes
number
Individual
as
this
religion
encourages
is
discrete
-
although
decimal
on
values
given
to
independent
of
times
a
person
changes
job
by
b
− 31
e
y
= −
given,
age
of
30
on
+ 156
y
b
=
x
y
=
13
x
1
+
c
y
= −
8
29
x
−
7
7
y
=
2x
+ 71
f
y
= −
1067
x
+
64
+ 15
2
b
a
y
y
50
70
y
y
=
5.1x
–
=
3
–6.2x
+
47
or
40
60
y
=
47
–
6.2x
30
50
20
40
10
30
20
x
0
1
2
3
4
5
6
10
x
0
2
4
6
8
10
12
14
c
d
y
y
110
30
100
28
26
90
y
=
–2.1x
+
149
24
80
y
22
70
60
20
50
18
=
0.88x
+
10.4
16
40
x
20
30
40
50
you
average
5
x
25
3
the
64
8
a
are
variable
6
2
values
7
x
of
giving
2
−3 x
=
nature
response
11
y
charitable
Positive
8
d
the
times
1
a
has
causation
response
depends
Direction:
1
temperature
Non-linear
Strength:
Practice
is
The
Myanmar
60
14
x
10
12
14
16
18
20
Answers
22
379
4
Individual
straight
a
(easiest
way
is
to
choose
a
scatter
scitamehtam
scitamehtam
800
700
600
500
gniyduts
gniyduts
400
300
emit
emit
200
lato
T
lato
T
6
this
b
100
6
6.5
Individual
y
do
900
7
7.5
8
8.5
Age
c
to
)sruoh(
)sruoh(
5
response
plot
where
all
data
points
lie
on
the
same
response
= 154.429 x
d
Individual
e
1502
f
Individual
to
y
9.5
10 10.5 11
900
800
700
600
500
400
300
200
100
6
6.5
7
7.5
=
8
8.5
Age
(years)
157x
9
9.5
10 10.5 11
(years)
828
− 814.476
response
the
e.g.
9
nearest
e.g.
The
whole
response
e.g.
result
using
technology
will
have
more
signicant
gures
number
There
is
no
data
for
15-year-old
a
students
so
you
cannot
be
sure
that
the
trend
will
b
1.2
1.1
emit
gnidaer
egarevA
0.6
tneps
0.5
0.6
s ruoh(
sruoh(
0.7
0.8
re p
tneps
re p
0.8
1
)yad
gnidaer
)yad
1
0.9
0.4
0.2
0
10
20
30
40
50
60
70
80
emit
–0.2
0.4
Age
(years)
egarevA
0.3
0.2
0.1
10
20
30
40
Age
Individual
c
y
=
exactly
line)
response
50
e.g.
y
response
e.g.
As
response
Not
0.0131 x
Individual
e
75
f
Individual
g
The
h
Individual
response
e.g.
i
Individual
response
(e.g.
3 8 0
70
80
=
0.01x
0.2
− 0.233
d
older
60
(years)
they
are,
the
the
very
condent:
longer
Answers
Older
no,
correlation
on
the
average
people
they
are
was
it
was
70-year-old
they
who
just
weak
are
spend
person
reading
retired
more
not
likely
have
to
easy
in
per
more
have
to
the
draw
survey
the
line
read
for
of
best
1.1
t
hours
a
day
time
spare
in
time
the
in
day
to
which
read
they
can
read)
day
continue
7
a
b
36
36
gniklaw
32
34
32
leef
ohw
26
egatnecreP
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24
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20
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20
20
30
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50
60
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Age
Age
c
y
=
−0.235 x
3
e
Individual
response
depending
f
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response
g
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h
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the
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and
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e
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i
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less
is
compare
=
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on
age
likely
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debatable)
the
was
e.g.
not
the
are
e.g.
very
to
be
you
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weak
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weak
c
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g
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j
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strong
0.8
c
0.7
g
i
0
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0.9
Individual
Reponses,
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Expected
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a
case
it
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home
out
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to
draw
predicted
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to
line
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best
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night
survey
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your
class
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c
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older
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will
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strong
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0.6
d
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weak
h
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moderate
d
0.5
h
0.75
the
the
more
number
an
assignment,
in
general,
the
higher
grade
you
are
likely
to
be
awarded
of
immunized
people,
the
fewer
cases
you
would
expect
to
see
correlation
person,
a
the
siblings
negative
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older
on
correlation
more
probable
it
is
that
at
some
point
in
their
past
they
have
been
relationship
show
Individual
response
the
of
a
of
means
that
the
available
food
needs
to
be
shared
between
more
children,
correlation
correlation
adolescent,
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of
of
for
person
of
height
hours
a
strong;
moderate
c
the
greater
the
expected
number
of
neural
connections
in
the
brain
of
years
and
e
moderate;
d
no
correlation
or
weak;
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no
correlation
strength
between
in
0.8
and
0.5)
education,
the
lower
the
unemployment
rate
example
and
the
outside
number
sales
bought
watching
sweets
person
total
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sweets
spent
(identical)
of
b
(e.g.
(identical)
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a,
number
temperature
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the
responses,
age
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e.g.
correlation
greater
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or
correlation
spend
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romantic
responses
b
1:
is
correlation
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you
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correlation:
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longer
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in
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1:
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which
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easy
example:
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involved
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not
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was
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f
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it
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results
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strong
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y
correlation
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b
80
3
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person,
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+
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a
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2
e.g.
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older
70
(years)
(years)
d
year
1
+ 39.077
60
80
and
of
hot
and
TV
bought
the
of
the
at
and
person’s
books
they
have
read
beverages
total
home
the
expenditure
and
on
that
achievement
amount
of
money
in
you
type
a
of
mock
have
sweet
exam
left
available
to
spend
weight
Answers
3 81
in
Practice
1
a
4
0.543
Positive correlation,
b
Negative
c
moderate
0.943
correlation,
Positive correlation,
d
a
The
y
straight
51 −
coecient
each
d
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the
e
a
b
f
a
y
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=
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of
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d
e
1,
implying
that
the
data
set
is
perfectly
correlated;
the
data
points
all
value
other
response:
check
they
that
Dependent
of
x
gives
correlation
the
corresponding
coecients
are
value
exactly
of
1
1)
give
variable:
Substitute
the
are
of
the x
values
from
the
table
into
your
equation
for
the
line
of
values
Year
variable:
we
some
corresponding y
Marriage
observing
rate
the
change
in
the
marriage
rate
as
the
year
the
same
(if
quantity
X
decreases
as
Y
increases,
then
quantity
Y
0.96
the
original
data:
y
=
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+ 747.02
14.5
)stnatibahni
14
13.5
001
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rep
rebmun(
12.5
12
etar
11.5
egair raM
11
10.5
2006
2007
2008
2009
201 0
2011
Year
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the
3 8 2
the
y
(or
increases
0.96
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on
4.417x
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Exactly
lie
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+
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strong
line
c
c
3
=
correlation,
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same
b
strong
1.000
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2
strong
0.976
switched
data:
y
Answers
=
−2.6 x
+ 2044.1
2012
2 013
2014
2015
2016
decreases
as
X
increases)
best
t
and
2016
2015
2014
2013
2012
r a eY
2011
2010
2009
2008
2007
2006
10.5
11
11.5
Mar riage
f
Individual
response
meaningful
4
Individual
Practice
None!
2
a
Individual
b
No
c
Individual
Correlation
a
The
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5
it
is
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pe r
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100
good
14
14.5
in hab itan ts )
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to
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are
response
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(e.g.
(e.g.
[for
is
data
sets
selected
causation.
hoping
wealth:
if
the
poor
subjected
response
raining
she
resources
suggests
response
imply
referring
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learning
individuals
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imply
without
from
response
conclusion
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does
response
better
b
c
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dependent
response
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provide
the
(e.g.
12.5
5
1
3
rate
conclusions
responses,
12
it
is
if
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for
family
their
by
moving
has
does
a
larger
believe
balance,
or
misrepresentations
spent
socializing
then
someone
to
not
a
larger
imply
house
house,
her
reading
ability
will
improve)
causation)
then
they
are
likely
to
be
wealthier
and
therefore
children)
researchers
raining,
example,
that
correlation
work-life
to
time
to
the
could
in
of
is
causation
technology
other
is
being
used
for
the
purposes
of
social
media,
lives).
person)
plants
be
there
if
outside
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9.8
10
Consumption
per
of
10.2
10.4
mozzarella
person
10.6
10.8
11
cheese
(pounds)
Answers
3 83
not
b
0.924
c
Form:
Linear
Direction:
Strength:
d
Equation
e
No:
f
Individual
Unit
1
i
Positive
Strong
is
y
+
52.5x;
not
15.4
imply
pounds
causation
per
and
person
these
variables
are
unlikely
to
be
linked
Review
a
Form:
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Strength:
b
Individual
a
Form:
a
Linear
Strength:
response
v
Individual
a
Form:
Negative
Direction:
b
a
No
b
Individual
correlation
response
response
Non-linear
Strength:
response
iii
Positive
Weak
b
Strong
Individual
Form:
Direction:
Linear
Strength:
b
ii
Positive
Moderate
Direction:
2
8.25
does
response
Direction:
iv
=
correlation
Negative
Strong
Individual
response
a
Variables
1
Number
a
3
a
4
a
of
times
you
laugh
Dependent
2
Variable
1
Age
1
Outside
2
Inside
1
Number
of
languages
2
Number
of
nationalities
temperature
1
2
learnt
2
1
adequate
Son’s
Father’s
height
2
1
with
response
with
response
with
justication
Individual
height
1
Reaction
2
Age
adequate
time
2
1
response
response
b
direct,
inverse
c
stronger,
weaker
y
21
20
19
18
17
16
15
14
13
12
x
15
Answers
20
25
30
35
d
possible,
with
justication
response
1
with
justication
Individual
adequate
Individual
3 8 4
response
justication
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in
plot
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adequate
scatter
tree
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of
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temperature
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and
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2
1
in
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day
family
b
Independent
causation
b
y
60
55
50
45
40
35
30
25
x
2
2.5
3
3.5
4
4.5
5
5.5
d
c
6
6.5
7
y
y
90
45
85
40
80
35
75
30
70
25
65
20
x
80
85
90
95
100
105
60
55
50
x
20
5
a
22
24
26
28
30
32
34
36
38
y
81.5
)sraey(
I
Swi
ycnatcepxe
efil
Japan
J
Switzerland
Swi
Singapore
S
Australia
A
Spain
Sp
Iceland
I
Italy
It
Israel
Is
Sweden
Swe
France
F
South
SK
A
81
80.5
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J
It
80
C
Sp
S
79.5
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Canada
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78.5
x
84
84.5
Female
b
No
c
Comparisons
85
life
85.5
86
expectancy
86.5
87
(years)
correlation
between
countries
are
easier
to
make
using
the
table
Answers
3 8 5
6
a
(a)
y
21
20
19
18
17
16
15
14
13
12
x
15
10
(b)
20
25
30
35
y
60
55
50
45
40
35
30
25
x
2
2.5
3
3.5
4
4.5
5
5.5
(c)
6
6.5
7
(d)
y
y
90
50
85
45
80
40
75
35
70
30
65
25
60
20
x
80
85
90
95
100
55
105
50
x
20
b
Individual
c
(a)
(c)
y
y
3 8 6
=
=
22
24
26
28
response
0.377 x
+ 9.175
−0.921 x
+ 120.239
Answers
(b)
y
=
−1.154 x
+ 46.662
(d)
y
=
−1.350 x
+ 105.663
30
32
34
36
38
7
Individual
and
iced
results
is
also
also
a
How
The
c
(e.g.
here;
there
it
tempting
many
sum
is
the
people
daily
response
Yes,
a
much
results
is
a
that
to
indicate
likely
weak
coee
inhabitants
of
has
is
likely
for
Individual
it
The
Causation
correlation
addiction
b
sales.
indicate
likely
negative
8
response
tea
is
here;
negative
fewer
purchase
there
are
in
higher
than
a
strong
given
purchase
density
of
the
drink
hot
of
the
correlation
cold
outside
drinks
for
between
when
temperatures
beverages
correlation
when
iced
tea
it
–
is
it
and
hot
this
outside
is
is
hot
coee
outside.
likely
temperature
outside.
to
sales.
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be
The
Causation
that
due
to
the
caeine
temperature)
land
number
than
to
between
positive
area
by
positive
prefer
regardless
divided
population
will
the
coee
a
is
will
correlation
people
weaker
temperatures
there
people
the
of
other
days
in
a
cointries
year
that
have
a
similar
average
annual
temperature
d
No
e
No
f
9
correlation
Individual
response
e.g.
people
Individual
response
e.g.
the
a
don't
cost
of
always
living,
choose
where
healthcare,
they
political
live;
it
is
where
they
are
born
stability
y
14
13
)sruoh(
12
peels
11
fo
10
tnuomA
9
8
7
6
x
0
10
20
30
40
Age
Form:
Strength:
c
=
the
line
represents
d
Individual
response
e
3.4
hours
1.3
to
hours
a
Individual
b
Form:
using
a
line
of
best
t,
d.p.
the
e.g.
the
average
hypothetical
the
equation
change
average
predicts
in
number
number
that
a
of
of
hours
hours
15-year-old
of
of
sleep
sleep
person
at
per
age
would
increase
in
age
of
one
year
zero
have
11
hours
sleep
no
1d.p.
not
make
response
sense,
(should
and
be
hence
supports
the
idea
that
this
is
a
non-linear
relationship
yes)
Non-linear
Strength:
=
1
to
would
Direction:
11
relationship
represents
This
10
of
this
+ 12.758
y-intercept
f
(years)
Negative
−0.117 x
Slope
70
moderate
Approximating
y
60
Non-linear
Direction:
b
50
c
y
d
323;
230x
e
Individual
f
Line
g
Individual
no,
of
Individual
Positive
Moderate
3400
this
would
response
best
t
be
too
(e.g.
predicts
response
high
so
the
relationship
extrapolation
approximately
e.g.
the
amount
of
likely
7400;
to
the
trac,
be
is
likely
actual
the
to
be
non-linear
inaccurate)
population
amount
of
is
about
industry,
the
6000
climate
response
Answers
3 87
6
Geometric
Practice
1
a
10
1
units
(x, y )
2
a
transformations
a
left,
(x
1
unit
− 10, y
down
b
− 1)
8
units
(x, y )
left,
a
(x
5
units
− 8, y
i
down
− 5)
ii
image
preimage
preimage
image
iii
3
b
Yes.
c
i
Repeatedly
(x, y )
to
4
The
(x, y )
a
transformation
a
−
apply
(x
obtain
(x
2, y
the
given
is
equivalent
− 5)
ii
to
(x, y )
the
a
simpler
(x
+ 3, y
+
transformation
3
6)
(x, y )
iii
units
down,
a
(x
+
− 3)
transformation
subsequent
ring
(so
a
in
total
apply
the
transformation
three
times)
b
y
y
9
(–2,
(–7,
13
8)
(–4,
(1,
12)
8
12
7
11
6
10
5
9
7)
4
3
7
2
6
1
5
4
0
–7
–6
–4
x
–2
(–6,
3)
–1
3
–2
2
–2)
–3
1
(–3,
0)
–4
x
(–6,
–7
–5)
–5
3 8 8
units
4, y )
each
(–9,
2
2, y
Answers
–6
–5
–4
–3
–2
–1
0
1
13)
right
y
c
y
d
5
10
(7,
9)
(–1,
4)
4
9
(2,
(–6,
8)
3)
8
3
7
2
6
1
5
0
–8
–7
–6
x
–4
4
–1
3
–2
2
–3
1
–4
–5
0
x
1
2
3
5
6
7
8
(–8,
–6)
–6
–1
(0,
–1)
–2
–7
–3
–8
–9
–4
(3,
–4)
(–5,
–9)
–10
–5
5
a
(12, 6 )
b
(5,
6
a
(
b
(17,
7
1L,
8
9
10
2,14 )
3U,
11L,
a
Upwards
b
Yes:
the
3D,
20 m;
a
Horizontally
Yes:
a
Individual
b
There
Practice
1
a
2
a
12U,
formed
230 mm;
pattern
are
5R,
sideways
pattern
b
the
3U,
several
12 )
5L,
the
the
e.g.
by
the
5U,
2D,
length
of
diamond
vertically
formed
response
by
by
3)
has
parallelograms
tessellation
patterns
the
or
a
no
and
(7,
c
( 4,
2U,
5D,
horizontal
pieces
152 mm;
bricks
25R,
c
has
no
11)
1)
5L,
diagonal
overlaps
combination
overlaps
triangles
11D,
and
of
no
5R,
of
and
3D
the
no
115 mm
diamond
gaps
horizontally
and
38 mm
vertically
gaps
translated
horizontally
visible
2
Individual
response
b
Individual
response
c
Individual
response
b
y
y
A
C ′
B ′
C ′
D
D ′
A
x
B
B ′
x
A′
D
D ′
C
B
C
A′
Answers
3 8 9
c
d
y
y
A
B
A
C
x
x
D ′
B
D
B ′
A′
A′
C
C ′
C ′
3
4
a
Blue
to
yellow:
Blue
to
black:
Blue
to
green:
Blue
to
red:
response,
Individual
response
a
(
c
(8, 0 )
9, 5)
( 6,
(0, 0 )
90°
units
units
black
from
b
of
2.5, y
(0,
anticlockwise
15)
Rotation
e.g.
b
of
2
10
e.g.
+
180°
−
2)
+ 5, y )
translation
Individual
a
(x
(x
translation
c
c
a
a
b
Rotation
5
(x, y )
(x, y )
(
down,
7.5
units
right ( x , y )
to
green
blue
to
(x,
y)
black:
(x
+
2.5,
rotation
y
center
about
the
2)
(
2.5,
0)
13,11)
(or
equivalently
clockwise)
response
b
Individual
response
7
a
Individual
response
b
Individual
response
3
b
Answers
−
origin
Individual
3 9 0
+ 7.5, y
2)
anticlockwise
a
(x
2)
180°
a
1
a
right
6
Practice
B ′
about
the
origin
c
Individual
response
2
a
Either
the
b
reect
diagram
Rotation
of
the
in
left/right
the
the
hand
horizontal
top/bottom
part
dotted
half
of
of
the
diagram
in
the
vertical
dotted
line
or
reect
the
top/bottom
part
the
diagram
through
180
°
about
the
point
where
the
two
lines
of
reection
intersect.
3
a
Reection
b
The
left/right
eur-de-lys
translated
4
of
have
hand
been
horizontally
half
of
diagram
translated
and
of
line
in
dotted
horizontally
vertical
and
the
line
symbols
in
the
quarters
of
the
shield
have
been
vertically
a
b
d
e
c
f
J ′
B
I
G
A
C
I ′
A′
H ′
D
H
D ′
K
J
K ′
C ′
B ′
G ′
Answers
3 91
Practice
1
4
a
y
b
y
A
B
D
C
C ′
x
x
B ′
D ′
y
=
–2
A
A′
C ′
D ′
D
B
A′
B ′
C
c
d
y
y
A
A
y
=
D
–x
C ′
B ′
x
x
y
=
–1
D
B
B
D ′
C
C
C ′
A′
D ′
A′
B ′
2
a
Reection
in
the
line
y
=
b
Reection
in
the
line
y
=
− x
c
Reection
in
the
line
y
=
0
d
Reection
in
the
line
=
−3
3
a
(5,
4
a
( 6,15)
5
a
Individual
6
Hexagons
7
8
9)
response
and
x
a
Squares,
Individual
rectangles
response
c
Individual
response
(i.e.
the
x
-axis)
b
(
b
( −13, −11)
b
Individual
rhombuses;
b
2
and
2, 0 )
Individual
response
response
trapezia
a
(x, y )
a
(
x, y )
i.e.
reection
in
y-axis
b
(x, y )
a
(
x, y )
i.e.
reection
in
y-axis
c
(x, y )
a
(
x, y )
i.e.
reection
in
y-axis
Practice
1
(12,
2
(
3
(16, 0 )
4
Scale
5
28)
4, 9 )
factor
3 9 2
3
Answers
c
(0, 8)
c
(0, 0 )
e.g.
the
symmetry
1
5
a
Scale
b
A
factor
4
negative
scale
factor
indicates
a
dilation
and
a
180°
degree
rotation
about
the
centre
of
dilation
3
6
Scale
factor
7
(4) ( x , y )
10
a
(3 x , 3 y )

(5)(
1
x, y ) a


(x, y) a


y
4


3

(6)
1
x
4
−

3

x ,−
10
y


10
y
8
A′
A
B ′
B
x
D
D ′
C
C ′
9
y
A
A′
B ′
x
B
C ′
C
10
11
a
The
b
Stars
smallest
a
Triangles,
b
Yes
of
stars
dierent
are
squares
because
congruence
sizes
there
are
and
is
trapezia
an
transformations
similarity
overall
are
transformations
translated,
basic
pattern
rotated
that
and
repeats
reected
without
overlapping
1
12
a
Repeated
b
No,
dilations
of
approximately
scale
factor
4
the
pattern
cannot
be
repeated
without
overlapping
Answers
3 9 3
Unit
Review
1
Point
Translation
up
3
Translation
units
left
5
Translation
units
2
units
7
(0,
0)
(0,
3)
(6,
1)
(6,
4)
(3,
1)
(3,
2)
(
7)
(
2
4,
8,
5)
(
(1,
(
(
4,
10)
(
8,
2)
a
5,
(
(13,
2)
9,
(10,
7)
13,
(3,
5)
(
factor
1,
2)
by
of
the
1)
4
4)
(12,
5)
(
7)
(
scale
0)
(24,
4)
a
about
origin
(0,
16,
8)
28)
32,
20)
y
b
y
Dilation
right
units
(7,
1)
2,
(
0)
down
and
B
C
C ′
B ′
A
D ′
B ′
x
x
A ′
C
C ′
B
D
A ′
A
3
4
No,
it
does
not
consist
a
Translation
5
b
Rotation
90°
c
Reection
d
Dilation
e
Reection
5
(60,
45)
6
Scale
factor
of
in
of
units
to
the
the
same
left,
anticlockwise
the
scale
in
of
the
line
x
factor
line
x
=
2
0
6
pattern
units
about
(i.e.
the
centered
y
repeated
up
the
origin
-axis)
about
the
origin
= 1
1
3
7
a
b
y
preimage
y
image
x
x
preimage
image
3 9 4
Answers
8
Point
(0,
0)
(8,
2)
Reection
Reection
in x-axis
in y-axis
in y = x
(0,
0)
(8,
(0,
2)
6)
(
1)
(
4,
1)
10)
(
5,
10)
4,
5,
(0,
(9,
(7,
(
(7,
(
9
Reection
3)
0)
6)
(
0)
8,
2)
7,
6)
(4,
(5,
(2,
8)
6,
(1,
10)
(0,
3)
(0,
3)
(9,
0)
(
0)
a
0)
(
1)
9,
(0,
(
10,
(
in y =
(0,
(
2,
in y =
0)
(0,
8)
(8,
(6,
7)
(7,
4)
(
1,
4)
(
(10,
5)
0)
(3,
9)
(0,
(
9)
Reection
3
in x =
6)
(
4)
(
12)
4,
5,
0)
b
y
Reection
–x
7)
5)
3,
(0,
Reection
(
0)
16,
2)
15,
5)
(
16)
8,
–4
(
6)
4,
3,
1)
10)
(0,
9)
(
8,
3)
(9,
6)
(
17,
0)
y
B ′
C ′
A′
A′
D ′
x
x
D
A
C
A
C
B ′
C ′
10
B
B
a
Individual
b
Each
square
response
contains
c
Each
square
has
reections
been
translated
and
a
rotations
distance
equal
to
its
diagonal
horizontally
and
vertically
11
Point
Rotation
of
degrees
(0,
0)
(8,
2)
(7,
6)
(
1)
(
4,
5,
(0,
(9,
12
(0,
8)
6,
(1,
10)
(
3)
(
0)
7)
4)
10,
3,
(0,
a
Rotation
180
5)
0)
9)
of
degrees
0)
(2,
(
90
(0,
(
7,
1)
(5,
10)
(
(
6)
(4,
(0,
(0,
2)
(10,
3)
9,
1,
0)
Rotation
360
0)
2,
(6,
(
of
degrees
0)
8,
(
Rotation
270
(0,
8)
(8,
7)
4)
5)
(3,
0)
(0,
9)
(
of
Rotation
degrees
−90
0)
(0,
2)
(
0)
2,
6)
(
1)
(
1,
4)
10)
(10,
5)
5,
(0,
(9,
(6,
8)
(7,
4,
3)
(3,
0)
of
degrees
(0,
7)
0)
9)
b
y
y
D
A
D ′
y
=
x
D
B
C
C ′
D ′
x
A
x
A′
A′
B ′
B
C
B ′
C ′
x
=
–3
Answers
3 9 5
13
14
a
(x, y )
a
(14,
c
(3,0)
a
(x
a
(9,0)
c
(
17
19
of
a
Individual
Reections
c
Individual
No,
the
Individual
a
Rotation
Reect
20
a
Individual
7
Linear
Practice
x
=
(
(0.5 x , 0.5 y )
c
(x, y )
a
(− x , − y )
d
(x, y )
a
(x,
y )
5,12)
about
(4,
the
origin
14)
about
the
origin
response
and
rotations
response
is
the
transformation ( x , y )
a
(− y , − x )
whereas
the
rotation
is
the
transformation
(− x , − y )
a
a
b
clockwise
180°
reection
a
b
1
90°
(x, y )
3)
b
(x, y )
18
of
b
b
5,
Rotation
16
− 7)
7)
Rotation
15
− 5, y
response
of
in
180°
the
b
about
line
y
=
response
0
Individual
the
response
c
Individual
c
x
response
origin
i.e.
the
x
-axis
b
Individual
b
x
response
systems
1
−2
=
−2
=
32
d
x
= 10
h
x
=
±13
6x
1
e
x
= 18
f
x
=
21
g
x
= −
2
i
x
=
−2
j
2
Individual
response
3
Individual
response
4
Individual
response
Practice
x
=
−1
2
1
1
a
y
=
−2 x
+ 4
b
y
=
2
x
− 2
y
c
=
2
1
e
y
=
3x
+ 9
f
y
=
1
y
=
+ 3
d
y
=
x
+ 2
h
y
=
1
x
− 4
y
g
=
6
i
x
+ 10
3
1
2
x
8
2
x
+
6
j
y
= 12 x
− 9
=
3
20
2
a
x
= 5, y
=
b
x
=
6, y
f
x
=
24, y
b
x
= 1, y
f
x
=
−3
c
x
=
−6, y
=
3
d
x
=
−4, y
= 12
c
x
=
−6, y
=
−1
d
x
=
−2, y
=
g
x
=
4, y
3
3
e
x
=
−2, y
a
x
=
9, y
= 16
= 12
=
−3
= 1
1
e
x
= −
, y
= 4
4, y
=
2
=
3
2
4
a
Number
3 9 6
of
movies
Standard
rental,
$
Loyalty
option
1,
$
Loyalty
option
1
6
35
14
2
12
35
18
3
18
35
22
4
24
35
26
5
30
35
30
Answers
2,
$
−6
y
b
Loya lty
option
1
35
30
25
)$(
tsoC
20
a
y
o
L
15
10
5
0
x
1
2
3
Number
c
Option
d
If
a
e
you
month
($10
it
a
One
Innitely
g
No
j
No
2
a
k
3
a
=
$4/movie)
than
cheaper
with
7
movies
Option
a
month
it
is
cheaper
with
Option
2,
but
if
you
download
b
No
c
No
e
Innitely
many
solutions
f
One
solution
solutions
h
Innitely
many
solutions
i
One
solution
solutions
k
One
l
No
solutions
−2
b
Individual
x
Individual
For
1
response:
is
c
cost
number
response
month,
$30.
If
if
you
days
based
you
use
used
For
you
cost
month,
used
the
$100,
so
e
Individual
f
Membership
g
Make
a
Linear,
cost
if
you
≠
or
more
movies
y
4
two
is
solutions
options
must
satisfy
equations
y
=
0.75 x
+ 90
and
y
=
5x
respectively,
part a
to
6
for
so
facilities
times
need
solution
solutions
cost
facilities
$120,
the
solutions
−2
the
facilities
for
would
for
on
cost
use
facilities
k
and
the
the
would
1
All
values
of
‘pay-as-you-go’
you
7
1.
response
many
b
b
plus
less
movies
solution
where
4
fee
of
5
3
d
d
is
Individual
Practice
1
2
download
4
a
times,
6
it
4
times
would
times,
month
use
membership
a
the
month
take
4
for
5
months,
facilities
cost
months,
$94.50
and
membership
‘pay
as
would
you
cost
go’
would
$108
but
months.
membership
for
would
4
for
would
cost
membership
more
than
5
$93
and
would
months
for
‘pay
cost
as
you
$102
but
membership
go’
would
cost
$20.
‘pay-as-you-go’
to
be
the
better
option.
response
would
per
one
day
be
the
same
better
with
or
option
as
without
it
will
joining
always
be
cheaper.
fee
solution
y
4
–6x
+
4y
=
–4
3
2
1
0
x
1
3x
+
2y
=
2
2
3
c
−6 x
− 4 y
=
−4
This
is
the
same
equation
as 3 x
+
2 y
=
2
if
you
divide
each
number
by
2
and
rearrange.
Answers
If
would
3 97
d
y
4
3
2
1
3x
+
2y
=
2
0
x
1
2
3
4
1
−6x
−
4y
=
−4
2
3
5
Individual
response
6
Individual
response
Practice
4
1
1
a
x
= 1, y
=
−3
b
x
=
7, y
= 1
c
x
=
6
, y
= 2
d
x
=
5
e
x
=
2, y
a
x
= 1, y
=
2
f
x
=
−5, y
= 1
b
x
=
−1, y
=
1
2
= 1
4
c
x
=
6
, y
= −
7
7
3
, y
=
d
2
x
=
−2, y
= 1
2
1
e
x
=
−2, y
=
−2
x
f
= 2, y
=
2
3
a
You
earn
each
b
4
$60
case
for
costs
each
$30
to
case
you
make
so
sell
y
=
so
44
Individual
x
=
2, y
=
−1
b
x
=
=
x
=
4, y
= 1
f
x
=
2, y
=
−1
x
=
4, y
= 1
j
x
=
2, y
=
0
a
Individual
b
Individual
the
could
Individual
response
Individual
response;
13
3
$600
4
3
5
45
6
solutions
a
2
at
8%,
sheep,
7
$6000
at
8
Adults
9
$3000
14
$400
the
17
at
9%
girls
beehives
9%,
£5,
3 9 8
ones
be
6
classrooms,
10
number
of
cases
sold;
The
initial
cost
is
$1300
and
d
After
selling
44
covers
c
x
=
−2, y
g
x
=
4, y
=
0
d
Innitely
−6
h
No
many
solutions
=
solution
response
response;
b
18
the
19
i
1
is
5
, y
e
Practice
x
5
many
3
where
response
19
2
60 x
2640
22
a
=
+ 1300
c
Practice
1
y
30 x
$2000
children
at
10%
£2
Answers
where
conrmed
ones
elimination
by
where
graphing
the
suggested
because
coecient
that
there
of x
or
y
there
would
is
1
were
not
are
no
be
solutions
any
easier
to
point
solve
or
where
where
by
the
there
two
were
lines
substitution.
innitely
intersect.
10
Adults
11
a
6400
12
2x
− 9 y
13
(3, 4 )
14
$40,
=
a
2.13
hectares
d
3.50
hectares;
Unit
a
b
Rs
b
29
3 040
000
they
making
would
a
need
to
000
increase
the
THB
c
number
of
hectares
of
their
Individual
farms
in
order
response
to
prot.
review
x
e
$8
−41
continue
1
teenagers
y
=
2
b
=
−42
f
x
y
=
−6
2
a
x
=
3, y
= 1
b
x
=
3
a
x
=
3, y
=
b
x
= −
0, y
=
x
=
c
No
−12
d
x
=
5
solution
d
x
=
2, y
−6
5
−5
c
= 10
164
, y
=
11
11
1
4
x
a
= −
, y
= 4
b
a
=
4, b
=
2
=
3
2
5
e
x
=
−3, y
a
One
6
x
7
−6, −5
8
Square;
9
x
10
11
=
−5
solution
b
Innitely
many
solutions
all
=
coins;
the
sides
are
30 cm
solution
long.
9
nickels,
18
dimes
and
Monthly
3
quarters
subscripti on
Pay
y
)$(
12
11
11
10
10
9
9
8
8
7
7
)$(
rep
tsoC
shtnom
12
6
song
6
tsoC
5
5
4
4
3
3
2
2
1
1
0
x
1
2
3
4
No.
9
c
12
6
teachers,
13
AU$15000
14
910
meals
per
y
0
15 16
No
8
a
b
c
−5
= 14, y
30
=
36
at
5
of
6
7
8
9
10
x
1
months
2
3
4
No.
Individual
5
of
6
7
8
9
10
months
response
students
3%,
that
are
AU$5000
paid
for
at
8%
and
800
free
meals
so
a
total
of
1710
meals.
years
16 100
nets,
40
solar
kits;
60
medicine
kits,
120
blankets.
Answers
3 9 9
Index
A
D
3D
shapes
cones
154–201
165–73,
cylinders
data
174,
157–65,
pyramids
spheres
addition
angles,
in
73,
scientic
measuring
apex/vertex
apothem
192
192
192
31–2,
39
types
208
of
decisions
of
surface
aspect
area
area
68
see
mean
scatter
axioms
surface
with
321,
of
between
distance
formula
210,
212,
220
236–7
298–9
scatter
distance
350
208,
independent
290–4,
direction
28
variables
swapping
170
see
ratios
averages
axes,
55,
38
notation
equations
dilations
circles
9–11,
251
318
dependent
177
204–52
211–12,
7–8,
scientic
degree
175
area
of
of
in
notation
data
reliability
decimals
89
84–5
165,
175,
191,
191,
AA (Angle-Angle)
bivariate
173–4,
175–82,
182–90,
192
diagrams
points
209
63–7
64
division
diagram
207,
208,
210
exponents
53
in
B
in
scientic
19–21
notation
32–3,
39
E
base
of
3D
cones
shapes
and
175
see
exponents
positive
bias
elimination
and
11,
negative
18,
38
13,
data
15
scatter
brackets
broken
227–40
plots
in
319,
of
linear
of
lines
linear
320
119,
of
lines
lines
112,
137–8,
best
quadratic
137–8,
of
lines
of
119,
139
120–36
systems
vertical
208
350
relationships
horizontal
207–43
equations
axes
350
dilations
cubic
204–52
analysing
also
333–9,
290
equations
211–12
bivariate
method
enlargement
pyramids
bases
192
165
t
319,
140
316–57
140
220–2,
320–1,
223–6
350
C
quartic
Cartesian
319
plane
solving
Pythagoras’
theorem
in
types
reections
rotations
see
also
on
318–22
63–7
of
319
278–80
on
equilateral
triangles
equivalent
equations
exponents/powers
and
correlation
237–40,
of
dilation
290–4,
centre
of
rotation
267,
19–21,
of
20–1,
55,
84–5,
rotations
coefcients
28,
area
volume
scientic
see
also
170–3,
192
192
reections;
congruent
shapes
congruent
triangles
coordinate
174,
transformations
conjectures
53,
261,
rotations;
261,
298
translations
267
69–70,
89
54
notation
negative
nite
decimals
9,
nite
numbers
9
xed
points
form
of
26–33
14–17,
19
four-square
in
diagrams
theorem
fractional
equations
fractional
exponents
7,
frustrums
theorem
10
182
scatter
fractions
plane/grid
Pythagoras’
and
F
39
166–9,
congruence
39
267–8
165–73
surface
38,
38
97–8
zero
clockwise
28,
17–19,
170
and
clinometers
22–4,
268
multiplying
of
38
38
299
laws
area
11–25,
243
dividing
centre
cones
334
graphs/graphing
causation
circles,
286
268–9
9–11,
208–9
55
320
38
38
167
63–7
G
solving
linear
systems
322–6
gradient-intercept
transformations
in
268–9,
lines
see
also
form
120–4,
125–9,
130,
131–2,
278–80
of
best
t
220–1,
223,
224
graphs/graphing
gradient/slope
111–12,
120–4,
125
coordinates
and
and
equations
and
linear
of
lines
correlation
equations
see
also
relationships
105–9,
130–2
of
linear
relationships
of
perpendicular
causation
237–40,
diagrams
rate
of
change
207,
coefcients
227–40,
vertical
and
horizontal
82–3,
gradient-intercept
gradient/slope
cubic
319,
cylinders
surface
217,
157–65,
area
solar
volume
161–5,
form
111–12,
120–4
113
systems
322–9,
and
Pythagoras’
350
theorem
and
scatter
and
transformations
diagrams
63–7
207
192
163,
268–9,
192
also
scatter
diagrams
173–4
H
192
compared
to
cones
compared
to
spheres
4 0 0
on
242
157–61,
panels
137–8,
350
see
of
112–14
137–40
267–8
linear
tting
lines
120–36,
89
rotations
equations
105–9,
243
and
ratio
counterclockwise
curve
lines
110
242
graphs/graphing
cosine
139
parallel
243
of
scatter
correlation
and
227–40
and
on
from
139
points
correlation
and
228
129–32
166
186–7
Index
height
measurements
heptagons,
tessellation
97–8
of
286
278–80
140
140
hexagonal
hexagons,
pyramids
175,
tessellation
horizontal
lines
hypotenuse
of
136–8,
56,
177,
179,
181
286
140
negative
rotations
negative
scale
nets
89
170,
180,
non-linear
267
factors
299
192
relationships
non-periodic
decimals,
110,
119,
innite
208–9
10
I
numbers
image
262,
4–47
298
decimal
independent
variables
208,
210,
212,
see
decimals
220
exponents
swapping
with
dependent
11–25
236–7
rational/irrational
innite
decimals
10
innite
numbers
9,
real
numbers
scientic
initial
conditions
intercepts
see
also
123,
point
38
notation
26–33,
36–7,
38
134
O
x-intercept;
irrational
129,
114–18
intersection,
6–11,
6
10
y-intercept
of
323,
numbers
6–8,
324,
octagons,
350
ordered
38
tessellation
pairs
outliers
L
219,
of
286
336
232–4
P
lateral
laws
in
legs
area
of
of
cones
scientic
of
best
line
of
reection
t
correlation
linear
regression
of
scatter
see
also
linear
point
102–50
118–35
distance
220–2
243
lines
112,
classifying
graphing
number
to
to
of
problem
316–57
326–7,
elimination
338
solve
solve
substitution
to
350
322–6
326–9
with
339–46
solve
horizontal
lines
of
lines
best
vertical
329–33,
350
gradient
positive
relationships
positive
rotations
of
power
of
a
product
power
of
a
of
105–9,
horizontal
line
of
linear
t
of
perpendicular
see
also
lurking
112,
112–14,
vertical
137–40
139,
242
316–57
222,
linear
161,
rules
225,
226
72
165
compared
18,
53–4,
Tower
56,
20,
175–6,
192
38
59
similarity
of
71–3
225,
226,
242
surface
area
191,
181,
60–3,
and
ag
and
maps
191,
191,
theorem
applying
34
192
178–82,
175–8,
Pythagoras’
238–40
12–13,
175–82,
volume
relationships
57
Hanoi
pyramids
224,
similarity
Pythagorean
350
136–8
variables
of
220–1,
puzzles
112–14
lines
38
298
pyramids
proofs
24,
23
diagrams
155,
product
23,
23
234–5
262,
triangle
217–26,
systems
regression
120–36,
140
136–8
best
parallel
and
137–8,
243
exponents/powers
scatter
prisms
rule
quotient
see
227–31,
72–3
power
principle
209,
267
a
137–8
111–15,
53,
139
of
preimage
223–6
13
power
predictions
120–36
220–2,
140
182
bases
powers
137–8
t
lines
gradient
graphs
119,
350
286–8
positive
and
of
324,
131–2,
129–32
positive
105–9
equations
323,
63–7
from
points
postulates
333–9,
solutions
solving
139
134–5
125–9,
between
equations
xed
219,
form
243
38
112–14,
intersection
polygons
227–31,
10,
231–7,
286
points
110–18
diagrams
of
of
9,
lines
of
relationships
point-slope
105–9
systems
lines
234
350
coefcient
decimals
piecewise
119
representing
strength
227–37
139,
113
tessellation
perpendicular
t
of
112–14,
correlation
periodic
298
representations
recognising
on
Pearson
242
279,
222,
best
characteristics
lines
parallelograms
39
coefcients
relationships
algebraic
38
pentagons,
275,
linear
linear
28,
217–26,
line
22–4,
56
line
also
parallel
20–1,
notation
(triangles)
see
170
exponents
192
192
56–68,
89
68
design
87–8
M
maps
63–7
mean
219,
63–7
Pythagorean
220–1,
of
angles
of
heights
metric
mirror
line
63
Q
84–5
60–3,
units
triples
243
measurements
quadratic
97–8
16–17,
275–6,
degree
21
quartic
298
equations
of
equations
quotient
319,
320–1
350
rules
319
19–21,
24,
38
models
R
and
decision-making
and
linear
318,
339
r-values
equations
129,
133,
correlation
moderate
relationships
228,
228,
231–7,
243
135–6
see
moderate
also
correlation
coefcients
229
209,
radius
and
rate
change
cylinders
162
243
of
105,
110–12,
123,
129
multiplication
rational
and
powers
17–19,
and
scientic
ratios
notation
32–3,
6–11,
38
inverse
see
trigonometric
ratios
39
real
multiplicative
numbers
20
numbers
6
19
see
N
also
irrational
reciprocals
negative
bases
negative
exponents/powers
13
negative
gradient
negative
relationships
rectangular
14–17,
19,
139
20,
24,
38
recurring
227–31,
243
see
also
24,
numbers;
rational
numbers
139
pyramids
decimals
reductions
209,
15,
177,
179,
192
8
290
dilations
Index
4 01
reections
on
the
line
274–82,
Cartesian
of
275,
regression
see
also
regular
279,
lines
line
298
best
polygons
278–80
tessellations
298
222,
of
terminating/nite
plane
analyzing
224,
225,
226,
242
tetrahedrons
relationships
see
and
bivariate
and
correlation
describing
linear
see
strength
data
227–40,
linear
of
of
data
right
cones
right
triangles
53,
of
251
theorem
tessellations
184,
12–13,
congruence
dilations
56–68
ratios
89
261,
274–82,
298
trend
see
trends,
261–7,
261–7,
lines
also
298
298
290
translations
188
298
298–9
267–74,
translations
23
34
256–313
290–4,
similarity
77–87,
pyramids
55–6
Hanoi
rotations
267–74,
287–8
triangular
reections
exponents
rounding
also
Tower
10
298
175
transformations
8,
272,
259
see
211–12,
trigonometric
rotations
tiles
165
Pythagoras’
as
243
227–31
decimals
roots
242,
relationships
repeating
also
theorems
227–31
reliability
and
204–52
270,
286–8
semi-regular
286–8
9
266–7,
283–9
polygons
t
numbers
259–61,
217,
line
of
scatter
271–2,
271–2,
221,
best
222,
298
298
242
t
diagrams
for
208
S
triangles
SAS
(Side-Angle-Side)
73,
50–98
89
congruent
scale
factors
290–4,
69–70,
diagrams/plots
correlation
286
207–43
and
and
227–40,
ag
design
line
of
best
t
theorem
56–68
217–26
similar
notation
26–33,
70–6,
with
31–2,
98
ag
design
with
32–3,
of
286
39
and
tessellations
87–8
39
tessellation
multiplication/division
89,
39
and
addition/subtraction
semi-regular
87–8
242–3
Pythagoras’
scientic
89
298
equilateral
scatter
trigonometric
ratios
77–87,
89
287–8
triangular
pyramids
175,
177,
179,
181
shapes
see
3D
also
tetrahedrons
154–201
trigonometric
polygons
ratios
trigonometry
259–61,
266–7,
270,
272,
see
height
measurement
cones
97–8
173
triangles
similar
images
similar
triangles
U
290
applications
ag
89
89
283–9
truncated
and
77–88,
298
and
analyzing
triangles
77–87,
286–8
tessellations
70–6,
of
89
73–6,
design
units,
98
metric
univariate
16–17
data
207
data
211–12,
87–8
V
similarity
transformations
290
validity
see
also
of
251
dilations
variables
sine
ratio
77–82,
89
and
slant
height
165,
bivariate
see
data
207,
242
170
correlation
slope
between
227–40
gradient/slope
and
slope-intercept
form
see
gradient-intercept
causation
237–40,
243
form
dependent/independent
solutions,
numbers
of
326–9,
and
spheres
182–90,
linear
area
183–6,
191,
186–90,
191,
pyramids
solar
cells
equations
for
between
see
319,
322,
scatter
350
165,
175
191
area
178–81,
182,
lines
136–8,
140
192
volume
volume
177,
178,
181,
192
cones
squares
(shape)
113,
166–9,
(Side-Side-Side)
73,
form
125–9,
161–5,
130,
131–2,
161,
of
best
t
225,
creating
steepness
see
also
straight
strength
strong
181,
192
191
W
gradient/slope
of
175–8,
186–90,
284–5
110
lines
192
192
226
spheres
stars,
175–6,
140
pyramids
lines
166,
89
prisms
standard
192
286
cylinders
SSS
see
weak
lines
relationships
correlation
relationships
websites,
209–10,
227–31,
227–31,
243
words,
reliability
describing
209–10,
of
227–31,
211–12,
relationships
243
251
with
139
237
X
strong
relationships
209,
243
x-intercept
substitution,
subtraction
surface
solving
and
systems
scientic
with
notation
area
cones
329–33,
31–2,
39
127,
cylinders
pyramids
125–9,
equations
178–82,
163,
131–2,
139
Y
192
157–61,
128,
350
y-intercept
170–3,
192
and
initial
from
139
120–4,
conditions
130–2
134
192
Z
and
solar
spheres
system
see
of
also
cells/panels
183–6,
linear
linear
188,
173–4
of
tangent
values
ratio
112,
322
systems
4 0 2
105–9,
82–4,
zero
exponents/powers
zero
power
191
equations
T
tables
236–7
diagrams/plots
175
vertical
surface
220,
238–40
of
vertex/apex
as
212,
192
relationships
square
210,
105–9
192
numbers
volume
relationships
192
lurking
surface
208,
350
139
89
Index
rule
38
14–17,
19,
20,
38
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