G8

Homework Name: ___________________________
Date: ___________________________
Unit 2 · Lesson 6: Convert Negative Exponents

Directions: Solve 2. 1
6−2
1. 3−5

3. 1 4. 9−2
7−3

5. −2 1

6. −4

Copyright © Swun Math Grade 8 Unit 2 Lesson 6 C TE 146

Answer Key

Homework

1. 3−5 = 1 2. 1 = 62
35 6−2

3. 1 = 73 4. 9−2 = 1
7−3 92

5. −2 = 1 6. 1 = 4
2 −4

147 Copyright © Swun Math Grade 8 Unit 2 Lesson 6 C TE

Notes

Copyright © Swun Math Grade 8 Unit 2 Lesson 6 C TE 148

Zero and Negative Exponents MPs Applied MP

Procedural Lesson * Embedded MP
Grade 8 · Unit 2 · Lesson 7
12345678
MC: 8.EE.1
* * * * *

Problem of the Day Student Journal Pages

Objective: I will evaluate expressions with zero and negative exponents. 82-87

Vocabulary Teacher Resources

Negative Exponent: indicates the number of Considerations:
times to divide by the base
Show chart with pattern of exponents starting with
x−2
positive exponents and ending at zero exponent.
The negative sign says to do the inverse.
Dividing is the inverse of multiplying. Show chart with pattern of exponents starting with

positive exponents and ending with a negative

exponent. Some students will recognize that the

power will “land” in the denominator if the larger
x4
exponent is in the denominator. Example: x6 = 1
x2

Reciprocal: what to multiply a value by to get 1; Steps:
multiplicative inverse 1. Write the expression in expanded form.
2. Simplify by canceling common factors.
A negative exponent is equivalent to the inverse of 3. Rewrite exponent as a positive exponent.
the positive of the same number. OR
x−3 is the reciprocal of x3 1. Subtract exponents of like bases.

4−5 is the reciprocal of 45 2. Write answer using positive exponent.

Application of MPs:

Properties of Exponents MP1: How are negative exponents and reciprocals

related?

Zero Exponent a0 = 1 Negative exponents and reciprocals are

related because .

Negative Exponent a−n = 1 MP3: Why is the reciprocal of − , ?
an


The reciprocal of 5−4 is 1 because .
54

149 Copyright © Swun Math Grade 8 Unit 2 Lesson 7 P TE

Input/Model

(Teacher Presents)

Directions: Simplify the expression. Write the solution using a positive exponent.

1. 6−8 ⋅ 68

Solution:

1st Strategy
• 6−8+8 (keep in mind the rules for adding exponents)
• 60 = 1

2nd Strategy

● Change each number so they have positive exponents.
1
● 6688 ⋅ 68
●
68
● 60 = 1

2. 3 5⋅ −2
4

Solution:
1st Strategy

• Since they have the same base, add exponents in the numerator. Then subtract with the exponents that

have the same base in the denominator.

• 3 3 = 3 ⋅ 3−4 (Students might want to write 4 – 3)
4
3
• 3 ⋅ −1 =

2nd Strategy
• Use expanded form and cancel the common factors.

• 3⋅ ⋅ ⋅ ⋅ ⋅ ⋅ −1⋅ −1 the two negative exponents so they are positive.
⋅ ⋅ ⋅
change

• 3 ∙ ∙ ∙ ∙ ∙ ∙ 1 → 3 ∙ 1
1 1

• 3 ⋅ 1 = 3
1

Copyright © Swun Math Grade 8 Unit 2 Lesson 7 P TE 150

Structured Guided Practice

(A/B Partners Practice)

Directions: Simplify the expression. Write the solution using a positive exponent.
1. (−8.1)4 ⋅ (−8.1)−4

Solution:
The bases are the same, so add the exponents.
(−8.1)0 = 1

2. 1 ⋅ 1
57 5−4

Solution:

The bases are the same, so add the exponents.
11

57+(−4) = 53

151 Copyright © Swun Math Grade 8 Unit 2 Lesson 7 P TE

Final Check for Understanding

(Teacher Checks Work)

Directions: Simplify the expression. Write the solution using a positive exponent.
1. ( )−3 ⋅ ( )3

Solution:
0 = 1

9 −3

2. 5

Solution:
9 ⋅ −3−5
1
9 ⋅ −8 = 9 −8 make the exponent positive so write the reciprocal of −8 which is 8

9 −8 = 9
8

Copyright © Swun Math Grade 8 Unit 2 Lesson 7 P TE 152

Student Practice Name: ___________________________
Date: ___________________________
Unit 2 · Lesson 7: Zero and Negative Exponents

Directions: Simplify the expression. Write the solution using a positive exponent.

1. 47⋅ 4−2 2. 2 4
48 5

Solution: Solution:
2 4
47+(−2) 5 = 2 4−5

45 48 45−8 (Students might want to write 5 – 8) 2 −1 make the exponent positive so −1 = 1
48 =

4−3 1 2
43
=

3. 1 ⋅ 1 4. 6−5 ⋅ 6−5
4 −4

Solution: Solution:
1 1
6−10= 610
4+(−4)
11 6. 7 −9
0 = 1 = 1 −5

5. −3 ⋅ 3 Solution:
7 −9
Solution: −5 = 7 −9−(−5)
−3+3
0 = 1 7 −4 make exponent positive so −4which is 1
4

7 −4 = 7
4

153 Copyright © Swun Math Grade 8 Unit 2 Lesson 7 P TE

Challenge Problems

Directions: Solve.

1. Simplify the following expression using 2. Find and explain the error.
– 3 –4
two different methods.
8 • –5 – 3 • – 3 • – 3 • – 3
6

What do you notice about the solutions? 1
– 12
Solution:
Various ways to simplify the expression.

1st Strategy

The bases are the same, so add the exponents in the

numerator and then subtract the exponents with the

denominator.

• = 8+(–5) 3
6
6
3
• 6 = 3−6

• −3

2nd Strategy

Use expanded form and cancel the common factors.
• • • • –1 • –1 –1 –1 –1
• • • • • • • • • • • • • change the five

negative exponents so they are positive.

• • • • • • • • (now cancel common factors) Solution:
• • • • • • • • • •
1 The exponent belongs to the variable only. The
• • • • • • • • • • • • • • • 3
• • • = expression should have been written the following

• The solutions are the same. way:

−3 − 3
4 4

Extension Activity

* MP1: Make sense of the problem and persevere in solving it.

* MP4: Apply mathematics in everyday life.

Using the digits 0-9, find at least 3 solutions to the problem below.

 ⋅ −
 = 1

Copyright © Swun Math Grade 8 Unit 2 Lesson 7 P TE 154

Closure

Reaching Consensus

*MP3: Do you agree or disagree with your classmate? Why or why not?

Student Presentations
*MP1: What steps in the process are you most confident about?
*MP6: Explain how you might show that your solution answers the problem.

Closure

Recap today’s lesson with one or more of the following questions:

MP1: How are negative exponents and reciprocals related?
1
MP3: Why is the reciprocal of 5−4, 54 ?

Closure

155 Copyright © Swun Math Grade 8 Unit 2 Lesson 7 P TE

Homework Name: ___________________________
Date: ___________________________
Unit 2 · Lesson 7: Zero and Negative Exponents

Objective: I will evaluate expressions with zero and negative exponents.

Vocabulary Steps:

Negative Exponent: indicates the number of 1. Write the expression in expanded form.
times to divide by the base 2. Simplify by canceling common factors.
3. Rewrite exponent as a positive exponent.
−2
OR
The negative sign says to do the inverse. 1. Subtract exponents of like bases.
Dividing is the inverse of multiplying.
2. Write answer using positive exponent.
Reciprocal: what to multiply a value by to get 1;
multiplicative inverse

A negative exponent is equivalent to the inverse of the
positive of the same number.

x−3 is the reciprocal of x3

4−5 is the reciprocal of 45

Properties of Exponents
Zero Exponent: a0 = 1
1
Negative Exponent: a−n = an

Example # 1 Example # 2

Directions: Simplify the expression. Write solution using a positive exponent.

6−8 ⋅ 68 3 5 ⋅ −2

Solution: 4

1st Strategy

The bases are the same, so add the exponents in the numerator

Solution and then subtract with the exponents that have the same base

1st Strategy in the denominator.
• 6−8+8(Keep in mind the rules for adding integers.) 3 3
• 4 = 3 ⋅ 3−4(Students might want to write 4 – 3)
• 60 = 1
• 3 ⋅ −1 = 3


2nd Strategy 2nd Strategy
Use expanded form and cancel the common factors.
● Change each number so they have positive

exponents.
1
● 6688 ⋅ 68 • 3⋅ ⋅ ⋅ ⋅ ⋅ ⋅ −1⋅ −1 Change the two negative exponents so they
● ⋅ ⋅ ⋅

68 are positive.

● 60 = 1 • 3 ∙ ∙ ∙ ∙ ∙ ∙ 1 → 3 ∙ 1
1 1
3 1 3
• 1 ⋅ =

Copyright © Swun Math Grade 8 Unit 2 Lesson 7 P TE 156

Homework

Unit 2 · Lesson 7: Zero and Negative Exponents

Directions: Simplify the expression. Write the solution using a positive exponent.

1. 6−2 ⋅ 62 2. 51
5−2

3. 86⋅8−9 4. −2 3
8−5 4 7

5. 1 ⋅ 1 6. 9−2⋅94
83 8−3 95

Explain the steps you used to solve problem number _______.

______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________

157 Copyright © Swun Math Grade 8 Unit 2 Lesson 7 P TE

Answer Key

Extension Activity 2. 51 = 51−(−2)
5−2
Answers will vary. Sample responses:
5 ⋅ −3 53

2 = 1 4. a−2−4 ⋅ b3−7 1
9 ⋅ −4 6 4

5 = 1
8 ⋅ −5

3 = 1

Homework

1. 6−2 ⋅ 62 = 6(−2)+(2)
60 = 1

3. 8−3+(5) = 82

−6 ⋅ −4 =

5. 1 = 1 = 1 6. 9−3 = 1
80 1 93

Copyright © Swun Math Grade 8 Unit 2 Lesson 7 P TE 158

Explore Exponents MPs Applied MP

Math Task * Embedded MP

Grade 8 · Unit 2 · Lesson 8 12345678

MC: 8.EE.1 * ** * *

Student Journal Pages

88-89

Objective: I will demonstrate my understanding of the properties of exponents.

Student Practice

• Explain the problem and ask students to independently solve the problem on their
recording sheet.

• While students are solving the problem and reaching consensus, look for 3 - 4
student examples to review during student presentation.

Reaching Consensus

• Students listen to other members explain how they solved the task.
• Students reach consensus on the answer, not on the solution process.

Student Presentations

• When selecting students to give presentations, choose those who will provide the
greatest teaching opportunity.

• Students who completed the Student Practice incorrectly and received help during
Reaching Consensus should present their mistake and how they corrected it.

159 Copyright © Swun Math Grade 8 Unit 2 Lesson 8 T TE

Final Check for Understanding

• Are students able to articulate metacognition?
• Are students able to articulate what they would do differently to solve the problem?
• How did you solve the problem?
• Are the students’ answers reasonable?
• How would you describe the problem in your own words?
• In what way does this problem connect to other mathematical concepts?
• What approach are you considering trying first?

Reflective Closure

• Were students able to demonstrate their knowledge of exponents correctly?
• Were students able to verbalize and use mathematical language?

Answer Key

Homework

Answers may vary. Sample response: When raising exponential expressions to powers, multiply the
exponents. Jessie is correct.

Copyright © Swun Math Grade 8 Unit 2 Lesson 8 T TE 160

Math Task Name: ___________________________
Date: ___________________________
Unit 2 · Lesson 8: Explore Exponents

Directions: Explain why each statement is correct or incorrect. If needed, explain the error.

1. Patty and Danny are discussing how to multiply expressions with the same base.

Patty says: 43 • 45 = 48, because 3 + 5 = 8 and the base stays the same.
Danny says: 43 • 45 = 415, because 3 • 5 = 15 and the base stays the same.

Does either student know how to multiply expressions with the same base? Give a
clear explanation of how to work with exponents to resolve the discussion between
Patty and Danny.

Solution:
Answers will vary. Sample response:
Patty is correct because 43 • 45 can be rewritten using expanded notation as

(4 • 4 • 4) • (4 • 4 • 4• 4 • 4), which is the same as 48.

2. Joey, Julian, and Austin are discussing how to multiply expressions with the same
exponent.

Joey says: 23 • 43 = 86, because 2 • 4 = 8 and 3 + 3 = 6.
Julian says: 23 • 43 = 89, because 2 • 4 = 8 and 3 • 3 = 9.
Austin says: 23 • 43 = 83, because 2 • 4 = 8 and the exponent doesn’t change.

Do any of the students know how to multiply expressions with the same exponent?
Give a clear explanation of how to work with exponents to resolve the discussion
between Joey, Julian, and Austin.

Solution:
Answers will vary. Sample response:
Austin is correct because 23 • 43 can be rewritten using the distributive property (2 •4)3, which is 83.

3. State rules for multiplying expressions with the same base and multiplying expressions
with the same exponent.

Solution:
Answers will vary. Sample response:
When multiplying expressions with the same base, keep the base the same and add the exponents.

x2 • x3 = (x • x) • (x • x • x) = x5

When multiplying expressions with the same exponent, multiply the bases and keep the exponent the same.
x3 • y3 = (x • y)3 = (xy)3

161 Copyright © Swun Math Grade 8 Unit 2 Lesson 8 T TE

Homework Name: ___________________________
Date: ___________________________
Unit 2 · Lesson 8: Explore Exponents

Directions: Explain why each statement is correct or incorrect. If needed, explain the error.

Sammy, Jessie, and Elizabeth are discussing how to raise exponential expressions to
powers.

Sammy says: (52)3= 55 because 2 + 3 = 5
Jessie says: (52)3= 56 because 2 • 3 = 6
Elizabeth says: (52)3= 58 because 23 = 8

Do any of the students know how to raise exponential expressions to powers? Give a clear
explanation of how to work with exponents to resolve the discussion between Sammy,
Jessie, and Elizabeth.

Copyright © Swun Math Grade 8 Unit 2 Lesson 8 T TE 162

Square Roots MPs Applied MP

Conceptual Lesson * Embedded MP

Grade 8 · Unit 2 · Lesson 9 1234 5 678

MC: 8.EE.2 *  *

Problem of the Day Student Journal Pages

90-93

Objective: I will solve equations using square roots.

Vocabulary Teacher Resources

Square Numbers (Perfect Square): the number Considerations:
after multiplying an integer by itself
Consider using a multiplication chart to support
students who need help identifying perfect squares.
Practice squaring numbers as a warm-up as needed.

Compare the concept of squaring and square rooting
as opposites that cancel each other out such as
addition/subtraction and multiplication/division. Be
sure to point out why the answer can be positive or
negative.

A calculator can be used to find an approximate value.

Square Root: the square root of a number is a Steps:
value that, when multiplied by itself, gives the
number 1. Isolate the variable.
2. Take the square root of both sides of the
Note: There is an understanding that a 2 sits outside the
radical to show it undoes a square 2√ . √ and x2 are equation.
inverse operations. 3. Identify the perfect square when possible.
4. Identify the 2 possible values for the
Rational Solution: solutions to equations that are
integers or fractions unknown.
5. Identify if the solution is rational or
2 = 9 → 2� 2 = 2√9 = 3
Irrational Solution: solutions to equations that irrational.
cannot be written as a simple fraction
Application of MPs:
2 = 11 → 2� 2 = 2√11 → = ±√11
MP5: What will be useful for solving these types of
problems?
It would be useful to ______________________.

MP6: What symbols or mathematical notations are
important in these types of problems?
The important notation/symbols in these
types of problems is ______________________.

163 Copyright © Swun Math Grade 8 Unit 2 Lesson 9 C TE

Input/Model

(Teacher Presents)

Directions: Solve. Determine if the solution is a rational or irrational number.
1. 2 = 25

Solution:
Step 1: 2 = 25
Step 2: √ 2 = √25
Step 3: √ ⋅ = √5 ⋅ 5 or √ ⋅ = �(−5) ⋅ (−5)
Step 4  Solution 1: = 5

Solution 2: = −5
Step 5: Solution is = ±5 and it is a rational number.

Talking points:
• Begin by identifying the unknown.
• Then, understand that it is a squared problem indicated by the exponent.
• To undo this power, take the square root both of sides of the equation to keep the equation equivalent.
• Since a square number and the square root of the number are inverse, they “undo” or cancel each other out.
• Solve using the steps.

2. 2 = 11

Solution: Grade 8 Unit 2 Lesson 9 C TE 164
Step 1: 2 = 11
Step 2: √ 2 = √11
Step 3:√ ⋅ = √11 11 is not a perfect square
Step 4: = ±√11
Step 5: The solution is = ±√11 ; an irrational number.

Copyright © Swun Math

Structured Guided Practice

(A/B Partners Practice)
Directions: Solve. Determine if the solution is a rational or irrational number.

1. 2 = 64

Solution:
= ±8 ; rational

2. 2 = 10

Solution:
= ±√10 ; irrational

165 Copyright © Swun Math Grade 8 Unit 2 Lesson 9 C TE

Final Check for Understanding

(Teacher Checks Work)

Directions: Solve. Determine if the solution is a rational or irrational number.
1. 2 = 121

Solution:
= ±11 ; rational

2. 2 = 13

Solution:
= ±√13 ; irrational

Closure

Recap today’s lesson with one or more of the following questions:
MP5: What will be useful for solving these types of problems?
MP6: What symbols or mathematical notations are important in these types of

problems?

Copyright © Swun Math Grade 8 Unit 2 Lesson 9 C TE 166

Homework Name: ___________________________
Date: ___________________________
Unit 2 · Lesson 9: Square Roots

Objective: I will solve equations involving squared roots.

Vocabulary Steps:

Square Numbers (Perfect Square): the number 1. Isolate the variable.
after multiplying an integer by itself 2. Take the square root of both sides of the

equation.
3. Identify the perfect square when possible.
4. Identify the 2 possible values for the

unknown.
5. Identify if the solution is rational or

irrational.

Square Root: the square root of a number is a
value that, when multiplied by itself, gives the
number

Note: There is an understanding that a 2 sits outside the
radical to show it undoes a square 2√ . √ and x2 are
inverse operations.

Rational Solution: solutions to equations that are
integers or fractions

2 = 9 → 2� 2 = 2√9 =

Irrational Solution: solutions to equations that
cannot be written as a simple fraction

2 = 11 → 2� 2 = 2√11 → = ±√11

167 Copyright © Swun Math Grade 8 Unit 2 Lesson 9 C TE

Homework

Unit 2 · Lesson 9: Square Roots

Example # 1 Example # 2

Directions: Solve for the equation and determine if the solution is a rational or irrational number.

2 = 25 2 = 11

Solution: Solution:
Step 1: 2 = 25 Step 1: 2 = 11
Step 2: √ 2 = √11
Step 2: √ 2 = √25 Step 3:√ ⋅ = √11 and 11 is not a perfect square
Step 4: = ±√11
Step 3: √ ⋅ = √5 ⋅ 5 or √ ⋅ = �(−5) ⋅ (−5) Step 5: Solution is = ±√11; an irrational number
Step 4  Solution 1: = 5

Solution 2: = −5

Step 5: Solution is = ±5 ; a rational number.

Copyright © Swun Math Grade 8 Unit 2 Lesson 9 C TE 168

Homework

Unit 2 · Lesson 9: Square Roots

Directions: Solve. Determine if the solution is a rational or irrational number.

1. 2 = 45 2. ℎ2 = 16

3. 2 = 33 4. 2 = 36

5. 2 = 12 6. 2 = 81

169 Copyright © Swun Math Grade 8 Unit 2 Lesson 9 C TE

Answer Key 2.
ℎ = ±4; rational
Homework
4.
1. = ±6; rational
= ±√45 = ±3√5; irrational
6.
3. = ±9; rational
= ±√33; irrational

5.
= ±√12 = ±2√3; irrational

Copyright © Swun Math Grade 8 Unit 2 Lesson 9 C TE 170

Cube Roots MPs Applied MP

Conceptual Lesson * Embedded MP
Grade 8 · Unit 2 · Lesson 10
123 456 7 8
MC: 8.EE.2
*  *

Problem of the Day Student Journal Pages

Objective: I will solve equations using cube roots. 94-97

Vocabulary Teacher Resources

Cube: the result of multiplying a base by itself Considerations:
three times
Share with students that taking the cube root is
2.1³ = 9.261 commonly used in equations involving volume.
Compare the concept of cubing and cube rooting as
Perfect Cube: the result after multiplying an opposites that cancel each other out.
integer three times by itself; also called a cubed
number When a number is irrational and cannot be simplified,
try to find another number close to the number
8 is a perfect cubed number described in the problem statement to approximate
because 2 • 2 • 2 or 23 = 8 the cube root.

Cube Root: is a special value that, when used in 3√30 is between the 3√27 and 3√64
multiplication three times, gives the original
number so the cube root of 30 is between 3 and 4

2 cubed is 8, so the cube root of 8 is 2 Discuss that when taking the cube root, a negative in
the cube root does not mean that the final answer is
cube irrational, because multiplying an odd amount of
negative numbers will result in a negative answer.
28

cube root Steps:
1. Isolate the variable.
√ and 2are inverse operations 2. Take the cube root of both sides of the
3√ and 3 are inverse operations equation.
3. Identify the perfect cube root when possible.
Increase Exponentially: 4. Identify the value of the unknown.
5. Decide if the solution is rational or irrational.
Notice on the table Number Squared Cubed
how the size increases 5 25 125 Application of MPs:
as the power increases. 6 36 216 MP2: How are squared values related to cubed
7 49 343
64 512 values?
This is called 8 81 729 They are related because _________________.
100 1000
increasing exponentially. 9 121 1331 MP5: How does this problem compare to square
144 1728 roots?
10 169 2197 This problem is similar/different to square
196 2744 roots by ___________________.
11 225 3375

12

Negative Numbers: 13

14

53 = 5 ⋅ 5 ⋅ 5 = 125 15

(−5)3 = −5 ⋅ −5 ⋅ −5 = −125

 Multiplying an odd amount of negative numbers will
equal a negative answer.

171 Copyright © Swun Math Grade 8 Unit 2 Lesson 10 C TE

Input/Model

(Teacher Presents)

Directions: Solve. Determine if the solution is rational or irrational.
1. 3 = 8

Solution:
Step 1: 3 = 8
Step 2: 3√ 3 = 3√8
Step 3: 3√ ⋅ ⋅ = √3 2 ⋅ 2 ⋅ 2
Step 4: = 2
Step 5: The solution is = 2; a rational number.

Talking points:
• Begin by identifying the unknown.
• Notice the exponent indicates it is a cubic problem.
• To undo this power, take the cubic root of both sides of the equation.
• Since a cubed number and the cubed root of a number are inverse operations, they “undo” or cancel each

other out.
• Solve using steps.

2. 3 = 13

Solution: Grade 8 Unit 2 Lesson 10 C TE 172
Step 1: 3 = 13
Step 2: 3√ 3 = 3√13
Step 3: 3√ ⋅ ⋅ = 3√13 (no perfect cube can be found)
Step 4: = 3√13
Step 5: The solution is 3√13 ; irrational.

Copyright © Swun Math

Structured Guided Practice

(A/B Partners Practice)
Directions: Solve. Determine if the solution is rational or irrational.

1. 3 = 216

Solution:
p=6; rational number

2. 3 = 25

Solution:
= 3√25; irrational

173 Copyright © Swun Math Grade 8 Unit 2 Lesson 10 C TE

Final Check for Understanding

(Teacher Checks Work)
Directions: Solve. Determine if the solution is rational or irrational.

1. 3 = 27

Solution:
t=3; rational number

2. 3 = 11

Solution:
= 3√11; irrational

Closure

Recap today’s lesson with one or more of the following questions:
MP2: How are squared values related to cubed values?
MP5: How does this problem compare to square roots?

Copyright © Swun Math Grade 8 Unit 2 Lesson 10 C TE 174

Homework Name: ___________________________
Date: ___________________________
Unit 2 · Lesson 10: Cube Roots

Objective: I will solve equations using cube roots. Steps
Vocabulary

Cube: the result of multiplying a base by itself 1. Isolate the variable.
three times
2. Take the cube root of both sides of the
2.1³ = 9.261 equation.

Perfect Cube: the result after multiplying an integer 3. Identify the perfect cube root when
three times by itself; also called a cubed number possible.

8 is a perfect cubed number 4. Identify the value of the unknown.
because 2 • 2 • 2 or 23 = 8
5. Decide if the solution is rational or
Cube Root: is a special value that, when used in irrational.
multiplication three times, gives the original number
Remember:
2 cubed is 8, so the cube root of 8 is 2  Multiplying an odd number of negative
numbers will result in a negative product.
cube

28

cube root

√ and 2are inverse operations
3√ and 3 are inverse operations

Increase Exponentially: Number Squared Cubed
Notice on the table 5 25 125
how the size increases 6 36 216
as the power increases. 7 49 343
This is called 8 64 729
increasing exponentially. 9 81 729
10 100 1000
Negative Numbers: 11 121 1331
12 144 1728
53 = 5 ⋅ 5 ⋅ 5 = 125 13 169 2197
−53 = −5 ⋅ −5 ⋅ −5 = −125 14 196 2744
15 225 3375

Example # 1 Example # 2

Directions: Solve. Determine if the solution is rational or irrational.

3 = 8 3 = 13

Solution: Solution:
Step 1: 3 = 8 Step 1: 3 = 13

Step 2: 3√ 3 = 3√8 Step 2: 3√ 3 = 3√13

Step 3: 3√ ⋅ ⋅ = √3 2 ⋅ 2 ⋅ 2 Step 3: 3√ ⋅ ⋅ = 3√13 (no perfect cube can be found)

Step 4: = 2 Step 4: = 3√13
Step 5: The solution is = 2; a rational number. Step 5: The solution is 3√13 ; irrational.

175 Copyright © Swun Math Grade 8 Unit 2 Lesson 10 C TE

Homework

Unit 2 · Lesson 10: Cube Roots

Directions: Solve. Determine if the solution is rational or irrational.

1. 3 = 49 2. 512 = 3

3. 3 = 64 4. ℎ3 = 15
5. 3 = 100 6. 3 = 1

Copyright © Swun Math Grade 8 Unit 2 Lesson 10 C TE 176

Answer Key 2. 512 = 3
3√512 = 3� 3
Homework 3√8 ⋅ 8 ⋅ 8 = 3√ ⋅ ⋅
8 = ; rational
1. 3 = 49
3� 3 = 3√49 4. ℎ3 = 15
3√ ⋅ ⋅ = 3√49 3�ℎ3 = 3√15
= 3√49; irrational 3√ℎ ⋅ ℎ ⋅ ℎ = 3√15
ℎ = 3√15; irrational
3. 3 = 64
3� 3 = 3√64 6. 3 = 1
3� ⋅ ⋅ = 3√4 ⋅ 4 ⋅ 4 3� 3 = 3√1
= 4; rational 3� ⋅ ⋅ = 3√1
= 1; rational
5. 3 = 100
3� 3 = 3√100
3√ ⋅ ⋅ = 3√100
= 3√100; irrational

177 Copyright © Swun Math Grade 8 Unit 2 Lesson 10 C TE

Notes

Copyright © Swun Math Grade 8 Unit 2 Lesson 10 C TE 178

Square & Cube Roots MPs Applied MP

Procedural Lesson * Embedded MP

Grade 8 · Unit 2 · Lesson 11 12345678

MC: 8.EE.2 *  * * * *

Problem of the Day Student Journal Pages

98-103

Objective: I will solve equations using square and cube roots.

Vocabulary Teacher Resources

Square Root: the square root of a number is a Considerations:
value that, when multiplied by itself, gives the
number Remind students that when a number is irrational
and cannot be simplified, find another number close
to the number described in the problem to
approximate the cube root.

Share with students that when taking the cube root,
a negative in the cube root does not mean the final
answer is irrational. This is because multiplying an
odd amount of negative numbers will result in a
negative answer.

Cube Root: is a special value that, when used Steps:
in multiplication three times, gives the 1. Isolate the variable.
original number 2. Take the square root of both sides of the
equation.
2 cubed is 8, so the cube root of 8 is 2 3. Identify the perfect square when possible.
4. Identify the 2 possible values for the unknown.
cube 5. Identify if the solution is rational or irrational.
OR
2 8 1. Identify the perfect cube root when possible.
cube root 2. Identify the value of the unknown.
3. Identify if the solution is rational or irrational.
Inverse operations:
� and 2are inverse operations Application of MPs:
3� and 3 are inverse operations MP2: How did you decide whether the solution is

Rational Solution: solutions to equations that rational?
are integers or fractions I decided the solution is rational because ______.
MP6: Why do equations with a squared variable get ±
2 = 9 → 2� 2 = 2√9 = 3 signs and cubed variables do not?
The squared variables get a ± because _______
Irrational Solution: solutions to equations that but a cubed variable does not because _________.
cannot be written as a simple fraction

2 = 11 → 2� 2 = 2√11 → = ±√11

179 Copyright © Swun Math Grade 8 Unit 2 Lesson 11 P TE

Input/Model

(Teacher Presents)
Directions: Solve. Determine if the solution is a rational or irrational number.

1. 2 = 36

Solution:
= ±6, a rational number.

2. ℎ3 = 64

Solution: Grade 8 Unit 2 Lesson 11 P TE 180
ℎ = 4, a rational number.

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Structured Guided Practice

(A/B Partners Practice)

Directions: Solve. Determine if the solution is a rational or irrational number.
1. 2 = 54

Solution:
= ±√54, an irrational number

2. 3 = 26

Solution:
= 3√26 , an irrational number.

181 Copyright © Swun Math Grade 8 Unit 2 Lesson 11 P TE

Final Check for Understanding

(Teacher Checks Work)

Directions: Solve. Determine if the solution is a rational or irrational number.
1. 2 = 97

Solution:
= ±√97, an irrational number.

2. 3= 125

Solution: Grade 8 Unit 2 Lesson 11 P TE 182
= 5, a rational number.

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Student Practice Name: ___________________________
Date: ___________________________
Unit 2 · Lesson 11: Square & Cube Roots

Directions: Solve. Identify the rational solutions.

1. 2 = 15 2. 3 = 1000

Solution: Solution:
= ±√15 = 10; rational

3. 2 = 49 4. 2 = 77

Solution: Solution:
= ±7; rational = ±√77

5. 3 = 64 6. 3 = 30

Solution: Solution:
= 4; rational = 3√30

183 Copyright © Swun Math Grade 8 Unit 2 Lesson 11 P TE

Challenge Problems

Directions: Solve. Determine if the solution is a rational or irrational number.

1. Why is ─2 not a solution to z³=8 ? 2. Find a solution for each equation using
the numbers below. Each number may
only be used once.

3, 4, 8, 9, 27, 64

�= ___
3�= ___

Solution: Solution:
Answers will vary. Sample response: ─2 is not a Answers will vary. Sample responses: √9 =3; 3√64= 4

solution because (– 2)(– 2)(– 2) = – 8 not +8.

Extension Activity

* MP1: Make sense of the problem and persevere in solving it.
* MP4: Apply mathematics in everyday life.

Solve a multi-step equation involving cubes.
2x³ = 54

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Closure

Reaching Consensus
*MP3: Do you agree or disagree with your classmate? Why or why not?

Student Presentations
*MP1: What steps in the process are you most confident about?
*MP6: Explain how you might show that your solution answers the problem.

Closure

Recap today’s lesson with one or more of the following questions:
MP2: How did you decide if the solution is rational?
MP6: Why do equations with a squared variable get ± signs and cubed variables do

not?

185 Copyright © Swun Math Grade 8 Unit 2 Lesson 11 P TE

Homework Name: ___________________________
Date: ___________________________
Unit 2 · Lesson 11: Square & Cube Roots

Objective: I will solve equations using square roots and cubic roots.

Vocabulary Steps:

Square Root: the square root 1. Isolate the variable.
of a number is a value that, 2. Take the square root of both sides of the
when multiplied by itself,
gives the number equation.
3. Identify the perfect square when possible.
Cube Root: is a special value 4. Identify the 2 possible values for the
that, when used in
multiplication three times, unknown.
gives the original number 5. Identify if the solution is rational or

2 cubed is 8, so the cube root of 8 is 2 irrational.
OR
cube
4. Identify the perfect cube root when possible.
2 8 5. Identify the value of the unknown.

cube root Identify if the solution is rational or
irrational.
Inverse Operations:
Number Squared Cubed
� and 2are inverse operations 5 25 125
3� and 3 are inverse operations 6 36 216
7 49 343
Rational Solution: solutions to equations that are 8 64 512
9 81 729
integers or fractions 10 100 1000
2 = 9 → 2� 2 = 2√9 = 3 11 121 1331
12 144 1728
Irrational Solution: solutions to equations that 13 169 2197
cannot be written as a simple fraction 14 196 2744
15 225 3375
2 = 11 → 2� 2 = 2√11 → = ±√11

Examples ℎ3 = 64
Directions: Solve. Determine if the solution is a rational or irrational number.

2 = 36

Solution: Solution:
Step 1: 2 = 36
Step 1: ℎ3 = 64
Step 2: � 2 = √36 Step 2: 3√ℎ3 = 3√64
Step 3: � ⋅ = √6 ⋅ 6 Step 3: 3√ℎ ⋅ ℎ ⋅ ℎ = 3√4 ⋅ 4 ⋅ 4
Step 4  Solution 1: = 6 Step 4: ℎ = 4
Step 5: The solution is ℎ = 4, a rational number.
Solution 2: = −6
Step 5: The solution is = ±6, a rational number.

Copyright © Swun Math Grade 8 Unit 2 Lesson 11 P TE 186

Homework

Unit 2 · Lesson 11: Square & Cube Roots

Directions: Solve. Determine if the solution is a rational or irrational number.

1. 2 = 36 2. 3 = 343

3. 2 = 28 4. 3 = 729

187 Copyright © Swun Math Grade 8 Unit 2 Lesson 11 P TE

Homework

Unit 2 · Lesson 11: Square & Cube Roots

5. 2 = 40 6. ℎ2 = 110

Explain the steps you used to solve problem number _______.

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Copyright © Swun Math Grade 8 Unit 2 Lesson 11 P TE 188

Answer Key

Extension Activity

• 2 3 = 54
2 3
• 2 = 54
2
• 3 = 27

• 3√ 3 = 3√27

• 3√ ⋅ ⋅ = 3√3 ⋅ 3 ⋅ 3
• = 3

Homework 2. 3 = 343
3� 3 = 3√343
1. 2 = 36 3√ ⋅ ⋅ = 3√7 ⋅ 7 ⋅ 7
� 2 = √36 = 7
√ ⋅ = √6 ⋅ 6 The solution is rational.
= ±6
The solution is rational. 4. 3 = 729
3� 3 = 3√729
3. 2 = 28 3� ⋅ ⋅ = 3√9 ⋅ 9 ⋅ 9
� 2 = √28 = 9
√ ⋅ = √28; notice 28 is not a perfect square The solution is rational.
= ±√28 or ±2√7
The solution is irrational. 6. ℎ2 = 110
�ℎ2 = √110
5. 2 = 40 √ℎ ⋅ ℎ = √110 ; notice 110 is not a perfect square
� 2 = √40 ℎ = ±√110
� ⋅ = √40; notice 40 is not a perfect square The solution is irrational.
= ±√40 or ±2√10
The solution is irrational.

189 Copyright © Swun Math Grade 8 Unit 2 Lesson 11 P TE

Notes

Copyright © Swun Math Grade 8 Unit 2 Lesson 11 P TE 190

Solve Square & Cubic Equations MPs Applied MP 8

Math Task * Embedded MP *
Grade 8 · Unit 2 · Lesson 12
1234567
MC: 8.EE.2
* ** *
Objective: I will solve equations involving square and cubic roots.
Student Journal Pages
Student Practice
104-105

• Explain the problem and ask students to independently solve the problem on their
recording sheet.

independently.
• While students are working, look for 3 - 4 student examples to review during

presentations.

Reaching Consensus

• Students listen to other members explain how they solved the task.
• Students reach consensus on the answer, not on the solution process.

Student Presentations

• When selecting students to give presentations, choose those who will provide the
greatest teaching opportunity.

• Students who completed the Student Practice incorrectly and received help during
Reaching Consensus should present their mistake and how they corrected it.

191 Copyright © Swun Math Grade 8 Unit 2 Lesson 12 T TE

Final Check for Understanding

• Are students able to articulate metacognition?
• Are students able to articulate what they would do differently to solve the problem?
• How did you solve the problem?
• Are the students’ answers reasonable?
• How would you describe the problem in your own words?
• In what way does this problem connect to other mathematical concepts?
• What approach are you considering trying first?

Reflective Closure

• Were students able to apply what they learned about square roots and cube roots
when solving equations?

• Were students able to verbalize and use mathematical language?

Answer Key

Homework 2. y = 3
4. Yes, ─6 is a solution because (─6)²=(─6)(─6)=36
1. The equation 3 = 566 − 54 has a greater value
for ; = 8
For the equation 2 = 928; = ±7

3. z=64

Copyright © Swun Math Grade 8 Unit 2 Lesson 12 T TE 192

Math Task Name: ___________________________
Date: ___________________________
Unit 2 · Lesson 12: Solve Square & Cubic Equations

Directions: Solve. 2. 4x²=1024

1. Circle the equation in which has the
greater value.
3 = 160 − 35
2 = 25 − (−11)

Solution:
• 3 = 160 − 35

3 = 125

3� 3 = 3√125
3√ ∙ ∙ = 3√5 ∙ 5 ∙ 5
= 5

• 2 = 25 − (−11) Solution:
2 = 36 • 4 2 = 1024
� 2 = √36 4 2
√ ∙ = √6 ∙ 6 • 4 = 1024
= ±6 4
• 2 = 256
• In the second equation, has a greater value when
• √ 2 = √256
the it equals positive 6.
• The value of in first equation is greater only • √ ∙ = √16 ⋅ 16
• = ±16
when the value of in the second equation is –6.
4. Why is ─5 not a solution to x³=125?
3. √ 2 = 36

Solution: Solution:
x³=125 has a solution of 5. When ─5 is cubed, the result
� 2 = 36 is ─125, not 125.
= ±36
Grade 8 Unit 2 Lesson 12 T TE
193 Copyright © Swun Math

Homework

Unit 2 · Lesson 12: Solve Square & Cubic Equations

Directions: Solve. 2. y³ ─ 8 = 19

1. Circle the equation in which has the
greater value.

2 = 98
2

3 = 566 − 54

3. 3√ 3 = 64 4. Is ─6 a solution to x² = 36? Explain.

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Unit 2 ∙ Constructed Response Name: ____________________________
Exponents
Date: ______________________________
Task 1: Combining Exponents
Student Journal Pages
Directions: Read and solve each problem.
106-111

1. Deana incorrectly solved this problem on a test. This is her work:

a. Solve the problem correctly. Show your work.

b. Identify Deana’s error(s). Explain a strategy Deana could use to check her work.

195 Copyright © Swun Math Grade 8 Unit 2 Constructed Response TE