Teacher’s Resource: Volume 1

ISBN: 978-1-62743-490-4

Grade 8 Teacher’s Resource – Volume 1

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Grade 8

Volume 1

Table of Contents

Unit 1: Rational vs. Irrational Numbers 1

Lesson # Lesson Type Lesson Title MC MP Applied Page

1C Decimal Expansion 8.NS.1 2, 5 3

8.NS.1 2, 8 11

2C Rational vs. Irrational Numbers 8.NS.1 5, 8 19

5, 7

3P Rational vs. Irrational Numbers 8.NS.1 29

3, 7

4C Convert Repeating Decimals to 8.NS.1 37

Fractions 5, 6

8.NS.2 6, 7 49

5P Convert Repeating Decimals to 8.NS.2 2, 6 57

Fractions 8.NS.2 67

8.NS.1

6C Estimate Irrational Numbers 8.NS.2 77

7P Estimate Irrational Numbers 81

89

8P Compare & Order Irrational Numbers

9 MT Classify Numbers

Constructed Responses

Constructed Response Scoring Rubrics

Unit 2: Exponents 93

101 Lesson Type Lesson Title MC MP Applied Page

1C Multiply Exponents 8.EE.1 2, 4 95

8.EE.1 4, 6 103

2P Properties of Exponents:

Multiplication 8.EE.1

8.EE.1

3C Divide Exponents 8.EE.1 2, 4 115

2, 6 123

4P Properties of Exponents: Division 3, 8 133

5C Properties of Exponents: Zero and

Negatives

Copyright © Swun Math Grade 8 Volume 1 Table of Contents TE i

Grade 8 • Volume 1

Table of Contents, continued

Unit 2: Exponents, continued

Lesson # Lesson Type Lesson Title MC MP Applied Page

6C Convert Negative Exponents 8.EE.1 1, 7 141

8.EE.1 1, 3 149

7P Zero and Negative Exponents 8.EE.1 159

8.EE.2 5, 6 163

8 MT Explore Exponents 8.EE.2 2, 5 171

8.EE.2 2, 6 179

9C Square Roots 8.EE.2 191

195

10 C Cube Roots 201

11 P Square & Cube Roots

12 MT Solve Square and Cubic Equations

Constructed Responses

Constructed Response Scoring Rubrics

Unit 3: Scientific Notation 203

Lesson # Lesson Type Lesson Title MC MP Applied Page

1C Scientific Notation 8.EE.3 6, 8 205

8.EE.3

2P Scientific Notation 8.EE.4 6, 7 213

8.EE.4

3C Scientific Notation: Add & Subtract 8.EE.4 1, 2 223

8.EE.4

4P Scientific Notation: Add & Subtract 8.EE.4 2, 6 231

8.EE.4

5C Scientific Notation: Multiply & Divide 2, 8 243

6P Scientific Notation: Multiply & Divide 6, 8 251

7P Scientific Notation: Mixed Practice 1, 2 263

8 MT Scientific Notation 275

Constructed Responses 279

Constructed Response Scoring Rubrics 283

Unit 4: Linear Equations 285

Lesson # Lesson Type Lesson Title MC MP Applied Page

1C Model Equations 8.EE.7 4, 5 287

8.EE.7

2P One-Step Equations 1, 7 295

ii Copyright © Swun Math Grade 8 Volume 1 Table of Contents TE

Grade 8 • Volume 1

Table of Contents, continued

Unit 4: Linear Equations, continued

Lesson # Lesson Type Lesson Title MC MP Applied Page

3C Collect Like Terms 8.EE.7b 1, 6 307

8.EE.7b 6, 7 315

4P Two-Step Equations

5C Simplify Expressions: Distributive 8.EE.7b 7, 8 327

Property

6C Solve Equations: Distributive Property 8.EE.7b 2, 7 335

8.EE.7b 7, 8 345

7C Equations with Variables on Both Sides 8.EE.7b 7, 8 353

8.EE.7a 2, 8 363

8P Equations with Rational Numbers 8.EE.7a 1, 7 371

8.EE.7b 4, 6 381

9C Equations: No Solution or Many 8.EE.7 393

398

10 P Equations: No Solution or Many 406

11 P Write Linear Equations

12 MT Linear Equations

Constructed Responses

Constructed Response Scoring Rubrics

Glossary 410

Resource Pages 428

indicates a major standard; unmarked standards are additional or supporting

Note of caution: Neglecting material, whether it is found in the major or additional/supporting clusters, will leave

gaps in students’ skills and understanding and will leave students unprepared for the challenges they face in later

grades. (Grade-Seven Chapter of the Mathematics Framework for California Public Schools, 2015, pg. 3)

Copyright © Swun Math Grade 8 Volume 1 Table of Contents TE iii

Grade 8

Unit 1: Rational vs. Irrational

Numbers

Table of Contents

Lesson # Lesson Type Lesson Title MC MP Applied Page

1C Decimal Expansion 8.NS.1 2, 5 3

2C Rational vs. Irrational Numbers 8.NS.1 2, 8 11

3P Rational vs. Irrational Numbers 8.NS.1 5, 8 19

4C 8.NS.1

5P Convert Repeating Decimals to 8.NS.1 5, 7 29

6C Fractions 8.NS.2

Convert Repeating Decimals to 3, 7 37

Fractions

5, 6 49

Estimate Irrational Numbers

7P Estimate Irrational Numbers 8.NS.2 6, 7 57

8P Compare & Order Irrational Numbers 8.NS.2 2, 6 67

9 MT Classify Numbers 8.NS.1 77

8.NS.2

indicates a major standard; unmarked standards are additional or supporting

1 Copyright © Swun Math Grade 8 Unit 1 Table of Contents TE

Grade 8 • Unit 1

Table of Contents, continued

Constructed Response

Task Lesson Title MC Page

1 Absolute Value and Distances 8.NS.1 81

8.NS.2 84

2 Estimate and Approximate Irrational Numbers 89

Constructed Response Scoring Guide

Materials Needed

Lesson Materials

1-7 Calculator (optional)

6 Graph paper

6-8 Number lines

indicates a major standard; unmarked standards are additional or supporting

Copyright © Swun Math Grade 8 Unit 1 Table of Contents TE 2

=Decimal Expansion MPs Applied MP

Conceptual Lesson * Embedded MP

Grade 8 · Unit 1 · Lesson 1 1234 5 67 8

MC: 8.NS.1 * * * * *

Problem of the Day Student Journal Pages

2-5

Objective: I will convert fractions to decimals and determine if it is terminating or repeating.

Vocabulary Teacher Resources

Terminating decimal: a decimal with digits that Considerations:

come to an end

This is a review lesson from Grades 6 and 7. However,

1 it is beneficial for students to have scientific

8 = 0.125 calculators to double-check answers quickly.

Repeating Decimal: a decimal number that has Bar graphs and number lines are great ways to show

digits that repeat forever that decimals and fractions are similar.

1 = 0.3333 = 0. 3� Discuss whether every approximation must be as

3 close to the exact number as possible. In reality, no,

but most of the time, an approximation rounded to the

nearest thousandth or ten thousandths is very usable.

* The 3 is a repeating digit, and is marked with a bar Steps:

notation.

Method 1

1. Divide to convert the fraction to a decimal.

2. Determine if the decimal is repeating or

terminating.

3. If it’s repeating, identify the pattern.

Method 2

1. Determine if fraction can be changed to an

equivalent fraction with 10, 100, 1,000 etc. in

the denominator.

2. Convert base-ten fractions to decimals.

Application of MPs:

MP5: What approach are you considering trying first

when it comes to converting between fractions and

decimals?

When changing a fraction into a decimal I can first

use

.

MP8: How can you predict which rational number can be

represented by a terminating decimal?

I can prove that the number is rational by .

3 Copyright © Swun Math Grade 8 Unit 1 Lesson 1 C TE

Input/Model

(Teacher Presents)

Directions: Convert each fraction to a decimal. Determine if repeating or terminating. If it’s

repeating, describe the pattern.

1. 1

4

Solution:

1

• 4 is the same as 1 ÷ 4

• Rewrite in long division form and divide.

• Continue 6 places

• 1 = 0.25; it is terminating decimal

4

• Therefore, 1 is rational.

4

2. 3

11

Solution:

• Do the long division for 3 ÷ 11.

• Notice the 27 is repeating, so stop the dividing after 6 places to the right of the decimal.

• 3 = 0.272727... or 0.27

11

• Repeating pattern is 27

• Therefore, 3 is rational.

11

Copyright © Swun Math Grade 8 Unit 1 Lesson 1 C TE 4

Structured Guided Practice

(A/B Partners Practice)

Directions: Convert each fraction to a decimal. Determine if repeating or terminating. If it’s

repeating, describe the pattern.

1. 1

3

Solution:

1 = 0.33333... 0.3

3 or

repeating pattern is 3

2. 11

16

Solution: Grade 8 Unit 1 Lesson 1 C TE

11 = 0.6875

16

terminating decimal

5 Copyright © Swun Math

Final Check for Understanding

(Teacher Checks Work)

Directions: Convert each fraction to a decimal. Determine if repeating or terminating. If it’s

repeating, describe the pattern.

1. 11

9

Solution:

11 = 1.2222...

9 or 1.2

repeating pattern is 2

2. 7

8

Solution:

7 = 0.875

8

terminating decimal

Closure

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

MP5: What approach are you considering trying first when it comes to converting

between fractions and decimals?

MP8: How can you predict which rational numbers can be represented by a

terminating decimal?

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

Homework Name: _____________________________

Date: _____________________________

Unit 1 · Lesson 1: Decimal Expansion

Objective: I will convert fractions to decimals and determine if it is terminating or repeating.

Vocabulary Steps:

Terminating decimal: a decimal with digits that Method 1

come to an end 1. Divide to convert the fraction to a decimal.

2. Determine if the decimal is repeating or

1 = 0.125

8 terminating.

3. If it’s repeating, identify the pattern.

Repeating decimal: a decimal number that has Method 2

digits that repeat forever

1. Determine if fraction can be changed to an

1 = 0.3333 = 0. 3� equivalent fraction with 10, 100, 1,000 etc. in

3 the denominator.

* The 3 is a repeating digit, and is marked with a bar 2. Convert base-ten fraction to decimal form.

notation.

Example # 1 Example # 2

Directions: Convert each fraction to a decimal. State whether it’s repeating or terminating. If it’s a repeating

decimal, describe the pattern.

13

4 11

Solution: Solution:

• 1 = 0.25 • 3 = 0.272727... or 0.27

4 11

• terminating decimal • Repeating pattern is 27

• Therefore, 1 is rational. • Therefore, 3 is rational.

4 11

7 Copyright © Swun Math Grade 8 Unit 1 Lesson 1 C TE

Homework

Unit 1 · Lesson 1: Decimal Expansion

Directions: Convert each fraction to a decimal. Determine if repeating or terminating. If it’s

repeating, describe the pattern.

1. 2 2. 11

3 9

3. 6 4. 3

11 8

5. 5 6. 5

6 11

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

Answer Key

Homework 2. 11 = 1. 2

9

1. 2 = 0.666666… or 0.6 repeating; pattern is 6

repeating decimal

3

4. 3 = 0.375

3. 6 = 0.5454... or 0.54

8

11

terminating decimal

repeating pattern is 5454

5. 5 = 0.8333... or 0.83 6. 5 = 0.4545... or 0.45

6 11

repeating pattern 3 repeating pattern is 45

9 Copyright © Swun Math Grade 8 Unit 1 Lesson 1 C TE

Notes

Copyright © Swun Math Grade 8 Unit 1 Lesson 1 C TE 10

Rational vs. Irrational Numbers MPs Applied MP

Conceptual Lesson * Embedded MP

Grade 8 · Unit 1 · Lesson 2

123 456 7 8

MC: 8.NS.1

* *

Problem of the Day Student Journal Pages

6-9

Objective: I will identify rational and irrational numbers and the differences between them.

Vocabulary Teacher Resources

Whole Numbers: the natural numbers, including 0 Considerations:

0, 1, 2, 3, …

Students can draw a representation of rational

Integers: the set of all counting numbers, including numbers or come up with a scenario where a rational

0, and their opposites number would be needed. Have student share

findings with one another and explain their thinking.

… ─3, ─2, ─1, 0, 1, 2, 3, …

Review with students how to take the square root of a

Squared Numbers (Perfect Square): the result after number from Grade 8, Unit 1. Perfect squares are

multiplying an integer by itself rational numbers. Calculators will help students see

the actual value of an irrational number. Continue to

encourage students to use the terms rational numbers

and irrational numbers in the right context.

Real Numbers: both rational and irrational numbers Steps:

1. Determine if the number can be expressed as a

Rational Numbers: whole numbers, fractions, and

decimals represented as a ratio of two integers; can fraction.

be represented in fractional form: If so, the number is rational.

where are integers and y ≠ zero. Check for terminating or repeating

decimals.

3

1.5 = 2 = ratio Look for perfect squares.

If not, the number is irrational

Irrational Numbers: the set of all numbers that 2. Justify your answer.

cannot be written as a ratio; cannot be written as a

simple fraction because the decimal is non- Application of MPs:

terminating and non-repeating

MP2: How are irrational numbers related to rational

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

Irrational numbers are ______ and rational

numbers are _______.

MP5: Is there a tool you can use to help you find

your answer?

A tool I can use to help me find my answer is

______________.

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

Input/Model

(Teacher Presents)

Directions: Determine if the number is rational or irrational. Justify your answer.

1. 0.74 2. 0.01001000100001… 3. √7 4.

Solution: Solution: Solution: Solution:

• The number seventy- • This is an irrational • I know that the square • is an irrational

four hundredths is a number. It is a root of seven is an number.

terminating decimal. decimal that has no irrational number.

Therefore, it is set pattern and it • When I punch π on

considered to be a doesn’t terminate. • The decimal form of the calculator, I get

rational number. this number is the decimal

• You cannot write this 2.64575131106… it 3.1415926535…

• It can be re-written as number as a ratio. never ends and it

doesn’t have a set • This decimal is

the fraction 37 pattern or repeat. neither terminating

nor has a set pattern.

50

Considerations:

• Have conversations with students stating the type of numbers that are considered rational and irrational.

• Create a table or circle map with the center being natural numbers and in the outer circles stating every

type of number that is considered rational. Give an example for each one.

• Do the same task, but this time with irrational numbers. Give examples of these numbers.

• Draw a rectangle to enclose both and label as “real numbers”.

• When sharing the above information, justify why the numbers are irrational or rational. Add them to the

circle map/poster you created.

rational irrational

integer

whole Real Numbers

v

Copyright © Swun Math Grade 8 Unit 1 Lesson 2 C TE 12

Structured Guided Practice

(A/B Partners Practice)

Directions: Determine if the number is rational or irrational. Justify your answer.

1. √16 2. 2 3. –3 4. 0.262626262626…

3

Solution: Solution: Solution: Solution:

Rational number; 16 is a Rational number; –3 is an Rational number; it is a

perfect square Rational number; the integer repeating decimal.

2

The √16 is 4 and 4 is an decimal form of 3 is 0.6

integer, a whole number,

a natural number and it All repeating decimals It can be written in the All repeating decimals

can be written in a ratio are rational numbers. form of a ratio: - 3 are rational.

4/1.

1

Considerations:

• As students are working with partners, have them justify why the number is rational or

irrational.

• Add the above examples to the circle map/poster you created.

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

Final Check for Understanding

(Teacher Checks Work)

Directions: Determine if the number is rational or irrational. Justify your answer.

1. √6 2. √9 3. 4.

0.18181818…. 0.02002000200002…

Solution: Solution: Solution: Solution:

Irrational; on a Rational; 9 is a perfect Rational; it is a repeating Irrational; even though it

calculator, the answer is square. decimal. looks like a pattern, it is

2.44948974… which is a not a set pattern.

non-terminating Therefore, it cannot be

decimal. It does not have considered a rational

a set pattern. Also, 6 is number.

not a perfect square.

Considerations:

• Most often, irrational numbers are recorded to the nearest thousandths place.

• Ask and have a conversation of where irrational numbers can be seen in the real-world.

Closure

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

MP2: How are irrational numbers related to rational numbers?

MP5: Is there a tool you can use to help you find your answer?

Copyright © Swun Math Grade 8 Unit 1 Lesson 2 C TE 14

Homework Name: ___________________________

Date: ___________________________

Unit 1 · Lesson 2: Rational vs. Irrational Numbers

Objective: I will identify rational and irrational numbers and the differences between them.

Vocabulary Steps:

Whole Numbers: the natural numbers, including 1. Determine if the number can be expressed as

0

a fraction.

0, 1, 2, 3, … If so, the number is rational.

Integers: the set of all counting numbers,

including 0, and their opposites Check for terminating/repeating

… ─3, ─2, ─1, 0, 1, 2, 3, … decimals.

Look for perfect squares.

Squared Numbers (Perfect Square): the result If not, the number is irrational.

after multiplying an integer by itself

2. Justify your answer.

Real Numbers: rational and irrational numbers

Rational Numbers: whole numbers, fractions,

and decimals represented as a ratio of two

integers; can be represented in fractional form:

where are integers and y ≠ zero.

3

1.5 = 2 = ratio

Irrational Numbers: the set of all numbers that

cannot be written as a ratio; cannot be written as

a simple fraction because the decimal is non-

terminating and non-repeating

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

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

Homework

Unit 1 · Lesson 2: Rational vs. Irrational Numbers

Example # 1

Directions: Determine if the number is rational or irrational. Justify your answer.

0.74 0.01001000100001… √7

Solution: Solution: Solution: Solution:

• I know that the square • is an irrational

• The number seventy-four • This is an irrational

root of seven is an number.

hundredths is a number. It is a decimal irrational number.

• When I punch π on the

terminating decimal. that has no set pattern • The decimal form of this calculator, I get the

number is decimal 3.1415926535…

Therefore, it is and it doesn’t terminate. 2.64575131106… it never

ends and it doesn’t have • This decimal is neither

considered to be a • You cannot write this a set pattern or repeat. terminating nor has a

rational number. number as a ratio. set pattern

• It can be re-written as

the fraction 37

50

Copyright © Swun Math Grade 8 Unit 1 Lesson 2 C TE 16

Homework

Unit 1 · Lesson 2: Rational vs. Irrational Numbers

Directions: Read and solve. Justify your answers.

1. Rational or irrational? 2. Which of the following real numbers is

11 not a rational number?

5

a. √48

b. 66

c. 3.77

d. 9.8

3. Which of the following real numbers is 4. Rational or irrational?

an irrational number? √81

a. √16

b. 5335.11

c. √51

d. 86

5. Which of the following real numbers is a 6. Why is considered to be an irrational

rational number?

number?

a. 3.14159265…

b. –7

c. √10

d. 0.13579111315…

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

Answer Key

Homework

1. Rational; it is a ratio that has an integer in the 2. Solution is a

denominator and the numerator and the Answers may vary. It is irrational because 48 is

denominator ≠ 0. not a perfect square.

3. Solution is c 4. Rational; it is a perfect square.

Answers may vary. It is irrational because it

cannot be written in decimal form that neither

repeats nor terminates

5. Solution is b 6. Π is an irrational number because it cannot be

Answers may vary. It is an integer and all integers written as a decimal that neither repeats nor

are rational numbers. terminates.

Copyright © Swun Math Grade 8 Unit 1 Lesson 2 C TE 18

Rational vs. Irrational Numbers MPs Applied MP

Procedural Lesson * Embedded MP

Grade 8 · Unit 1 · Lesson 3

1 2 345 67 8

MC: 8.NS.1

* * * * *

Problem of the Day Student Journal Pages

10-15

Objective: I will identify rational and irrational numbers and the difference between them.

Vocabulary Teacher Resources

Squared Numbers (Perfect Square): the result Considerations:

after multiplying an integer by itself

Review with students how to take the square roots of

numbers. It would be helpful if they made a list of

perfect squares to keep handy. Calculator will help

students see the actual value of an irrational number.

Continue to encourage students to use the terms

“rational numbers” and “irrational numbers” in the

right context.

Rational Numbers: whole numbers, fractions, Steps:

and decimals represented as a ratio of two 1. Determine if the number can be expressed as

integers; can be represented in fractional form: a fraction.

If so, the number is rational.

integers and y ≠ zero

Check for terminating/repeating

3

2 decimals.

Look for perfect squares.

If not, the number is irrational.

2. Justify your answer.

1.5 = = ratio

Irrational Numbers: the set of all numbers that Application of MPs:

cannot be written as a ratio of two integers; it

cannot be written as a simple fraction because MP2: How are irrational numbers related to rational

the decimal is non-terminating and non- numbers?

repeating Irrational numbers are ______________ and

rational numbers are ______________.

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

MP8: How would you prove that you are dealing with

a rational number?

Proof that I am dealing with a rational number

is ______________.

Copyright © Swun Math Grade 8 Unit 1 Lesson 3 P TE

19

Input/Model

(Teacher Presents)

Directions: Read and solve.

1. Which of the following is an irrational number and why?

a. 7

b. 10278

c. √3

d. 99

Solution:

• c is not a rational number

• The decimal expansion of this number does not terminate or repeat.

2. True or False? 0.333… is an irrational number. Justify your answer.

Solution:

• False

• Understand that this is a common rational number 1 , expressed in decimal form.

3

• Even though the decimal expansion may seem irrational because it never ends, the fact that it has a

pattern of a repeating digit makes it a rational number

Copyright © Swun Math Grade 8 Unit 1 Lesson 3 P TE 20

Structured Guided Practice

(A/B Partners Practice)

Directions: Read and solve.

1. Which of the following is not a rational number and why?

a. √4

b. √6

c. 27

d. 17

Solution:

b; the decimal expansion of this number does not terminate or repeat.

2. True or False? 1.5 is a rational number. Justify your answer.

Solution:

True; the decimal converts to one and a half. Its fraction form is 3 .

2

Copyright © Swun Math Grade 8 Unit 1 Lesson 3 P TE

21

Final Check for Understanding

(Teacher Checks Work)

Directions: Read and solve.

1. Which of the following is a rational number and why?

a. .657891203…

b. √111

c. √81

d. .09098765…

Solution:

c; it is a perfect square. The square root of 81 is 9.

2. True or false? √49 is an irrational number. Justify your answer.

Solution: Grade 8 Unit 1 Lesson 3 P TE 22

False; 49 is a perfect square.

Copyright © Swun Math

Student Practice Name: ___________________________

Date: ___________________________

Unit 1 · Lesson 3: Rational vs. Irrational Numbers

Directions: Read and solve. Justify your answer.

1. Is √9 a rational number? 2. Is 7,892,564 a rational number?

Solution: Solution:

Yes

Yes

This is a big and complex looking number, but it is still an

We know that 9 is equal to 3 times 3. This makes the integer and integers are rational numbers.

equation read: √3 ⋅ 3 Since it can be rewritten as 7892564/1, and both 7892564

and 1 are integers, then yes, it is a rational number.

9 is equal to 3 which is an integer. All integers are rational.

Since 3 is the same as writing 3 , and both 3 and 1 are

1

integers, then yes, this is a rational number.

3. Is √79 rational number? 4. Is 9.2 a rational number?

Solution: Solution:

No Yes

Examine 79, it is not a perfect square number under the

radical. The number 9.2 can easily be converted to fraction form by

Square roots of perfect squares are classified as rational

numbers because they are integers. making it 92 .

10

5. Which of the following is/are rational

numbers? Justify your answer. Since both 92 and 10 are integers, then this is a rational

a. 2.5 number.

b. √2

c. √81 Since 9.2 is a terminating decimal, it is called a rational

d. 29.5826593…

number.

6. Which of the following is/are irrational

numbers? Justify your answer.

a. √7

b. 1.973258...

c. 10.5

d. 1

3

Solution: Solution:

(a) because 2.5 can be written as a fraction (a) because seven is not a perfect square

(b) since it is a non-terminating decimal that lacks and

(c) 81 is a perfect square; 81 = 9 and 9 can be written as a cannot be written in a fraction from.

ratio in the form of 9 .

1

23 Copyright © Swun Math Grade 8 Unit 1 Lesson 3 P TE

Challenge Problems

Directions: Read and solve.

1. True or false? 1 is a rational number. 2. Label each real number as rational or

0 irrational. Justify your answer.

Justify your answer.

√25 √5

22

______ ______

Solution: Solution:

False

Remember, the rules of fractions require a non-zero √25 is rational because it can be rewritten as 5

integer to be in the denominator position. 2 2

This number doesn’t exist; therefore, it is not rational. √5

2 is irrational because 5 is not a perfect square which

makes √5 not an integer. This number cannot be

rewritten in fraction form with an integer in the

numerator.

Extension Activity

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

* MP4: Apply mathematics in everyday life.

Directions: Using the integers 1-9, fill in the boxes to satisfy each number type.

� = a whole number

� = a rational number

� = a rational number

� = a rational number

� = an irrational number

� = an irrational number

Copyright © Swun Math Grade 8 Unit 1 Lesson 3 P TE 24

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 are irrational numbers related to rational numbers?

MP8: How would you prove that you are dealing with a rational number?

Closure Grade 8 Unit 1 Lesson 3 P TE

Copyright © Swun Math

25

Homework Name: ___________________________

Date: ___________________________

Unit 1 · Lesson 3: Rational vs. Irrational Numbers

Objective: I will identify rational and irrational numbers and the difference between them.

Vocabulary Steps:

Squared Numbers (Perfect Square): the result after 1. Determine if the number can be

multiplying an integer by itself expressed as a fraction.

If so, the number is rational.

• Check for terminating or

repeating decimals.

• Look for perfect squares.

If not, the number is irrational.

2. Justify your answer.

Rational Numbers: whole numbers, fractions, and decimals

represented as a ratio of two integers; can be represented in

fractional form:

integers and y ≠ zero.

3

1.5 = 2 = ratio

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 , , √10, √1.6, −√123

Example # 1

Directions: Determine if the given real number is rational or irrational. Justify your answer.

Which of the following is an irrational number and why? True or False? 0.333… is an irrational number. Justify your

a. 7 answer.

b. 10278

c. √3 Solution:

d. 99 False; Understand that this is a common rational number

1 , expressed in decimal form. Even though the decimal

Solution: 3

Irrational; the decimal expansion of this number does not expansion may seem irrational because it never ends, the

terminate or repeat. fact that it has a pattern of a repeating digit makes it a

rational number

Copyright © Swun Math Grade 8 Unit 1 Lesson 3 P TE 26

Homework

Unit 1 · Lesson 3: Rational vs. Irrational Numbers

Directions: Determine if the number is rational or irrational. Justify your answer.

1. √18 2. 0.415415415…

3. 27 4. √100

3 6. 3. 1���4�

5. 0.12343752….

Explain the steps you used to solve problem number _______.

______________________________________________________________________________________________________________________

______________________________________________________________________________________________________________________

______________________________________________________________________________________________________________________

______________________________________________________________________________________________________________________

______________________________________________________________________________________________________________________

27 Copyright © Swun Math Grade 8 Unit 1 Lesson 3 P TE

Answer Key

Extension Activity

Answers will vary. Sample responses:

a) 25

b) 4

c) 9

d) 36

e) 7

f) 18

Homework 2. Rational; it is repeating with a set pattern.

1. Irrational

Answers may vary. It is irrational because 18 is not

a perfect square so the √18 is not an integer.

3. Rational 4. Rational; it is a perfect square.

Answers may vary. It is a rational number because

the fraction has an integer in the numerator and

an integer in the denominator.

5. Irrational; the decimal is not a terminating 6. Rational; it is a decimal with a set pattern. The

decimal and it is not a decimal with a set pattern. repeating pattern is 14

Copyright © Swun Math Grade 8 Unit 1 Lesson 3 P TE 28

Convert Repeating Decimals MPs Applied MP

to Fractions

* Embedded MP

Conceptual Lesson

12 3 45678

Grade 8 · Unit 1 · Lesson 4

* *

MC: 8.NS.1

Problem of the Day Student Journal Pages

16-19

Objective: I will convert a repeating decimal into a fraction.

Vocabulary Teacher Resources

Rational Numbers: whole numbers, fractions, Considerations:

and decimals represented as a ratio; can be

represented in fractional form: Consider having students recall how multiplying a

decimal by multiples of ten affects the decimal. Use a

where are integers and y ≠ zero. calculator to demonstrate the equivalency between

the fraction and the repeating decimal. Have students

3 work in pairs or groups to discuss/compare the

1.5 = 2 = ratio example problems and find the patterns.

Irrational Numbers: the set of all numbers that Have students practice the steps with smaller

cannot be written as a ratio of two integers; it numbers if they are struggling with large patterns.

cannot be written as a simple fraction because

the decimal is non-terminating and non- Steps:

repeating 1. Determine the repeating digit(s).

2. Write an equation where x = the repeating

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

decimal.

Repeating Decimal: a decimal number that has 3. Label this Equation #1.

digits that repeat indefinitely; the repeating digit 4. Multiply both sides of the equation by the

is marked with a bar notation

power of 10 equal to the number of repeating

digits.

5. Label this Equation #2. Keep a set of the

repeating digits to the right of the decimal.

6. Subtract Equation #1 from Equation #2.

7. Solve for x by dividing each side of the

equation by the coefficient.

8. Simplify.

1 = 0.3333 = 0. 3� Application of MPs:

3

MP5: What tool can help you convert a repeating

Coefficient: a number used to multiply a variable decimal into a fraction?

7m; 7 is the coefficient A tool I can use to help convert a repeating

decimal into a fraction is ________________.

MP7: What pattern was used when converting a

repeating decimal into a fraction?

A pattern I used when converting a repeating

decimal into a fraction is _________________.

29 Copyright © Swun Math Grade 8 Unit 1 Lesson 4 C TE

Input/Model

(Teacher Presents)

Directions: Convert the decimal into a fraction using both methods.

1.

0. 4�

Solution:

Create two equations:

• Equation #1

o x = 0.4 (single digit pattern).

• Determine the number of repeating digits (1) and

multiply by that power of 10.

• Equations #2

o 10x = 4.4 (multiply by 10 it is a single digit

pattern).

• Subtract the first equation from the second.

• 10x − x = 4.4 − 0.4

9x = 4

• x = 4

9

2.

0. 2�

Solution:

x = 0. 2�

Create two equations:

• Equation #1

o x = 2.2

• Equation #2

o 10x = 2.2…

Solve:

10x – x = 2.2… – 0.2…

9x = 2

99

x =2

9

Answer: 2

9

Copyright © Swun Math Grade 8 Unit 1 Lesson 4 C TE 30

Structured Guided Practice

(A/B Partners Practice)

Directions: Convert the decimal into a fraction.

1.

0. 1���8�

Solution:

18 2

x = 99 = 11

2. 0. 1�

Solution:

1

x = 9

31 Copyright © Swun Math Grade 8 Unit 1 Lesson 4 C TE

Final Check for Understanding

(Teacher Checks Work)

Directions: Convert the decimal into a fraction.

1.

0. 5�

Solution: 0. �1�2��3�

x =5

9

2.

Solution: 41

123 333

x = 999 =

Closure

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

MP5: What tool can help you convert a repeating decimal into a fraction?

MP7: What pattern was used when converting a repeating decimal into a fraction?

Copyright © Swun Math Grade 8 Unit 1 Lesson 4 C TE 32

Homework Name: ___________________________

Date: ___________________________

Unit 1 · Lesson 4: Convert Repeating Decimals to

Fractions

Objective: I will convert a repeating decimal into a fraction.

Vocabulary Steps:

Rational Numbers: whole numbers, fractions, 1. Determine the repeating digits.

and decimals represented as a ratio of two 2. Write an equation where x = the repeating

integers; can be represented in fractional form:

decimal.

where are integers and y ≠ zero. 3. Label this Equation #1.

4. Multiply both sides of the equation by the

3

1.5 = 2 = ratio power of 10 equal to the number of repeating

digits.

Irrational Numbers: the set of all numbers that 5. Label this Equation #2. Keep a set of the

cannot be written as a ratio of two integers; it repeating digits to the right of the decimal.

cannot be written as a simple fraction because 6. Subtract Equation #1 from Equation #2.

the decimal is non-terminating and non- 7. Solve for x by dividing each side of the

repeating equation by the coefficient.

8. Simplify.

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

Repeating Decimal: a decimal number that has

digits that repeat indefinitely; the repeating digit

is marked with a bar notation

1 = 0.3333 = 0. 3�

3

Coefficient: a number used to multiply a variable

7m; 7 is the coefficient

33 Copyright © Swun Math Grade 8 Unit 1 Lesson 4 C TE

Homework

Unit 1 · Lesson 4: Convert Repeating Decimals to

Fractions

Examples

Directions: Convert the decimal into a fraction. 0. 2�

0. 4�

Solution: Solution:

x = 0. 2�

Create two equations:

• Equation #1 Create two equations:

• Equation #1

o x = 0.4 (single digit pattern).

• Determine the number of repeating digits (1) and o x = 2. 2�

• Equation #2

multiply by that power of 10.

• Equations #2 o 10x = 2.2…

o 10x = 4.4 (multiply by 10 it is a single digit Solve:

pattern). 10x – x = 2.2… – 0.2…

• Subtract the first equation from the second. 9x = 2

• 10x − x = 4.4 − 0.4 99

x =2

9x = 4

9

• Answer: x = 4

Answer: x= 2

9

9

Copyright © Swun Math Grade 8 Unit 1 Lesson 4 C TE 34

Homework

Unit 1 · Lesson 4: Convert Repeating Decimals to

Fractions

Directions: Convert the decimal into a fraction.

1. 0.7� 2. 0. 6�

3. 0. 5� 4. 0. 3�

5. 0. 3��4��5� 6. 0. �2��7�

35 Copyright © Swun Math Grade 8 Unit 1 Lesson 4 C TE

Answer Key

Homework

1. 0.777… 2. 0.666…

10x = 7.777… 10x = 6.666…

─x = ─0.777… ─x = ─0.666…

9x = 7 9x = 6

x = 7 x = 6

9 9

3. 5 4. 1

9 3

5. 115 6. 3

333 11

Copyright © Swun Math Grade 8 Unit 1 Lesson 4 C TE 36

Convert Repeating Decimals MPs Applied MP

to Fractions

* Embedded MP

Procedural Lesson

Grade 8 · Unit 1 · Lesson 5 12 3 4567 8

MC: 8.NS.1 * * * **

Problem of the Day Student Journal Pages

20-25

Objective: I will convert repeating decimals into fractions.

Vocabulary Teacher Resources

Rational Numbers: whole numbers, fractions, Considerations:

and decimals represented as a ratio of two

integers; can be represented in fractional form: Consider having students recall how multiplying a

decimal by multiples of ten affects the decimal.

where are integers and y ≠ zero. Students have been multiplying and dividing by 10 to

move the decimal since 5th grade. Consider using a

3 calculator to demonstrate the equivalency between

1.5 = 2 = ratio the fraction and the repeating decimal. Have students

work in pairs or groups to discuss/compare the

example problems and find the patterns.

Irrational Numbers: the set of all numbers that Steps:

cannot be written as a ratio of two integers; it 1. Determine the repeating digits.

cannot be written as a simple fraction because 2. Write an equation where x = the repeating

the decimal is non-terminating and non-

repeating decimal.

3. Label this Equation #1.

0.101101110… , �32 , π, √10, √1.6, −√123 4. Multiply both sides of the equation by the

Repeating Decimal: a decimal number that has power of 10 equal to the number of repeating

digits that repeat indefinitely; the repeating digit digits

is marked with a bar notation 5. Label this Equation #2. Keep a set of the

repeating digits to the right of the decimal.

6. Subtract Equation #1 from Equation #2.

7. Solve for x by dividing each side of the

equation by the coefficient.

8. Simplify.

1 = 0.3333 = 0. 3� Application of MPs:

3

MP3: How did you test if your methods worked?

Coefficient: a number used to multiply a variable I tested to see if my method worked by

7m; 7 is the coefficient ____________.

MP7: What pattern was used when converting a

repeating decimal into a fraction?

A pattern I used when converting a repeating

decimal into a fraction is _________________.

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

37

Input/Model

(Teacher Presents)

Directions: Convert the decimals into fractions.

1.

1.222…

Solution:

• Determine the repeating pattern: 2

• x = 1.2�

Create two equations:

• Equation 1: x = 1.2�

• Equation 2: 10x = 12.2�

Solve:

• 10x ─ x = 12.2 ─ 1.2

• 9x = 11

11

• x = 9

2.

0.1243243…

Solution:

Determine the repeating pattern: 243

x = 0.1�2�4��3�

Create two equations: * Remember the goal is to get the repeating numbers to the right of the decimal in both

equations, so choose powers of ten that will help you to do so.

• 10x = 1.�2�4��3� (keep the repeating pattern right after the decimal)

• 10000x = 1243. �2�4��3�

Solve:

• 10000x – 10x = 1243.243… – 1.243243…

9990x = 1242

x= 1242 = 621 =56593

9990 4995

Answer: 69

553

Copyright © Swun Math Grade 8 Unit 1 Lesson 5 P TE 38

Structured Guided Practice

(A/B Partners Practice)

Directions: Convert the decimals into fractions.

1.

1.888…

Solution:

17 8

9 =1 9

2.

1.03292929…

Solution:

5113 or 1 163

4950 4950

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

Final Check for Understanding

(Teacher Checks Work)

Directions: Convert the decimals into fractions.

1.

1.3030…

Solution:

129 = 43

99 33

2.

0.13636…

Solution:

27 = 3

198 22

Copyright © Swun Math Grade 8 Unit 1 Lesson 5 P TE 40

Student Practice Name: ___________________________

Date: ___________________________

Unit 1 · Lesson 5: Convert Repeating Decimals to

Fractions

Directions: Convert the decimals into fractions.

1. 0.222… 2. 0.5959…

Solution: Solution:

2

9 59

99

3. 0.3777…

4. 0.58383…

Solution: Solution:

7

17 12

45

6. 4.206206…

5. 0.8369369…

Solution: Solution:

8361 =1912190 4202

9990 999

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

41

Challenge Problems

Directions: Solve. 2. What is the previous problem telling you?

Can you think of another way of showing

1. Convert 0.999… into a fraction. Consider that this is true?

rounding. Also, remember the fractions

with a base nine--how is this example

different?

Solution: Solution:

Create two equations: 1 + 1 + 1 =1 and 0.333… + 0.333… + 0.333… = 0.999…

● 1x = 0.999… 333

● 10x = 9.99… Since 1 + 1 + 1 =1

Solve: 333

10x – 1x = 9.99… - 0.999…

and 0.333… + 0.333… + 0.333… = 0.999

9x = 9 then, 1 = 0.999…

x = 9 = 1 Answer: x = 0.999

9

Answer: x=1

Extension Activity

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

* MP4: Apply mathematics in everyday life.

Imagine you are a scientist working on a project in a lab, and you need to make a calculation

using a repeating decimal. Provide a pro and a con for estimating the value of the repeating

decimal to a specific place value versus converting to an equivalent fraction.

Copyright © Swun Math Grade 8 Unit 1 Lesson 5 P TE 42

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:

MP3: How did you test if your methods worked?

MP7: What pattern was used when converting a repeating decimal into a fraction?

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

Homework Name: ___________________________

Date: ___________________________

Unit 1 · Lesson 5: Convert Repeating Decimals to

Fractions

Objective: I will convert repeating decimals into fractions.

Vocabulary Steps:

Rational Numbers: whole numbers, fractions, and 1. Find the repeating digits.

decimals represented as a ratio of two integers; can be 2. Write an equation where x = the

represented in fractional form:

repeating decimal. Label this

where are integers and y ≠ zero. Equation #1.

3. Multiply both sides of the equation

3 by the power of 10 equal to the

1.5 = 2 = ratio number of repeating digits. Label

this Equation #2. Keep a set of the

Irrational Numbers: the set of all numbers that cannot be repeating digits to the right of the

written as a ratio of two integers; it cannot be written as a decimal.

simple fraction because the decimal is non-terminating 4. Subtract Equation #1 from Equation

and non-repeating. #2.

5. Solve for x by dividing each side of

0.101101110… , �32 , π, √10, √1.6, −√123 the equation by the coefficient.

6. Simplify.

Repeating Decimal: a decimal number that has digits that

repeat indefinitely; the repeating digit is marked with a

bar notation

1 = 0.3333 = 0. 3�

3

Coefficient: a number used to multiply a variable

7m; 7 is the coefficient

Copyright © Swun Math Grade 8 Unit 1 Lesson 5 P TE 44

Homework

Unit 1 · Lesson 5: Convert Repeating Decimals to

Fractions

Example # 1

Directions: Convert the decimals into fractions.

Based on your experience with conversions, what can you predict about the answer?

0.222…

Solution:

• Using horizontal subtraction, create two equations:

• x = 0.222…

• 10x = 2.22…

Solve:

10x – x = 2.22… – 0.222…

9x = 2

99

x = 2

9

Answer: 2

9

Possible answer: I can predict that the denominator will be 9 since the decimal has a one digit-repeating pattern.

Example # 2

The following decimal has a triple-digit repeating pattern. How does this affect the outcome of its fraction

form?

0.1243243…

Solution:

• Use vertical subtraction.

• x = 0.1243243…

Create two equations:

• 10x = 1.243243…

• 10000x – 10x = 1243.243243…

Solve:

• 10000x – 10x = 1243.243… – 1.243243…

9990x = 1242

x = 1242 = 621

9990 4995

69

Answer: 553

Possible answer: Since the decimal contains one digit plus a repeating pattern containing 3 digits, the unsimplified fraction

has a 4-digit denominator.

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