MATHEMATICS 10C
REAL NUMBERS
High School collaborative venture with
Harry Ainlay, Jasper Place, McNally, Queen Elizabeth,
Ross Sheppard and Victoria Schools
Harry Ainlay: David Cunningham, Christine Dibben
Jasper Place: Linda Aschenbrenner, Shelaine Kozakavich, Nic Ryan
Ross Sheppard: Tim Gartke, Jeremy Klassen, Don Symes
Victoria: Kevin Bissoon, Elisha Pinter
Facilitator: John Scammell (Consulting Services)
Editor: Rosalie Mazurok (Contracted)
2009 - 2010
Mathematics 10C
Real Numbers
Page 2 out of 61
TABLE OF CONTENTS
STAGE 1
DESIRED RESULTS
PAGE
Big Idea
4
Enduring Understandings
4
Essential Questions
5
Knowledge
6
Skills
7
Stage 2
ASSESSMENT EVIDENCE
Teacher Notes For Transfer Tasks
8
Transfer Task
A Radical Board Game
Teacher Notes for A Radical Board Game and Rubric
Transfer Task
Rubric
9
10 - 12
13 - 14
The Golden Ratio in a Face
Teacher Notes for The Golden Ratio in a Face and Rubric
Transfer Task
Rubric
Possible Solution
15
16 - 22
23 - 24
25 - 32
Stage 3 LEARNING PLANS
Lesson #1
Factors and Multiples
33 - 35
Lesson #2
Square Roots and Cube Roots
36 - 39
Lesson #3
Estimating Radicals
40 - 41
Lesson #4
Working with Radicals
42 - 44
Lesson #5
Rational Exponents
45 - 48
Lesson #6
Negative Exponents
49 - 51
Lesson #7
Irrational Numbers – Classifying and Ordering
52 - 54
Lesson #8
Working with Exponent Laws
55 - 56
APPENDIX - Handouts
Real Numbers Unit Handouts
Mathematics 10C
58 - 60
Real Numbers
Page 3 out of 61
Mathematics 10C
Real Numbers
STAGE 1
Desired Results
Big Idea:
Real Numbers provide students with a foundation upon which they build an
understanding of the different ways to represent and order real quantities. This
understanding will enable students to solve problems related to all disciplines.
Implementation note:
Post the BIG IDEA in a prominent
place in your classroom and refer
to it often.
Enduring Understandings:
The students will understand…

numeracy as it relates to real numbers.
o that the set of real numbers is continuous and is made up of rational and
irrational numbers.

that there are various ways of representing numbers including exponents,
fractions, and radicals.
Mathematics 10C
Real Numbers
Page 4 out of 61
Essential Questions:

When and why should we use exact values?

What is a real number?
o What are the different ways of representing real numbers?
o How can real numbers be classified?
o What strategies can you use to order real numbers appropriately?

What is the meaning of continuous?
o When are there gaps in a number line?
o How many numbers are there?
Implementation note:
Ask students to consider one of the
essential questions every lesson or two.
Has their thinking changed or evolved?
****
Mathematics 10C
Real Numbers
Page 5 out of 61
Knowledge:
Enduring
Understanding
Specific
Outcomes
Students will know…
Students will understand…

numeracy as it relates
to real numbers.



*AN 2
o that the set of real
numbers is
continuous and is
made up of
rational and
irrational numbers.

that there are various
ways of representing
numbers including
exponents, fractions,
and radicals.
that numbers can be ordered.
that numbers can be approximated.
the relationship between sets of numbers
(union of sets).
that the number line is infinitely continuous.
Students will know…
Students will understand…

Knowledge that applies to this
Enduring Understanding
*AN 1
*AN 2
*AN 3






that numbers can be approximated.
what an exponent is.
what integral and rational exponents
mean.
the exponent laws.
components of radicals.
*AN = Algebra and Number
*
Mathematics 10C
Real Numbers
Page 6 out of 61
Skills:
Enduring
Understanding
Specific
Outcomes
Students will be able to…
Students will understand…

numeracy as it relates
to real numbers.
o that the set of real
numbers is
continuous and is
made up of
rational and
irrational numbers.
Skills that apply to this Enduring
Understanding
*AN 1
*AN 2
*AN 3








Students will be able to…
Students will understand…

that there are various
ways of representing
numbers including
exponents, fractions,
and radicals.
sort real numbers into categories.
approximate irrational numbers.
order real numbers.
apply exponent laws.
solve problems involving real numbers.
solve problems involving real numbers.
express radicals in mixed and entire forms
and convert between forms.
express a number as a product of its prime
factors.
*AN 1
*AN 2
*AN 3








sort real numbers into categories.
approximate irrational numbers.
order real numbers.
apply exponent laws.
solve problems involving real numbers.
solve problems involving real numbers.
express radicals in mixed and entire forms
and convert between forms.
express a number as a product of its prime
factors.
*
*AN = Algebra and Number
Implementation note:
Teachers need to continually ask
themselves, if their students are
acquiring the knowledge and skills
needed for the unit.
Mathematics 10C
Real Numbers
Page 7 out of 61
STAGE 2
1
Assessment Evidence
Desired Results Desired Results
A Radical Board Game or The Golden Ratio in a Face
Teacher Notes
There are two transfer tasks to evaluate student understanding of the concepts relating to
slope. The teacher (or the student) will select one for completion. Photocopy-ready versions
of the two transfer tasks and rubric are included in this section.
Implementation note:
Students must be given the transfer task & rubric at
the beginning of the unit. They need to know how
they will be assessed and what they are working
toward.
Each student will:
(A Radical Board Game)

demonstrate their understanding of the exponent laws.

demonstrate their understanding of vocabulary related to real numbers.
(The Golden Ratio in a Face)

demonstrate an appreciation for the beauty of mathematics.

create an approximate value for phi and understand that it is
a special irrational
number.
Mathematics 10C
Real Numbers
Page 8 out of 61
Teacher Notes for A
Radical Board Game Transfer Task
Board game considerations should be tailored – only share as much as you feel is necessary
(differentiated instruction). Teachers should feel free to add any suggestions that may move
students along. For example, chance cards could be created that would require students to use
(an additional) law or power rule before moving.
Teacher Notes for Rubric

No score is awarded for the Insufficient/Blank column , because there is no evidence of
student performance.

Limited is considered a pass. The only failures come from Insufficient/Blank.

When work is judged to be Limited or Insufficient/Blank, the teacher makes decisions
about appropriate intervention to help the student improve.
Implementation note:
Teachers need to consider what performances and
products will reveal evidence of understanding?
What other evidence will be collected to reflect
the desired results?
Mathematics 10C
Real Numbers
Page 9 out of 61
A Radical Board Game - Student Assessment Task
Task: You have been hired by HAZBRO to create an award-winning mathematical
board game focusing on exponents and radicals. You are expected to present your
game idea and a working prototype to the board of directors. The following concepts
need to be included in your design:
Name
Definition
Example
Completed
Zero Exponent Law
x0  1
70  1
Product of Powers
x m  x n  x mn
x 2  x3  x 23  x5
Quotient of Powers
x m  x n  x mn
x 4  x 2  x 42  x 2
Power of a Power
( x m )n  x mn
( x 2 )3  x 23  x6
Power of a Product
( xy)n  x n y n
( xy)3  x3 y3
Power of a Quotient
 x   xn 
   n 
 y  y 
Negative Exponent
Rational Exponents
n
xn 
1
xn
m
2
 x   x2 
   2 
 y  y 
x 3 
1
x3
2
x n  n xm
x 3  3 x2
Converting from
Mixed to Entire
A radical with a
coefficient of 1
2 5  20
Converting from
Entire to Mixed
Product of a rational
number and a
radical
Number Line
Placing radicals in
order on a number
line
Rational and
Irrational Numbers
Real Numbers
40  2 10
 5, 0,  , 11, 2 6
Rational:
4
5
Irrational: 
1.
2.
3.
Board Game Considerations
You may want to include the following:

Board design (Do you need a game board?)
o Example: Locate some cardboard that can be used to form the playing board for
your math board game. You can use whatever you have on hand, as long as one
side contains no writing. Use a black marker and a meter stick to mark evenly
spaced squares around the perimeter.

Playing cards

Game pieces
o Instead of using standard dice, create one where each side’s value is a radical.
You may choose a die with 8, 12 or 20 sides. For Example, if you rolled 5 ,
whose value of that is  2.236067977… so you would move 2 spaces.
Essentially, you would always round to the nearest whole number.
o Moving pieces

Rules

How to start

How to win

How to move or score points
o Example: Roll the dice--the player who rolls the highest roll goes first. Take
turns rolling the dice and moving game pieces around the board. Each time you
land, your opponent will read a math problem from a card that matches the
space you have landed on. If you answer correctly, you get the points assigned
to that colour. If you answer incorrectly, you do not get any points. The first
player to reach 100 points wins!
Struggling?

Make a long list of math problems and come up with the solutions - every problem
you include may represent a game card, or a board space.

Look for inspiration for your game. Feel free to use ideas from other games that you
have played in the past (e.g., Monopoly, Snakes and Ladders, Sorry!, Trivial Pursuit,
Cranium, etc.).
Still stuck?...

Use colour markers to assign a point value to each square on your math game
board. For example, use red to denote spaces that are worth ten points. Use yellow
to denote spaces that are worth five points. Try not to assign points to every
square, you can add some fun to the board by including lose your turn spaces, roll
again spaces, free points spaces, a bet-your-own points space, chance cards etc.

Equally divide the math problems on your list into categories that match the colour
point values. Make sure that the most challenging math problems are placed into the
category with the most points assigned to it and that all others are grouped
accordingly, as well.

Write the math problems onto cards. If you can find colour note cards to match the
various point categories, use them. If you cannot, just colour the edges of the note
cards with a marker for identification. Group the note cards into piles.
Assessment
Mathematics 10C
Real Numbers
Rubric
Level
Criteria
Excellent
4
Proficient
3
Adequate
2
Limited*
1
Insufficient /
Blank*
No score is
awarded
because there
is no evidence
of student
performance.
No data is
presented.
Performs
Calculations
Performs
precise and
explicit
calculations.
Performs
focused and
accurate
calculations.
Performs
appropriate
and generally
accurate
calculations.
Performs
superficial
and irrelevant
calculations.
Presents Data
Presentation of
data is
insightful and
astute.
Presentation
of data is
logical and
credible.
Presentation of
data is
simplistic and
plausible.
Presentation of
data is vague
and
inaccurate.
Explains
Choice
Shows a
solution for the
problem;
provides an
insightful
explanation.
Shows a
solution for
the problem;
provides a
logical
explanation.
Communicates
findings
Develops a
compelling and
precise
presentation
that fully
considers
purpose and
audience; uses
appropriate
mathematical
vocabulary,
notation and
symbolism.
Develops a
convincing
and logical
presentation
that mostly
considers
purpose and
audience;
uses
appropriate
mathematical
vocabulary,
notation and
symbolism.
Shows a
solution for the
problem;
provides
explanations
that are
complete but
vague.
Develops a
predictable
presentation
that partially
considers
purpose and
audience; uses
some
appropriate
mathematical
vocabulary,
notation and
symbolism.
Shows a
solution for the
problem;
provides
explanations
that are
incomplete or
confusing.
Develops an
unclear
presentation
with little
consideration
of purpose and
audience; uses
inappropriate
mathematical
vocabulary,
notation and
symbolism.
No explanation
is provided.
No findings are
communicated.
Glossary
accurate – free from errors
astute – shrewd and discerning
appropriate – suitable for the circumstances
compelling – convincing and persuasive
complete – including every necessary part
convincing – impressively clear or definite
credible – believable
explicit – expressing all details in a clear and obvious way
focused – concentrated on a particular thing
incomplete – partial
inaccurate – not correct
inappropriate – not suitable
insightful – a clear perception of something
irrelevant – not relevant or important
logical - based on facts, clear rational thought, and sensible reasoning
precise - detailed and specific
plausible – believable
predictable - happening or turning out in the way that might have been expected
simplistic – lacking detail
superficial - having little significance or substance
unclear – ambiguous or imprecise
vague - not clear in meaning or intention
Teacher Notes for The Golden Ratio in a Face Transfer Task
This task leads students through discovering the golden ratio. Students will use pictures of
faces, measure set dimensions and calculate ratios to approximate the golden ratio.
The introduction to this project comes from the following website:
http://www.markwahl.com/index.php?id=22
The calculation of the golden ratio lends itself quite nicely to an excel application.
Avoid the implication that beauty can be measured by the proximity of your proportions to the
golden ratio. For example, Julia Roberts has a wide mouth and big lips, but these are
considered her most beautiful and distinguishing feature.
Teacher Notes for Rubric

No score is awarded for the Insufficient/Blank column , because there is no evidence of
student performance.

Limited is considered a pass. The only failures come from Insufficient/Blank.

When work is judged to be Limited or Insufficient/Blank, the teacher makes decisions
about appropriate intervention to help the student improve.
Implementation note:
Teachers need to consider what performances and
products will reveal evidence of understanding?
What other evidence will be collected to reflect
the desired results?
Mathematics 10C
Real Numbers
Page 15 out of 61
The Golden Ratio in a Face - Student Assessment Task
Statues of human bodies that the ancient Greeks considered most “perfect” embodied many
Golden Ratios. It turns out that the “perfect” (to ancient Greeks) human face has a whole
flock of Golden Ratios as well.
In this task, you will measure distances on the face of a famous Greek statue (despite its
broken nose) by using the instructions on this page. Before you start, notice that near the
face on the second page are names for either a location on the face or a length between
two places on the face. Lines mark those lengths or locations exactly.
Using your cm/mm ruler and the face picture on the next page, find each measurement
below to the nearest millimetre that is tenth of a cm or .1cm (___._ cm). Remember, you are
measuring the distance or length between the two locations mentioned. You can use the
marking lines to place the ruler for your measurements. Complete the table.
a = Top-of-head to chin = ___ . __ cm
b = Top-of-head to pupil = ___ . __ cm
c = Pupil to nose tip = ___ . __ cm
d = Pupil to lip = ___ . __ cm
e = Width of nose = ___ . __ cm
f = Outside distance between eyes = ___ . __ cm
g = Width of head = ___ . __ cm
h = Hairline to pupil = ___ . __ cm
i = Nose tip to chin = ___ . __ cm
j = Lips to chin = ___ . __ cm
k = Length of lips = ___ . __ cm
I = Nose tip to lips = ___ . __ cm
Now use these letters and go on to the next page to compute ratios with them with your
a
calculator. Remember: the first ratio means divide measurement a by measurement g ;
g
round your answers to 3 decimal places.
Studies suggest that Shania Twain may have one of the most beautifully proportioned
faces.
Head Width
Eye Nose
Eye
Top of the Head
Hairline
Pupils
Nose Tip
Lips
Chin
Lips
a = Top-of-head to chin = _____cm
h = Hairline to pupil = _____cm
b = Top-of-head to pupil = _____cm
i = Nose tip to chin = _____cm
c = Pupil to nose tip = _____cm
j = Lips to chin = _____cm
d = Pupil to lip = _____cm
k = Length of lips = _____cm
e = Width of nose = _____cm
I = Nose tip to lips = _____cm
f = Outside distance between eyes = _____cm
g = Width of head = _____ cm
How closely proportioned to the golden ratio is Johnny Depp’s face?
Head Width
Eye Nose Eye
Top of the Head
Hairline
Pupils
Nose Tip
Lips
Chin
Lips
a = Top-of-head to chin = _____ cm
g = Width of head = _____ cm
b = Top-of-head to pupil = _____ cm
h = Hairline to pupil = _____ cm
c = Pupil to nose tip = _____ cm
i = Nose tip to chin = _____ cm
d = Pupil to lip = _____ cm
j = Lips to chin = _____ cm
e = Width of nose = _____ cm
k = Length of lips = _____ cm
f = Outside distance between eyes = _____ cm
I = Nose tip to lips = _____ cm
Place the measurements for Shania Twain and Johnny Depp here, and calculate the
ratios.
Shania
Average ratio = __________
Johnny
Average ratio = __________
Take a photo of yourself straight on or find a straight on headshot from a magazine or
the internet. Identify the same ratios to see how closely you (or your chosen picture)
match the golden ratio.
Your name here: ____________________
Average ratio = __________
What you have is an approximation of the golden ratio. Originally the golden ratio was
developed using the following ratio.
1
0
1 x
x
The ancient Greeks set up the following ratio, where x represented the Golden ratio.
1
x

x 1 x
We can approximate many different types of constants with something called nested
radicals and continued fractions. Nested radicals are radicals within radicals and
continued fractions are fractions within fractions both of which continue without end.
For the golden ratio, , the continued fraction looks like...
1
  1
1
1
1
The nested ratio for
1
1  ...
is...
Use the pattern for the continued fraction and the nested radical to determine the
value of
to four decimal places. You will know that you have done it correctly when
consecutive terms no longer change the value of the 4 th decimal place.
How closely do the Greek statue, Shania, Johnny, and yourself match the value of the
golden ratio calculated using continued fractions and nested radicals?
Assessment
Mathematics 10C
Real Numbers
Rubric
Level
Criteria
Excellent
4
Proficient
3
Adequate
2
Limited*
1
Insufficient /
Blank*
No score is
awarded
because there
is no evidence
of student
performance.
No data is
presented.
Performs
Calculations
Performs
precise and
explicit
calculations.
Performs
focused and
accurate
calculations.
Performs
appropriate
and generally
accurate
calculations.
Performs
superficial
and irrelevant
calculations.
Presents Data
Presentation of
data is
insightful and
astute.
Presentation
of data is
logical and
credible.
Presentation of
data is
simplistic and
plausible.
Presentation of
data is vague
and
inaccurate.
Explains
Choice
Shows a
solution for the
problem;
provides an
insightful
explanation.
Shows a
solution for
the problem;
provides a
logical
explanation.
Communicates
findings
Develops a
compelling and
precise
presentation
that fully
considers
purpose and
audience; uses
appropriate
mathematical
vocabulary,
notation and
symbolism.
Develops a
convincing
and logical
presentation
that mostly
considers
purpose and
audience;
uses
appropriate
mathematical
vocabulary,
notation and
symbolism.
Shows a
solution for the
problem;
provides
explanations
that are
complete but
vague.
Develops a
predictable
presentation
that partially
considers
purpose and
audience; uses
some
appropriate
mathematical
vocabulary,
notation and
symbolism.
Shows a
solution for the
problem;
provides
explanations
that are
incomplete or
confusing.
Develops an
unclear
presentation
with little
consideration
of purpose and
audience; uses
inappropriate
mathematical
vocabulary,
notation and
symbolism.
No explanation
is provided.
No findings are
communicated.
Glossary
accurate – free from errors
astute – shrewd and discerning
appropriate – suitable for the circumstances
compelling – convincing and persuasive
complete – including every necessary part
convincing – impressively clear or definite
credible – believable
explicit – expressing all details in a clear and obvious way
focused – concentrated on a particular thing
incomplete – partial
inaccurate – not correct
inappropriate – not suitable
insightful – a clear perception of something
irrelevant – not relevant or important
logical - based on facts, clear rational thought, and sensible reasoning
precise - detailed and specific
plausible – believable
predictable - happening or turning out in the way that might have been expected
simplistic – lacking detail
superficial - having little significance or substance
unclear – ambiguous or imprecise
vague - not clear in meaning or intention
The Golden Ratio in a Face –Possible Solution
Statues of human bodies that the ancient Greeks considered most “perfect” embodied
many Golden Ratios. It turns out that the “perfect” (to ancient Greeks) human face has a
whole flock of Golden Ratios as well.
In this task, you will measure distances on the face of a famous Greek statue (despite its
broken nose) by using the instructions on this page. Before you start, notice that near the
face on the second page are names for either a location on the face or a length between
two places on the face. Lines mark those lengths or locations exactly.
Using your cm/mm ruler and the face picture on the next page, find each measurement
below to the nearest millimetre that is tenth of a cm or .1cm (___._ cm). Remember, you
are measuring the distance or length between the two locations mentioned. You can use
the marking lines to place the ruler for your measurements. Complete the table.
a = Top-of-head to chin = 7.4 cm
b = Top-of-head to pupil = 3.7 cm
c = Pupil to nose tip = 1.4 cm
d = Pupil to lip = 2.2 cm
e = Width of nose = 1.4 cm
f = Outside distance between eyes = 3.8 cm
g = Width of head = 4.9 cm
h = Hairline to pupil = 2.2 cm
i = Nose tip to chin = 2.3 cm
j = Lips to chin = 1.5 cm
k = Length of lips = 2.3 cm
I = Nose tip to lips = 0.8 cm
Now use these letters and go on to the next page to compute ratios with them with your
a
calculator. Remember: the first ratio, means divide measurement a by measurement g ;
g
round your answers to 3 decimal places.
Mathematics 10C
Real Numbers
Page 25 out of 61
7.4
4.9
1.510
3.7
2.2
1.682
2.3
1.5
1.533
2.3
1.4
1.643
1.4
0.8
1.750
3.8
2.2
1.727
2.3
1.4
1.643
Average ratio = __________
1.641
Mathematics 10C
Real Numbers
Page 26 out of 61
Studies suggest that Shania Twain may have one of the most beautifully proportioned
faces.
Head Width
Eye
Nose
Eye
Top of the Head
Hairline
Pupils
Nose Tip
Lips
Chin
Lips
a = Top-of-head to chin = 9.1 cm
g = Width of head = 6.2 cm
b = Top-of-head to pupil = 4.5 cm
h = Hairline to pupil = 3.1 cm
c = Pupil to nose tip = 1.6 cm
i = Nose tip to chin = 3.0 cm
d = Pupil to lip = 2.9 cm
j = Lips to chin = 1.6 cm
e = Width of nose = 1.5 cm
k = Length of lips = 2.9 cm
f = Outside distance between eyes = 4.2 cm
I = Nose tip to lips = 1.4 cm
Mathematics 10C
Real Numbers
Page 27 out of 61
How closely proportioned to the golden ratio is Johnny Depp’s face?
Head Width
Eye Nose
Eye
Top of the Head
Hairline
Pupils
Nose Tip
Lips
Chin
Lips
a = Top-of-head to chin =
9.9
cm
g = Width of head = 5.6 cm
b = Top-of-head to pupil = 5.3 cm
h = Hairline to pupil = 2.9 cm
c = Pupil to nose tip = 1.7 cm
i = Nose tip to chin = 2.9 cm
d = Pupil to lip = 2.8 cm
j = Lips to chin = 1.7 cm
e = Width of nose = 1.2 cm
k = Length of lips = 2.2 cm
f = Outside distance between eyes = 3.6 cm
I = Nose tip to lips = 1.1 cm
Mathematics 10C
Real Numbers
Page 28 out of 61
Place the measurements for Shania Twain and Johnny Depp here, and calculate the
ratios.
Johnny Depp
Shania Twain
1.768
9.1
6.2
1.468
4.5
2.9
1.552
5.3
2.8
1.893
3.0
1.6
1.875
2.8
1.7
1.647
3.0
1.6
1.875
2.9
1.7
1.706
1.5
1.4
1.071
1.2
1.1
1.091
4.2
3.1
1.355
3.6
2.9
1.241
2.9
1.5
1.933
2.2
1.2
1.833
9.9
5.6
1.597
Average ratio = __________
1.590
Average ratio = __________
Mathematics 10C
Real Numbers
Page 29 out of 61
Take a photo of yourself straight on or find a straight on headshot from a magazine or
the internet. Identify the same ratios to see how closely you (or your chosen picture)
match the golden ratio.
Eye
Head width
Eye
Nose
Top of the head
Hairline
Pupils
Tip of the Nose
Lips
Chin
Mouth
Mathematics 10C
Real Numbers
Page 30 out of 61
Your name here: _Myself__
a = Top-of-head to chin = 7.8 cm
7.8
5.4
b = Top-of-head to pupil = 3.2 cm
c = Pupil to nose tip = 1.5 cm
d = Pupil to lip = 2.7 cm
e = Width of nose = 1.7 cm
f = Outside distance between eyes = 3.5 cm
g = Width of head = 5.4 cm
h = Hairline to pupil = 2.7 cm
i = Nose tip to chin = 3.1 cm
j = Lips to chin = 1.9 cm
k = Length of lips = 2.1 cm
I = Nose tip to lips = 1.2 cm
1.444
3.2
2.7
1.185
3.1
1.9
3.1
1.5
1.632
1.7
1.2
1.417
3.5
2.7
1.296
2.1
1.7
1.235
2.067
1.468
Average ratio = __________
What you have is an approximation of the golden ratio. Originally the golden ratio was
developed using the following ratio.
1
0
1 x
x
The ancient Greeks set up the following ratio, where x represented the Golden ratio.
1
x

x 1 x
Mathematics 10C
Real Numbers
Page 31 out of 61
We can approximate many different types of constants with something called nested
radicals and continued fractions. Nested radicals are radicals within radicals and
continued fractions are fractions within fractions both of which continue without end.
For the golden ratio, , the continued fraction looks like...
1
  1
1
1
1
The nested ratio for
1
1  ...
This eventually
equals 1.618.
is...
This also eventually
equals 1.618.
How closely do the Greek statue, Shania, Johnny, and yourself match the value of the
golden ratio calculated using continued fractions and nested radicals?
In order we come in at:
Me – 1.468
Shania – 1.590
Johnny – 1.597
Phi – 1.618
Statue – 1.641
Now with reference to Phi I will calculate the percentage error from the exact value for
everyone.
 1.618  1.468 
Me: 
 100  9.3%
1.618


 1.618  1.590 
Shania: 
 100  1.7%
1.618


 1.618  1.597 
Johnny: 
 100  1.3%
1.618


 1.618  1.641 
Statue: 
 100   1.4%
1.618


Mathematics 10C
Real Numbers
Page 32 out of 61
STAGE 3
Learning Plans
Lesson 1
Factors and Multiples
STAGE 1
BIG IDEA:
Real Numbers provide students with a foundation upon which they build an understanding of the
different ways to represent and order real quantities. This understanding will enable students to solve
problems related to all disciplines.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
The students will understand…


numeracy as it relates to real numbers.
o that the set of real numbers is
continuous and is made up of
rational and irrational numbers.
there are various ways of representing
numbers including exponents, fractions,
and radicals.

When and why should we use exact
values?

What is a real number?
o What are the different ways of
representing real numbers?
o How can real numbers be
classified?
o What strategies can you use to
order real numbers appropriately?
.
KNOWLEDGE:
SKILLS:
Students will know…
Students will be able to…


that numbers can be ordered.
that numbers can be approximated.

express a number as a product of its
prime factors.
Implementation note:
Each lesson is a conceptual unit and is not intended to
be taught on a one lesson per block basis. Each
represents a concept to be covered and can take
anywhere from part of a class to several classes to
complete.
Mathematics 10C
Real Numbers
Page 33 out of 61
Lesson Summary
Students will review whole number factors and multiples. (Note: these were last
addressed in grade 6.)
Lesson Plan
This lesson is review for students (previously learned in grade 6) and will likely not take an
entire class. This topic is also covered in Polynomials, and so may have been previously
taught.
Activate Prior Knowledge
Quick discussion of Multiples vs. Factors.
Play the “Buzz”.
Have students stand in a circle – the teacher says a number, and the students count.
When the count gets to a multiple of the starting number the student says “buzz”
rather than the number. Students who make a mistake sit down. The winner is the
last person standing. For a more advanced game, use two or more numbers as the
“buzz” factors.
Factors Activity
Students build rectangles to explore the factors of a given number. This can be done
as a class using a projector or an interactive whiteboard or in a computer lab where
students work independently.
http://www.shodor.org/interactivate/activities/FactorizeTwo/
Introduce Greatest Common Factor and Least Common Multiple
Provide definitions of Greatest Common Factor (GCF) and Least Common Multiple
(LCM).
Provide groups of students with several pairs of numbers and ask them to find both
GCF and LCM. Have students describe their strategies on posters and share with the
rest of the class.
Mathematics 10C
Real Numbers
Page 34 out of 61
Check for Understanding
Quick Check: put two numbers on the board and ask the students to individually find
the GCF and LCM.
Practise new learning
Assign selected exercises from text.
Assess learning
Exit slip at the end of the class.
Give the students sets of numbers and have them find GCF and LCM (this should be
completed individually).
Going Beyond
Strong students should be given a larger set of numbers (find GCF and LCM of 3 or 4
numbers)
Resources
Math 10 (McGraw Hill: sec 5.2)
Foundations and Pre-calculus Mathematics 10 (Pearson: sec 3.1)
Interactive whiteboard or Projector or class set of computers
Glossary
greatest common factor (GCF) – the largest or most complex factor that a set of terms have
in common
least common multiple (LCM) – the smallest or least complex multiple that a set of
terms have in common
Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary
(http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in
more than one division. Some terms have animations to illustrate meanings.
Mathematics 10C
Real Numbers
Page 35 out of 61
Lesson 2
Square Roots and Cube Roots
STAGE 1
BIG IDEA:
Real Numbers provide students with a foundation upon which they build an understanding of the
different ways to represent and order real quantities. This understanding will enable students to solve
problems related to all disciplines.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
The students will understand…


numeracy as it relates to real numbers.
o that the set of real numbers is
continuous and is made up of
rational and irrational numbers.
that there are various ways of
representing numbers including
exponents, fractions, and radicals.

When and why should we use exact
values?

What is a real number?
o What are the different ways of
representing real numbers?
o How can real numbers be
classified?
o What strategies can you use to
order real numbers appropriately?
.
KNOWLEDGE:
SKILLS:
Students will know…
Students will be able to…

components of radicals.


solve problems involving real numbers.
Identify square roots, cube roots, perfect
squares and perfect cubes.
Lesson Summary
Students will identify perfect squares and perfect cubes, and then determine square
roots and cube roots.
Mathematics 10C
Real Numbers
Page 36 out of 61
Lesson Plan
Activate Prior Knowledge/Experience
Provide students with a set of 1 unit square algebra tiles.

Ask students to make a rectangle of area 15 and identify the side lengths.

Ask students if we can make a square with area 15.

Ask students to make rectangles with the following areas and identify which can
be made into a square.
o 4
o 6
o 8
o 9
o 12
Use Perfect Squares and Cubes notebook file (see notebook file in the share site) to
model on the board if necessary.
Discuss perfect squares and square roots based on the investigations.
Diagnostic Assessment of Prior Knowledge
Ask students to identify perfect squares and square roots of larger numbers without
the use of algebra tiles.
Present New Learning
Provide students with a set of snap cubes.

Ask students to build a 3 dimensional shape of volume 8.

Ask students if they can use the 8 pieces to make a cube. What is the side
length of the cube?

Ask students to make rectangular prisms (if possible) with the following
volumes and identify which can be made into a cube.
o 12
o 18
o 25
o 27
Mathematics 10C
Real Numbers
Page 37 out of 61
Discuss perfect cubes and cube roots based on the investigation.
Define a radical (and components of), specifically in terms of perfect squares
and perfect cubes.
Make a connection between radicals and prime factorization (link to Lesson #4)
9  3 3  3
Examples:
3
27  3 3  3  3  3
3
64  3 2  2  2  2  2  2  3 2  2  2  3 2  2  2  2  2  4
Check for Understanding
Ask students to identify perfect squares and square roots of larger numbers without
the use of algebra tiles.
Practise New Learning
Assign selected exercises from text.
Resources
Foundations and Pre-calculus Mathematics 10 (Pearson: sec 3.2)
Math 10 (McGraw Hill: sec 4.1)
Perfect Squares and Cubes notebook file (in share site)
algebra tiles
snap cubes
interactive whiteboard or a projector
Mathematics 10C
Real Numbers
Page 38 out of 61
Glossary
cube – the result of a number or term being multiplied by itself twice more
cube root – a number that when multiplied by itself twice more will result in the original
number
index – the order of root being taken (e.g. 2 is the index for square root and 3 is the
index for a cube root)
radical – the radical symbol or the symbol together with the index and the radicand
radicand – the number or expression of which the root is being taken
square – the result of a number or term being multiplied by itself
square root (of a non-negative number) – a non-negative number that when
multiplied by itself results in the original number
Mathematics 10C
Real Numbers
Page 39 out of 61
Lesson 3
Estimating Radicals
STAGE 1
BIG IDEA:
Real Numbers provide students with a foundation upon which they build an understanding of the
different ways to represent and order real quantities. This understanding will enable students to solve
problems related to all disciplines.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
The students will understand…


numeracy as it relates to real numbers.
o that the set of real numbers is
continuous and is made up of
rational and irrational numbers.
that there are various ways of
representing numbers including
exponents, fractions, and radicals.

When and why should we use exact
values?

What is a real number?
o What are the different ways of
representing real numbers?
o How can real numbers be
classified?
o What strategies can you use to
order real numbers appropriately?

What is the meaning of continuous?
o When are there gaps in a number
line?
o How many numbers are there?
.
KNOWLEDGE:
SKILLS:
Students will know…
Students will be able to…



that numbers can be ordered.
that numbers can be approximated.
components of radicals.



approximate irrational numbers.
express radicals in mixed and entire forms
and convert between forms.
express a number as a product of its
prime factors.
Lesson Summary
Students will estimate and order radicals (using a number line).
Mathematics 10C
Real Numbers
Page 40 out of 61
Lesson Plan
Activate Prior Knowledge/Experience
Review of perfect squares and perfect cubes – given a list of radicals that are perfect
squares and cubes and whole numbers, put them in order.
36, 4,3, 3 27, 3 64, 144,16, 6
Present new learning
Discussion of what is happening between the values above.
What methods can we use to figure out where
Try to place on a number line:
18 would go?
10, 24, 3 24, 3 16, 3 48, 90, 2 3, 4 2,3 3 3
Practice new learning
Assign selected exercises from text.
Review and consolidate learning
Number line game – each student is given a card with a radical or whole number –
they need to put themselves in a line from least to greatest without using calculators
(and without talking for an extra challenge).
Resources
Math 10 (McGraw Hill: sec 4.4)
Foundations and Pre-calculus Mathematics 10 (Pearson: sec 4.1)
Mathematics 10C
Real Numbers
Page 41 out of 61
Lesson 4
Working with Radicals
STAGE 1
BIG IDEA:
Real Numbers provide students with a foundation upon which they build an understanding of the
different ways to represent and order real quantities. This understanding will enable students to solve
problems related to all disciplines.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
The students will understand…


numeracy as it relates to real numbers.
o that the set of real numbers is
continuous and is made up of
rational and irrational numbers.
that there are various ways of
representing numbers including
exponents, fractions, and radicals.

When and why should we use exact
values?

What is a real number?
o What are the different ways of
representing real numbers?
o How can real numbers be
classified?
o What strategies can you use to
order real numbers appropriately?
.
KNOWLEDGE:
SKILLS:
Students will know…
Students will be able to…

components of radicals.


express radicals in mixed and entire forms
and convert between forms.
express a number as a product of its
prime factors.
Lesson Summary
Students will learn to convert between mixed and entire radicals.
Mathematics 10C
Real Numbers
Page 42 out of 61
Lesson Plan
Activate Prior Knowledge/Experience
Review prime factorization. Put three or four examples on the board, have students
try them.
Review factors of a number from Lesson #1
(Example: 36 = 1x36, 2x18, 3x12, 4x9, 6x6)
Review definition of a radical.
Present New Learning
Expand the definition of a radical to include numbers other than 2 or 3 as your index.
Estimate the value of 18 . Estimate the value of 3 2 . What do you notice? Why is
that?
Method 1
Prime factorization
18  3  3  2  3  3  2  3 2
Method 2
What are the factors of 18?
118
29
3 6
Discuss that the second option includes
9 3 .
Handout “Simplifying Radicals” Worksheet (see Appendix). Have students complete
the table. Then follow with a class discussion.
Expand lesson to include moving from mixed to entire radicals.
Mathematics 10C
Real Numbers
Page 43 out of 61
Practise new learning
Assign selected exercises from text.
Self / Peer Assessed Practice Work
“Matching Game” – in partners, given a set of cards with one radical per card – half
are entire radicals, half are mixed radicals – have students match up the cards that
are equal. It could be in the form of a jigsaw puzzle.
Going Beyond
Try simplifying radicals with indexes equal to 4 and 5. Discuss
9,  9 and 9.
Resources
Foundations and Pre-calculus Mathematics 10 (Pearson: sec 4.3)
Math 10 (McGraw Hill: sec 4.4)
Glossary
entire radical – an expression where an entire term is under a radical sign or symbol
mixed radical – an expression where part of the term is outside the radical and part is
under the radical
Mathematics 10C
Real Numbers
Page 44 out of 61
Lesson 5
Rational Exponents
STAGE 1
BIG IDEA:
Real Numbers provide students with a foundation upon which they build an understanding of the
different ways to represent and order real quantities. This understanding will enable students to solve
problems related to all disciplines.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
The students will understand…


numeracy as it relates to real numbers.
o that the set of real numbers is
continuous and is made up of
rational and irrational numbers.
that there are various ways of
representing numbers including
exponents, fractions, and radicals.

When and why should we use exact
values?

What is a real number?
o What are the different ways of
representing real numbers?
o How can real numbers be
classified?
o What strategies can you use to
order real numbers appropriately?
.
KNOWLEDGE:
SKILLS:
Students will know…
Students will be able to…



what an exponent is.
what integral and rational exponents
mean.
the exponent laws.


apply exponent laws.
solve problems involving real numbers.
Lesson Summary
Students will review the exponent laws from math 9 and expand their knowledge set to
include rational exponents.
Mathematics 10C
Real Numbers
Page 45 out of 61
Lesson Plan
Activate Prior Knowledge / Experience
Students have seen the exponent laws in Math 9 but were limited to natural number
exponents.
Hand out worksheet ‘Laws of Exponents Review’ (see Appendix). Have students work
in group to complete the worksheet, using the examples provided and their past
knowledge
Diagnostic Assessment of Prior Knowledge
Discuss results of worksheet as a class to ensure everyone came to the same
conclusions.
Present New Learning
1
1
Use the exponent laws to have students discover the value of a 2 and a 3 .
Example #1
1
1
1 1

2
(a) 4 2  4 2  4 2
 41
1
2
What must 4 be equal to so that the number multiplied by itself is 4?
1
1
1 1

2
(b) 9 2  9 2  9 2
 91
1
2
What must 9 be equal to so that the number multiplied by itself is 9?
1
What do you think the value of 25 2 is? Check your answer with a graphing
calculator.
Mathematics 10C
Real Numbers
Page 46 out of 61
Example #2
1
3
1
3
1
3
(a) (8)  (8)  (8)  (8)
1 1 1
 
3 3 3
 (8)1
1
3
What must (8) be equal to so that the number multiplied by itself three
times is -8?
1
3
1
3
1
3
(b) 27  27  27  27
1 1 1
 
3 3 3
 271
1
3
What must 27 be equal to so that the number multiplied by itself three
times is 27?
1
3
What do you think the value of (64) is? Check your answer with a calculator.
Discuss why cube roots can have negative bases, but square roots cannot when in
brackets.
Hand out worksheet – “Laws of Exponents Extended” (see Appendix). Discuss the
strategy used and the ‘rule’
Define radicals – including index and radicand.
How could you use the definition to write the following expressions in radical form?
(a) 27
1
3
1
(e) 16 4
(b) 64
1
2
(c) 9
1
(f) (243) 5
3
2
(d) 8
2
3
3
6
(g) 256 4
(h) 32 5
Provide example with variable bases.
Practise New Learning
Assign selected exercises from text.
Mathematics 10C
Real Numbers
Page 47 out of 61
Resources
Math 10 (McGraw Hill: sec 4.3)
Foundations and Pre-calculus Mathematics 10 (Pearson: sec 4.4)
Mathematics 10C
Real Numbers
Page 48 out of 61
Lesson 6
Negative Exponents
STAGE 1
BIG IDEA:
Real Numbers provide students with a foundation upon which they build an understanding of the
different ways to represent and order real quantities. This understanding will enable students to solve
problems related to all disciplines.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
The students will understand…


numeracy as it relates to real numbers.
o that the set of real numbers is
continuous and is made up of
rational and irrational numbers.
that there are various ways of
representing numbers including
exponents, fractions, and radicals.

When and why should we use exact
values?

What is a real number?
o What are the different ways of
representing real numbers?
o How can real numbers be
classified?
o What strategies can you use to
order real numbers appropriately?
.
KNOWLEDGE:
SKILLS:
Students will know…
Students will be able to…



what an exponent is.
what integral and rational exponents
mean.
the exponent laws.


apply exponent laws.
solve problems involving real numbers.
Lesson Summary
Students will be able to simplify expressions with negative exponents using
reciprocals.
Mathematics 10C
Real Numbers
Page 49 out of 61
Lesson Plan
Activate Prior Knowledge/Experience
Review dividing fractions (multiplying by reciprocal of the divisor).
Diagnostic Assessment of Prior Knowledge
Quick check from previous lesson prior to starting (provide students with power
expressions with rational exponents).
Present new learning
Investigate.
Given

22
23
Simplify using factorization:
22
2 2
22
1



3
2
2 2 2 2  2  2 2

Simplify using exponent laws.
22
 22  23  223  21
23

Can we say that 2 1 
1
?
2
Try simplifying the following using both methods above:
2
(a)
33
35
Mathematics 10C
1
 
2
(b)   4
1
 
2
3
2
 
3
(c)  4
2
 
3
Real Numbers
Page 50 out of 61
Use the pattern you see above to evaluate the following:
1
(b)  
3
(a) 4 2
(d) 16

3
2
2
 1
(e)   
 8
3
(c)  
2

2
3
4
4
(f)  
9

5
2
Express the rule(s) you used as a general statement.
Provide examples with variable bases.
Practise New Learning
Assign selected exercises from text.
Resources
Foundations and Pre-calculus Mathematics 10 (Pearson: sec 4.5)
Math 10 (McGraw Hill: sec 4.2)
Mathematics 10C
Real Numbers
Page 51 out of 61
Lesson 7
Irrational Numbers – Classifying and Ordering
STAGE 1
BIG IDEA:
Real Numbers provide students with a foundation upon which they build an understanding of the
different ways to represent and order real quantities. This understanding will enable students to solve
problems related to all disciplines.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
The students will understand…


numeracy as it relates to real numbers.
o that the set of real numbers is
continuous and is made up of
rational and irrational numbers.
there are various ways of representing
numbers including exponents, fractions,
and radicals.

When and why should we use exact
values?

What is a real number?
o What are the different ways of
representing real numbers?
o How can real numbers be
classified?
o What strategies can you use to
order real numbers appropriately?

What is the meaning of continuous?
o When are there gaps in a number
line?
o How many numbers are there?
.
KNOWLEDGE:
SKILLS:
Students will know…
Students will be able to…





that numbers can be ordered.
that numbers can be approximated.
the relationship between sets of numbers
(union of sets).
that the number line is infinitely
continuous.
what an exponent is.



sort real numbers into categories.
approximate irrational numbers.
order real numbers.
Lesson Summary
Students will practise classifying and ordering rational and irrational numbers
presented in different forms.
Mathematics 10C
Real Numbers
Page 52 out of 61
Lesson Plan
Activate Prior Knowledge / Experience
Review of ordering numbers on a number line.
Present New Learning
Taken from Pearson Foundations and Pre-Calculus 10
Given a table with a selection of rational and irrational numbers, what generalizations
can you make about rational and irrational numbers based on the values provided
below.
Rational
 4
0.36
3
8
3
4
16
25

5
Irrational Numbers
7
3
 8
64
6
2
3
27

1
2
24

0.6
Define real numbers, integers, whole numbers and natural numbers.
Example
Given the following numbers:
5,
2
,
3
 3 24 ,  7 ,
49 .

Classify each number as being rational or irrational.

Order the numbers from lowest to highest.

Locate them on a number line.
Mathematics 10C
Real Numbers
Page 53 out of 61
Check for understanding
Have the students create a number line with  8 and  as the extreme values. The
students need to find 7 different numbers with values between those extremes and
place them on their number line.
Practise new learning
Assign selected exercises from text.
Resources
Math 10 (McGraw Hill: sec 4.4)
Foundations and Pre-calculus Mathematics 10 (Pearson: sec 4.2)
Mathematics 10C
Real Numbers
Page 54 out of 61
Lesson 8
Working with Exponent Laws
STAGE 1
BIG IDEA:
Real Numbers provide students with a foundation upon which they build an understanding of the
different ways to represent and order real quantities. This understanding will enable students to solve
problems related to all disciplines.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
The students will understand…


numeracy as it relates to real numbers.
o that the set of real numbers is
continuous and is made up of
rational and irrational numbers.
that there are various ways of
representing numbers including
exponents, fractions, and radicals.

When and why should we use exact
values?

What is a real number?
o What are the different ways of
representing real numbers?
o How can real numbers be
classified?
o What strategies can you use to
order real numbers appropriately?
.
KNOWLEDGE:
SKILLS:
Students will know…
Students will be able to…

the exponent laws.

apply exponent laws.
Lesson Summary
Students will use their knowledge of exponent laws to apply an appropriate and
efficient strategy to simplify a variety of expressions.
Mathematics 10C
Real Numbers
Page 55 out of 61
Lesson Plan
Activate Prior Knowledge/Experience
Review adding and subtracting fractions.
Present new learning
Separate the students into groups. Give each group an example of a multi-step
question (see examples below). Each group comes up with an approach to simplify
each expression, arriving at the correct answer based on the information from the
previous lessons. Once groups have verified that their answer is correct, they will
‘teach’ the example to the class. Questions should be given at varying levels to
accommodate student levels.
Simplify each of the following:
 3 23 
x x 


1
2
a 2   ab 
 3x y   2 xy 
2
3 2
2 2
 5 a 3b 

2 
 3ab 
 2m2 n5 
72
49 2
3
3
27
5
3
5
 2 
 2 1 
 3x y 
3
33  6  32
3
3
Check for understanding
Provide students with further examples to try individually, and then go over.
Practice new learning
Assign selected exercises from text.
Resources
Foundations and Pre-calculus Mathematics 10 (Pearson: sec 4.6)
Math 10 (McGraw Hill: sec 4.3 and 4.4)
Mathematics 10C
Real Numbers
Page 56 out of 61
Appendix
Handouts
Mathematics 10C
Real Numbers
Page 57 out of 61
Simplifying Radicals
Complete the following table.
Entire Radical
Form
32
Prime Factorization Method
2 2 2 2 2
= 2 2  2 2  2
= 2 2 2
Factor Form
16  2
or
48
2 8
 2 4 2
 2 2 2
27
3
16
2 25
Mixed
Radical Form
4 2
Laws of Exponents Extended
Complete the following table (some have been completed for you)
Exponential
form
Simplified
Form (Single
base)
Numerical
Value
1
2
(83 )2  22
83
16
Expanded Form
3
2
5
(27) 3
16
5
2
1
 
9
 12
1
 1
 92

5
2
3
 9 2
 
4
4
 8 3
 
 27 
5

5
 1
  3 


4
Laws of Exponents Review
Complete the following table (some have been completed for you)
Exponential
form
Expanded Form
28
2 2 2 2 2 2 2 2
Simplified
Form (Single
base)
Numerical
Value
256
34
5 5 5
42  43
(4  4)  (4  4  4)
45
1024
4 4 4
4 4
41
4
34  32
62  61  63
43  42 
43
42
35
32
5 5 5 5 5
5 5
(33 ) 2
(3  3  3)2  (3  3  3)  (3  3  3)
36
(2  3)  (2  3)  (2  3)  2  2  2  3  3  3
23  33
4 4
  
3 3
2
 4 4  4



2
 3 3  3
729
(22 )3
(43 )3
(2  3)3
8  27  216
(4  2  3)2
4
 
3
2
2
 
5
3
16
9
ACKNOWLEDGEMENTS
Pages 15 – 22 and 31-32
Wahl, Mark, A Golden Ratio Activity, http://www.markwahl.com/index.php?id=22
Pictures or Digital Images
Page 11
1. http://www.toolsforeducators.com/boardgames/loop.php
2. http://www.zazzle.com/game_board_poster-228628221447007502
3. http://www.toolsforeducators.com/boardgames/loop.php
Pages 17, 26
http://www.markwahl.com/index.php?id=22
Pages 18, 27
http://images.smh.com.au/2009/12/22/994337/shania_twain_420-420x0.jpg
Pages 19, 28
http://www.southshields-sanddancers.co.uk/photos_posters/johnny_depp_calendar_photo.jpg
Page 30
Photograph supplied by Jeremy Klassen
Mathematics 10C
Real Numbers
Page 61 out of 61