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Lesson 2�7 115

Advance PreparationFor the Math Message, prepare 3 chart-paper-size name-collection boxes. Label the name-collection boxes 1, 10,

and 100, respectively. Position these in separate locations around the classroom, or post them together on the board.

Teacher’s Reference Manual, Grades 4–6 pp. 256–264

Key Concepts and Skills• Use place value to make magnitude

estimates for products.

[Number and Numeration Goal 1]

• Make magnitude estimates for problems.

[Operations and Computation Goal 6]

• Round numbers to make magnitude

estimates for multiplication problems.


Key ActivitiesStudents use rounding to estimate the place

value of products and mark their estimates

on a magnitude bar. Students practice

estimating products by playing Multiplication

Bull’s-Eye.

Ongoing Assessment: Informing Instruction See page 117.

Key Vocabularymagnitude estimate

MaterialsMath Journal 1, p. 47

Student Reference Book, pp. 5 and 323

Study Link 2�6

Class Data Pad � per partnership: 4 each of

number cards 0–9 (from the Everything Math

Deck, if available), 1 six-sided die, calculator

Experimenting with SpinnersMath Journal 1, p. 48

Students apply probability concepts

by designing spinners.

Math Boxes 2�7Math Journal 1, p. 49

Students practice and maintain skills

through Math Box problems.

Ongoing Assessment: Recognizing Student Achievement Use Math Boxes, Problems 3 and 4. [Operations and Computation Goal 1]

Study Link 2�7Math Masters, p. 53

Students practice and maintain skills

through Study Link activities.

READINESS

Practicing Extended FactsStudent Reference Book, p. 18

Math Masters, p. 54

4 each of number cards 0–9 (from the

Everything Math Deck, if available)

Students investigate a shortcut for solving

extended multiplication facts.

EXTRA PRACTICE

5-Minute Math5-Minute Math™, pp. 19, 95, and 182

Students practice estimation.

ENRICHMENTUsing Multiplication PatternsStudent Reference Book, p. 5

Math Masters, p. 66A

Students analyze why there are patterns in

the number of zeros in the factors and

products.

Teaching the Lesson Ongoing Learning & Practice Differentiation Options

�� Estimating ProductsObjective To provide experiences with making and using

magnitude estimates for products of multidigit numbers,

including decimals.

eToolkitePresentations Interactive Teacher’s

Lesson Guide

Algorithms Practice

EM FactsWorkshop Game™

AssessmentManagement

Family Letters

CurriculumFocal Points

Common Core State Standards

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Magnitude Estimates for ProductsLESSON

2 � 7

Date Time

A magnitude estimate is a very rough estimate of the answer to a problem. A

magnitude estimate will tell you whether the exact answer is in the tenths, ones, tens,

hundreds, thousands, and so on.

For each problem, make a magnitude estimate. Ask yourself, “Is the answer in the

tenths, ones, tens, hundreds, thousands, or ten-thousands?” Circle the appropriate box.

Then write a number sentence to show how you estimated. Do not solve the problems.

Example: 14 � 17 1. 56 � 37

How I estimated How I estimated

2. 7 � 326 3. 95 � 48


4. 5 � 4,127 5. 46 � 414


6. 4.5 � 0.6 7. 7.6 � 9.1


8. 160 � 2.9 9. 0.8 � 0.8


1 � 1 � 1200 � 3 � 600

8 � 9 � 725 � 1 � 5

50 � 400 � 20,00010 � 4,000 � 40,000

100 � 50 � 5,00010 � 300 � 3,000

60 � 40 � 2,40010 � 20 = 20010s 100s 1,000s 10,000s 10s 100s 1,000s 10,000s

10s 100s 1,000s 10,000s 10s 100s 1,000s 10,000s

10s 100s 1,000s 10,000s 10s 100s 1,000s 10,000s

0.1s 1s 10s 100s

0.1s 1s 10s 100s

0.1s 1s 10s 100s

0.1s 1s 10s 100s

Math Journal 1, p. 47

Student Page

116 Unit 2 Estimation and Computation

Getting Started

Mental Math and Reflexes

Write the problems on the board or the Class Data Pad so students can visually recognize the patterns.

Math MessageUse the numbers 10, 6, 9, 8, and 5 to make expressions that are equivalent names for 1, 10, and 100. Use addition, subtraction, multiplication, division, or exponents, and try to use all 5 numbers.

Record your expressions on the class name-collection box for that number.

Study Link 2�6 Follow-Up Briefly review the answers.

8 ∗ 4 32

8 ∗ 40 320

80 ∗ 400 32,000

60 ∗ 2 120

60 ∗ 20 1,200

60 ∗ 200 12,000

30 [10s] 300

300 [10s] 3,000

300 [100s] 30,000

1 Teaching the Lesson

▶ Math Message Follow-Up WHOLE-CLASSDISCUSSION

Allow students time to discuss and compare the expressions in the three name-collection boxes. Survey the class for expressions that individual students had not considered. Ask students to compare the differences between strategies they used to rename 1 and strategies they used to rename 10. Have students compare strategies they used to rename 10 to strategies they used to rename 100. One approach is to look for ways to make multiples. For example, any expression that names 10 can be squared to name 100.

▶ Estimating Products WHOLE-CLASSDISCUSSION

(Math Journal 1, p. 47)

Through much of their work with arithmetic, students are encouraged to make an estimate prior to solving a problem. As students work with larger numbers, particularly using multiplication or division, making magnitude estimates allows them to know whether the solution to a problem is reasonable. This is true whether they are working with a calculator, or using paper and a pencil.

Remind students that a magnitude estimate is a very rough estimate that answers questions such as: Is the solution in the tens? Hundreds? Thousands?

Pose the following problem: Is the result of 14 ∗ 17 in the tens? Hundreds? Thousands? Ask students to justify their answers. The product of 14 ∗ 17 must be greater than 10 ∗ 10 = 100; it must be less than 20 ∗ 20 = 400. So the product is in the hundreds.

Explain that because magnitude estimates predict answers in terms of multiples of 10, one approach is to round the factors first.

ELL

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Tom rolls a 3, so the target range of the product is from 1,001 to 3,000.He turns over a 5, a 7, a 9, and a 2.

Tom uses estimation to try to form 2 numbers whose product falls within thetarget range—for example, 97 and 25.

He then finds the product on the calculator: 97 * 25 � 2,425.

Since the product is between 1,001 and 3,000, Tom has hit the bull's-eye andscores 1 point.

Some other possible winning products from the 5, 7, 2, and 9 cards are: 25 * 79, 27 * 59, 9 * 257, and 2 * 579.

Multiplication Bull’s-Eye

Materials � number cards 0–9 (4 of each)� 1 six-sided die� 1 calculator

Players 2Skill Estimating products of 2- and 3-digit numbersObject of the game To score more points. Directions

1. Shuffle the deck and place it number-side down on the table.

2. Players take turns. When it is your turn:

♦ Roll the die. Look up the target range of the product in the table at the right.

♦ Take 4 cards from the top of the deck.

♦ Use the cards to try to form 2 numbers whose productfalls within the target range. Do not use a calculator.

♦ Multiply the 2 numbers on your calculator to determinewhether the product falls within the target range. If itdoes, you have hit the bull’s-eye and score 1 point. If itdoesn’t, you score 0 points.

♦ Sometimes it is impossible to form 2 numbers whoseproduct falls within the target range. If this happens, you score 0 points for that turn.

3. The game ends when each player has had 5 turns.4. The player scoring more points wins the game.

Games

Number Target Rangeon Die of Product

1 500 or less

2 501 – 1,000

3 1,001 – 3,000

4 3,001 – 5,000

5 5,001 – 7,000

6 more than 7,000

Student Reference Book, p. 323

Student Page

Lesson 2�7 117

To support English language learners, discuss the common meaning of round and its meaning in this context. Discuss the rounding approach to estimate a product of two numbers:

1. Round both factors to the nearest multiple of a power of 10.

Ask a volunteer to count by a multiple of a power of 10. Repeat until students have counted by 10, 100, and 1,000. Encourage students to count by larger multiples as well. Ask: What is the nearest multiple of a power of 10 for x? Ask questions using several sample numbers. Suggestions: 76; 220; 4,892. Restate student responses. For example, The nearest multiple of a power of 10 for 76 is 80 because 76 rounded to the nearest 10 is 80. Similarly, 220 rounded to the nearest 100 is 200; 4,892 rounded to the nearest 1,000 is 5,000.

2. Then find the product of the rounded numbers.

For example, to estimate 14 ∗ 17, round 14 to 10 and 17 to 20. Since 10 ∗ 20 = 200, 14 ∗ 17 is in the hundreds.

Emphasize the use of “friendly numbers,” that is, numbers that are close to the numbers being multiplied and easy to work with. A friendly number is a number that students can use in their heads for mental arithmetic. Write these problems on a transparency or the board: 420 ∗ 43,892; 6,748 ∗ 3,480; and 88,889 ∗ 4,965. Ask a volunteer to round both factors to the nearest multiple of a power of 10. 400 ∗ 40,000; 7,000 ∗ 3,000; and 90,000 ∗ 5,000 Ask: Which friendly numbers would you multiply mentally to find the product of the rounded numbers? Sample answers: 4 ∗ 4; 7 ∗ 3; 9 ∗ 5 Point out that because the goal is to make use of mental arithmetic when making magnitude estimates, one student’s friendly numbers might be different from another’s.

Ongoing Assessment: Informing Instruction

Watch for students who do not recognize place-value columns as multiples of

powers of 10. Refer them to the Place-Value Chart on page 205 in the journal.

Remind students that when they make estimates, they are not attempting to find the exact answers to the problems.

Ask students to use rounding to make a magnitude estimate of the product in Problem 1 (56 ∗ 37) on journal page 47. They should write the number sentence they used for their estimate, showing the rounded factors and solution, and then use their solution to circle the magnitude of the answer. Ask volunteers to explain what they did. (See estimate diagram below.) Sample answers: Round 56 to 60. Round 37 to 40. Then multiply 60 ∗ 40 and write 60 ∗ 40 = 2,400 on the line. Finally, circle thousands because the magnitude estimate indicates that 56 ∗ 37 is in the thousands.

10s 100s 1,000s 10,000s

60 ∗ 40 = 2, 400

How I estimated

NOTE Magnitude estimates for products

and quotients are made in the same way.

Students will practice making magnitude

estimates for quotients in Unit 3.

NOTE Make a classroom display of the

steps for this rounding approach to making

magnitude estimates to serve as a quick,

available reference for students.


9. Did your prediction match your result? Explain on a different piece of paper why youthink it was the same or different.

Spinner ExperimentsLESSON

2� 7

Date Time

You can make a spinner by dividing acircle into different-color parts andholding a large paper clip in placewith the point of a pencil.

1. Divide the spinner at the right into 3 parts.Color the parts red, blue, and green so the paper clip has

� a �13� chance of landing on red;

� a �12� chance of landing on blue; and

� a �16� chance of landing on green.

12

657

48

39

210

111

Use the words and phrases from the Word Bank.Describe the chance that the spinnerwould land on... Sample answers:2. red.

3. blue.

4. green. very unlikely50-50 chanceunlikely

Suppose you spin the paper clip 90 times.About how many times would you expectit to land on...

5. red?

6. blue?

7. green? 154530

8. Spin a paper clip on your spinner 90 times. Tally the results in the table.

Word Bank

certain extremely likely very likely 50–50 chanceimpossible extremely unlikely very unlikely unlikely

Color Tallies

red

green

blue


Student Page

Math Boxes LESSON

2 �7

Date Time

4. Subtract.

a. 322� 199

123

1. Write the repeated-factor notations.a. 34 � 3 � 3 � 3 � 3

b. 53 �

c. 74 �

d. 25 �

e. 103 �10 � 10 � 102 � 2 � 2 � 2 � 2

7 � 7 � 7 � 75 � 5 � 5

2. Estimate. 247 � 974

a. Write your estimate as a number sentence:

b. How I estimated.

3. Add.

a. 3,672� 1,319

4,991

6

13 14 15–17

6. When rolling a pair of dice, is there a better chance of rolling a 7 or a 9? Explain.

5. Solve.

a. 18.95� 6.07

12.88

34–36 129

18 219247–249

b. 1,654� 2,020

3,674b. 602

� 483

119

b. 215.29� 38.75

254.04

� �

200 � 1,000 � 200,000

Round down 247 andround up 974; then multiply 200 and 1,000.

Sample answer: There is abetter chance of rolling a 7because you can roll 7three different ways but you can roll a 9 only twodifferent ways.


Student Page

118 Unit 2 Estimation and Computation

Direct students to Problem 6, and explain that the same strategy can be used to make magnitude estimates for products of decimals. Ask volunteers to tell how they would estimate, and record their responses on the board or a transparency. Sample answer: Round 4.5 to 5; round 0.6 to 1; multiply 5 ∗ 1 = 5; circle ones because the magnitude estimate indicates that 4.5 ∗ 0.6 is in the ones.

Students finish the problems on their own. When most students have finished, write some of the problems on the board and have students describe how they estimated. Allow them to present any different strategies. Conclude by emphasizing that magnitude estimates are usually reserved for work with very large or very small numbers, but that the process of rounding to make an estimate can be used in a variety of situations.

▶ Playing Multiplication Bull’s-Eye PARTNER ACTIVITY

(Student Reference Book, p. 323)

The purpose of this game is to provide practice in estimating products. While the rules of the game call for two players, the game can also be played by one player. If a student plays alone, the game ends after 10 turns, and the goal of the game becomes to top previous scores. Go over the rules of the game on page 323 in the Student Reference Book. Play a few rounds with the class.

2 Ongoing Learning & Practice

▶ Experimenting with Spinners PARTNER ACTIVITY


Students make a spinner to match given criteria. Then they use the language of probability to describe events. When most students have finished, ask: Suppose you spin the paper clip 36 times. About how many times might it land on a color other than blue? 18 times About how many times might it land on red or blue? 30 times

▶ Math Boxes 2�7

INDEPENDENT ACTIVITY


Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 2-5. The skill in Problem 6 previews Unit 3 content.

Writing/Reasoning Have students write a response to the following: Explain how you solved Problem 5b. Include in your explanation the strategies and reasoning used. Answers vary.


STUDY LINK

2�7 Magnitude Estimates

Name Date Time

A magnitude estimate is a very rough estimate. It tells whether the exact answer falls in the tenths, ones, tens, hundreds, thousands, and so on. For each problem, make a magnitude estimate. Ask yourself: Is the answer in the tenths, ones, tens, hundreds, thousands, or ten-thousands?Circle the appropriate box. Do not solve the problems.

Example: 18 º 21

How I estimated

2. 12 º 708

How I estimated

4. 17 º 2.2

How I estimated

1. 73 º 28

How I estimated

3. 98 º 105

How I estimated

5. 2.6 º 3.9

How I estimated3 º 4 � 12

0.1s 1s 10s 100s

100 º 100 � 10,00010s 100s 1,000s 10,000s

70 º 30 � 2,10010s 100s 1,000s 10,000s

20 º 2 � 4010s 100s 1,000s 10,000s

10 º 700 � 7,00010s 100s 1,000s 10,000s

20 º 20 � 40010s 100s 1,000s 10,000s

Try This

6. Use the digits 4, 5, 6, and 8. Make as many factor pairs as you can that havea product between 3,000 and 5,000. Use a calculator to solve the problems. Sample answers: 45 º 68 � 3,060; 684 º 5 � 3,420; and 864 º 5 � 4,320

250

Math Masters, p. 53

Study Link Master

Name Date Time

LESSON

2�7 Using Multiplication Patterns

Find information about Powers of 10 on page 5 of your Student Reference

Book. Study the example below. Then try to use the same strategy to solve

Problems 1 and 2.

20 ∗ 300 = (2 ∗ 10) ∗ (3 ∗ 100) Write each factor in expanded form.

= 2 ∗ 10 ∗ 3 ∗ 100 Remove the parentheses.

= 2 ∗ 3 ∗ 10 ∗ 100 Use the Commutative Property so that the

powers of 10 are together.

= (2 ∗ 3) ∗ (10 ∗ 100) Multiply the basic fact, and multiply the

powers of 10.

= 6 ∗ 1,000 Multiply the partial products.

= 6,000

Solve the problems. Show your work.

1. 900 ∗ 70 = 2. 500 ∗ 6,000 =

3. Explain why you think counting zeros works in solving multiplication problems

involving powers of 10.

4. Use what you know about counting zeros in multiplication to help you figure out

the missing numbers below.

4,200 ∗ = 840,000

∗ 40 = 2,000,000

250 ∗ = 50,000,000

5. On the back of this page, write two problems of your own that can be solved by

counting zeros.

Sample answer: Each time you multiply by 10, you are

attaching another 0. When you rewrite the numbers in

expanded form, you can see that the zeros you attach

come from the powers of 10.

200

50,000

200,000

63,000 3,000,000

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Math Masters, p. 66A

Teaching Master

Lesson 2�7 119

Ongoing Assessment: Math Boxes

Problems

3 and 4 �

Recognizing Student Achievement

Use Math Boxes, Problems 3 and 4 to assess students’ ability to add and

subtract whole numbers. Students are making adequate progress if they

demonstrate successful strategies for solving these problems.


▶ Study Link 2�7


(Math Masters, p. 53)

Home Connection Students practice making magnitude estimates. Remind them to round each number and then multiply the products. Stress that students should not find exact answers.

3 Differentiation Options

READINESS


▶ Practicing Extended Facts 5–15 Min

(Student Reference Book, p. 18; Math Masters, p. 54)

To explore multiplication patterns, have students use their knowledge of basic facts and multiplication patterns to help them solve extended fact problems. Have students describe or write about patterns they see in their problems. Ask students to refer to Student Reference Book, page 18, if necessary.

EXTRA PRACTICE

SMALL-GROUP ACTIVITY

▶ 5-Minute Math 5–15 Min

To offer students more experience with estimation, see 5-Minute Math pages 19, 95, and 182.

ENRICHMENT


▶ Using Multiplication Patterns 5–15 Min

(Student Reference Book, p. 5; Math Masters, p. 66A)

To apply their understanding of patterns in the number of zeros in the product when multiplying by powers of 10, students analyze the connections between the patterns and rewrite factors as numbers in expanded form.

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Name Date Time

LESSON

2�7 Using Multiplication Patterns

Find information about Powers of 10 on page 5 of your Student Reference

Book. Study the example below. Then try to use the same strategy to solve

Problems 1 and 2.

20 ∗ 300 = (2 ∗ 10) ∗ (3 ∗ 100) Write each factor in expanded form.

= 2 ∗ 10 ∗ 3 ∗ 100 Remove the parentheses.

= 2 ∗ 3 ∗ 10 ∗ 100 Use the Commutative Property so that the

powers of 10 are together.

= (2 ∗ 3) ∗ (10 ∗ 100) Multiply the basic fact, and multiply the

powers of 10.

= 6 ∗ 1,000 Multiply the partial products.

= 6,000

Solve the problems. Show your work.

1. 900 ∗ 70 = 2. 500 ∗ 6,000 =

3. Explain why you think counting zeros works in solving multiplication problems

involving powers of 10.

4. Use what you know about counting zeros in multiplication to help you figure out

the missing numbers below.

4,200 ∗ = 840,000

∗ 40 = 2,000,000

250 ∗ = 50,000,000

5. On the back of this page, write two problems of your own that can be solved by

counting zeros.

Copyright

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