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52 Unit 1 Number Theory

Advance PreparationFamiliarize yourself with the use of the square root key on your students’ calculators.

Teacher’s Reference Manual, Grades 4–6 pp. 74, 75, 79–83, 94–98, 267–269

Unsquaring NumbersObjective To introduce the concept of square roots and the

use of the square-root key on a calculator.

Key Concepts and Skills• Use exponential notation to name square

numbers, and explore the relationship

between square numbers and square roots. 

[Number and Numeration Goal 4]

Key ActivitiesStudents investigate “unsquaring” numbers

without using the square-root key on a

calculator and use the square-root key to

test their answers. They explore properties

of square numbers and their square roots.

Key Vocabularyunsquaring a number � square root � 

square-root key

MaterialsMath Journal 1, p. 23

Study Link 1�7

calculator � overhead calculator (optional) � 

16 counters (optional)

Playing Multiplication Top-It(Extended-Facts Version)Student Reference Book, p. 334

Math Masters, p. 493

per partnership: 4 each of number

cards 1–10 (from the Everything Math

Deck, if available), calculator

Students use their knowledge of

extended facts to form and compare

numbers.

Ongoing Assessment: Recognizing Student Achievement Use Math Masters, page 493. [Operations and Computation Goal 2]

Math Boxes 1�8Math Journal 1, p. 24

Students practice and maintain skills

through Math Box problems.

Study Link 1�8Math Masters, p. 22

Students practice and maintain skills

through Study Link activities.

ENRICHMENTComparing Numbers with Their SquaresMath Masters, p. 23

calculator

Students investigate the relationship between

numbers and their squares.

EXTRA PRACTICE

5-Minute Math5-Minute Math™, p. 108

slate or paper

Students practice using the square root sign.

Teaching the Lesson Ongoing Learning & Practice

132

4

Differentiation Options

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eToolkitePresentations Interactive Teacher’s

Lesson Guide

Algorithms Practice

EM FactsWorkshop Game™

AssessmentManagement

Family Letters

CurriculumFocal Points

Common Core State Standards

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Lesson 1�8 53

Getting Started

Mental Math and ReflexesPose the following problems. Have students write an expression as you describe the calculation. Students are not expected to calculate the solution. Answers may vary.

Math MessageFind the numbers that make these statements true.

∗

= 4

2

= 81

Study Link 1�7 Follow-Up Have partners compare answers and resolve differences.

1 Teaching the Lesson

▶ Math Message Follow-Up WHOLE-CLASSDISCUSSION

Algebraic Thinking Ask a volunteer to read the first problem. Some number times some number equals 4. What numbers could the placeholders in this problem represent? The factors of 4; 1 and 4; 2 Write the first problem on the board or a transparency, replacing both placeholders with the letter n. Ask students what number the letter n (the variable) represents. 2 Explain that the variable can only represent one number if the number sentence is true. Tell students that unknown numbers will be represented using variables as the placeholders. Rewrite the second problem using the variable m. m2 = 81 Ask a volunteer to read the problem. m squared equals 81; m ∗ m equals 81 What is the number m that makes this number sentence true? 9

▶ “Unsquaring” Numbers

WHOLE-CLASSACTIVITY

Begin this activity by explaining that solving problems like the Math Message problems requires unsquaring a number. We needed to undo the operation that squared the number. If students square a number, they multiply it by itself to get the product. Given the product of a squared number, they have to undo the multiplication in order to identify the number that was squared.

4 ∗ 4 = p Square the number 4 to find p.

n ∗ n = 16 Unsquare the number 16 to find n.

The difference between squaring and unsquaring a number

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The sum of 9 and 8 9 + 8

7 less than 7 7 - 7

The quotient of 24 divided by 6 24 / 6

7 less than the product of 2 and 9 (2 ∗ 9) - 7

Double 8 and then add 0 more (2 ∗ 8) + 0

0 times the sum of 8 and 2 0 ∗ (8 + 2)

8 less than the sum of 10 and 5 (10 + 5) - 8

3 more than triple 3 3 + (3 ∗ 3)

10 less than triple 10 (3 ∗ 10) - 10

Interactive whiteboard-ready

ePresentations are available at

www.everydaymathonline.com to

help you teach the lesson.

Adjusting the Activity

Have students use counters to build the

square array for 16. Note that when 16 is

unsquared, the result is the same as the

number of rows, or the number of columns,

of the original square.

A U D I T O R Y � K I N E S T H E T I C � T A C T I L E � V I S U A L

ELL

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Unsquaring NumbersLESSON

1�8

Date Time

You know that 62 � 6 � 6 � 36. The number 36 is called the square of 6. If you

unsquare 36, the result is 6. The number 6 is called the square root of 36.

1. Unsquare each number. The result is its square root. Do not use the key

on your calculator.

Example: � 144 The square root of 144 is .

a.2

� 225 The square root of 225 is .

b.2

� 729 The square root of 729 is .

c.2

� 1,600 The square root of 1,600 is .

d.2

� 361 The square root of 361 is .

2. Which of the following are square numbers? Circle them.

576 794 1,044 4,356 6,400 5,770

List all factors of each square number. Make a factor rainbow to check your work. Then

fill in the missing numbers.

3. 49: 2

� 49 The square root of 49 is .

4. 64: 2

� 64 The square root of 64 is .

5. 81: 2

� 81 The square root of 81 is .

6. 100: 2

� 100

The square root of 100 is .10

1010 20 25 50 1005421

999 27 8131

888 16 32 64421

777 491

1919

4040

2727

1515

12122

Math Journal 1, p. 23

Student Page

54 Unit 1 Number Theory

Ask: What number, multiplied by itself, is equal to 289? Give students a few minutes to find the number. They may use their calculators if they wish.

After a few minutes, survey the class for their solution strategies. Most students will have used one of the following approaches:

� The random method: Some students might have tried various numbers without using a system to guide their choices.

� The “squeeze” method: Some students might have tried various numbers, each time using the result to help select their next choice. To unsquare 289, you might:

● Try 10: 102 = 100; much less than 289

● Try 20: 202 = 400; more than 289

Then try numbers between 10 and 20, probably closer to 20 than to 10.

● Try 18: 182 = 324; still too large, but closer.

● Try 17: 172 = 289; the answer is 17.

� Endings and products: When students established an interval, such as the interval from 10 to 20, some might have reasoned that since 17 ends in 7, and 7 ∗ 7 = 49; then 17 should be the next choice because 289 also ends in 9.

If students mention using the square-root key on their calculator, acknowledge that this is an efficient way of unsquaring a number, but the focus on this portion of the lesson is to help them understand the process of squaring and unsquaring numbers before they use the calculator function.

Give students a few more square numbers to unsquare. (See margin.) Challenge them to use as few guesses as possible.

Tell students that when they unsquare a number, they have found the square root of the number. What number squared is 64? 8 So what is 64 unsquared? 8, because 8 ∗ 8 = 64 What is the square root of 64? 8

▶ Finding the Square Root PARTNERACTIVITY

of Numbers(Math Journal 1, p. 23)

Allow partners a few minutes to complete Problems 1 and 2. Survey the class for suggestions for checking the answers in Problem 1. Most students will respond with the following possibilities:

� Multiply the square root of a number by itself.

� Use the square-root key, for example , on the calculator to find the square root of a number.

To support English language learners, write the following on the board: 82 = 8 ∗ 8 = 64. The square of 8 is 64. The square root of 64 is 8.

ELL

196 14 10,000 100

1,024 32 676 26

7,225 85 3,136 56

441 21 900 30

1,849 43 5,041 71

Suggested square numbers for students to “unsquare”

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Name Date Time

Top-It Record Sheet

Round Player 1 >, <, = Player 2

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Round Player 1 >, <, = Player 2

Sample

1

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Name Date Time

Top-It Record Sheet

132

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132

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EM3cuG5MM_U03_067-101.indd 493 11/10/10 4:36 PM

Math Masters, p. 493

Game Master

Date Time

5. Subtract. Show your work.

a. b. c. d.

1. Write < or >.

a. 3.8 > 0.83

b. 0.4 > 0.30

c. 6.24 > 6.08

d. 0.05 < 0.5

e. 7.12 < 7.2

3. List all the factors of 64.

1, 2, 4, 8, 16, 32, 64

4. In the morning, I need 30 minutes to

shower and dress, 15 minutes to eat, and

another 15 minutes to ride my bike to

school. School begins at 8:30 A.M. What is

the latest time I can get up and still get to

school on time?

7:30 A.M.

2. Round each number to the

nearest thousand.

a. 8,692 9,000

b. 49,573 50,000

c. 2,601,458 2,601,000

d. 300,297 300,000

e. 599,999 600,000

777

− 259

518

555

− 125

430

5,009

− 188

4,821

8,435

− 997

7,438

9 32 33

10 12 244 245

4 249

15–17

Math BoxesLESSON

1�8

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Math Journal 1, p. 24

Student Page

Lesson 1�8 55

Explain that in the same way that the class has used a calculator to test the result of other computations, they will use a calculator to test that they have accurately found the square-root of a number. If available, use an overhead calculator to demonstrate how to use the square-root function key. Model for students how to test the answers in Problem 2. Emphasize the following points:

� If the display shows a whole number, then the original number is a square number. For example, 576 is a square number because using the square-root key displays a whole number—24.

� If the display shows a decimal, then the original number is not a square number. For example, 794 is not a square number because using the square-root key displays a decimal—28.178006 (rounded to 6 decimal places).

Ask students to check the remaining numbers in Problem 2. Partners complete the remaining problems on journal page 23.

2 Ongoing Learning & Practice

▶ Playing Multiplication Top-It PARTNERACTIVITY

(Extended-Facts Version)(Student Reference Book, p. 334; Math Masters, p. 493)

Students apply their knowledge of basic multiplication facts to extended facts by playing Multiplication Top-It (Extended-Facts Version). Students use the same rules as described on Student Reference Book, page 334; however, they attach a zero to the first card drawn and multiply by the second card drawn. For example, suppose 5 is the first number drawn; 7 is the second number drawn. The student would compute: 5 ∗ 10 = 50; 50 ∗ 7 = 350.

Ongoing Assessment: Recognizing Student Achievement

Math Masters

Page 493 �

Use the Top-It Record Sheet (Math Masters, page 493) to assess students’

ability to solve and compare multiplication extended fact problems. Have the

class record and compare 70 ∗ 8 and 50 ∗ 9 for the sample record. Partners

record their first five rounds. Students are making adequate progress if they

correctly solve and compare all five extended facts. Some students may be able

to solve and compare problems with both factors multiplied by 10: 70 ∗ 80.

[Operations and Computation Goal 2]

▶ Math Boxes 1�8

INDEPENDENT ACTIVITY

(Math Journal 1, p. 24)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 1-6. The skill in Problem 5 previews Unit 2 content.

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STUDY LINK

1�8

Name Date Time

Factor Rainbows, Squares, and Square Roots

1. List all the factors of each square number. Make a factor rainbow to check

your work. Then fill in the missing numbers.

Reminder: In a factor rainbow, the

product of each connected factor pair

should be equal to the number itself.

For example, the factor rainbow for

16 looks like this:

Example:

4:

2

� 4 The square root of 4 is .

25:

2

� 25 The square root of 25 is .

9:

2

� 9 The square root of 9 is .

36:

2

� 36 The square root of 36 is .66

33

55

2. Do all square numbers have an odd number of factors?

Unsquare each number. The result is its square root. Do not use the

square root key on your calculator.

3.

2

� 121 4.

2

� 2,500

The square root of 121 is . The square root of 2,500 is .

5. 4,318 6. 36 7. 2,852

� 1,901 � 85 � 5

8. 50 � 6 ∑ 9. 333 � 291 � 428 R2

14,2603,0606,219

5011

5011

Yes

1, 2, 4

1, 5, 25

2 2

1, 3, 9

1, 2, 3, 4, 6, 9, 12, 18, 36

4 8 1621

2413 91

5 251 6 9 12 18 364321

271

Practice

1 º 16 � 16 2 º 8 � 16 4 º 4 � 16

Math Masters, p. 22

Study Link Master

LESSON

1�8

Name Date Time

Comparing Numbers with Their Squares

1. a. Unsquare the number 1.

2

� 1

b. Unsquare the number 0.

2

� 0

2. a. Is 5 greater than or less than 1?

b. 52�

c. Is 52 greater than or less than 5?

3. a. Is 0.50 greater than or less than 1?

b. Use your calculator. 0.502�

c. Is 0.502 greater than or less than 0.50?

4. a. When you square a number, is the result

always greater than the number you started with?

b. Can it be less?

c. Can it be the same?

5. Write 3 true statements about squaring and unsquaring numbers.

Answers vary.

YesYes

No

Less than0.25

Less than

Greater than25

Greater than

01

Math Masters, p. 23

Teaching Master

56 Unit 1 Number Theory

Writing/Reasoning Have students write a response to the following: Was Jason correct when he said that 64 is a prime number in Problem 3? Explain your answer.

Sample answer: Jason was not correct. The factors of 64 are 1, 2, 4, 8, 16, 32, and 64. Because it has more than two factors, it is a composite number. A prime number has only two factors.

▶ Study Link 1�8

INDEPENDENT ACTIVITY

(Math Masters, p. 22)

Home Connection Students list all the factors of the first 4 square numbers, write numbers in exponential notation, and identify square roots.

3 Differentiation Options

ENRICHMENT PARTNER ACTIVITY

▶ Comparing Numbers with 15–30 Min

Their Squares(Math Masters, p. 23)

To further explore factoring numbers, have students investigate the relationship between numbers and their squares. Ask students to think about the following question as they work the problems for this activity: When you square a number, will the result be greater than, less than, or equal to the number?

Guide students to recognize that squaring a number does not necessarily result in a number that is greater than the original number. For example, both 0 and 1 are equal to their squares. (See Problem 1.) Ask students what they noticed about the numbers and relationships they found in Problem 3. The number 0.50 is a decimal; the square was smaller. Explain that the square of a number that is greater than 0, but less than 1, is always less than the original number. Ask volunteers to suggest other numbers between 0 and 1 for partners to square.

EXTRA PRACTICE

SMALL-GROUPACTIVITY

▶ 5-Minute Math 5–15 Min

To offer students more experience with using the square root sign, see 5-Minute Math, page 108.

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Copyright

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right

Gro

up/M

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raw

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493

Name Date Time

Top-It Record Sheet

Round Player 1 >, <, = Player 2

Sample

1

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5

Round Player 1 >, <, = Player 2

Sample

1

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Name Date Time

Top-It Record Sheet

132

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EM3cuG5MM_U03_067-101.indd 493EM3cuG5MM_U03_067-101.indd 493 11/10/10 4:36 PM11/10/10 4:36 PM