MATHEMATICS 20-1
Radical Expressions and Equations
High School collaborative venture with
M. E LaZerte, McNally, Queen Elizabeth, Ross Sheppard,
Strathcona and Victoria
M. E. LaZerte: Teena Woudstra
Queen Elizabeth: David Underwood
Ross Sheppard: Dean Walls
Strathcona: Christian Digout
Victoria: Steven Dyck
McNally: Neil Peterson
Facilitator: John Scammell (Consulting Services)
Editor: Jim Reed (Contracted)
2010 - 2011
Mathematics 20-1
Radical Expressions and Equations
Page 2 of 53
TABLE OF CONTENTS
STAGE 1
DESIRED RESULTS
PAGE
Big Idea
4
Enduring Understandings
4
Essential Questions
4
Knowledge
5
Skills
6
STAGE 2
ASSESSMENT EVIDENCE
Transfer Task
Captain Red Ickle’s Booty
Teacher Notes for Transfer Task
Transfer Task
Rubric
Possible Solution
7
9
15
16
STAGE 3 LEARNING PLANS
Lesson #1
Reviewing Radicals
17
Lesson #2
Adding and Subtracting
23
Lesson #3
Multiplying
27
Lesson #4
Division and Rationalizing
34
Lesson #5
Domain of Radicals
40
Lesson #6
Solve Radical Equations
45
Appendix – Worksheets/Keys
Mathematics 20-1
49
Radical Expressions and Equations
Page 3 of 53
Mathematics 20-1
Radical Expressions and Equations
STAGE 1
Desired Results
Big Idea:
Radical numbers allow us to use exact values in real life situations like measurement,
distance and surveying. Radical number operations are the fundamental skills useful in
other mathematical topics like trigonometry and coordinate geometry.
Implementation note:
Post the BIG IDEA in a prominent
place in your classroom and refer to
it often.
Enduring Understandings:
Students will understand …




That operations performed on radicals are similar to other number systems and
algebraic operations.
Radicals with even indices are limited to non-negative radicands, while odd indices
have no restrictions on the radicands.
Solving radical equations can yield extraneous roots.
There are conventions for simplifying answers after performing radical operations.
Essential Questions:




Where are radicals used in real life?
When is it appropriate to round or approximate values? When are exact values
necessary?
What does the square root of a negative number represent?
How are operations performed on radicals similar or different to those performed on
other number families?
Implementation note:
Ask students to consider one of the
essential questions every lesson or two.
Has their thinking changed or evolved?
Mathematics 20-1
Radical Expressions and Equations
Page 4 of 53
Knowledge:
Enduring
Understanding
List enduring
understandings (the
fewer the better)
Students will understand…







There are conventions
for simplifying answers
after performing
radical operations.
8888
I*AN =
when to simplify a radical
what like terms are
when to perform the operations +, - , x, ÷
radicals can be expressed in different forms
a radical yields a numerical approximation
Students will know …
AN.2, AN.3


AN.3
AN.2
the radicand must be a non-negative number
when the index is even
the radicand can be a positive or negative
number when the index is odd
Students will know …


Solving radical
equations can yield
extraneous roots.
Students will understand…

Students will know …
Radicals with even
indices are limited to
non-negative
radicands, while odd
indices have no
restrictions on the
radicands.
Students will understand…
Description of
Knowledge
The paraphrased outcome that the group is
targeting
*AN.2, AN.3
That operations
performed on radicals
are similar to other
number systems and
algebraic operations.
Students will understand…

Specific
Outcomes
List the reference
# from the
Alberta Program
of Studies
to check each root for validity
why some roots are extraneous
Students will know …


when to rationalize denominators
when to simplify radicals
Algebra and Number
Mathematics 20-1
Radical Expressions and Equations
Page 5 of 53
Skills:
Enduring
Understanding
List enduring
understandings (the
fewer the better)
Students will understand…

Specific
Outcomes
List the reference
# from the
Alberta Program
of Studies
Description of
Skills
The paraphrased outcome that the group is
targeting
Students will be able to…
*AN.2, AN.3



That operations
performed on radicals
are similar to other
number systems and
algebraic operations.


Students will understand…



AN.3
There are conventions
for simplifying answers
after performing
radical operations.
*AN = Algebra and Number
Mathematics 20-1
AN.2
determine the domain of radical expressions
and equations
Students will be able to…


Solving radical
equations can yield
extraneous roots.
Students will understand…

Students will be able to…
AN.2, AN.3
Radicals with even
indices are limited to
non-negative
radicands, while odd
indices have no
restrictions on the
radicands.
Students will understand…
add, subtract, multiply and divide radicals
simplify entire and mixed radicals
compare and order radical expressions with
numerical radicands in a given set
express an entire radical with a numerical
radicand as a mixed radical
express a mixed radical with a numerical
radicand as an entire radical
identify extraneous roots through verification
solve radical equations algebraically
Students will be able to…

rationalize monomial or binomial denominators
Implementation note:
Teachers need to continually ask
themselves, if their students are
acquiring the knowledge and skills
needed for the unit.
Radical Expressions and Equations
Page 6 of 53
STAGE 2
1
Assessment Evidence
Desired Results Desired Results
Captain Red Ickle’s Booty
Teacher Notes
There is one transfer task to evaluate student understanding of the concepts relating to
radical expressions, operations and equations. A photocopy-ready version of the transfer
task is 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:


Solve problems that involve operations on radicals and radical expressions with
numerical and variable radicands.
Solve problems that involve radical equations (limited to square roots).
Map
Master map student template
Master map no solutions
Mater map with student solutions
 Files were added to the EPSB Understanding by Design share site
Mathematics 20-1
Radical Expressions and Equations
Page 7 of 53
Teacher Notes for Captain Red Ickle’s Booty Transfer Task
Glossary
conjugates – Two binomial factors whose product is the difference of two squares [Math
20-1 (McGraw-Hill Ryerson: page 587)]
entire radical - A radical with a coefficient of 1 [Math 20-2 (Nelson: page 515)]
equation - A statement of equality between two expressions
expression – A general term that ultimately represents a number. An expression can
consist of numbers, variables and operations on these.
extraneous root – A number obtained in the process of solving an equation that does not
satisfy the equation
like radicals – Radicals with the same radicand and index [Math 20-1 (McGraw-Hill
Ryerson: page 273)]
mixed radicals - A radical with a coefficient other than 1
operation – Associates two or more members of a set with one of the members of the set.
The basic operations in mathematics are addition, subtraction, multiplication, division and
exponentiation.
radical –
n
x - Includes radical sign, radical index and radicand.
radical sign – n
radical index – n
radicand – x
radical equation – An equation with a variable within a radicand
rationalize – A procedure for converting to a rational number without changing the value of
the expression [Math 20-1 (McGraw-Hill Ryerson: page 590)]
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.
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 20-1
Radical Expressions and Equations
Page 8 of 53
Captain Red Ickle’s Booty - Student Assessment Task
After years of piracy on the high seas, Captain Red Ickle has finally been
captured. Alone he waits in a cold dark cell; waiting for the trial that will
sentence him to life in the dungeon, or to his death. The excitement of his
capture and the anticipation of his judgement day have created quite a stir
among the citizens. You feel anxious to know the outcome, but feel unsure of
which you hope for. Piracy is unforgiveable, but you imagine yourself sailing
port to port on an endless adventure and feel a strange sense of sympathy.
You decide, against your better judgment, to sneak down to the holding cell
to meet the Captain. But what would a good person like you want to learn
from a thieving pirate?
You arrive at the prison shortly after sunset. A small bribe is all is takes to
get past the night guard and suddenly you are standing in front of the
Captain’s cell. He walks towards the bars to size you up. His mouth curls up
into a sly grin, as if he recognizes you.
“Aye Peter, ye’ve come fer me spoils, have ye?” He has obviously mistaken
you for someone else, but you feel that now is not the time to correct him.
You nod slowly.
“Yarr,it pains me to give it up after all these years, but I fear me ship be
approaching her final port. When ye find me treasure chest, don’t share the
booty with anybody.” Reaching into his boot, he draws a small roll of
parchment. You know at once what this is: a map.
The Captain says no more as he returns to the dark corner of his cell. You
leave feeling confused. Who is Peter? Where does this map lead? And what
is this treasure?
Destiny has spoken. You know what must be done. The following morning
you set out with nothing but two weeks rations and your map. You find a
merchant ship willing to take you to your destination. The captain of the
merchant ship lets out a deep chuckle as he tells you: “Foolish lad, if death is
what you seek, I’ll take you there – for a fee of course. But I warn you that no
one comes back from Skwaroot Island!”
As the merchant ship nears the island, you are forced to jump ship and swim
ashore. Walking up the beach, dripping wet, you wonder if this was such a
good idea. “No time for doubt now,” you tell yourself, “I’ve got a treasure to
find.”
Captain Red Ickle’s Booty
Part I
1. Looking more closely at the map, you notice that Captain Ickle has
designated a path along which you will find the key to his treasure chest.
You can see from his legend that you must pass by five landmarks, in the
order specified. A note at the bottom tells you the correct path should have
a distance of 12 10 + 6 5 + 5 2 (assume that you always walk directly
from one landmark to the next). Noticing the scale on the map, you see
that each square of the map’s grid has side length of 2 km. Along which
route will you find the key? Indicate your chosen path and show the work
supporting your choice.
2. Wherever there is a fork in the road, you will have at least two options. If
you always went right, how long would the path be to the treasure? If you
always went left, how long would the path be to the treasure? Which route
is longer and by how much? Show all your work and express your answer
in simplest form.
3. At long last, you have arrived at the treasure. Judging by the Sun, you
estimate the time spent searching to be 5 5 hours. Given that the path
length was 12 10 + 6 5 + 5 2 km, was your average walking speed?
total distance
average speed =
.
total time
4. Along your way, you noticed a beautiful lake in the shape of a
parallelogram and were impressed by its size. Calculate the approximate
area of this lake. (Area = base x height).
Captain Red Ickle’s Booty
Part II
Think of a situation that would require you to hide something. Write a brief
explanation of at least one paragraph of why you’ve hidden this thing and
why it must now be found. Create a map to guide someone to your secret
hiding spot. Your map must include the following:
 A title
 A start point and a finish point
 At least 3 different routes; each route must have at least 4
segments
 A legend which indicates: a scale for the map in which each square
has an irrational side length, a desired path to the hiding spot, and a
clue related to the distance of the path
You will be asked to submit two copies of your map: an original map
and a solution map.
Your project must also include:


Questions about:
o Total distance along a route
o The difference between the distances of two routes
o The average speed of travel (given the time)
Detailed solutions to each question
Radicals are not limited to numerical radicands. You will therefore be
asked to find the distances for each segment for any scale. Do this by
using x km as the side length of your grid’s squares.
 Calculations showing the distance of each segment and
total distance of each route if each grid square had a side
length of x km.
Captain Red Ickle’s Booty




●
Key on Path: ■ ▲
Distance of correct path:
9 10 + 10 5 + 10 2
Scale: each grid square
2 km ´ 2 km
Captain Red Ickle’s Booty
Glossary
conjugates – Two binomial factors whose product is the difference of two squares
[Math 20-1 (McGraw-Hill Ryerson: page 587)]
entire radical - A radical with a coefficient of 1 [Math 20-2 (Nelson: page 515)]
equation - A statement of equality between two expressions
expression – A general term that ultimately represents a number. An expression can
consist of numbers, variables and operations on these
extraneous root – A number obtained in the process of solving an equation that does
not satisfy the equation
like radicals – Radicals with the same radicand and index [Math 20-1 (McGraw-Hill
Ryerson: page 273)]
mixed radicals - A radical with a coefficient other than 1
operation – Associates two or more members of a set with one of the members of the
set. The basic operations in mathematics are addition, subtraction,
multiplication, division and exponentiation.
radical –
n
x - Includes radical sign, radical index and radicand.
radical sign – n
radical index – n
radicand – x
radical equation – An equation with a variable within a radicand
rationalize – A procedure for converting to a rational number without changing the
value of the expression [Math 20-1 (McGraw-Hill Ryerson: page 590)]
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.
Assessment
Mathematics 20-1
Radical Expressions and Equations
Captain Red Ickle’s Booty Rubric
COMPONENT
Description of Requirements
Assessment
Part II – Design Your Own Secret Map
Required
Components







Questions
Clarity of Work









Calculations


Visual/Artistic
Presentation &
Creativity




The map has a title to indicate the name of the island
The map has a start point and a finishing point
There are at least 3 routes with at least 4 line segments on each route
The map’s legend is clearly indicated and includes: a scale, a desired path, and a
clue related to the distance of desired path
Project includes a description of at least one paragraph explaining the context of
the map
The project includes an appropriate question about total distance
The project includes an appropriate question about a difference between two
distances
The project includes an appropriate question involving distance, speed and time
Questions include a solution key showing detailed calculations
Start point, finishing point, and landmarks are clearly indicated
Routes on the solution map are drawn using a straight edge
Routes on the solution map have distances clearly indicated for each segment
Descriptive paragraph is written using proper spelling and grammar
Calculations are included for each question
Calculations are done correctly
Calculations show simplification of entire radicals to mixed radicals in simplest
form
Calculations demonstrate an understanding of like radicals
Calculations demonstrate an understanding of simplifying numerical and variable
radicands
The limits of the map are clearly defined
The map has a grid overlaid (should students create their own canvas)
The map is visually appealing
The project showcases the student’s creative talents
IN 1 2 3 4
IN 1 2 3 4
IN 1 2 3 4
IN 1 2 3 4
IN 1 2 3 4
Assessment: IN – Inadequate 1 – Limited 2 – Adequate 3 – Proficient 4 – Excellent
Possible Solution to Captain Red Ickle’s Booty
4 10
8 5
4 5
5 10
5 10
10 5
10 2
●
Key on Path: ■ ▲
Distance of correct path:
3 26
3 10
6 5
12 10 + 6 5 +10 2
Scale: each grid square
2 km ´ 2 km
Mathematics 20-1
Radical Expressions and Equations
Page 16 of 53
STAGE 3
Learning Plans
Lesson 1
Reviewing Radicals
STAGE 1
BIG IDEA: Radical numbers allow us to use exact values in real life situations like measurement,
distance and surveying. Radical number operations are the fundamental skills useful in other
mathematical topics like trigonometry and coordinate geometry.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …




That operations performed on radicals are
similar to other number systems and algebraic
operations.
Radicals with even indices are limited to nonnegative radicands, while odd indices have no
restrictions on the radicands.
Solving radical equations can yield extraneous
roots.
There are conventions for simplifying answers
after performing radical operations.
 Where are radicals used in real life?
 When is it appropriate to round or approximate
values? When are exact values necessary?
 What does the square root of a negative
number represent?
 How are operations performed on radicals
similar or different to those performed on other
number families?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …
 radicals can be expressed in different forms
 a radical yields a numerical approximation



compare and order radical expressions with
numerical radicands in a given set
express an entire radical with a numerical
radicand as a mixed radical
express a mixed radical with a numerical
radicand as an entire radical
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 20-1
Radical Expressions and Equations
Page 17 of 53
Lesson Summary



Achievement Indicators: 2.1, 2.2, 2.3
2.7, 2.9 and 3.5 will be embedded in the lesson
Students will estimate radicals and convert between entire and mixed radicals
Lesson Plan
Consider the following structure:
Hook
Show students a Number 9 clock. With the class, discuss the different operations and
symbols that are used to build the clock. See if they can come up with other
possibilities.
9
An example is: 9 + 9 - = 5
9
Get students to build their own # 4 clock. Students can work in groups.
http://www.flickr.com/photos/ohrphan/1373936341/
Lesson Goal
To activate prior knowledge of what they know about comparing and ordering radicals
and converting between mixed and entire radicals.
Mathematics 20-1
Radical Expressions and Equations
Page 18 of 53
Lesson
Complete the following activity with students:
Create a Pythagorean spiral.
http://www.geom.uiuc.edu/~demo5337/Group3/spiral.gif
Have students create the first four triangles of the Pythagorean spiral. Encourage a
discussion about the need to represent distance as exact values.
Mathematics 20-1
Radical Expressions and Equations
Page 19 of 53
The following interactive shows the solution:
http://members.shaw.ca/jreed/math20-1/pythagoreanSpiral.htm
Instructions:
Things You'll Need:



paper
pencil, pen, or other writing utensil
protractor or other object that has a 90° angle
http://www.ehow.com/how_4621697_spiral-pythagorean-theorem.html
After students have completed the first 4 triangles give them a printout of Figure 1.
Directions:


Cut out the triangles from the printout.
Make a number line from 1 to 10 using the referent of unit 1 from the first
triangle of Figure 1.

Using the length of
number line:
Mathematics 20-1
2 (a from Figure 1) put the following lengths on the
2, 2 2, 3 2, ...
Radical Expressions and Equations
Page 20 of 53

Using the length of 3 (b from Figure 1) put the following lengths on the

number line: 3, 2 3, 3 3, ...
Continue the process with triangles c through m.
Generate discussions on equivalent lengths expressed in various forms.
ie: mixed, entire
4 = 2, 2 2 = 8
The following interactive may be useful to reinforce some of these equivalent lengths.
http://members.shaw.ca/jreed/math20-1/radical.htm
Going Beyond
Resources
Supporting
Assessment
Mathematics 20-1
Radical Expressions and Equations
Page 21 of 53
Glossary
equation - A statement of equality between two expressions
expression – A general term that ultimately represents a number. An expression can
consist of numbers, variables and operations on these.
operation – Associates two or more members of a set with one of the members of the
set. The basic operations in mathematics are addition, subtraction, multiplication,
division and exponentiation.
radical –
n
x - Includes radical sign, radical index and radicand.
radical sign – n
radical index – n
radicand – x
radical equation – An equation with a variable within a radicand
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.
Other
Mathematics 20-1
Radical Expressions and Equations
Page 22 of 53
Lesson 2
Adding and Subtracting
STAGE 1
BIG IDEA: Radical numbers allow us to use exact values in real life situations like measurement,
distance and surveying. Radical number operations are the fundamental skills useful in other
mathematical topics like trigonometry and coordinate geometry.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …




That operations performed on radicals are
similar to other number systems and algebraic
operations.
Radicals with even indices are limited to nonnegative radicands, while odd indices have no
restrictions on the radicands.
Solving radical equations can yield extraneous
roots.
There are conventions for simplifying answers
after performing radical operations.
 Where are radicals used in real life?
 When is it appropriate to round or approximate
values? When are exact values necessary?
 What does the square root of a negative
number represent?
 How are operations performed on radicals
similar or different to those performed on other
number families?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …









when to simplify a radical
what like terms are
when to perform the operations + and the radicand must be a non-negative number
when the index is even
the radicand can be a positive or negative
number when the index is odd
check each root for validity
when to simplify radicals
add and subtract radicals
simplify entire and mixed radicals
Lesson Summary


Achievement Indicators: 2.4
2.7, 2.9 and 3.5 will be embedded in the lesson
Mathematics 20-1
Radical Expressions and Equations
Page 23 of 53
Lesson Plan
Hook
Give students a couple of familiar questions and have them add and subtract and note
how and why they do this. In particular, make sure students know what like terms are.
1 1
+
2 3
2x + 5x2 – 3x2 + y
Lesson Goal

Students learn to efficiently add and subtract expressions with radicals.
Activate Prior Knowledge
Have students simplify several radicals:
8, 96, 3 363 .
Lesson
Have students explore how to add and subtract radicals. Have them use a calculator
to find the decimal approximation of each of the following. Ask students to
compare/discover equivalent expressions.
a.
f.
2- 2
b. 2 2
c.
g. 0 2
h.
3- 3
d. 2 3
e.
i. 0 3
j.
3+ 2
3- 2
Have students come up with a conclusion and what “like terms” are for radicals and
compare this to like terms for adding algebraic expressions.
Have students explore how to add and subtract radicals. Have them use a calculator
to find the decimal approximation of each of the following.
a.
e.
b.
3
2- 2
3
f.
4
5
2+ 2
2- 2
c. 2 3 2
3
g. 0 2
d.
6
2
h.
3
2
Have students come up with a conclusion and an explanation of what “like terms” are
for radicals and compare this to like terms for adding algebraic expressions.
Mathematics 20-1
Radical Expressions and Equations
Page 24 of 53
Is each of the following pairs of expressions ‘like’ radicals?
a.
2 and 8
b.
3 and
5
b.
3 and
6
Have a further discussion with students about “like terms” and how the radicals must
be simplified to know if you have like terms. Students should also note that the indices
have to be the same.
Alternate Lesson:
Use activity attached to review concepts of like terms. Use similar activity to expand
to the operation of addition and subtraction of radicals.
Going Beyond
Resources
Math 20-1 (McGraw-Hill Ryerson: sec 5.1)
Supporting
Assessment
Glossary
entire radical - A radical with a coefficient of 1 [Math 20-2 (Nelson: page 515)]
equation - A statement of equality between two expressions
expression – A general term that ultimately represents a number. An expression can
consist of numbers, variables and operations on these.
like radicals – Radicals with the same radicand and index [Math 20-1 (McGraw-Hill
Ryerson: page 273)]
mixed radicals - A radical with a coefficient other than 1
Mathematics 20-1
Radical Expressions and Equations
Page 25 of 53
operation – Associates two or more members of a set with one of the members of the
set. The basic operations in mathematics are addition, subtraction, multiplication,
division and exponentiation.
radical –
n
x - Includes radical sign, radical index and radicand.
radical sign – n
radical index – n
radicand – x
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.
Other
Mathematics 20-1
Radical Expressions and Equations
Page 26 of 53
Lesson 3
Multiplying
STAGE 1
BIG IDEA: Radical numbers allow us to use exact values in real life situations like measurement,
distance and surveying. Radical number operations are the fundamental skills useful in other
mathematical topics like trigonometry and coordinate geometry.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …




That operations performed on radicals are
similar to other number systems and algebraic
operations.
Radicals with even indices are limited to nonnegative radicands, while odd indices have no
restrictions on the radicands.
Solving radical equations can yield extraneous
roots.
There are conventions for simplifying answers
after performing radical operations.
 Where are radicals used in real life?
 When is it appropriate to round or approximate
values? When are exact values necessary?
 What does the square root of a negative
number represent?
 How are operations performed on radicals
similar or different to those performed on other
number families?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …






when to simplify a radical
what like terms are
when to perform the operations +, - , x, ÷
when to simplify radicals
add, subtract, multiply and divide radicals
simplify entire and mixed radicals
Lesson Summary


Achievement Indicators: 2.4
2.7, 2.9 and 3.5 will be embedded in the lesson
Mathematics 20-1
Radical Expressions and Equations
Page 27 of 53
Lesson Plan
Hook
Review perimeter and then discuss how to find the area of the polygon in order to
determine what students instinctively know about multiplying radicals.
In the diagram, AB, BC, CD, DE, EF, FG, GH, and HK all have length 4, and all angles
are right angles, with the exception of the angles at D and F.
Determine the perimeter of ABDFHK.
Solution:
P = 4(4) + 4(4√2) = 16 + 16√2
Mathematics 20-1
Radical Expressions and Equations
Page 28 of 53
Lesson Goal
Multiplying monomial x monomial
Multiplying monomial x binomial
Multiplying binomial x binomial
Activate Prior Knowledge
Already activated in Hook.
Determine the area of the rectangle ABHK.
Solution:
A = 4(4+ 4√2 + 4) = 4(8+ 4√2) = 32 + 16√2
Lesson
In groups of 4, have the students complete the Multiplying Polynomial and Radical
Practice Worksheet. Have the students make comparisons between the
multiplication operations of polynomials and radicals. Each group should record their
solutions on chart paper. Record the polynomials on one piece of paper and radicals
on a second piece. Once all the groups are finished they can hang them on the walls
of the classroom. As a large group you can discuss the similarities between the
multiplication operations of both polynomials and radicals. Finally as a class you can
write the rules for multiplying radicals.
Going Beyond
Mathematics 20-1
Radical Expressions and Equations
Page 29 of 53
Resources
Math 20-1 (McGraw-Hill Ryerson: sec 5.2)
Supporting
Lesson 3 Worksheet: “Multiplying Polynomial and Radical Practice Worksheet”
Multiplying Polynomial and Radical Practice Worksheet
 copy was added to Appendix
Mathematics 20-1
Radical Expressions and Equations
Page 30 of 53
Multiplying Polynomial and Radical Practice
A. Multiply the following Polynomials:
( )
1. 2x 3 4x 2
( )
2. -3x 3 4x 3
(
3. 2x 3 x + 5x 2
)
(
4. 4xy - xy - x 2
(
)(
(
)(
)
)
5. x + 5 x + 4
)
6. 2x - 7 3x + 6
B. Multiply the following Radicals:
1.
( 5)( 6)
2.
(2 3 ) (-3 5 )
3.
(3 6 ) ( 2 + 5 3 )
4. -8
Recall
18 = 9 2
=3 2
( 6 ) (1+ 4 10 )
5.
(
)(
6.
(5 + 6 ) ( 2 - 6 )
3 +4 3+ 3
)
How are multiplying radicals similar to those on polynomials?
How are multiplying radicals different from those on polynomials?
Answers:
A. 1. 8x 5
2. -12x 6
3. 2x 4 +10x 5
4. -4x 2 y 2 - 4x 3 y
5. x 2 + 9x + 20
6. 6x 2 - 9x - 42
B. 1.
30
2. -6 15
3. 6 6 + 45 2
4. -8 6 - 64 15
5. 7 3 + 15
6. -3 6 + 4
How are multiplying radicals similar to those on polynomials?
 Multiplying performed on radicals and on polynomials both follow the same
principles of distribution.
How are operations on radicals different from those on polynomials?
 There are conventions for simplifying answers after performing radical operations.
 Radicals with even indices are limited to non-negative radicands, while odd indices
have no restrictions on the radicands.
 Solving radical equations can yield extraneous roots.
Mathematics 20-1
Radical Expressions and Equations
Page 32 of 53
Assessment
Glossary
conjugates – Two binomial factors whose product is the difference of two squares [Math
20-1 (McGraw-Hill Ryerson: page 587)]
entire radical - A radical with a coefficient of 1 [Math 20-2 (Nelson: page 515)]
extraneous root – A number obtained in the process of solving an equation that does not
satisfy the equation
like radicals – Radicals with the same radicand and index [Math 20-1 (McGraw-Hill
Ryerson: page 273)]
mixed radicals - A radical with a coefficient other than 1
operation – Associates two or more members of a set with one of the members of the set.
The basic operations in mathematics are addition, subtraction, multiplication, division and
exponentiation.
radical –
n
x - Includes radical sign, radical index and radicand.
radical sign – n
radical index – n
radicand – x
rationalize – A procedure for converting to a rational number without changing the value of
the expression [Math 20-1 (McGraw-Hill Ryerson: page 590)]
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.
Other
Mathematics 20-1
Radical Expressions and Equations
Page 33 of 53
Lesson 4
Division and Rationalizing
STAGE 1
BIG IDEA: Radical numbers allow us to use exact values in real life situations like measurement,
distance and surveying. Radical number operations are the fundamental skills useful in other
mathematical topics like trigonometry and coordinate geometry.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …




That operations performed on radicals are
similar to other number systems and algebraic
operations.
Radicals with even indices are limited to nonnegative radicands, while odd indices have no
restrictions on the radicands.
Solving radical equations can yield extraneous
roots.
There are conventions for simplifying answers
after performing radical operations.
 Where are radicals used in real life?
 When is it appropriate to round or approximate
values? When are exact values necessary?
 What does the square root of a negative
number represent?
 How are operations performed on radicals
similar or different to those performed on other
number families?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …










when to simplify a radical.
what like terms are
when to perform the operations +, - , x, ÷
the radicand must be a non-negative number
when the index is even
the radicand can be a positive or negative
number when the index is odd
when to rationalize denominators
when to simplify radicals



add, subtract, multiply and divide radicals
simplify entire and mixed radicals
determine the domain of radical expressions
and equations
identify extraneous roots through verification
solve radical equations algebraically
rationalize monomial or binomial
denominators
Lesson Summary


Achievement Indicators: 2.4, 2.5, 2.6
2.7, 2.9 and 3.5 will be embedded in the lesson
Mathematics 20-1
Radical Expressions and Equations
Page 34 of 53
Lesson Plan
Hook
1. Sometimes we need to perform operation on or compare fractions. Have students try:
1 9
+
6 27
9
1
to .
27
3
1 1
+
6 3
Then find an equivalent fraction with the lowest common denominator by multiplying
1
2
the
by
before adding.
3
2
1 1 2
1 2
3
1
+ · ® + ® ®
6 3 2
6 6
6
2
Discuss how it is advantageous to reduce
2. This leads to the convention of rationalizing the denominator. Have students try:
1 1
+
2
2
Discuss the similarity with the above example.
Lesson Goal
Students will be able to divide expressions with radicals and leave all responses with
rational denominators.
Activate Prior Knowledge
Review the concept of difference of squares, distributive property, simplifying fractions
Mathematics 20-1
Radical Expressions and Equations
Page 35 of 53
Lesson
Exploration 1:
8
,
2
8
,
2
4,
2 2
,
2
16
,2
2
Discuss with students that there are multiple ways to simplify the expression
can divide first
multiply
8
by
2
8
=
2
2
4 , simplify to
2 2
8
. They
2
or they may see right away that they can
2
. This gives students 6 different representations of the same value.
2
Exploration 2:
6
,2
2
6 2 6
,
and 3
8
8
In the form,
2 6
, students cannot divide
6 and
8 evenly, so they will have to try
8
different things:

change
8 to 2 2 and then simplify

change
6 to
2 3 and divide
Students will need to make some connection on how they want to do this. Discuss the
method(s) students prefer.
Exploration 3:
7
2
Students are faced with the dilemma that they cannot divide the two numbers evenly.
Discuss rationalization of denominators.
Exploration 4:
10
20
Students will simplify this differently and you can discuss which method is most efficient.
At this point you can stress to students what simplest form is, but they can construct their
own way to get there.
Mathematics 20-1
Radical Expressions and Equations
Page 36 of 53
Exploration 5:
5 3
and
3 6
5
2 12
Tell students their answer should be in simplest form.
 The second example will give students a couple of different options and you can
discuss the pros and cons of each.
Exploration 6:
22 15
363
At this point we can move onto a harder example where we want them to simplify first.
If a student chooses to rationalize the denominator first they will get,
22 5445
.
363
Many students will have difficulty reducing 5445 . Use this opportunity to stress
simplifying first when possible to do so.
22 15
22 15
22 15 2 15 æ 3 æ 2 45 2 9 · 5 6 5
=
=
=
=
=
= 2 5 compared to
æ æ=
3
3
3
363
3 ·121 11 3
3 æ 3æ
22 15
363
=
22 15
3 * 121
=
22 15
=2 5
11 3
Exploration 7: Binomial numerators and denominators:
Expand the lessons and rules learned to include expressions with binomials in numerators
and/or denominators.
1+ 2
2
Have students simplify and then compare their responses with their peers. As a group,
discuss the correct answer and the process to achieve this.
Exploration 8: Conjugates
Have the students expand and simplify.
(1+ 2 ) (1- 2 ), (1+ 2 ) (1+ 2 ), (1- 2 ) (1- 2 )
Discuss the similarity and differences in the 3 products. Introduce the term conjugates.
Have the students expand and simplify.
(3 3 + x ) (3 3 - x ) , (3 3 + x ) (3 3 + x ) , (3 3 - x ) (3 3 - x )
Mathematics 20-1
Radical Expressions and Equations
Page 37 of 53
Have students expand, simplify and then compare responses with their peers.
(3 7 + 4 x ) (3 7 - 4 x ) and 2 3 +2
2
Going Beyond
Investigate the concept of rationalizing a numerator or rationalize an expression with an
index of 3 or 4. Students would most likely need direction, but if they have some exposure
to factoring sum and difference of cubes, you can relate rationalizing cube roots with
rationalizing binomial square roots with difference of squares.
Resources
Math 20-1 (McGraw-Hill Ryerson: sec 5.2)
Supporting
Assessment
Glossary
conjugates – Two binomial factors whose product is the difference of two squares [Math
20-1 (McGraw-Hill Ryerson: page 587)
rationalize denominator – create an equivalent expression that does not have a radical in
the denominator.
Other
Like Terms Worksheet
 copy was added to Appendix
Mathematics 20-1
Radical Expressions and Equations
Page 38 of 53
Like Terms
Examine all the different terms below. Place each into the box with the matching like
term at the bottom of this page. Note: Like terms for radicals can only be determined
when the radicals are reduced to simplest form, like terms have the same index and
radicand, much like polynomials where like terms have the same variables with the
same exponents.
75
50
3
80
100
-3 2
8x2
72
3x2
-8
18
5a
32
5
3 48
3
20
8
27
8x
-2x2
2a
16
16
-a
3
8ax2
-x
8
128
You must state why each group is a like term.
3
3
5
x
ax
2
a
2
x2
constant
2
Mathematics 20-1
Radical Expressions and Equations
128
Page 39 of 53
Lesson 5
Domain of Radicals
STAGE 1
BIG IDEA: Radical numbers allow us to use exact values in real life situations like measurement,
distance and surveying. Radical number operations are the fundamental skills useful in other
mathematical topics like trigonometry and coordinate geometry.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …




That operations performed on radicals are
similar to other number systems and algebraic
operations.
Radicals with even indices are limited to nonnegative radicands, while odd indices have no
restrictions on the radicands.
Solving radical equations can yield extraneous
roots.
There are conventions for simplifying answers
after performing radical operations.
 Where are radicals used in real life?
 When is it appropriate to round or approximate
values? When are exact values necessary?
 What does the square root of a negative
number represent?
 How are operations performed on radicals
similar or different to those performed on other
number families?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …



the radicand must be a non-negative number
when the index is even
the radicand can be a positive or negative
number when the index is odd
determine the domain of radical expressions
and equations
Lesson Summary

Achievement Indicators: 2.8, 3.1
Mathematics 20-1
Radical Expressions and Equations
Page 40 of 53
Lesson Plan
Hook
Have students find 3 values that can be substituted for x in each of the following
x,- x
, and - x and evaluate the 3 numbers to the nearest hundredth. Students will discuss in
small groups the smallest and largest possible number for each of the above expressions
and answer the question: Is there a number that can be substituted into all three
expressions that would create a valid response?
The teacher can then show the graphs of each expression and have students identify
which graph is for which expression based on the domains and make sure that students
understand that the value of the radicand needs to be greater or equal to zero.
Lesson Goal

Understand the restriction to the domain of a radical function/expression.
Activate Prior Knowledge
Review the concept of domain and range as discussed in 10C and 20-1 (quadratics).
Lesson
Teacher Note:
A file was created using GeoGebra to illustrate the transformation of radicals.
L5 BasicRadicalTransformations.ggb
 file was added to the EPSB Understanding by Design share site
To use this file you will need to download GeoGebra. At the time this document was prepared the
program could be downloaded for free from: http://www.geogebra.org/cms/en/download.
Mathematics 20-1
Radical Expressions and Equations
Page 41 of 53
Exploration 1:
Continue the discussion (with a software graphing tool) of what happens to the domain
when the expression is x + 2 . Students could explore this with their own graphing
calculator or as a class on the board with students observing the graph moves 2 units to
the left compared to the graph of f ( x ) = x . Students can then make the connection that
+ 2 changes the domain of the radical expression. You can further explore what happens
with
x - 2 and
2- x.
From here we want students to have a good grasp that the domain of a radical expression
is the values of the variable that will make the radicand greater or equal to zero.
Exploration 2:
Solve the radicand to determine the domain without the graph.
2 - 3x
x 2 +1
x2
x 2 + 5x + 6
At this point, we want to stress that the radicand cannot be negative for square roots, so
we are always solving the radicand to be greater or equal to zero no matter what the
expression looks like inside the radical.
Students can also look at the graphs for further verification and to discuss the domain
restrictions.
Exploration 3:
A square root has an index of 2. Have students investigate higher order indices with the
following:
3
x
3
x +3
4
x
5
x
6
x
7
x
Have students graph these expressions and find their domains. The following questions
can be investigated.
1. Are there any restrictions on cube roots?
2. Are there any restrictions on 4th roots?
3. Are there any restrictions on 5th roots?
4. What rule in general relates to restrictions involving radicals of higher indices?
With a graphing calculator, students can graph these quickly and should see the pattern of
even and odd indices.
Mathematics 20-1
Radical Expressions and Equations
Page 42 of 53
Going Beyond
Look at restrictions if there is a radical in the denominator of a fraction such as:
1
x
You can also investigate imaginary numbers as ways to deal with domain restrictions that
may occur when performing calculations with radicals.
Resources
Math 20-1 (McGraw-Hill Ryerson: sec 5.1)
Supporting
Assessment
Glossary
conjugates – Two binomial factors whose product is the difference of two squares [Math
20-1 (McGraw-Hill Ryerson: page 587)]
entire radical - A radical with a coefficient of 1 [Math 20-2 (Nelson: page 515)]
equation - A statement of equality between two expressions
expression – A general term that ultimately represents a number. An expression can
consist of numbers, variables and operations on these.
extraneous root – A number obtained in the process of solving an equation that does not
satisfy the equation
like radicals – Radicals with the same radicand and index [Math 20-1 (McGraw-Hill
Ryerson: page 273)]
mixed radicals - A radical with a coefficient other than 1
Mathematics 20-1
Radical Expressions and Equations
Page 43 of 53
operation – Associates two or more members of a set with one of the members of the set.
The basic operations in mathematics are addition, subtraction, multiplication, division and
exponentiation.
radical –
n
x - Includes radical sign, radical index and radicand.
radical sign – n
radical index – n
radicand – x
radical equation – An equation with a variable within a radicand
rationalize – A procedure for converting to a rational number without changing the value of
the expression [Math 20-1 (McGraw-Hill Ryerson: page 590)]
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.
Other
Mathematics 20-1
Radical Expressions and Equations
Page 44 of 53
Lesson 6
Solve Radical Equations
STAGE 1
BIG IDEA: Radical numbers allow us to use exact values in real life situations like measurement,
distance and surveying. Radical number operations are the fundamental skills useful in other
mathematical topics like trigonometry and coordinate geometry.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …




That operations performed on radicals are
similar to other number systems and algebraic
operations.
Radicals with even indices are limited to nonnegative radicands, while odd indices have no
restrictions on the radicands.
Solving radical equations can yield extraneous
roots.
There are conventions for simplifying answers
after performing radical operations.
 Where are radicals used in real life?
 When is it appropriate to round or approximate
values? When are exact values necessary?
 What does the square root of a negative
number represent?
 How are operations performed on radicals
similar or different to those performed on other
number families?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …












when to simplify a radical.
what like terms are
when to perform the operations +, - , x, ÷
the radicand must be a non-negative number
when the index is even
the radicand can be a positive or negative
number when the index is odd
check each root for validity.
why some roots are extraneous
when to rationalize denominators
when to simplify radicals



add, subtract, multiply and divide radicals
simplify entire and mixed radicals
determine the domain of radical expressions
and equations
identify extraneous roots through verification
solve radical equations algebraically
rationalize monomial or binomial
denominators
Lesson Summary


Achievement Indicators: 3.2, 3.3, 3.4
2.7, 2.9 and 3.5 will be embedded in the lesson
Mathematics 20-1
Radical Expressions and Equations
Page 45 of 53
Lesson Plan
Hook: A cable television company is laying cable in an area with underground utilities.
Two subdivisions are located on opposite sides of Willow Creek, which is 500 m wide. The
company has to connect points P and Q with cable, where Q is on the north bank, 1200
metres east of R. It costs $40/m to lay cable underground and $80/m to lay cable
underwater.
What is the least expensive way to lay the cable?
1200
R
S
Q
x
500
P
Lesson
Exploration 1:
Reactivate prior learning of solving linear equations:
-3 = x + 2
x = -5
The equation should still be true, if we square both sides.
(-3)2 = (x + 2) 2
square both sides
9 = x2 + 4x + 4
solve resulting equation
0 = x2 + 4x - 5
( x + 5) ( x - 1) = 0
x = -5 and x = 1
()
x = 1 is an extraneous root, it does not verify, because -3 ¹ 1 + 2 .
x = -5
When would it be useful to square both sides?
Mathematics 20-1
Radical Expressions and Equations
Page 46 of 53
Exploration 2:
Start with a couple of straightforward examples with students in small groups:
x = 3
x =4
x = -4
With the first example, it may also be useful to note that x = 4 and x - 4 = 0 are the
same equation. This change may be necessary for more complicated equations when we
have to isolate the radical.
Use the last example to emphasize the need of verifying. It may also be helpful to remind
students that the square root of a number is always positive. There will be no solution for
x = -4. Consider using this opportunity to remind students that x2 = 16 could have
x = -4 as a solution.
At this point you can also use the graphing calculator as a tool to show the solution is
where the lines intersect on the graph. If you used the graphs with domain and range,
students will already be familiar with the shape of a radical graph.
Exploration 3:
At this point students may have been able to work out a solution by guessing and
checking. We will investigate equations where guessing and checking is an inefficient way
to find the solution. It may be a good idea to make sure that all students have an
understanding of the process of squaring each side.
x -1 = x - 7
Exploration 4:
Move onto more complicated example(s):
x + 2x + 1 = 2x -1; x - 4x + 8 = 2x + 8 ; 7 + 3x = 5x + 4 + 5
At this point if students struggle you may want to have a class discussion about what
processes occur in every question:
 Isolate radical. If there are two radicals, isolate one of them.
 Square both sides. Remind students about the distributive property (ex.
( x + 2)



2
¹ x 2 + 22 ). Have students investigate this and see that they do not find the
solution if they do not square the whole side.
o You may need to review some basic equation solving rules or have students
come up with their own and put on chart paper.
If there were originally two radicals, there should now be only one. Isolate this
radical and square both sides.
Solve the resulting equation
Verify the solution in the ORIGINAL equation.
Mathematics 20-1
Radical Expressions and Equations
Page 47 of 53
o
Again, you could have students verify in the original and in an intermediate and
check the results by graphing and see which one is correct.
Going Beyond
Resources
Math 20-1 (McGraw-Hill Ryerson: sec 5.3)
Supporting
Assessment
Glossary
radical –
n
x - Includes radical sign, radical index and radicand.
radical sign – n
radical index – n
radicand – x
operation – Associates two or more members of a set with one of the members of the set.
The basic operations in mathematics are addition, subtraction, multiplication, division and
exponentiation.
expression – A general term that ultimately represents a number. An expression can
consist of numbers, variables and operations on these.
equation - A statement of equality between two expressions
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.
Other
Mathematics 20-1
Radical Expressions and Equations
Page 48 of 53
Appendix
Appendix 1: Multiplying Polynomial and Radical Practice Worksheet and Key
Appendix 2: Like Terms Worksheet and Key
Mathematics 20-1
Radical Expressions and Equations
Page 49 of 53
Appendix 1: Multiplying Polynomial and Radical Practice
A. Multiply the following Polynomials:
( )
1. 2x 3 4x 2
( )
2. -3x 3 4x 3
(
3. 2x 3 x + 5x 2
)
(
4. 4xy - xy - x 2
(
)(
(
)(
)
)
5. x + 5 x + 4
)
6. 2x - 7 3x + 6
B. Multiply the following Radicals:
1.
( 5)( 6)
2.
(2 3 ) (-3 5 )
3.
(3 6 ) ( 2 + 5 3 )
4. -8
Recall
18 = 9 2
=3 2
( 6 ) (1+ 4 10 )
5.
(
)(
6.
(5 + 6 ) ( 2 - 6 )
3 +4 3+ 3
)
How are multiplying radicals similar to those on polynomials?
How are multiplying radicals different from those on polynomials?
Answers:
A. 1. 8x 5
2. -12x 6
3. 2x 4 +10x 5
4. -4x 2 y 2 - 4x 3 y
5. x 2 + 9x + 20
6. 6x 2 - 9x - 42
B. 1.
30
2. -6 15
3. 6 6 + 45 2
4. -8 6 - 64 15
5. 7 3 + 15
6. -3 6 + 4
Appendix 2: Like Terms
Examine all the different terms below. Place each into the box with the matching
like term at the bottom of this page. Note: Like terms for radicals can only be
determined when the radicals are reduced to simplest form, like terms have the
same index and radicand, much like polynomials where like terms have the same
variables with the same exponents.
75
50
3
80
100
-3 2
8x2
72
3x2
-8
18
5a
32
5
3 48
3
20
8
27
8x
-2x2
2a
16
16
3
8
3
5
x
ax
2
2
a
2
x2
constant
3
128
8ax2
-x
You must state why each group is a like term.
-a
128
Answers
3
3 48 ,
a
2
50 , 18 ,
27
32 ,
72 ,
5a , -a , 2a
8 , 128
3
5
2
x2
4 3 2 , 3 128 , - 3 2
8x 2 , 3x 2 , -2x 2
ax 2
x
Constant
8ax 2
8x , - x
75 ,
80 ,
Mathematics 20-1
20
100 , 5 , 8 , 16
Radical Expressions and Equations
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