G8

Homework

Unit 1 · Lesson 5: Convert Repeating Decimals to

Fractions

Directions: Convert the decimals into fractions.

1. 0.777… 2. 0.0833…

3. 0.0909 4. 1.1666…

Copyright © Swun Math Grade 8 Unit 1 Lesson 5 P TE 46

Homework

Unit 1 · Lesson 5: Convert Repeating Decimals to

Fractions

5. 0.343434… 6. 0.0151515…

Explain the steps you used to solve problem number _______.

______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________

47 Copyright © Swun Math Grade 8 Unit 1 Lesson 5 P TE

Answer Key

Extension Activity

Answers may vary. Sample response: Estimating the value of the decimal would give us an easier number to
deal with when calculating formulas. On the other hand, at times it is very important to be exact, for example a
non-exact calculation may result in a vaccination that is not as effective. In this case, a fractional
representation of a repeating decimal would be more effective.

Homework 2. Two equations to represent 0.0833…:
• 100x = 8.333…
1. Two equations to represent 0.777…: • 1000x = 83.33…
• x = 0.777…
• 10x = 7.77… Solve:
1000x – 100x = 83.33… - 8.333…
Solve: 900x = 75
10x – x = 7.777… - 0.777… x = 75 → 75 ÷ 75 → 1
9x = 7
x= 7 900 900 ÷ 75 12

9 Answer: 1

Answer: 7 12

9 4. 7

3. Two equations to represent 0.0909…: 6
• x = 0.0909…
• 100x = 9.09… Two equations to represent 1.1666…:
• 10x = 11.666…
Solve: • 100x = 116.66…
100x – 1x = 9.09… - 0.0909…
99x = 9 Solve:
x = 9 → 9÷9 → 1 100x – 10x = 116.66… - 11.666…
90x = 105
99 99 ÷ 9 11 x = 105 → 105 ÷ 15 → 7

Answer: 1 90 90 ÷ 15 6

11 Answer: 7 or 1 1

5. Two equations to represent 0.343434…: 66
• x = 0.343434…
• 100x = 34.3434… 6. Two equations to represent 0.0151515…:
• 10x = 0.151515…
Solve: • 1000x = 15.1515…
100x – 1x = 34.3434… - 0.343434…
99x = 34 Solve:
1000x – 10x = 15.1515… - 0.151515…
x = 34 990x = 15
x = 15 = 15÷15 = 1
99
990 990÷15 66
Answer: 34
Answer: 1
99
66

Copyright © Swun Math Grade 8 Unit 1 Lesson 5 P TE 48

Estimate Irrational Numbers MPs Applied MP 8
* Embedded MP
Conceptual Lesson 123 *
4567
Grade 8 · Unit 1 · Lesson 6 *

MC: 8.NS.2

Problem of the Day Student Journal Pages

26-29

Objective: I will use estimation to approximate the value of an irrational square root.

Vocabulary Teacher Resources

Square Numbers (Perfect Square): the result after Considerations:
multiplying an integer by itself
Use an area model to determine if the number is a
perfect square. If not, use the number line to
approximate and use the list of perfect squares as a
reference. Calculators may be used to verify the
reasonableness of their approximations.

√29

√16 √25 √36
4 56

Irrational Numbers: the set of all numbers that Steps:
cannot be written as a ratio; it cannot be written 1. Determine if the number is a perfect square.
as a simple fraction because the decimal is non- 2. If not, find the perfect square immediately
terminating and non-repeating above and below the number.
3. The answer will be between the square roots
0.101101110… , �23 , π, √10, √1.6, −√123 of these two numbers.
4. Estimate the decimal based on how close your
number is to each of the squares.

Square Root: the square root of a number is a Application of MPs:
value that, when multiplied by itself, gives the
number MP5: What approach are you considering trying
first?
√16 = 4
I am going to first try _____________________.
MP6: How are you estimating the quantities?

I estimate the quantities by first,
_________________________________________.

49 Copyright © Swun Math Grade 8 Unit 1 Lesson 6 C TE

Input/Model

(Teacher Presents)

Directions: Determine if the expression is a perfect square. If not, determine the two perfect
squares that the square root lies between.

1. √4

Solution:
• A square is a plane figure with four equal sides and angles. To prove that 4 is a perfect square, build a square that has

an area of 4.

• When creating a perfect square with an area of 4, the length needs to be 2 and the width needs to be 2.

• The integer multiplied by itself to have a product of four is 2.

• √4 is the same as √2 ⋅ 2. This indicates that the square root of 4 is 2.

• 22 = 2 ⋅ 2 = 4, therefore
• √4 = √2 ⋅ 2 = 2

Answer: Since 2 is a whole number, √4 is a rational number.

2. √20

Solution:
• 20 is not a perfect square and this can be proved since a square cannot be built with an area of 20

4

5
• Since √20 is not a perfect square, to best estimate its value, we need to identify what two perfect squares it lies in

between. squares

16 20 25

-1 0 1 2 3 4 5 6 7 8 9 10 11 12
roots

• 20 lies between 16 and 25, which are both perfect squares.

• Since √16 = 4 and √25 = 5, √20 must lie between 4 and 5.
Answer: Since √20 is an irrational number, √20 lies between 4 and 5.

Copyright © Swun Math Grade 8 Unit 1 Lesson 6 C TE 50

Structured Guided Practice

(A/B Partners Practice)

Directions: Determine if the expression is a perfect square. If not, determine the two perfect
squares that the square root lies between.

1. √9

Solution:
32 = 3 ⋅ 3 = 9, therefore
√9 = √3 ⋅ 3 = 3

When creating a perfect square with an area of 9, the measurement of the length needs to be 3 and
the measurement of the width needs to be 3.

2. √5

Solution:
5 is not a perfect square and this can be proved since a square cannot be built with an area of 5.

45 squares
9

-1 0 1 2 3 4 5 6 7 8 9 10 11 12
roots

Since √5 is not a perfect square, the answer √5 lies between 2 and 3.

51 Copyright © Swun Math Grade 8 Unit 1 Lesson 6 C TE

Final Check for Understanding

(Teacher Checks Work)

Directions: Determine if the expression is a perfect square. If not, determine the two perfect
squares that the square root lies between.

1. √16

Solution:
42 = 4 ⋅ 4 = 16, therefore
√16 = √4 ⋅ 4 = 4

2. √29

Solution:
29 is not a perfect square.

squares 36
25 29

-1 0 1 2 3 4 5 6 7 8 9 10 11 12
roots

29 lies between 25 and 36, which are both perfect squares.
Since √29 is not a perfect square, √29 lies between 5 and 6.

Closure

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

MP5: What approach are you considering trying first?
MP6: How are you estimating the quantities?

Copyright © Swun Math Grade 8 Unit 1 Lesson 6 C TE 52

Homework Name: ___________________________
Date: ___________________________
Unit 1 · Lesson 6: Estimate Irrational Numbers

Objective: I will use estimation to approximate the value of an irrational square root.

Vocabulary Steps:

Square Numbers (Perfect Square): the result after 1. Determine if the number is a perfect square.
multiplying an integer by itself
2. If not, find the perfect square immediately
above and below the number.

3. The answer will be between the square roots
of these two numbers.

4. Estimate the decimal based on how close your
number is to each of the squares

Irrational Numbers: the set of all numbers that
cannot be written as a ratio of two integers; it
cannot be written as a simple fraction because
the decimal is non-terminating and non-
repeating

0.101101110… , �32 , π, √10, √1.6, −√123

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

53 Copyright © Swun Math Grade 8 Unit 1 Lesson 6 C TE

Homework

Unit 1 · Lesson 6: Estimate Irrational Numbers

Example # 1

Directions: Determine if the expression is a perfect square. If not, determine the two perfect squares that the
square root lies between

√4

Solution:
• When creating a perfect square with an area of 4, the length needs to be 2 and the width needs to be 2.

• A square is a plane figure with four equal sides and angles. To prove that 4 is a perfect square, build a square that has an
area of 4.

• The integer multiplied by itself to have a product of four is 2.

• √4 is the same as √2 ⋅ 2. This indicates that the square root of 4 is 2.

• 22 = 2 ⋅ 2 = 4, therefore
• √4 = √2 ⋅ 2 = 2

Example # 2

√20

Solution:
• 20 is not a perfect square and this can be proved since a square cannot be built with an area of 20

4

5

• Since √20 is not a perfect square, find what two perfect squares it lies in between.

squares

16 20 25

-1 0 1 2 3 4 5 6 7 8 9 10 11 12
roots

• 20 lies between 16 and 25, which are both perfect squares.

• Since √16 = 4 and √25 = 5, √20 must lie between 4 and 5.
• Answer: Since √20 is not a perfect square, and √20 lies between 4 and 5.

Copyright © Swun Math Grade 8 Unit 1 Lesson 6 C TE 54

Homework

Unit 1 · Lesson 6: Estimate Irrational Numbers

Directions: Determine if the expression is a perfect square. If not, determine the two perfect
squares that the square root lies between.

1. √45 2. √36

3. √49 4. √17

5. √34 6. √100

55 Copyright © Swun Math Grade 8 Unit 1 Lesson 6 C TE

Answer Key

Homework 2. Perfect square; 6
4. Not a perfect square; between 4 and 5
1. Not a perfect square; between 6 and 7 6. Perfect square; 10
3. Perfect square; 7
5. Not a perfect square; between 5 and 6

Copyright © Swun Math Grade 8 Unit 1 Lesson 6 C TE 56

Estimate Irrational Numbers MPs Applied MP

Procedural Lesson * Embedded MP

Grade 8 · Unit 1 · Lesson 7 12345 6 78

MC: 8.NS.2 * * * *  *

Problem of the Day Student Journal Pages

30-35

Objective: I will estimate the value of an irrational square root.

Vocabulary Teacher Resources

Square Numbers (Perfect Square): the result after Considerations:
multiplying an integer by itself
Students may need to look at a list of perfect
squares as a reference. Using a number line will
help students visualize the decimal answer quickly.
Calculators may be used to verify the
reasonableness of their approximations.

Irrational Numbers: the set of all numbers that Steps:
cannot be written as a ratio of two integers; it 1. Find the perfect square immediately
cannot be written as a simple fraction because the above and below your number.
decimal is non-terminating and non-repeating 2. The answer will be between the square
roots of these two numbers.
0.101101110… , �32 , π, √10, √1.6, −√123 3. Estimate the decimal based on how
close your number is to each of the
squares.

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

MP6: How are you estimating the quantities?
I will estimate the quantities by first
___________________.

MP7: How can you use today’s pattern to estimate
a more accurate estimation?
I can use today’s pattern to estimate a more
accurate estimation by
______________________.

57 Copyright © Swun Math Grade 8 Unit 1 Lesson 7 P TE

Input/Model

(Teacher Presents)

Directions: Estimate the decimal value. Use a calculator to check.
1. −√75

Solution:
• −√75 can be thought of as (-1) • −√75
• −√75 can be approximated in the same manner and multiplied by –1
• When considering√75 on a number line, the perfect square less than 75 is 64
• √64 = 8

• The perfect square greater than 75 is 81.

• √81 = 9
• Now make the statement √64 < √75 < √81.

• Simplify 8 <√75 < 9

• Recall, −√75= (–1) •√75. Therefore, −√75 is between –8 and –9.

• Calculator: −√75 ≈ −8.7

2. √150
2

Solution:
• First, focus on √150

• On a number line, the perfect square less than 150 is 144.

• √144 = 12

• The perfect square greater than 150 is 169.

• √169 = 13
• Now, make the statement √144< √150 <√169.

• Simplify 12 <√150 < 13

• So, √150 is between 12 and 13.

• Looking at the original ratio, √150 I can now make the statement that, √150 is between 12 and 13 or 6 <
2 2 2 2

√150 < 6.5

• Calculator: √150 ≈ 6.1
2

Copyright © Swun Math Grade 8 Unit 1 Lesson 7 P TE 58

Structured Guided Practice

(A/B Partners Practice)

Directions: Estimate the decimal value. Use a calculator to check.
1. √40

Solution:
√40 is between 6 and 7 or 6 < √40 < 7.

Calculator: √40 ≈ 6.3

2. −√28
2

Solution:

Recall −√28 = (−1)2⋅√28, therefore −√28 will fall between -2 and -3 or −3 < −√28 < −2
2 2 2

Calculator: − √28 ≈ −2.6
2

59 Copyright © Swun Math Grade 8 Unit 1 Lesson 7 P TE

Final Check for Understanding

(Teacher Checks Work)

Directions: Estimate the decimal value. Use a calculator to check.
1. √78

Solution:
√78 is between 8 and 9.
Calculator: √78 ≈ 8.8

2. −√37

Solution:
√37 is between 6 and 7.
Recall, −√37 = (−1) ⋅ √37. Therefore, −√37 is between –6 and –7.
Calculator: −√37 ≈ −6.1

Copyright © Swun Math Grade 8 Unit 1 Lesson 7 P TE 60

Student Practice Name: ___________________________
Date: ___________________________
Unit 1 · Lesson 7: Estimate Irrational Numbers

Directions: Estimate the decimal value. Use a calculator to check.

1. √120 2. √52
2

Solution: Solution:
between 10 and 11.
Calculator: √120 ≈ 10.95 between 3.5 and 4.

3. −√19 Calculator: √52 ≈ 3.6
2

4. −√45

Solution: Solution:

between –4 and –5 between −7 and −6 or −7 < −√45 < −6.
Calculator: −√19 ≈ −4.4 Calculator: −√45 ≅ −6.71

5. −√130 6. √110
2 2

Solution: Solution:

between -5.5 and -6 or −6 < −√130 < −5.5 between 5 and 6.
2
−1√130 Calculator: √110 ≈ 5.2
Calculator: 2 ≈ −5.7 2

61 Copyright © Swun Math Grade 8 Unit 1 Lesson 7 P TE

Challenge Problems

Directions: Solve. 2. Find an irrational number whose decimal
approximation is between 15 and 16. What
1. Is 2√4 the same as 4√2? Why or why process did you use?
not?

Solution: Solution:
No, they are not equal. 2√4 simplifies to 2(2) = 4. We The square root of a number between 225 and 256 is
know that √2 is greater than √1 because 2 is greater irrational. The approximate square root will fall
than 1, thus 4√2 will be 4 times some number greater between 15 and 16. I multiplied 15 x 15 = 225 and 16 x 16
than 1, which will not equal 4. Thus, the two are not =256.

equal.

Extension Activity

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

Directions: Using the digits 0-9, fill in the boxes so that the square root falls either on or in
between the indicated number(s) on the number line.

�  � 

456 78 9
� 
�  � 

Copyright © Swun Math Grade 8 Unit 1 Lesson 7 P TE 62

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:
MP6: How are you estimating the quantities?
MP7: How can you use today’s pattern to estimate a more accurate estimation?

Closure

63 Copyright © Swun Math Grade 8 Unit 1 Lesson 7 P TE

Homework Name: ___________________________
Date: ___________________________
Unit 1 · Lesson 7: Estimate Irrational Numbers

Objective: I will estimate the decimal of an irrational square root.

Vocabulary Steps:

Square Numbers (Perfect Square): the result after 1. Find the perfect square immediately above
multiplying an integer by itself and below your number.

2. The answer will be between the square roots
of these two numbers.

3. Estimate the decimal based on how close
your number is to each of the squares.

Irrational Numbers: the set of all numbers that
cannot be written as a ratio of two integers; it cannot
be written as a simple fraction because the decimal is
non-terminating and non-repeating

0.101101110… , �23 , π, √10, √1.6, −√123

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

Example # 1

Directions: Estimate the decimal value. Use a calculator to check.

√40

Solution:

• When considering √40 on a number line, the perfect square less than 40 is 36.
• √36 = 6
• The perfect square greater than 40 is 49.
• √49 = 7
• Now I can make the statement√36 < √40 < √49.
• Simplify 6 <√40 < 7
• √40 is between 6 and 7 or 6 < √40 < 7
• Calculator: √40 ≈ 6.3

Example # 2

−√28

2

Solution:

• −√28 = (−1)⋅√28
2 2
√25 √28 √36
• 2 < 2 < 2

• 5 < √28 < 6
2 2 2
√28
• 2.5 < 2 < 3

• √28 will fall between 2 and 3.
2
−√28
• Since it is negative, it will fall between -2 and -3 or −3 < 2 < −2.= ─2.6

Copyright © Swun Math Grade 8 Unit 1 Lesson 7 P TE 64

Homework

Unit 1 · Lesson 7: Estimate Irrational Numbers

Directions: Estimate the decimal value. Use a calculator to check.

1. −√113 2. −√33
2

3. √45 4. −√42
2

5. √385 6. √21
2

Explain the steps you used to solve problem number _______.

______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________
______________________________________________________________________________________________________________________

65 Copyright © Swun Math Grade 8 Unit 1 Lesson 7 P TE

Answer Key

Extension Activity 30 49 52 70
28 53 78
Answers will vary. 27 80
78
16 9

45 6

Homework 2. −√ between –2.5 and –3

1. −√113 lies between ─10 and ─11
−√33 ≈ −2.9
−√113 ≈ −10.6 2

3. √45 between 6 and 7 4. −√42 between –3 and –3.5
√45 ≈ 6.7 2

−√42 ≈ −3.2
2

5. √ between 9.5 and 10 6. √21 between 4 and 5
√21 ≈ 4.6

√385 ≈ 9.8
2

Copyright © Swun Math Grade 8 Unit 1 Lesson 7 P TE 66

Compare & Order Irrational MPs Applied MP
Numbers
* Embedded MP
Procedural Lesson
1 2 345 6 78
Grade 8 · Unit 1 · Lesson 8
*  * * * *
MC: 8.NS.2

Problem of the Day Student Journal Pages

36-41

Objective: I will compare and order sets of rational and irrational numbers.

Vocabulary Teacher Resources

Irrational Numbers: the set of all numbers Considerations:
that cannot be written as a ratio of two
integers; cannot be written as a simple Some students may need a quick review of converting
fraction because the decimal is non- decimals to fractions and vice versa. Remind
terminating and non-repeating students the values on the number line decrease to
the left of the zero. Practice comparing with integers
0.101101110… , �32 , π, √10, √1.6, −√123 only, if needed.

Steps:
1. Estimate irrational number(s) and plot on a
number line.
2. Place rational number(s) on the number line.
3. Compare the numbers.

Application of MPs:

MP2: Why do we go to the same place value when
comparing numbers?
When comparing numbers, we find a common
denominator or go to the same place value
because_________________________________.

MP6: What mathematical language can you use to
describe the process of comparing irrational
numbers?
The mathematical language I would use to compare
irrational numbers would include
___________________________.

67 Copyright © Swun Math Grade 8 Unit 1 Lesson 8 P TE

Input/Model

(Teacher Presents)

Directions: Place the numbers on a number line. Then compare using an inequality.
1. √7, 3, 2, 2.5

Solution:
• *If needed, provide students the numbers to write below the notches on the number line.
• Step 1 - Analyze the irrational number to narrow down answer between two consecutive integers on a

number line:
Since I know that 22 = 4 and 32 = 9, √7 must be between these two numbers.

√7

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1

• Step 2 - Continue narrowing it down more precisely if necessary:
Since 2.52 = 6.25, and √7 is more than that, √7 will be between 2.5 and 3.

• Step 3 - Compare the numbers: conclude that the inequality will be 2 < 2.5 < √7 < 3.
Answer: 2 < 2.5 < √7 < 3

2. √23, 4, √19, 5

Solution:
• Step 1 - Analyze the irrational number to narrow down answer between two consecutive integers on a

number line:
Since I know that 42 = 16 and 52 = 25, √19 and √23 must be between these two numbers.

√19 √23

4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5 5.1

• Step 2 - Continue narrowing it down more precisely if necessary:
Since 4.52 = 20.25, and √19 is less than 4.5, and √23 will be between 4.5 and 5.

• Step 3 - Compare the numbers: conclude that the inequality will be 4 < √19 < √23 < 5
Answer: 4 < √19 < √23 < 5

Copyright © Swun Math Grade 8 Unit 1 Lesson 8 P TE 68

Structured Guided Practice

(A/B Partners Practice)

Directions: Place the numbers on a number line. Then compare using an inequality.
1. √15, 3, 4, 3.5

Solution:

√15

3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4 4.1

Answer: 3 < 3.5 < √15 < 4.

2. √43, 6, √47, 7

Solution:

√43 √47

6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7 7.1
Answer: 6 < √43 < √47 < 7

69 Copyright © Swun Math Grade 8 Unit 1 Lesson 8 P TE

Final Check for Understanding

(Teacher Checks Work)

Directions: Place the numbers on a number line. Then compare using an inequality.
1. √67, 8, √53, √73

Solution:

√53 √67 √73

7 7.2 7.4 7.6 7.8 8 8.2 8.4 8.6 8.8 9 9.2

Answer: √53 < 8 < √67 < √73

2. √87, 9, √107, 10

Solution: √87 √107
9
9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10 10.1 10.2 10.3

Answer: 9 < √87 < 10 < √107.

Copyright © Swun Math Grade 8 Unit 1 Lesson 8 P TE 70

Student Practice Name: ___________________________
Date: ___________________________
Unit 1 · Lesson 8: Compare & Order Irrational Numbers

Directions: Place the numbers on a number line. Then compare using an inequality.

1. 6, √29, √35, 7 2. √51, √67, √47, 8

Solution: Solution:
√29 < √35 < 6 < 7 √47 < √51 < 8 < √67

3. 12, √131, √123, 11 4. √2, √5, √1, √7, 2

Solution: Solution:
11 < √123 < √131 < 12 √1 < √2 < 2 < √5 < √7

5. 5, √23, √26, √19, √17 6. √159, √147, √171, 13, √167

Solution: Solution:
√17 < √19 < √23 < 5 < √26 √147 < √159 < √167 < 13 < √171

71 Copyright © Swun Math Grade 8 Unit 1 Lesson 8 P TE

Challenge Problems

Directions: Place the numbers on a number line. Then compare using an inequality.

1. 42, √289, 72, √361, 5.52, √307 2. 62, √529, 32, √729, 2, √631

Solution: Solution:
42 < √289 < √307 < √361 < 5.52 < 72 32 < 2 < √529 < √631 < √729 < 62

Extension Activity

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

* MP6: Attend to precision.

Directions: Using the digits 0-9, fill in the boxes so that the square root falls on or between
the numbers indicated on the number line.

� �  � 

0 1 2 34 5 678 9 10 11

� �  � 

Copyright © Swun Math Grade 8 Unit 1 Lesson 8 P TE 72

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: Why does going to the same place value help when comparing numbers?
MP6: What mathematical language would you use to describe the process of comparing

irrational numbers?

Presentations

Closure Grade 8 Unit 1 Lesson 8 P TE

73 Copyright © Swun Math

Homework Name: ___________________________
Date: ___________________________
Unit 1 · Lesson 8: Compare & Order

Irrational Numbers

Objective: I will compare and order sets of rational and irrational numbers.

Vocabulary Steps:

Irrational Numbers: the set of all numbers that 1. Estimate irrational number(s) and plot on
cannot be written as a ratio of two integers; a number line.
cannot be written as a simple fraction because
the decimal is non-terminating and non- 2. Place rational number(s) on the number
repeating line.

3. Compare the numbers.

0.101101110… , �32 , π, √10, √1.6, −√123

Example # 1 Example # 2

Directions: Place the numbers on a number line. Then compare using an inequality.

1. √7, 3, 2, 2.5 2. √23, 4, √19, 5

Solution: Solution:
2 < 2.5 < √7 < 3 4 < √19 < √23 < 5

Copyright © Swun Math Grade 8 Unit 1 Lesson 8 P TE 74

Homework Name: ___________________________
Date: ___________________________
Unit 1 · Lesson 8: Compare & Order

Irrational Numbers

Directions: Place the numbers on a number line. Then compare using an inequality.

1. √13, 3, √11, 4 2. √53, 8, √65, 7

3. √2, 2, √3, 3, √2.5 4. √63, √57 , √35, √41, √101
5. √17, 4, √27, 5, 6 6. √5, 22, √10, 12, √0.36

Explain the steps you used to solve problem number _______.

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75 Copyright © Swun Math Grade 8 Unit 1 Lesson 8 P TE

Answer Key

Extension Activity 2. 7 < √53, < 8 < √65
4. √35 < √41 < √57 < √63 < √101
Answers will vary. Sample response: 5. √0.36 < 12 < √5 < √10 < 22
Square root is between 1 and 3: 3 or 8
Square root is 3: 9
Square root is 5: 25
Square root is between 6 and 7: 40 or 41
Square root is between 7 and 8: 60, 61 or 63
Square root is between 8 and 9: 71 or 78

*Be sure that the digits 0-9 are only used once.

Homework

1. 3 < √11 < √13 < 4

3. √2 < √2.5 < √3 < 2 < 3

4. 4 < √17 < 5 < √27 < 6

Copyright © Swun Math Grade 8 Unit 1 Lesson 8 P TE 76

Classify Numbers MPs Applied MP 8

Math Task * Embedded MP *
Grade 8 · Unit 1 · Lesson 9
1234567
MC: 8.NS.1
* ** *
Objective: I will classify rational and irrational numbers.
Student Journal Pages

42-43

Student Practice

• Explain the problem and ask students to solve independently.

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

• 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.

77 Copyright © Swun Math Grade 8 Unit 1 Lesson 9 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 correctly classify the numbers based on the characteristics of
rational and irrational numbers?

• Were students able to verbalize and use mathematical language?

Answer Key

Homework 2. All the fractions and repeating or terminating
decimals must be listed outside of integers as
1. Rational numbers include integers, whole integers only include positive and negative whole
numbers, natural numbers, repeating and numbers.
terminating decimals, fractions.
They are all included as they can all be expressed

as a fraction or a ratio in the form of .

3. Natural numbers use the same set as whole 4. Answers will vary. Sample response: a number
numbers except for zero.
such as 4 or 6i. These are not possible within the
0

set of real numbers.

Copyright © Swun Math Grade 8 Unit 1 Lesson 9 T TE 78

Math Task Name: ___________________________
Date: ___________________________
Unit 1 · Lesson 9: Classify Numbers

Directions: Create a diagram to classify real numbers.

Sometimes a visual diagram can be created to help show the relationship between different
sets of numbers. For example, we can create a mini-diagram to show that the sets of even
and odd numbers fit within the set of whole numbers. All other numbers would be outside
that specific set.

Whole Numbers

Odd √3

Even

–2 .38

Using a ruler and a blank paper, create a diagram for rational and irrational numbers, which both fit
into an all-inclusive set called real numbers. Draw a large rectangle that is 7 inches by 5 inches and
label the rectangle real numbers on the upper left corner.

Consider the following sets of numbers:

• rational • irrational

• whole numbers • integers

Choose a shape for each set of numbers. All the sets above must fit within the real number set. Some
sets will fit inside other sets based on characteristics and definitions will not. Please provide 4
example numbers for each set and be prepared to justify why you chose your examples. Be sure to
label each set or subset along its top edge. Use different colors in each shape so the examples can be
easily read.

Solution: real numbers
Answers will vary.
Some students may − 2 rational numbers 1
choose to add 3 4
additional shapes integers
inside rational 2
numbers for –2
decimals and fractions.
whole numbers

03
√3
1 −√22
2
.453671123…
6

2 –10
.333… 0.25

79 Copyright © Swun Math Grade 8 Unit 1 Lesson 9 T TE

Homework Name: ___________________________
Date: ___________________________
Unit 1 · Lesson 9: Classify Numbers

Directions: Answer the following questions based on the diagram created.

1. What are all the sets of numbers you put 2. What numbers are included in the
inside rational numbers? Why? rational numbers that are not included in
the integers?

3. Why are natural numbers tucked inside 4. Can you think of other numbers that
the whole numbers category? What is the wouldn’t fit within the sets of numbers
difference between these two sets? that have been discussed? Do you
remember a number that is not possible
within the real numbers?

Copyright © Swun Math Grade 8 Unit 1 Lesson 9 T TE 80

Unit 1 ∙ Constructed Response Name: ____________________________
Rational vs. Irrational Numbers Date: ______________________________

Task 1: Decimal Representations and Expansions Student Journal Pages

44-51

Directions: Read and solve each problem.

1. Consider this equation:

—13 — y = 10

a. Solve the equation. Show your work in detailed and organized steps.

b. Analyze your solution. Under which categories of numbers does your solution fall?

c. Give three examples of irrational numbers. c.
a. b.

81 Copyright © Swun Math Grade 8 Unit 1 Constructed Response TE

Unit 1 ∙ Constructed Response
Rational vs. Irrational Numbers

2. Consider this statement:
24 out of 64 cups of sugar

a. Write the fraction and decimal expansion that best represent this statement.

b. Explain how you could show your answer to 2a is correct.

Copyright © Swun Math Grade 8 Unit 1 Constructed Response TE 82

Unit 1 ∙ Constructed Response
Rational vs. Irrational Numbers

3. Consider this decimal.

2.3939...

a. Convert the decimal into a fraction. In the table below, show and explain the process
necessary to find your solution.

Mathematical Process Explanation of Steps

Answer:

b. Create an illustration of the scenario to demonstrate that your answer from above is
correct.

83 Copyright © Swun Math Grade 8 Unit 1 Constructed Response TE

Unit 1 ∙ Constructed Response Name: ____________________________
Rational vs. Irrational Numbers Date: ______________________________

Task 2: Estimate and Approximate Rational Numbers

Directions: Read and solve each problem.
4. Consider this value:

a. Between which two integers is the value found?

b. Explain how you found the answer to 4a and what the answer means.

c. How would your process and answer change if the value was changed to ?

Copyright © Swun Math Grade 8 Unit 1 Constructed Response TE 84

Unit 1 ∙ Constructed Response
Rational vs. Irrational Numbers

5. Consider this value:
a. Estimate the value to the nearest hundredth. Explain your reasoning using words or
numbers.

b. What concepts helped you find the solution? How were those concepts useful?

85 Copyright © Swun Math Grade 8 Unit 1 Constructed Response TE

Unit 1 ∙ Constructed Response
Rational vs. Irrational Numbers

6. Consider this statement:

6

2 is slightly greater than

9

a. Is this statement true or false? Explain your reasoning using words or numbers.

b. Demonstrate your answer is correct by visually representing the comparison of the
numbers on a number line.

Copyright © Swun Math Grade 8 Unit 1 Constructed Response TE 86

Unit 1 ∙ Constructed Response
Rational vs. Irrational Numbers

7. Consider this expression:
a. Simplify the expression.

b. Calculate the value of the expression to the nearest hundredth. Explain your
reasoning using words or numbers.

87 Copyright © Swun Math Grade 8 Unit 1 Constructed Response TE

Unit 1 ∙ Constructed Response
Rational vs. Irrational Numbers

8. Consider this expression:
a. Simplify the expression.

b. Explain why it is a good idea to first simplify this type of expression before solving
it. Why not solve it without simplifying?

Copyright © Swun Math Grade 8 Unit 1 Constructed Response TE 88

Unit 1 ∙ Constructed Response Unit 1 ∙ Constructed
Rational vs. Irrational Numbers ResponseSCORING RUBRIC

TASK 1: Absolute Value and Distances

# Answer/Exemplar Standard Points

1a. —13 — y = 10 8.NS.1 1
—y = 23
y = —23

1b. —12 is an integer and a rational number. It is an integer because it 8.NS.1 1
is a whole number and a negative natural number. It is also a
rational number because it is an integer that can be written as a
ratio.

1c. Answers will vary. 8.NS.1 1

2a. Decimal Expansion: 0.375 8.NS.1 1
1
Fraction: 3 1
8
1
2b. The decimal form is a terminating, non-repeating decimal and 8.NS.1

rational number because 0.375 has no additional or repeating

numbers after the value in the hundredths place. It is a rational

number because it is a terminating integer that can be written as a

ratio.

3a. Mathematical Process: 8.NS.1
x = 2.3939
100x = 239.3939
100x — 2.3939 = 239.3939 — 2.3939
99x = 237

x= 237
99

Explanation of Steps: 8.NS.1
To find the solution I first set up the equation x = 2.3939. I
multiplied both sides of the equation by 100 so that the values in
each decimal place would be equal, and I can derive a whole
number after subtracting the value of x. The new equation is 100x
= 239.3939. When I subtract the value of x from both sides, I
maintain a balanced equation. The difference is a whole number.
Division is the final step

89 Copyright © Swun Math Grade 8 Unit 1 Constructed Response Rubric TE

Unit 1 ∙ Constructed Response Unit 1 ∙ Constructed
Rational vs. Irrational Numbers ResponseSCORING RUBRIC

TASK 1: Absolute Value and Distances, continued

# Answer/Exemplar Standard Points

3b. To show my answer is correct, I can convert the fraction into a 8.NS.1 1
decimal using division. If the quotient is 2.3939…, then my answer
is correct.

237 = 2.3939
99

Copyright © Swun Math Grade 8 Unit 1 Constructed Response Rubric TE 90

Unit 1 ∙ Constructed Response Unit 1 ∙ Constructed
Rational vs. Irrational Numbers ResponseSCORING RUBRIC

TASK 2: Estimate and Approximate Irrational Numbers

# Answer/Exemplar Standard Points

4a. The square root of 125 is between 11 and 12. 8.NS.2 1
1
4b. I used what I know about perfect squares. is between 8.NS.2 1
is 12, then
and . Since = 11 and 1

must be between 11 and 12.

4c. If the value was changed to , the process to solve would 8.NS.2

change because the value inside the radical would be multiplied

by 5. Since the value of is between and , or

5 × 11 and 5 × 12. So the value of is between 55 and 60.

5a. 9.43; 8.NS.2

5b. It was helpful to begin the estimation by comparing the square 8.NS.2 1

roots of perfect squares. I noticed is slightly closer to 9 than 1
1
10, so it will be most likely between 9.4 and 9.5. After calculating
the squares of 9.4 and 9.5, I saw 9.42 is closer to 89 than 9.52, which

helped me to make a reasonable estimate of the value in the

hundredths place.

6a. False; 2 6 is not larger than . The decimal equivalent of 2 6 is 8.NS.2
9 9

. I know . Since , then . Because

and , 2 6 cannot be greater than .
9

6b. 2 6 = 2.666 8.NS.2
9

01 23

91 Copyright © Swun Math Grade 8 Unit 1 Constructed Response Rubric TE

Unit 1 ∙ Constructed Response Unit 1 ∙ Constructed
Rational vs. Irrational Numbers ResponseSCORING RUBRIC

TASK 2: Estimate and Approximate Irrational Numbers, continued

# Answer/Exemplar Standard Points

7a. 8.NS.2 1
1
7b. 17.52 8.NS.2 1
1
8a. 8.NS.2

8b. Simplifying this type of expression is a good idea because when 8.NS.2
you combine like terms you are making the expression smaller
and easier to solve. If we did not simplify it, we would be
approximating the value of every term in the expression instead of
finding the approximation of fewer combined like terms.

Copyright © Swun Math Grade 8 Unit 1 Constructed Response Rubric TE 92

Grade 8

Unit 2: Exponents

Table of Contents

Lesson # Lesson Type Lesson Title MC MP Applied Page

1C Multiply Exponents 8.EE.1 2, 4 95
8.EE.1
2P Properties of Exponents: 8.EE.1 4, 6 103
Multiplication 8.EE.1
8.EE.1
3C Divide Exponents 8.EE.1 2, 4 115
8.EE.1
4P Properties of Exponents: Division 8.EE.1 2, 6 123
5C 8.EE.2
6C Properties of Exponents: Zero and 8.EE.2 3, 8 133
Negatives 8.EE.2
8.EE.2 1, 7 141
Convert Negative Exponents

7P Zero and Negative Exponents 1, 3 149

8 MT Explore Exponents 159

9C Square Roots 5, 6 163

10 C Cube Roots 2, 5 171

11 P Square & Cube Roots 2, 6 179

12 MT Solve Square and Cubic Equations 191

 indicates a major standard; unmarked standards are additional or supporting

93 Copyright © Swun Math Grade 8 Unit 2 Table of Contents TE

Grade 8 • Unit 2
Table of Contents, continued

Constructed Response

Task Lesson Title MC Page

1 Combining Exponents 8.EE.1 195
8.EE.2 198
2 Square and Cube Roots 8.EE.1 200
201
3 Zero & Negative Exponents

Constructed Response Scoring Guide

Materials Needed

Lesson Materials

2 Base ten blocks (optional)

2 Calculators (optional)

 indicates a major standard; unmarked standards are additional or supporting

Copyright © Swun Math Grade 8 Unit 2 Table of Contents TE 94

Multiply Exponents MPs Applied MP

Conceptual Lesson * Embedded MP
Grade 8 · Unit 2 · Lesson 1
1 23 4567 8
MC: 8.EE.1
*  *

Problem of the Day Student Journal Pages

54-57

Objective: I will use the expanded form to explore multiplying exponents.

Vocabulary Teacher Resources

Base: the repeated factor Considerations:

Exponent: indicates the number of times the Students began to explore exponential notation in
base is multiplied by itself; also known as grade 6. Using base ten blocks can help students
power visualize how quickly something grows by
multiplying powers.
Factors: numbers that are multiplied to create
a product Have students consistently write the expanded form
of an exponential expression (5 4 = 5 ∙5 ∙5 ∙5) before
Power: a product in which the factors are the focusing on the rules. Once students understand the
same concept of exponential form, teach the rule, but
remind them they can always fall back on writing the
5 • 5 • 5 • 5  54 expanded expression.

4 factors power When counting combinations of things, consider
something from daily life. For example, if there are 3
ties, 3 pants, and 3 shirts, there are 33 possible outfits
combinations.

Say as, “4 copies of 5 multiplied together” Steps:
n3= 3 copies of n, multiplied together= n ⋅ n ⋅ n
1. Identify the base(s) and exponent(s).
Exponential Form: expressions written with 2. Write in expanded form.
exponents 3. Simplify.
4. Check using the properties of exponents.

Properties of Exponents Application of MPs:
Power to Power (an)m = anm (Multiply)
Multiply (Like Bases) am ⋅ an = am+n (Add) MP2: What is the relationship between the
Power of a Product anbn = (ab)n (Keep) exponent and the base?
The exponent indicates ______________. The
base indicates ______________.

MP4: What rule might apply in this situation?
A rule that may apply can be ______________.

95 Copyright © Swun Math Grade 8 Unit 2 Lesson 1 C TE