MATHEMATICS 20-2
Radicals
High School collaborative venture with
Harry Ainlay, McNally, M. E LaZerte, Ross Sheppard, Scona,
and W.P. Wagner
Harry Ainlay: Colin Veldkamp
Harry Ainlay: Debby Sumantry
Harry Ainlay: Mathias Stewart
Harry Ainlay: Meriel Hughes
McNally: Enchantra Gramlich
M. E. LaZerte: Monique Merchant
Ross Sheppard: Jeremy Klassen
Ross Sheppard: Tim Gartke
Scona: Joe Johnston
W. P. Wagner: Kiki Brisebois
Facilitator: John Scammell (Consulting Services)
Editor: Jim Reed (Contracted)
2010 – 2011
Mathematics 20-2
Radicals
Page 2 of 125
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 Tasks
1. How do you find the area of a triangle? (let me count the ways)
Teacher Notes for Transfer Task
Transfer Task
Rubric
Possible Solution
7
9
15
18
2. Spiralling Out of Control
Teacher Notes for Transfer Task
Transfer Task
Rubric
Possible Solution
44
45
52
55
2. Radical Race
Teacher Notes for Transfer Task
Transfer Task
Rubric
Possible Solution
65
69
85
87
STAGE 3 LEARNING PLANS
Lesson #1
What Should I Remember?
90
Lesson #2
What’s with all the Letters?
94
Lesson #3
Let’s Play Operation (Operations on Radicals)
97
Lesson #4
Radical Issues (Solving Radical Equations)
103
Appendix – Worksheets/Keys
Mathematics 20-2
108
Radicals
Page 3 of 125
Mathematics 20-2
Radicals
STAGE 1
Desired Results
Big Idea:
Understanding radicals will further develop student’s sense of exact values, and
enhance their ability to simplify expressions and solve equations.
Implementation note:
Post the BIG IDEA in a prominent
place in your classroom and refer to
it often.
Enduring Understandings:
Students will understand …





There are restrictions on values for the variable in a radical.
Some equations have extraneous roots, and why.
There are appropriate forms of communications in mathematics.
Radicals are more precise efficient, concise and accurate.
Operations can be performed on radicals, as with other numbers and algebraic
expressions.
Essential Questions:


Why are extraneous roots produced?
Why is there a restriction on the domain?



Why does your calculator say ERROR on
?
When is an exact solution required?
Under what circumstances would a mixed radical or an entire radical be
appropriate?


When is x > y ?
When would you use radicals?
Implementation note:
Ask students to consider one of the
essential questions every lesson or two.
Has their thinking changed or evolved?
Mathematics 20-2
Radicals
Page 4 of 125
Knowledge:
Enduring
Understanding
List enduring
understandings (the
fewer the better)
Specific
Outcomes
List the reference
# from the
Alberta Program
of Studies
Students will know …
Students will understand…

There are restrictions
on values for the
variable in a radical.
*NL 3.6
There are appropriate
forms of
communications in
mathematics.
NL 3.2, 3.3, 3.5
Radicals are more
precise, efficient,
concise and accurate.
NL 4.5
Operations can be
performed on radicals,
as with other numbers
or algebraic
expressions.
I*NL

when the expression is simplified

when it is appropriate to use an exact value
versus a rounded decimal
Students will know …
Students will understand…

the square root of a negative number is not a
real number
Students will know …
Students will understand…


Students will know …
Students will understand…

Description of
Knowledge
The paraphrased outcome that the group is
targeting
NL 3.4


what like terms are
the rules for operations on radicals
= Number and Logic
Mathematics 20-2
Radicals
Page 5 of 125
Skills:
Enduring
Understanding
List enduring
understandings (the
fewer the better)
Specific
Outcomes
List the reference
# from the
Alberta Program
of Studies
Students will be able to…
Students will understand…

There are restrictions
on values for the
variable in a radical.
*NL 3.6
Some equations have
extraneous roots, and
why.
NL 4.1, 4.3
There are appropriate
forms of
communications in
mathematics.


NL 3.2, 3.3, 3.4,
3.5, 4.5



Radicals are more
precise, efficient,
concise and accurate.
determine any restrictions on values for the
variable in a radical equation
verify, by substitution, that the values
determined in solving a radical equation are
roots of the equation
simplify radical expressions with numerical or
variable radicands
rationalize the monomial denominator of a
radical expression
model a situation with a radical equation
Students will be able to…
Students will understand…

identify values of the variable for which the
radical expression is defined
Students will be able to…
Students will understand…


Students will be able to…
Students will understand…

Description of
Skills
The paraphrased outcome that the group is
targeting

answer questions with exact values
NL3, NL4
Students will understand…
Students will be able to…

Operations can be
performed on radicals,
as with other numbers
or algebraic
expressions.
I*NL
NL 3.4, 4.2


perform operations with radicals
solve radical equations to determine the roots
= Number and Logic
Implementation note:
Teachers need to continually ask
themselves, if their students are
acquiring the knowledge and skills
needed for the unit.
Mathematics 20-2
Radicals
Page 6 of 125
STAGE 2
1
Assessment Evidence
Desired Results Desired Results
Teacher Notes
There are three transfer tasks to evaluate
student understanding of the concepts relating
to radicals. A photocopy-ready version of each
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.
The first is not as open ended as many of the other transfer tasks have been.
Glossary
absolute value – Represents how far the number is from zero
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
mixed radicals – A radical with a coefficient other than 1 [Math 20-2 (Nelson: page
516)]
perfect square – A number, from a given number system, that can be expressed as
the square of a number from the same number system
principal square root – The positive number square root
rationalize the denominator – The process used to write a radical expression that
contains a radical denominator into an equivalent expression with a rational
denominator [Math 20-2 (Nelson: page 517)]
restrictions – The values of a the variable in an expression that ensure it to be
defined [Math 20-2 (Nelson: page 517)]
secondary square root – The negative number square root
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 20-2
Radicals
Page 7 of 125
Transfer Task 1: How do you find the area of a triangle? (let me count the
ways)

When students choose three squares to create a triangle, the intention is that
students may reuse a square if they wish. They do not have to use three
different squares. (Student assessment Part C)

Some combinations will not produce a triangle, i.e. 12, 12, 48 or 12, 12, 72.

Students may need to be reminded to show their work in finding the
semiperimeter (half the perimeter) when they use Heron’s formula.

Students will need to review the product of binomial conjugates
(ax - b) (ax + b) = a x
2
2
- b2 )
This will save them significant amounts of time, when they recognize it in the
problem.
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 20-2
Radicals
Page 8 of 125
How do you find the area of a triangle? - Student Assessment Task
There are many different ways to find the area of a triangle. You may recall that the
1
area of a triangle is A = bh , but this is only one of several different formulae for
2
achieving the same value. In fact it is often easier to use a different formula when the
information you have is not conducive to the regular formula.
A.
To start, find the area of the following equilateral triangles.
2
2
4
4
2
4
To use the formula A =
1
bh you will first need to find its altitude. Use the following
2
diagram as your guide.
2
2
h
1
1
How do you find the area of a triangle? - Student Assessment Task
Based on your above work, if the length of the side of the equilateral triangle
is 2n, then find expressions for both the height and the area of the triangle.
Verify this with an equilateral triangle with a side length of 6 or 8.
B. Many other formulas have been developed to find the area of a triangle. The
following two formulas use just the side lengths of a triangle to determine its area.
(
)(
)(
)
A = s s - a s - b s - c , where s =
a
c
1 2 2 æ a2 + c 2 - b2 ö
A=
a c -ç
÷ø
2
2
è
a + b +c
2
2
The first equation given above is called Heron’s equation and it was published in
60 AD. In this equation s is called the semiperimeter (half of the perimeter of the
triangle). The second formula was developed independently by the Chinese and was
published in 1247 AD. Both the equations are useful, especially for oblique triangles
because all you need to find the area of the triangle is the length of its sides.
http://en.wikipedia.org/wiki/Heron%27s_formula
How do you find the area of a triangle? - Student Assessment Task
Verify these equations with one of the triangles you worked with originally.
Now, if you are given that the area of a triangle is 14 5 units2, then use the formula
1 2 2 æa2 +c 2 - b2 ö
A=
a c -ç
÷ø
2
2
è
2
to find the possible exact values of the lengths of side b given that a = 12 and c = 7.
The first step has been completed for you.
b
7
12
12
1 2 2 æ 72 + 122 - b 2 ö
14 5 =
7 12 - ç
÷ø
2
2
è
2
How do you find the area of a triangle? - Student Assessment Task
C. Consider four squares with areas of 12, 27, 48 and 72 u2.
If we take three of these squares and connect their vertices as shown below…
48 u 2
12 u 2
72 u 2
The lengths of sides of each of the squares can be determined, hence allowing us to
determine the area of the triangle contained between.
Show that the area of the triangle in the above diagram is 3 15 u 2, using Heron’s
Formula.
How do you find the area of a triangle? - Student Assessment Task
Given the area, determine the smallest altitude in that triangle in simplest mixed
1
radical form, rationalizing the denominator when necessary. (Hint: use A = bh )
2
D. Now using any three of the above squares (duplicating squares if you wish)
enclose a triangle of your own making and determine its area using Heron’s
formula, and its smallest altitude. You must connect the vertices to create a
triangle. Some combinations won’t work.
Note: this means that students can reuse a square if they wish to, but not all
combinations work together to create a triangle.
How do you find the area of a triangle? - Student Assessment Task
Glossary
absolute value – Represents how far the number is from zero
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
mixed radicals – A radical with a coefficient other than 1 [Math 20-2 (Nelson: page
516)]
perfect square – A number, from a given number system, that can be expressed as
the square of a number from the same number system
principal square root – The positive square root
rationalize the denominator – The process used to write a radical expression that
contains a radical denominator into an equivalent expression with a rational
denominator [Math 20-2 (Nelson: page 517)]
restrictions – The values of a the variable in an expression that ensure it to be
defined [Math 20-2 (Nelson: page 517)]
secondary square root – The negative square root
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 20-2
How do you find the area of a triangle?
Rubric
Component
Description of Requirements
- Part A
- Altitude and Area are determined for the two
given triangles
- Algebraic expressions for the altitude and area
are determined
- Expression is verified
-
Mathematical
Content
Assessment
IN 1 2 3 4
Part B
Original area is verified using the two new
formulas
Determines the possible lengths of the sides of
the triangle
IN 1 2 3 4
-
Part C
Determines semiperimeter value
Shows how area is 3√15𝑢2
Determines smallest altitude in proper form
IN 1 2 3 4
-
Part D
Creates a valid triangle
Determines area and smallest altitude
IN 1 2 3 4
-
Level
Excellent
Criteria
4
Math
All required
Content
elements
Part A
are present
and correct
Proficient
3
All required
elements
are present
but may
contain
minor
errors
Math
Content
Part B
All required
elements
are present
and correct
All required
elements
are present
but may
contain
minor
errors
Math
Content
Part C
All required
elements
are present
and correct
All required
elements
are present
but may
contain
minor
errors
Math
Content
Part D
All required
elements
are present
and correct
All required
elements
are present
but may
contain
minor
errors
Adequate
2
Some
required
elements
are
missing, or
contain
major
errors
Some
required
elements
are
missing, or
contain
major
errors
Some
required
elements
are
missing, or
contain
major
errors
Some
required
elements
are
missing, or
contain
major
errors
Limited
1
Most
required
elements
are
missing or
incorrect
Insufficient
Blank
No score is
awarded as
there is no
evidence given
Most
required
elements
are
missing or
incorrect
No score is
awarded as
there is no
evidence given
Most
required
elements
are
missing or
incorrect
No score is
awarded as
there is no
evidence given
Most
required
elements
are
missing or
incorrect
No score is
awarded as
there is no
evidence given
When work is judged to be limited or insufficient, the teacher makes decisions about
appropriate intervention to help the student improve
Glossary
entire radical – A radical with a coefficient of 1 [Math 20-2 (Nelson: page 515)]
mixed radicals – A radical with a coefficient other than 1 [Math 20-2 (Nelson: page
516)]
perfect square – A number, from a given number system, that can be expressed as the
square of a number from the same number system
principal square root – The positive square root
rationalize the denominator – The process used to write a radical expression that
contains a radical denominator into an equivalent expression with a rational
denominator [Math 20-2 (Nelson: page 517)]
restrictions – The values of a the variable in an expression that ensure it to be defined
[Math 20-2 (Nelson: page 517)]
secondary square root – The negative square root
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.
Possible Solution to How do you find the area of a triangle?
There are many different ways to find the area of a triangle. You may recall that the
1
area of a triangle is A = bh , but this is only one of several different formulae for
2
achieving the same value. In fact it is often easier to use a different formula when the
information you have is not conducive to the regular formula.
A. To start, find the area of the following equilateral triangles.
2
2
4
4
2
4
1
bh you will first need to find its altitude. Use the
2
following diagram as your guide.
To use the formula A =
2
2
h
1
1
Sample Solution 1
Mathematics 20-2
Radicals
Page of 125
18
Sample Solution 2
Mathematics 20-2
Radicals
Page of 125
19
Based on your above work, if the length of the side of the equilateral triangle is
2n, then find expressions for both the height and the area of the triangle.
Verify this with an equilateral triangle with a side length of 6 or 8.
Sample Solution 1
Sample Solution 2
Mathematics 20-2
Radicals
Page of 125
20
B. Many other formulas have been developed to find the area of a triangle. The
following two formulas use just the side lengths of a triangle to determine its area.
(
)(
)(
)
A = s s - a s - b s - c , where s =
a
c
1 2 2 æa2 +c 2 - b2 ö
A=
a c -ç
÷ø
2
2
è
a + b +c
2
2
The first equation given above is called Heron’s equation and it was published in 60 AD.
In this equation s is called the semiperimeter (half of the perimeter of the triangle). The
second formula was developed independently by the Chinese and was published in
1247 AD. Both the equations are useful, especially for oblique triangles because all you
need to find the area of the triangle is the length of its sides.
http://en.wikipedia.org/wiki/Heron%27s_formula
Mathematics 20-2
Radicals
Page of 125
21
Verify these equations with one of the triangles you worked with originally.
Sample Solution 1
Sample Solution 2
Now, if you are given that the area of a triangle is 14 5 units2, then use the formula
1 2 2 æa2 +c 2 - b2 ö
A=
a c -ç
÷ø
2
2
è
2
to find the possible exact values of the lengths of side b given that a = 12 and c = 7.
The first step has been completed for you.
b
7
12
12
Sample Solution 1
Sample Solution 2
Mathematics 20-2
Radicals
Page of 125
23
D. Consider four squares with areas of 12, 27, 48 and 72 u2.
If we take three of these squares and connect their vertices as shown below…
48 u 2
12 u 2
72 u 2
The lengths of sides of each of the squares can be determined, hence allowing us to
determine the area of the triangle contained between.
Show that the area of the triangle in the above diagram is 3 15 u 2, using Heron’s
Formula.
Sample Solution 1
Mathematics 20-2
Radicals
Page of 125
24
Sample Solution 2
Mathematics 20-2
Radicals
Page of 125
25
Given the area, determine the smallest altitude in that triangle in simplest mixed
1
radical form, rationalizing the denominator when necessary. (Hint: use A = bh )
2
Sample Solution 1
Sample Solution 2
D. Now using any three of the above squares (duplicating squares if you wish)
enclose a triangle of your own making and determine its area using Heron’s
formula, and its smallest altitude. (you must create a triangle with some area,
which means some combinations won’t work)
Sample Solution 1
Mathematics 20-2
Inductive and Deductive Reasoning
Page 27 of 125
Sample Solution 2
Mathematics 20-2
Inductive and Deductive Reasoning
Page 28 of 125
Sample Solution 3
Mathematics 20-2
Inductive and Deductive Reasoning
Page 29 of 125
Sample Solution 4
Mathematics 20-2
Inductive and Deductive Reasoning
Page 30 of 125
Sample Solution 5
Mathematics 20-2
Inductive and Deductive Reasoning
Page 31 of 125
Sample Solution 6
Mathematics 20-2
Inductive and Deductive Reasoning
Page 32 of 125
Sample Solution 7
Mathematics 20-2
Inductive and Deductive Reasoning
Page 33 of 125
Sample Solution 8
Mathematics 20-2
Inductive and Deductive Reasoning
Page 34 of 125
Sample Solution 9
Mathematics 20-2
Inductive and Deductive Reasoning
Page 35 of 125
Sample Solution 10
Mathematics 20-2
Inductive and Deductive Reasoning
Page 36 of 125
Sample Solution 11
Mathematics 20-2
Inductive and Deductive Reasoning
Page 37 of 125
Sample Solution 12
Mathematics 20-2
Inductive and Deductive Reasoning
Page 38 of 125
Sample Solution 13
Mathematics 20-2
Inductive and Deductive Reasoning
Page 39 of 125
Sample Solution 14
Mathematics 20-2
Inductive and Deductive Reasoning
Page 40 of 125
Sample Solution 15
Mathematics 20-2
Inductive and Deductive Reasoning
Page 41 of 125
Sample Solution 16
Mathematics 20-2
Inductive and Deductive Reasoning
Page 42 of 125
Sample Solution 17
Note: this means that students can reuse a square if they wish to, but all combinations
work together to create a triangle.
Mathematics 20-2
Inductive and Deductive Reasoning
Page 43 of 125
Transfer Task 2: Spiralling Out of Control
Teacher Notes
Students may need graph paper provided for Part B.
Students may need some help in constructing the radical spiral. Remember that they
are adding isosceles triangles at every step, so this is not the same as the
Pythagorean spiral in which the outer edge always has a length of one.
The solution to the intersection of the line and circle in Part C involves solving a
quadratic equation by factoring. This is beyond the scope of the curriculum.
Glossary
absolute value – Represents how far the number is from zero
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
Fibonacci number sequence- The sequence obtained when each term is the sum of
the previous two terms and the first two terms are 0 and 1 (alternatively the first two
terms could be 1 and 1)
mixed radicals – A radical with a coefficient other than 1 [Math 20-2 (Nelson: page
516)]
perfect square – A number, from a given number system, that can be expressed as
the square of a number from the same number system
principal square root – The positive square root
rationalize the denominator – The process used to write a radical expression that
contains a radical denominator into an equivalent expression with a rational
denominator [Math 20-2 (Nelson: page 517)]
restrictions – The values of a the variable in an expression that ensure it to be
defined [Math 20-2 (Nelson: page 517)]
secondary square root – The negative square root
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 20-2
Inductive and Deductive Reasoning
Page 44 of 125
Spiralling Out of Control - Student Assessment Task
Take a moment to contemplate your math education. Do you feel, as our title
suggests, that it is spiralling out of control? If it does, you may find comfort in knowing
a bit more about spirals.
Radical Spirals
Part A. To introduce how this type of a
spiral can be drawn we will start with a
basic right triangle. We will use grid
paper to draw these, because it will be far
easier to count out our lengths and draw
in our perpendicular lines.
Notice that the two legs of this triangle are
both 1 unit in length. The hypotenuse is
2 units in length.
Now draw another isosceles right triangle
on top of this one using its hypotenuse as
one of the legs as shown.
Mathematics 20-2
Inductive and Deductive Reasoning
Page 45 of 125
Spiralling Out of Control - Student Assessment Task
In this particular task we will always be
adding isosceles right triangles no
matter what right triangle we begin with.
Now we repeat this process on the
triangle we just created as follows.
Remember that the right angle is always
created at the outer end of the previous
triangle.
And this continues on and on until you are no longer able to add triangles without
overlapping your previous work. For simplicity we will restrict our work so that no
triangle will lie adjacent to the original.
Mathematics 20-2
Inductive and Deductive Reasoning
Page 46 of 125
Spiralling Out of Control - Student Assessment Task
From the previous diagram determine a simplified expression for the length of the
spiral, where its length is the sum of the outer, bolded segments.
Part B. Now, using graph paper, draw your own initial right triangle. It can have side
lengths of 1 and 3, 2 and 3, 1 and 2… whatever you like; only don’t make it too big as
you will quickly run out of room. It will also help to orient your triangle so that the
longest leg is running vertically. Continue the process again, adding isosceles right
triangles on the hypotenuse of your initial triangle. Complete this process for at least
three more triangles.
Determine a simplified expression for the length of each of the spirals you generated.
Spiralling Out of Control - Student Assessment Task
Now, after considering the expressions from the previous diagrams make a
generalization about the length of a spiral.
Golden Spiral
Part C. Consider the Golden Spiral. Joining quarter-circles, whose radii are all
consecutive values in the Fibonacci sequence, as follows, creates it. Recall that a
Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, … Each value is the sum of the previous
two terms.
Spiralling Out of Control - Student Assessment Task
Now let’s sketch both of these graphs on the same grid.
If we now consider just a portion of the golden spiral, we can consider it the quarter of a
circle with the equation y = 25 - x 2 , x ³ 0.
The extension of one of these segments of the radical spiral, with the equation
y = x + 1, will intersect that portion of the circular part of the golden spiral as shown
below.
Spiralling Out of Control - Student Assessment Task
Now let’s look at these two curves specifically and extended.
To make things a little more straight forward let’s remove the restriction on the domain
of the equation of the circle and just look at the two functions y = 25 - x 2 , and
y = x + 1. You need to find the point of intersection of these two functions by solving
this system. The first step is done for you.
x + 1= 25 - x 2
Spiralling Out of Control - Student Assessment Task
Consider the graph provided above and compare it to your algebraic work. Do you
notice any discrepancies? Comment here.
Explain why your algebraic work may have provided more solutions than you are seeing
on the graph.
Assessment
Mathematics 20-2
Spiralling Out of Control
Rubric
.
Component
Description of Requirements
- Part A
- Determines length of radical spiral in proper
form
Assessment
IN 1 2
-
Part B
Draws 3 radical spirals
Determines total length of each of the spirals
Makes generalization based on results of
findings
IN 1 2 3 4
-
Part C
Determines solution to radical equation
IN 1 2 3 4
Presentation
of Data
-
Part B
Spirals are properly and neatly drawn
IN 1 2
Explanation
of Findings
-
Part C
Explains existence of and reasons for
extraneous roots
IN 1 2 3 4
Mathematical
Content
Level
Criteria
Math
Content
Part A
Excellent
4
Proficient
3
Math
Content
Part B
All required
elements
are present
and correct
All required
elements
are present
but may
contain
minor
errors
Adequate
2
All required
elements
are present
and correct
Some
required
elements
are
missing, or
contain
major
errors
Math
All required All required Some
Content elements
elements
required
Part C
are present are present elements
and correct but may
are
contain
missing, or
minor
contain
errors
major
errors
Presents
Presentation
Data
of data is
Part B
clear,
precise and
accurate
Explains Provides
Provides
Provides
Findings insightful
logical
explanations
Part C
explanations explanations that are
complete
but vague
Limited
1
Some
required
elements
are missing
or incorrect
Most
required
elements
are missing
or incorrect
Insufficient
Blank
No score is
awarded as there
is no evidence
given
Most
required
elements
are missing
or incorrect
No score is
awarded as there
is no evidence
given
No score is
awarded as there
is no evidence
given
Presentation Presentation of
of data is
data is
vague and
incomprehensible
inaccurate
Provides
No explanation is
explanations provided
that are
incomplete
or
confusing.
Glossary
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
mixed radicals – A radical with a coefficient other than 1 [Math 20-2 (Nelson: page
516)]
perfect square – A number, from a given number system, that can be expressed as
the square of a number from the same number system
principal square root – The positive number square root
restrictions – The values of a the variable in an expression that ensure it to be
defined [Math 20-2 (Nelson: page 517)]
secondary square root – The negative number square root
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.
Possible Solution to Spiralling Out of Control
Spiralling Out of Control - Key
Take a moment to contemplate your math education. Do you feel, as our title
suggests, that it is spiralling out of control? As it does so, you may find comfort in
knowing a bit more about spirals.
Radical Spirals
Part A. To introduce how this kind of a
spiral can be drawn we will start with a
basic right triangle. We will use grid
paper to draw these, because it will be far
easier to count out our lengths and draw
in our perpendicular lines.
Notice that the two legs of this triangle are
both 1 unit in length. The hypotenuse is
√2 units in length.
Now draw another isosceles right triangle
on top of this one using its hypotenuse as
one of the legs as shown.
Mathematics 20-2
Inductive and Deductive Reasoning
Page 55 of 125
Spiralling Out of Control - Key
In this particular task we will always be
adding isosceles right triangles no
matter what right triangle we begin with.
Now we repeat this process on the
triangle we just created as follows.
Remember that the right angle is always
created at the outer end of the previous
triangle.
And this continues on and on until you are no longer able to add triangles without
overlapping your previous work. For simplicity we will restrict our work to not even lie
adjacent to the original. Calculations:
Spiralling Out of Control - Key
From that picture determine a simplified expression for the length of the spiral, where
its length is the sum of the outer, bolded segments.
Part B. Now, using graph paper, draw your own initial right triangle. It can have side
lengths of 1 and 3, 2 and 3, 1 and 2… whatever you like; only don’t make it too big, as
you will quickly run out of room. It will also help to orient your triangle so that the
longest leg is running vertically. Continue the process again, adding isosceles right
triangles on the hypotenuse of your initial triangle. Complete this process for at least
three more triangles.
Determine a simplified expression for the length of each of the spirals you generated.
Starting first triangle with 2 & 3
Starting first triangle with 1 & 3
Starting first triangle with 1 & 4
Spiralling Out of Control - Key
Now, after considering the expressions you have written make a generalization about the
length of a spiral.
Golden Spiral
Part C. Now recall the Golden Spiral. Joining quarter-circles, whose radii are all
consecutive values in the Fibonacci sequence, as follows, creates it. Recall that a
Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, … Each value is the sum of the previous two
terms.
Spiralling Out of Control - Key
Now let’s sketch both of these graphs on the same grid.
If we now consider just a portion of the golden spiral, we can consider it the quarter of a
circle with the equation y = 25 - x 2 , x ³ 0.
The extension of one of these segments of the radical spiral, with the equation
y = x + 1, will intersect that portion of the circular part of the golden spiral as shown
below.
Spiralling Out of Control - Key
Now let’s look at these two curves specifically and extended.
Now, to make things a little more straight forward let’s remove the restriction on the
domain of the equation of the circle and just look at the two functions y = 25 - x 2 , and
y = x + 1. You need to find the point of intersection of these two functions by solving
this system. The first step is done for you.
Spiralling Out of Control - Key
Consider the graph provided above and compare it to your algebraic work. Do you
notice any discrepancies? Comment here.
Explain why your algebraic work may have provided more solutions than you are seeing
on the graph.
Transfer Task 3: Radical Race
Teacher Notes
The third transfer task is an option for the end of the unit on radicals (Outcomes 3 and
4). The goal of this task is to have the students practice all skills learned in the unit.
The race is composed of 11 question cards, each one in a multiple choice format. The
task the students are given is to answer the questions in a given order and choose the
next suitable location for subsequent clues, similar to a treasure hunt. The task is
designed to work at any school. The teacher will choose the 40 locations and write them
on the location list. This list, along with a map of the school will be given to the students.
Of the 40 locations, the teacher chooses 11 locations for the correct corresponding
answers. The teacher places the clues in a treasure hunt format. There is one clue per
card so that the students can clearly show their math processes for each question.
The task design allows completion by individuals or groups depending upon class
dynamics and teacher preferences.
It is advised that teachers build the treasure hunt carefully, ensuring that they clearly
record the locations for each clue on their version of the clue cards.
The correct answers are marked with a double asterisk. When building the race, choose
locations for the correct answers that you want students to access. The other 30
locations are designed as distracters.
Question card 11 requires solving a radical equation involving squaring a binomial.
The textbook has no questions like this but it was felt that with the skill learned in 10C
that this would be appropriate.
The rubric focuses on the calculations and the explanations supporting these
calculations.
Teacher Materials





Question Cards with space for solutions
Location list
School map
Answer key
Pencil or pens
Rules
 No running
 No loud noises coming from group
 Justify all answers mathematically
Mathematics 20-2
Inductive and Deductive Reasoning
Page 65 of 125
Assessment:


Correct answers recorded for all 11 questions with worked out solutions
Self assessment
Mathematics 20-2
Inductive and Deductive Reasoning
Page 66 of 125
Location List:
Location 1:
Location 11:
Location 21:
Location 31:
Location 2:
Location 12:
Location 22:
Location 32:
Location 3:
Location 13:
Location 23:
Location 33:
Location 4:
Location 14:
Location 24:
Location 34:
Location 5:
Location 15:
Location 25:
Location 35:
Location 6:
Location 16:
Location 26:
Location 36:
Location 7:
Location 17:
Location 27:
Location 37:
Location 8:
Location 18:
Location 28:
Location 38:
Location 9:
Location 19:
Location 29:
Location 39:
Location 10:
Location 20:
Location 30:
Location 40:
Mathematics 20-2
Inductive and Deductive Reasoning
Page 67 of 125
Teacher Notes on Questions Cards:
Card 1: Outcome 3: Achievement indicator 3.2
 Express an entire radical with a numerical radicand as mixed radical
Card 2: Outcome 3: Achievement indicator 3.1
 Comparing and ordering radical expression with numerical radicands
Card 3: Outcome 3: Achievement indicator 3.3
 Express a mixed radical with a numerical radicand as an entire radical.
Card 4: Outcome 3: Achievement indicator 3.4
 Perform one or more operations to simplify radical expressions with numerical or
variable radicands.
Card 5: Outcome 3: Achievement indicator 3.4 – It was felt that this aspect of the
curriculum warranted more than one question.
 Perform one or more operations to simplify radical expressions with numerical or
variable radicands.
Card 6: Outcome 3: Achievement indicator 3.5
 Rationalize the monomial denominator of a radical expression.
Card 7: Outcome 4: Achievement indicator 4.1
 Determine any restrictions on values for the variable in a radical equation.
Card 8: Outcome 4: Achievement indicator 4.2
 Determine algebraically, the roots of a radical equation
Card 9: Outcome 4: Achievement indicator 4.3
 Verify, by substitution, that the values determined in solving a radical equation
are roots of an equation.
Card 10: Outcome 4: Achievement indicator 4.5
 Solve problems by modeling a situation with a radical equation and solving the
equation.
Card 11: Outcome 4: Achievement indicator 4.4
 Explain why some roots determined in solving a radical equation are extraneous.
Note:
Outcome 3: Achievement indicator 3.6 was not included as it was felt that the concept
was sufficiently covered with achievement indicator 4.1
Mathematics 20-2
Inductive and Deductive Reasoning
Page 68 of 125
Glossary
absolute value – Represents how far the number is from zero
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
mixed radicals – A radical with a coefficient other than 1 [Math 20-2 (Nelson: page
516)]
perfect square – A number, from a given number system, that can be expressed as the
square of a number from the same number system
principal square root – The positive square root
rationalize the denominator – The process used to write a radical expression that
contains a radical denominator into an equivalent expression with a rational
denominator [Math 20-2 (Nelson: page 517)]
restrictions – The values of a the variable in an expression that ensure it to be defined
[Math 20-2 (Nelson: page 517)]
secondary square root – The negative square root
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 20-2
Inductive and Deductive Reasoning
Page 69 of 125
Transfer Task 3: Radical Race - Student Assessment Task
CARD 1:
Express each of the following as an entire radical.
i)
2 13
ii)
3 6
iii)
4 5
iv)
5 2
1.
26, 18, 20, 10
Go to location one
2.
15, 9, 9, 9
Go to location two
3.
11, 3, 1, -3
Go to location three
4.
52, 54, 80, 50
Go to location four
Transfer Task 3: Radical Race - Student Assessment Task
CARD 2:
Arrange the following in order from least to greatest.
i)
2 13
ii)
3 6
iii) 4 5
iv) 5 2
1.
2 13, 3 6, 4 5, 5 2
Go to location five
2.
2 13, 5 2, 3 6, 4 5
Go to location six
3.
5 2, 2 13, 3 6, 4 5
Go to location seven
4.
2 13, 4 5, 5 2, 3 6
Go to location eight
Transfer Task 3: Radical Race - Student Assessment Task
Card 3:
Convert the following mixed radicals to entire radical.
iii)
2 x3
a2 a
x 5 xy
iv)
3xy 3 3 2z 4
i)
ii)
1.
4 x 3 , a 3 , x 6 y , 3 27x 3 y 9z 4
Go to location nine
2.
4 x 3 , a 5 , x 11y , 3 27x 3 y 9z 4
Go to location ten
3.
4 x 3 , a 3 , x 11y , 3 27x 3 y 9z 4
Go to location eleven
4.
4 x 3 , a 5 , x 11y , 3 54x 3 y 9z 4
Go to location twelve
Transfer Task 3: Radical Race - Student Assessment Task
Card 4:
Simplify the following expression by combining like radicals
3
5
1
-5 128 +
848 +
50
4
4
2
1.
Go to location thirteen
2.
41 2
-36 2 - 5 3
3.
-44 2 - 5 3
Go to location fifteen
4.
-31 5
Go to location sixteen
Go to location fourteen
Transfer Task 3: Radical Race - Student Assessment Task
Card 5:
(
Expand and simplify 2 18 -
27
)
2
1.
99 - 36 6
Go to location seventeen
2.
4 18 - 27
Go to location eighteen
3.
27 - 4 18
Go to location nineteen
4.
36 16 - 99
Go to location twenty
Transfer Task 3: Radical Race - Student Assessment Task
Card 6:
When expressed in simplest form
18 2
3
1.
2 6 Go to location twenty-one
2.
6 5 Go to location twenty-two
3.
18 6 Go to location twenty-three
4.
6 6 Go to location twenty-four
equals
Transfer Task 3: Radical Race - Student Assessment Task
Card 7:
Given the equations:
I.
6x + 4 - 2 = 1
II.
5 + 8x + 12 = 6
III.
15x + 10 - 3 = 1
IV.
9x + 6 + 5 = 9
V.
2x + 3 - 3 = 5
VI.
6 + 30x + 20 = 10
The four equations that have the same restrictions for x are:
1.
2.
3.
4.
II, III, VI, V
I, II, IV, V
I, III, IV, VI
II, IV, V, VI
Go to location twenty-five
Go to location twenty-six
Go to location twenty-seven
Go to location twenty-eight
Transfer Task 3: Radical Race - Student Assessment Task
Card 8:
Determine, algebraically, the roots of the equation:
1. No solution
1
2. x =
4
1
3. x = 4
1
1
4. x = or x = 4
4
4x + 5 = 2
Go to location twenty-nine
Go to location thirty
Go to location thirty-one
Go to location thirty-two
Transfer Task 3: Radical Race - Student Assessment Task
Card 9:
Determine the correct combination of equations and answers from the following.
P.
Answers
x=7
1
3x - 2 = 2
4
Q.
x=4
3.
1
5x - 4 + 2 = 4
2
R.
x=6
4.
25 - 3x - 5 = -3
S.
No Solution
(
Equations
)
1.
3 2x + 1 = -3
2.
1.
2.
3.
4.
(
)
Equation 2 and answer P
Equation 3 and answer Q
Equation 1 and answer R
Equation 4 and answer S
Go to location thirty-three
Go to location thirty-four
Go to location thirty-five
Go to location thirty-six
Transfer Task 3: Radical Race - Student Assessment Task
Card 10: Two squares are shown.
The side length of the smaller square is 6 cm. The
perimeter of the larger square in simplified form is:
1.
12 2
2. 3 2
3. 6 2
4. 24 2
Go to location thirty-seven
Go to location thirty-eight
Go to location thirty-nine
Go to location forty
Transfer Task 3: Radical Race - Student Assessment Task
Card 11:
Solve the equation x + 1= x 2 - 25. Explain your result.
Exemplar: Location List Harry Ainlay High School
Location 1:
Location 12:
Exit 15
Location 23: South
entrance to the
West Court yard
Location 34: Office
Entrance
Location 2:
Exit 2
Location 13:
Exit 16
Location 24:
Wellness Centre
Desk
Location 35:
South entrance to
east courtyard
Location 3:
Exit 5
Location 14:
Exit 17
Location 25:
Dance studio
Entrance
Location 36:
Outside entrance to
the Auto
Location 4:
Exit 4
Location 15:
Outdoor class room
Location 37:Inside
Entrance to
Construction
Location 5:
Exit 6
Location 16:
Exit 14
Location 6:
Exit 8
Location 17:
Exit 13
Location 26:
Cosmetology
Parking Propane
tank storage
Location 27:
North east corner of
the temporary
library
Location 28:
Southeast corner of
the rotunda
Location 7:
Visitor Parking Stall
# 116
Location 18:
North East Parking
lot Student Bench
Location 29: Book
Room 304
Location 40:
Room 384
Location 8:
Exit 10
Location 19:
Exit 19
Location 30:
Drama room
Location 9:
Bike Racks
Location 20:
Exit 20
Location 31:
Theatre Entrance
Location 10:
Exit 11
Location 21:
South entrance to
student activities
Location 32:
Athletics Office
Location 11:
Exit 12
Location 22: North
East corner of the
Rotunda
Location 33:
Student Services
and Career Centre
Exit 1
Location 38:
Inside entrance to
Design Studies
Location 39:
Centre Table of the
Rotunda
Student checklist of skills: This can be done by reading out the skills and having
students think about how they would rank themselves.
Analysis of skills: On a scale of 1 to 10, where 1 is no confidence and 10 is you are
ready to teach this concept, rank yourself on the following:
Card 1: Express an entire radical as a mixed radical
Card 2: Comparing and ordering radical expressions
Card 3: Express a mixed radical as an entire radical.
Card 4: Simplify and add or subtract radicals.
Card 5: Multiply and simplify radicals.
Card 6: Rationalize the denominator of a radical expression.
Card 7: Determine any restrictions on values for the variable in a radical equation.
Card 8: Determine algebraically, the roots of a radical equation
Card 9: Verify, by substitution, that the values determined in solving a radical
equation are roots of an equation.
Card 10: Solve problems by modeling a situation with a radical equation and solving
the equation.
Card 11: Explain why some roots determined in solving a radical equation are
extraneous.
Radical Race - Student Assessment Task
Glossary
absolute value – Represents how far the number is from zero
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
mixed radicals – A radical with a coefficient other than 1 [Math 20-2 (Nelson: page
516)]
perfect square – A number, from a given number system, that can be expressed as the
square of a number from the same number system
principal square root – The positive square root
rationalize the denominator – The process used to write a radical expression that
contains a radical denominator into an equivalent expression with a rational
denominator [Math 20-2 (Nelson: page 517)]
restrictions – The values of a the variable in an expression that ensure it to be defined
[Math 20-2 (Nelson: page 517)]
secondary square root – The negative square root
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 20-2
Radical Race
Rubric
Level
Criteria
Excellent
3
Adequate
2
Limited
1
Performs
Calculations
Performs precise
and explicit
calculations.
Performs appropriate
and generally accurate
calculations.
Performs superficial
and irrelevant
calculations.
Explains
Choices
Shows a solution
for the problems;
provides an
insightful
explanation.
Shows a solution for the
problem; provides
explanations that are
complete but vague.
Shows a solution for
the problem but no
supporting
mathematics.
Glossary
absolute value – Represents how far the number is from zero
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
mixed radicals – A radical with a coefficient other than 1 [Math 20-2 (Nelson: page
516)]
perfect square – A number, from a given number system, that can be expressed as the
square of a number from the same number system
principal square root – The positive square root
rationalize the denominator – The process used to write a radical expression that
contains a radical denominator into an equivalent expression with a rational
denominator [Math 20-2 (Nelson: page 517)]
restrictions – The values of a the variable in an expression that ensure it to be defined
[Math 20-2 (Nelson: page 517)]
secondary square root – The negative square root
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.
Possible Solution to Radical Race
Questions Cards: Master List – Answers marked with a double asterisk
Card 1: Express each of the following as an entire radical.
i. 2 13
ii. 3 6
iii. 4 5
iv. 5 2
1.
26, 18, 20, 10
Go to location one
2.
15, 9, 9, 9
Go to location two
3.
11, 3, 1, -3
Go to location three
4.
52, 54, 80, 50
Go to location four **
Card 2: Arrange the following in order from least to greatest
i)
2 13
ii)
3 6
iii)
4 5
iv)
5 2
1.
2 13, 3 6, 4 5, 5 2
Go to location five
2.
2 13, 5 2, 3 6, 4 5
Go to location six
3.
5 2, 2 13, 3 6, 4 5
Go to location seven **
4.
2 13, 4 5, 5 2, 3 6
Go to location eight
Card 3: Convert the following mixed radicals to entire radical.
iii)
2 x3
a2 a
x 5 xy
iv)
3xy 3 3 2z 4
i)
ii)
1.
4 x 3 , a 3 , x 6 y , 3 27x 3 y 9z 4
Go to location nine
2.
4 x 3 , a 5 , x 11y , 3 27x 3 y 9z 4
Go to location ten
3.
4 x 3 , a 3 , x 11y , 3 27x 3 y 9z 4
Go to location eleven
4.
4 x 3 , a 5 , x 11y , 3 54x 3 y 9z 4
Go to location twelve **
Mathematics 20-2
Inductive and Deductive Reasoning
Page 87 of 125
Card 4: Simplify the following expression by combining like radicals
3
5
1
-5 128 +
848 +
50
4
4
2
1.
Go to location thirteen
2.
41 2
-36 2 - 5 3
3.
-44 2 - 5 3
Go to location fifteen
4.
-31 5
Go to location sixteen
Go to location fourteen **
(
Card 5: Expand and simplify 2 18 -
27
)
2
1.
99 - 36 6
Go to location seventeen **
2.
4 18 - 27
Go to location eighteen
3.
27 - 4 18
Go to location nineteen
4.
36 16 - 99
Go to location twenty
Card 6: When expressed in simplest form
18 2
equals
3
1.
2 6 Go to location twenty-one
2.
6 5 Go to location twenty-two
3.
18 6 Go to location twenty-three
4.
6 6 Go to location twenty-four **
Card 7: Given the equations:
I.
6x + 4 - 2 = 1
II.
5 + 8x + 12 = 6
III.
15x + 10 - 3 = 1
IV.
9x + 6 + 5 = 9
V.
2x + 3 - 3 = 5
VI.
6 + 30x + 20 = 10
The four equations that have the same restrictions for x are:
1.
2.
3.
4.
II, III, VI, V
I, II, IV, V
I, III, IV, VI
II, IV, V, VI
Mathematics 20-2
Go to location twenty-five
Go to location twenty-six
Go to location twenty-seven **
Go to location twenty-eight
Inductive and Deductive Reasoning
Page 88 of 125
Card 8: Determine, algebraically, the roots of the equation:
1. No solution
1
2. x =
4
1
3. x = 4
1
1
4. x = or x = 4
4
4x + 5 = 2
Go to location twenty-nine
Go to location thirty
Go to location thirty-one **
Go to location thirty-two
Card 9: Determine the correct combination of equations and answers from the following.
Equations
Answers
x=7
1.
P.
3 2x + 1 = -3
(
)
2.
1
3x - 2 = 2
4
Q.
x=4
3.
1
5x - 4 + 2 = 4
2
R.
x=6
4.
25 - 3x - 5 = -3
S.
No Solution
1.
2.
3.
4.
(
)
Equation 2 and answer P
Equation 3 and answer Q
Equation 1 and answer R
Equation 4 and answer S
Go to location thirty-three
Go to location thirty-four **
Go to location thirty-five
Go to location thirty-six
Card 10: Two squares are shown.
The side length of the smaller square is 6 cm. The perimeter
of the larger square in simplified form is:
1.
12 2 Go to location thirty-seven
2. 3 2 Go to location thirty-eight
3. 6 2 Go to location thirty-nine
4.
24 2 Go to location forty **
Card 11: Solve the equation x + 1= x 2 - 25. Explain your result.
x 2 + 2x + 1 = x 2 - 25
x = -13
(
)
Check: -13 + 1 =
( -13)
2
- 25
- 12 = 12 No Solution
Mathematics 20-2
Inductive and Deductive Reasoning
Page 89 of 125
STAGE 3
Learning Plans
Lesson 1
What Should I Remember?
STAGE 1
BIG IDEA: Understanding radicals will further develop student’s sense of exact values, and enhance
their ability to simplify expressions and solve equations.
ENDURING UNDERSTANDINGS:
Students will understand …


There are appropriate forms of
communications in mathematics.
Radicals are more precise efficient, concise
and accurate.
ESSENTIAL QUESTIONS:

Under what circumstances would a mixed
radical or an entire radical be appropriate?

When is
x > y?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …
 when the expression is simplified

simplify radical expressions with numerical or
variable radicands
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 20-2
Inductive and Deductive Reasoning
Page 90 of 125
Lesson Summary
This lesson will review prior knowledge needed in this unit.
 Definitions of square roots, cube roots, perfect squares, perfect cubes and radicals
 Estimate radicals
 Compare and order numbers
 Mixed to entire radicals
 Entire to mixed radicals
 Review operations on polynomials so as to connect those with operations on
radicals
Lesson Plan
Hook
What do you see? (Think Pair Share)
3
25
5


5
27
3
3
Have a Think-Pair-Share discussion involving the picture above in relation to
square roots, cube roots, perfect squares, perfect cubes, factors, etc.
After the Think-Pair-Share ask students to define the following
o Square Root
o Cube Root
o Perfect Square (List as many as you can)
o Perfect Cube (List as many as you can)
Lesson Goal

Activate prior knowledge
Mathematics 20-2
Inductive and Deductive Reasoning
Page 91 of 125
Lesson
Sorting Activity
 Create groups and handout a number card from the same set to each member of
the group. The group must sort themselves from lowest to highest. Each member
should try to determine the approximate value of his or her card. Not allowing a
calculator will help students refine their number sense.
 Possible sets of numbers
3
o
2, 1.9, 3 9, 8,
o
5, 2 3, 3.9, 2 7, 3 5, 6, 51
o
2 3 2,
3
27, 2 3 4,
3
27, 4.9,
26
64, 4.1, 3 3 3, 5
L1 Sorting Activity (3 sets of numbers)
 files were added to Appendix
 files were added to the EPSB Understanding by Design share site
Entire to Mixed / Mixed to Entire Puzzle Activity
 Groups of students receive the puzzle in pieces and perhaps a blank nine by nine
grid. A math joke is presented to them “What does the little mermaid wear?” and
they have to assemble the puzzle correctly by lining up the entire radical side with
its corresponding mixed radical side. The letters on the puzzle make up the answer
(the answer is “an algae-bra”). NOTE: the orientation of the letters has been
scrambled to avoid giving directional clues.
L1 Puzzle Activity
 file was added to Appendix
 file was added to the EPSB Understanding by Design share site
Going Beyond
Resources
Math 20-2 (Nelson: sec 4.1, page(s) 176-183)
Supporting
Mathematics 20-2
Inductive and Deductive Reasoning
Page 92 of 125
Assessment
Glossary
entire radical – A radical with a coefficient of 1 [Math 20-2 (Nelson: page 515)]
mixed radicals – A radical with a coefficient other than 1 [Math 20-2 (Nelson: page
516)]
perfect square – A number, from a given number system, that can be expressed as the
square of a number from the same number system
perfect cube – A number, from a given number system, that can be expressed as the
cube of a number from the same number system.
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 20-2
Inductive and Deductive Reasoning
Page 93 of 125
Lesson 2
What’s with all the Letters?
STAGE 1
BIG IDEA: Understanding radicals will further develop student’s sense of exact values, and enhance
their ability to simplify expressions and solve equations.
ENDURING UNDERSTANDINGS:
Students will understand …


There are restrictions on values for the
variable in a radical.
There are appropriate forms of
communications in mathematics.
ESSENTIAL QUESTIONS:


Why is there a restriction on the domain?
Why does your calculator say ERROR on

Under what circumstances would a mixed
radical or an entire radical be appropriate?
-2 ?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …
 the square root of a negative is not a real
number
 when the expression is simplified

Mathematics 20-2

identify values of the variable for which the
radical expression is defined
simplify radical expressions with numerical or
variable radicands
Inductive and Deductive Reasoning
Page 94 of 125
Lesson Summary





Warm Up Activity
Development
Binomial Radicands Restriction
Entire to Mixed Radicals with Variables
Mixed to Entire Radicals with Variables
Lesson Plan
Warm Up Activity
Option A: Determine two identical numbers that will give these products.
1
25 16
100, 49, 225, 0, - 36, - 16, , 54,
, - , 1.21, 27.04
4
9 49
Discuss what the restrictions would be in the value(s) of:
1. the product
2. the identical numbers
Option B: Sort these numbers according to some sort of criteria (can be done individually
or in groups)
16
, 325, 1.44, 0,
81
Ask students how they sorted these values.
100, 38, -36, 49, -
25
, 9.61
9
Questions to ask:
1. Are there answers for all of these? (Discuss what’s defined.)
2. Are there any interesting unexpected answers?
Option C: Have students generate a list of square roots where the radicand is positive or
negative, then evaluate these. Questions to ask:
1. Are there answers for all of these? (Discuss what’s defined.)
2. Are there any interesting unexpected answers?
To meet curricular objectives consider repeating using cubic roots for all options.
The intent of the warm-up would be for students to discover the restriction on square
roots (which doesn’t apply to cubic roots) is that the radicand must be greater than or
equal to zero. This should be summarized and recorded in some way. To enhance the
understanding of restrictions students should also see radicals of the form
Mathematics 20-2
Inductive and Deductive Reasoning
ax + b.
Page 95 of 125
DI Suggestions
Consider an extension to roots beyond cubic, looking for patterns
between odd an even indexed roots
Consider a graphing activity designed to explore the domain of square
and cubic roots
Development
Present students with the following list and ask them to identify the perfect squares:
x , x 2, x 3 , x 4, x 5, x 6
Using this knowledge try simplifying the following list:
x , x 2,
x 3, x 4, x 5, x 6
Note that the accurate simplification for
x 3 would be x
x ,but we will always assume
the principal root for the purposes of this course and will instead simplify
x 3 to x x .
It would be beneficial for students to be introduced to numerical coefficients combined
with variable radicands before having students practice this concept. For example,
2 18x 5 can be rewritten as 6x 2 2x .
An optional activity would be to have students take mixed radicals (with variable in the
coefficients) and convert these to entire radicals (ie working backwards from previous
method). However, there are no questions provided in this resource.
Sorting Activity
Provide students with a list of numbers and ask them to sort them according to criteria of
their choice. Guide them toward selecting defined and undefined.
Start with a list of positive and negative square roots. (Include 0)
Next make a list of positive and negative cube roots (Include 0).
DI Suggestion
Make a list of positive and negative
even and odd indexed roots.
DI Graphing Activities
Graph basic square root and cube root functions to
analyze the domain.
Extension: Have students plot the given numbers (x is
radicand and y is square root of radicand).
Mathematics 20-2
Inductive and Deductive Reasoning
Page 96 of 125
Binomial Radicands Restriction
Once students understand the basic restrictions on x, extend it to binomials. This was
previously mentioned in Warm Up Activity, Option C.
Entire to Mixed Radicals with Variables
Start with a list to see a pattern.
x , x 2,
x 3 , x 4 , ...
and then practice.
L2 Converting Entire to Mixed Worksheet
 file was added to Appendix
 file was added to the EPSB Understanding by Design share site
Mixed to Entire Radicals with Variables
Compare to the numerical strategy and practice.
L2 Converting Mixed to Entire Worksheet
 file was added to Appendix
 file was added to the EPSB Understanding by Design share site
Going Beyond
Resources
Math 20-2 (Nelson, page 211, #1, 2, 3, 11)
Worksheet
Supporting
Assessment
Mathematics 20-2
Inductive and Deductive Reasoning
Page 97 of 125
Glossary
entire radical – A radical with a coefficient of 1 [Math 20-2 (Nelson: page 515)]
mixed radicals – A radical with a coefficient other than 1 [Math 20-2 (Nelson: page
516)]
restrictions – The values of a the variable in an expression that ensure it to be defined
[Math 20-2 (Nelson: page 517)]
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 20-2
Inductive and Deductive Reasoning
Page 98 of 125
Lesson 3
Let’s Play Operation (Operations on Radicals)
STAGE 1
BIG IDEA: Understanding radicals will further develop student’s sense of exact values, and enhance
their ability to simplify expressions and solve equations.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …






There are restrictions on values for the
variable in a radical.
There are appropriate forms of
communications in mathematics.
Radicals are more precise efficient, concise
and accurate.
Operations can be performed on radicals, as
with other numbers or expressions.

When is an exact solution required?
Under what circumstances would a mixed
radical or an entire radical be appropriate?
When would you use radicals?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …
 when the expression is simplified
 what like terms are
 the rules for operations on radicals




Mathematics 20-2
identify values of the variable for which the
radical expression is defined
simplify radical expressions with numerical or
variable radicands
rationalize the monomial denominator of a
radical expression
perform operations with radicals
Inductive and Deductive Reasoning
Page 99 of 125
Lesson Summary

Students will explore and practice the operations on radicals.
Lesson Plan
Add / Subtract Radicals
Adding and Subtracting Radicals Discovery Activity
 This activity will ask students to discover and practice the rules for adding and
subtracting radicals.
 The false examples may be used to discuss common errors.
 Challenge questions extend concept to include variables and more complex
questions.
L3 Adding and Subtracting Radicals Discovery Activity
 file was added to Appendix
 file was added to the EPSB Understanding by Design share site
L3 Adding and Subtracting Radicals Key
 PDF file was added to the EPSB Understanding by Design share site
Visual Representation
 A visual representation may be used to reinforce the concept of adding and
subtracting like radicals. There is a file with examples of the visualization in use
(Example Add Subtract Visual) and then a file to be used to solve additional
questions (Interactive Add Subtract Visual).
L3 Visual Representation
 notebook files were added to the EPSB Understanding by Design share site
Reinforce concepts with practice from textbook.
Multiplying Radicals
Multiplying Radicals Discovery Activity
 This activity will ask students to discover and practice the rules for multiplying
radicals.
 The false examples may be used to discuss common errors.
 Challenge questions extend concept to include variables and more complex
questions.
Mathematics 20-2
Inductive and Deductive Reasoning
Page 100 of 125
L3 Multiplying Radicals Discovery Activity
 file was added to Appendix
 file was added to the EPSB Understanding by Design share site
L3 Multiplying Radicals Key
 PDF file was added to the EPSB Understanding by Design share site
Reinforce concepts with practice from textbook.
Dividing Radicals
Dividing Radicals Introduction
 Warm Up
o Review squaring radicals and equivalent fractions.
 Define and Discuss Rationalizing
o Reasons for rationalizing include consistency, communication, and
comparisons.
 Examples: Division & Rationalizing
o Discuss various strategies to solve the problem (division, simplifying, or
rationalize first)
o Nelson Principles of Mathematics 11 has an excellent example of multiple
strategies on pages 194-196.
Reinforce concepts with practice from textbook.
L3 Dividing Radicals Introduction
 file was added to Appendix
 file was added to the EPSB Understanding by Design share site
L3 Dividing Radicals Key
 PDF file was added to the EPSB Understanding by Design share site
Remember: when dealing with expressions with variables you should state the
restrictions on the variable.
Explain a Process Assignment
This type of assignment may be used for each operation on radicals. The assignment
simply involves students writing an explanation of how to do the operation and providing
an example of a solution.
This assignment may reinforce/assess understanding while working on communication
skills.
Mathematics 20-2
Inductive and Deductive Reasoning
Page 101 of 125
Going Beyond
Resources
Math 20-2 (Nelson: sec 4.3, page(s) 184-190)
Math 20-2 (Nelson: sec 4.3, page(s) 191-201)
Assessment
Glossary
absolute value – Represents how far the number is from zero
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
mixed radicals – A radical with a coefficient other than 1 [Math 20-2 (Nelson: page
516)]
perfect square – A number, from a given number system, that can be expressed as the
square of a number from the same number system
principal square root – The positive square root
rationalize the denominator – The process used to write a radical expression that
contains a radical denominator into an equivalent expression with a rational denominator
[Math 20-2 (Nelson: page 517)]
restrictions – The values of a the variable in an expression that ensure it to be defined
[Math 20-2 (Nelson: page 517)]
secondary square root – The negative square root
Other
Mathematics 20-2
Inductive and Deductive Reasoning
Page 102 of 125
Lesson 4
Radical Issues (Solving Radical Equations)
STAGE 1
BIG IDEA: Understanding radicals will further develop student’s sense of exact values, and enhance
their ability to simplify expressions and solve equations.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …





There are restrictions on values for the
variable in a radical.
Some equations have extraneous roots, and
why.
There are appropriate forms of
communications in mathematics.
Radicals are more precise efficient, concise
and accurate.
Operations can be performed on radicals, as
with other numbers or expressions.



Why are extraneous roots produced?
Why is there a restriction on the domain?
Why does your calculator say ERROR on


When is an exact solution required?
Under what circumstances would a mixed
radical or an entire radical be appropriate?
When would you use radicals?

-2 ?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …
 the square root of a negative is not a real
number
 all roots determined algebraically need to be
verified
 when it is appropriate to use an exact value
versus a rounded decimal
 what like terms are
 the rules for operations on radicals

Mathematics 20-2



determine any restrictions on values for the
variable in a radical equation
verify, by substitution, that the values
determined in solving a radical equation are
roots of the equation
model a situation with a radical equation
solve radical equations to determine the roots
Inductive and Deductive Reasoning
Page 103 of 125
Lesson Summary


Use a table with appropriate data to introduce a cubic root function.
Explore methods to solve radical functions.
Lesson Plan
Provide students with the following data table. Have students look for a pattern, to make
it easier for them you may want to inform them that it’s a cubic root function.
x
5
24
61
132
213
y
4
6
8
10
12
For students that are struggling you may want to inform that the function is of the form
y =a3 x +b
Solution:
y = 23 x + 3
Discuss the solution with respect to the domain of x. Are there any restrictions?
Here is a link to some application questions that could be used as examples or have
students work through together in an attempt to understand the solving process.
http://cnx.org/content/m21965/latest/
One example from the site:
Mathematics 20-2
Inductive and Deductive Reasoning
Page 104 of 125
Answers:
(a) 7.96
(b) 5
Other problems that could be searched:
 Using the distance formula
 Meteorology problems: length of time vs. diameter of storm
 Length of diagonal in a cube
 Given an expression representing the volume of a 3D object, calculate the surface
area (or vice versa)
 Numerous problems from Physics texts. (Questions involving periods, kinetic
energy, etc)
Discussion during the problem solving process should involve extraneous roots. To
assist in the discussion students must know how to verify their answers.
From here contrast between exact values vs. approximate value answers.
Example:
Calculate the side length of a cube that has a volume of 375 cm 3. From there calculate
the surface area of the cube.
Approximate Solution:
r = 375
3
= 7.2
( )
SA = 6 7.2
Exact Solution:
r = 3 375
= 3 125 ´ 3 3
= 53 3
2
= 311 cm2
( ) cm
= 150 ( 9 ) cm
SA = 6 5 3 3
2
3
2
2
= 312 cm2
Using this value for volume it is interesting to note that the answers differ by 1 cm 2. It
may be worthwhile having a discussion whether or not this is significant. What if the
volume was greater?
Mathematics 20-2
Inductive and Deductive Reasoning
Page 105 of 125
The remainder of the lesson should be spent on rearranging radical equations to isolate
the unknown. The focus should be on performing opposite operations. A comparison to
equations solved in previous years is suggested.
Linear
3x = 9
divide both sides by 3
x=3
3(x + 2) = 9
Divide both sides by 3
x+2=3
Subtract both sides by 2
x=1
3(x + 2) – 6 = 9
Add both sides by 6
3(x + 2) = 15
Divide both sides by 3
x+2=5
Subtract both sides by 2
x=3
Radical
x = 25
Square both sides
x = 625
x + 2 = 25
Square both sides
x + 2 = 625
Subtract both sides by 2
x = 623
x + 2 - 6 = 19
Add both sides by 6
x + 2 = 25
Square both sides
x + 2 = 625
Subtract both sides by 2
x = 623
After guiding students through a few of these it would be beneficial to present students
with a more complicated equation to work on individually or in groups. For example,
8 = 3 2x - 25 -13
x = 37
In order to assist in the transfer task students should be exposed to equations that are not
limited to variable as radicands only. For example
3 + x -1 = x
Going Beyond
Resources
Math 20-2 (Nelson, page 222, #1-17)
Mathematics 20-2
Inductive and Deductive Reasoning
Page 106 of 125
Supporting
Assessment
Glossary
absolute value – Represents how far the number is from zero
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
mixed radicals – A radical with a coefficient other than 1 [Math 20-2 (Nelson: page
516)]
perfect square – A number, from a given number system, that can be expressed as the
square of a number from the same number system
principal square root – The positive square root
rationalize the denominator – The process used to write a radical expression that
contains a radical denominator into an equivalent expression with a rational denominator
[Math 20-2 (Nelson: page 517)]
restrictions – The values of a the variable in an expression that ensure it to be defined
[Math 20-2 (Nelson: page 517)]
secondary square root – The negative r square root
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 20-2
Inductive and Deductive Reasoning
Page 107 of 125
Appendix
Copies of worksheets for Lessons 1 – 3 follow:
Lesson 1 Radicals Sorting Number Set 1
Lesson 1 Radicals Sorting Number Set 2
Lesson 1 Radicals Sorting Number Set 3
Lesson 1 Mixed to Entire Puzzle Activity & Answers
Lesson 2 Converting Entire to Mixed & Answers
Lesson 2 Converting Mixed to Entire Radical Worksheet & Answers
Lesson 3 Adding and Subtracting Radicals & Answers
Lesson 3 Multiplying Radicals & Answers
Lesson 3 Dividing Radicals Introduction & Answers
Lesson 3 Converting Mixed to Entire Radical Worksheet & Answers
Mathematics 20-2
Inductive and Deductive Reasoning
Page 108 of 125
M20-2 Lesson 1 Radicals Sorting Number Set 1
2
1.9
3
8
3
4.9
27
26
Mathematics 20-2
Inductive and Deductive Reasoning
Page 109 of 125
9
M20-2 Lesson 1 Radicals Sorting Number Set 2
5 2 3 3.9
6
2 7 3 5
51
Mathematics 20-2
Inductive and Deductive Reasoning
Page 110 of 125
M20-2 Lesson 1 Radicals Sorting Number Set 3
3
2 3
3
64
3
3
27 2 4
4.1
5
Mathematics 20-2
Inductive and Deductive Reasoning
Page 111 of 125
3
3 3
M20-2 Lesson 1 Mixed to Entire Puzzle Activity
Preparation: copy and cut along the gridlines.
Mathematics 20-2
Inductive and Deductive Reasoning
Page 112 of 125
M20-2 Lesson 1 Mixed to Entire Puzzle Activity – Student Answers
What does the little mermaid wear?”
Answer: AN ALGAE-BRA
Mathematics 20-2
Inductive and Deductive Reasoning
Page 113 of 125
M20-2 Lesson 2 Converting Entire to Mixed Worksheet (Variable
Radicands)
Simplify the following.
1. √𝑥 8
2. √49𝑥 2
3. √125𝑥 4
3
9. √−1000𝑥 3
3
27
10. √64 𝑥 6
3
11. √−125𝑥 8
4. √48𝑥 5
3
12. √363𝑥16
9
5. √4 𝑥10
6. √1.69𝑥 3
3
1
13. √8 𝑥12
3
14. √0.125𝑥 9
64
7. √ 9 𝑥 7
8. √4.41𝑥 9
3
15. 2√216𝑥 5
3
8
16. 10√125 𝑥 7
Extra Problem
3
√−3375𝑥 −12 𝑦 21
Mathematics 20-2
Inductive and Deductive Reasoning
Page 114 of 125
Answers:
1.
2.
3.
4.
5.
𝑥4
7𝑥
5𝑥 2 √5
4𝑥 2 √𝑥
3 5
𝑥
2
6. 1.3𝑥√𝑥
8
7. 3 𝑥 3 √𝑥
8. 2.1𝑥 4 √𝑥
9. −10𝑥
3
10. 4 𝑥 2
11.
12.
13.
14.
15.
16.
3
−5𝑥 2 √𝑥 2
3
11𝑥 5 √3𝑥
1 4
𝑥
2
0.5𝑥 3
3
12𝑥 √𝑥 2
3
4𝑥 2 √𝑥
Extra Problem
−15𝑥 −4 𝑦 7
Mathematics 20-2
Inductive and Deductive Reasoning
Page 115 of 125
M20-2 Lesson 2 Converting Mixed to Entire Radical Worksheet
1. 3𝑥√2𝑥
2.
3
𝑥 2 √𝑥
4
3. 5𝑥 3 √3
4.
1
3
𝑦 3 √2 𝑦
2
3
6. 2𝑥 ∙ √𝑥 2
7.
1
3
3
𝑥 2 ∙ √2𝑥
3
8. 1.2𝑥 ∙ √5
9. 2𝑦 ∙ 3√125𝑦
2
5. 5.1𝑦√7𝑦
10. 3.4𝑦 3 ∙ √5
Extra Problem
2 3 2 3 3 2
𝑥 𝑦 ∙ √ 𝑥𝑦
3
2
Mathematics 20-2
Inductive and Deductive Reasoning
Page 116 of 125
Answers:
1. √18𝑥 3
9
2. √16 𝑥 5
3. √75𝑥 6
3
4. √8 𝑦 7
5. √182.07𝑦 3
3
6. √8𝑥 5
3
2
7. √27 𝑥 7
3
8. √8.64𝑥 3
3
9. √1000𝑦 4
578
10. √4.624𝑦 6 OR √128 𝑦 6
Extra Problem
4
√ 𝑥10 𝑦 3
9
3
Mathematics 20-2
Inductive and Deductive Reasoning
Page 117 of 125
M20-2 Lesson 3 Adding and Subtracting Radicals
Discovery Activity
If these are true …………………………………………………………………… and these are false.
3 5 6 5 9 5
3 5  6 5  9 10
8 7 3 7  5 7
8 7 3 7  5 0
13  5 13  6 13
13  5 13  5 13
8 +3 2 =5 2
8  3 2  4 10
27  12  9 3  4 3
27  12  5 3
 13 3
then find the answer to these:
20 13  10 13 
4 11  11 
8  18 
2 12  48 
(Hint: Think about
addition &
subtraction of
polynomials.)
… and now see if you can do these!
7 5 3
3 27  5 12  75 
4 x 3 x 5 x 
3 x3  x x  3 
Mathematics 20-2
Inductive and Deductive Reasoning
Page 118 of 125
Answers:
Mathematics 20-2
Inductive and Deductive Reasoning
Page 119 of 125
M20-2 Lesson 3 Multiplying Radicals
Discovery Activity
If these are true …………………………………………………………………… and these are false.
 3  7  
( 3)( 7) =
21
3 2 4 5   12
 5  5  
3 2 4 5   7
10
 5  5  
25  5
 3  3   3
2 2  6   2
10
10
25  5
 3  3   9
2 2  6   2
12  4 3
12  24
then find the answer to these:
 2  11 
3 5 2 2  
 8  8  
5 6 3 3  
(Hint: Think about
addition &
subtraction of
polynomials.)
… and now see if you can do these!
2 x 5 x  
4 x 7

x3 
Mathematics 20-2
x


2x 3 5 x 4 
2 3 3

7 8 8 
Inductive and Deductive Reasoning
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Answers:
Mathematics 20-2
Inductive and Deductive Reasoning
Page 121 of 125
M20-2 Lesson 3 Dividing Radicals Introduction
Warm Up
Solve the following:
Write three equivalent fractions
for each of the following:
2  2  _______
3  3  _______
4  4  _______
5  5  _______
2

3
6

12
Rationalizing
What is rationalizing? Why do we rationalize?
Examples
a)
c)
5
2
6
3

b)

d)
6 7x
5 3x
2 10
4 2


and now…
3 12  7 6
3
3 5  9 24 x 5

Mathematics 20-2
3 x

Inductive and Deductive Reasoning
Page 122 of 125
Answers:
Mathematics 20-2
Inductive and Deductive Reasoning
Page 123 of 125
M20-2 Lesson 3 Converting Mixed to Entire Radical Worksheet
11. 3𝑥√2𝑥
3
3
16. 2𝑥 ∙ √𝑥 2
1
3
12. 4 𝑥 2 √𝑥
17. 3 𝑥 2 ∙ √2𝑥
13. 5𝑥 3 √3
18. 1.2𝑥 ∙ √5
1
3
14. 2 𝑦 3 √2 𝑦
3
19. 2𝑦 ∙ 3√125𝑦
2
15. 5.1𝑦√7𝑦
20. 3.4𝑦 3 ∙ √5
Extra Problem
2 3 2 3 3 2
𝑥 𝑦 ∙ √ 𝑥𝑦
3
2
Mathematics 20-2
Inductive and Deductive Reasoning
Page 124 of 125
Answers:
11. √18𝑥 3
9
12. √16 𝑥 5
13. √75𝑥 6
3
14. √8 𝑦 7
15. √182.07𝑦 3
3
16. √8𝑥 5
3
2
17. √27 𝑥 7
3
18. √8.64𝑥 3
3
19. √1000𝑦 4
578
20. √4.624𝑦 6 OR √128 𝑦 6
Extra Problem
4
√ 𝑥10 𝑦 3
9
3
Mathematics 20-2
Inductive and Deductive Reasoning
Page 125 of 125