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Technology Assessment Management SystemJournal page 50See the iTLG.

120 Unit 2 Estimation and Computation

Teaching the Lesson materials

Key ActivitiesStudents review the partial-products method for whole numbers. They use magnitude estimates to solve multiplication problems involving whole numbers and decimals.

Key Concepts and Skills• Make magnitude estimates.

[Operations and Computation Goal 3]• Use magnitude estimates to place the decimal point in products.

[Operations and Computation Goal 6]• Solve whole-number and decimal problems using the partial-products algorithm.

[Patterns, Functions, and Algebra Goal 2]

Key Vocabularypartial-products method • magnitude estimate • ballpark estimate

Ongoing Assessment: Informing Instruction See page 122.

Ongoing Assessment: Recognizing Student Achievement Use journal page 50. [Operations and Computation Goal 6]

Ongoing Learning & Practice materials

Students solve addition and subtraction number stories by writing and solving open number sentences.

Students practice and maintain skills through Math Boxes and Study Link activities.

Differentiation Options materials

Students use base-10 blocks to model thepartial-products method.

Students explore a calculation strategy formultiplying by 9.

� Teaching Masters (Math Masters,pp. 56 and 57)

� Per partnership: Transparencies(Math Masters, pp. 416 and 417);base-10 blocks; dry erase markerand eraser

See Advance Preparation

ENRICHMENTREADINESS

3

� Math Journal 1, pp. 49 and 50� Student Reference Book, p. 339� Study Link Master (Math Masters,

p. 55)

2

� Math Journal 1, pp. 50 and 51 � Student Reference Book,

pp. 19, 38, and 39� Study Link 2�7� Teaching Aid Master (Math Masters,

p. 415; optional)

See Advance Preparation

1

Additional InformationAdvance Preparation Make copies of the computation grid (Math Masters, p. 415) available for students’ use throughout Part 1. For Part 3, make transparencies of MathMasters, pages 416 and 417, and cut them out. Tape them together with clear tape.

Objectives To review the partial-products method for whole numbers and decimals.

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� Math Message Follow-UpAsk volunteers to share their estimates and number models. Ask:How does estimating products of decimals differ from estimatingproducts of whole numbers? Emphasize that when the decimal isless than 1, comparisons to other decimals can help students makean appropriate estimate. For example, they can compare thedecimal to 0.5 or 0.33 or compare it to 1.

� Reviewing the Partial-Products Method with Whole Numbers(Student Reference Book, p. 19)

The partial-products method for multiplication has beenstressed since Third Grade Everyday Mathematics. It is analgorithm that all students are expected to know because it helpsstudents develop a good understanding of place-value and multiplication concepts. It has the added benefit of facilitatingstudent use of mental arithmetic as they solve problems.

Refer students to page 19 of the Student Reference Book. With thepartial-products method, each part of one factor is multiplied byeach part of the other factor. Each partial product is written on aseparate line. These partial products are then added. This processis usually fairly simple and has the additional benefit of providingpractice with column addition.

Go through the following example with the class. Then distributecomputation grids (Math Masters, page 415) and have studentssolve the Check Your Understanding problems using this method.

WHOLE-CLASSDISCUSSION

WHOLE-CLASSDISCUSSION

1 Teaching the Lesson

Lesson 2�8 121

Getting Started

Math MessageEstimate the solution to this problem.Write a number sentence showing how you found your estimate.

3.7 � 6.2 4 � 6 = 24

Study Link 2�7 Follow-Up Have partners compare answers andcorrect any errors. Ask students to sharesolutions to Problem 6.

Mental Math and Reflexes Write problems on the board or the class data pad, so studentscan visually recognize the patterns. Suggestions:

3 � 8 24 6 � 5 30 7 � 9 633 � 80 240 60 � 5 300 70 � 9 63030 � 80 2,400 60 � 50 3,000 70 � 90 6,300300 � 80 24,000 600 � 50 30,000 700 � 90 63,000

NOTE Working from left to right is consistentwith the process of estimating products. Pointout for students that when they have foundthe partial product for the leftmost digits, theyhave a ballpark estimate for the product. Tosupport English language learners, discussthe meaning of ballpark estimate.

Multiplication Algorithms

The symbols � and * are both used to indicate multiplication.

In this book, the symbol * is used more often.

Partial-Products MethodIn the partial-products method, you must keep track of theplace value of each digit. It may help to write 1s, 10s, and 100sabove the columns. Each partial product is either a basicmultiplication fact or an extended multiplication fact.

Whole Numbers

Check Your UnderstandingCheck Your Understanding

Check your answers on page 433.

Multiply. Write each partial product. Then add the partial products.1. 265 * 3 2. 42 * 67 3. 40 * 58 4. 83 * 54 5. 372 * 50

ExampleExample 4 * 236 = ? 100s 10s 1s

2 3 6∗ 4

8 0 01 2 0

2 49 4 4

Think of 236 as 200 � 30 � 6.

Multiply each part of 236 by 4. 4 ∗ 200 ∑4 ∗ 30 ∑4 ∗ 6 ∑

Add the three partial products.

4 ∗ 236 � 944

ExampleExample 43 * 26 = ? 100s 10s 1s

2 6∗ 4 3

8 0 02 4 0

6 01 8

1 , 1 1 8

Think of 26 as 20 � 6.

Think of 43 as 40 � 3.

Multiply each part of 26 40 ∗ 20 ∑by each part of 43. 40 ∗ 6 ∑

3 ∗ 20 ∑3 ∗ 6 ∑

Add the four partial products.

43 ∗ 26 � 1,118

extended multiplication facts

basic multiplication fact

extended multiplication facts

basic multiplication fact

Student Reference Book, p. 19

Student Page

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Multiplication of DecimalsLESSON

2 � 8

Date Time

For each problem, make a magnitude estimate. Circle the appropriate box.

Do not solve the problems.

1. 2.4 � 63 2. 7.2 � 0.6

How I estimated How I estimated

3. 13.4 � 0.3 4. 3.58 � 2.1


5. 7.84 � 6.05 6. 2.8 � 93.6


7. Solve each problem above for which your estimate is at least 10. Show your work

on the grid below.

3 � 90 � 2708 � 6 � 48

4 � 2 � 810 � 0.3 � 3

7 � 1 � 72 � 60 � 1200.1s 1s 10s 100s

0.1s 1s 10s 100s

0.1s 1s 10s 100s

0.1s 1s 10s 100s0.1s 1s 10s 100s

0.1s 1s 10s 100s

Sample answers:

6 3

� 2 .4

1 2 0 0

6 0

2 4 0

1 2

1 5 1 .2

6 .0 5

� 7 .8 4

4 2 0 0 0 0

3 5 0 0

4 8 0 0 0

4 0 0

2 4 0 0

2 0

4 7 .4 3 2 0

9 3 .6

� 2 .8

1 8 0 0 0

6 0 0

1 2 0

7 2 0 0

2 4 0

4 8

2 6 2 .0 8

Math Journal 1, p. 51

Student Page

Multiplication of Whole NumbersLESSON

2 8

Date Time

For each problem, make a magnitude estimate. Circle the appropriate box.Do not solve the problems.

1. 6 543 2. 3 284


3. 46 97 4. 4 204


5. 25 37 6. 56 409


7. Solve each problem above for which your estimate is at least 1,000. Use thepartial-products method for at least one problem. Show your work on the grid.

60 400 24,00030 40 1,200

4 200 80050 100 5,000

3 300 90010 500 5,00010s 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

Sample answers: 3 72 5

6 0 01 4 01 5 0

3 59 2 5

4 0 95 6

20 0 0 04 5 0

2 4 0 05 4

22 9 0 4

9 74 6

36 0 02 8 05 4 0

4 244 6 2

4 3560030

2 4 081

5 832


Student Page

Example: 43 * 26 = ? 100s 10s 1s

Think of 26 as 20 + 6: 2 6Think of 43 as 40 + 3: � 4 3Multiply each part of 26 40 * 20 8 0 0by each part of 43: 40 * 6 2 4 0

3 * 20 6 03 * 6 1 8

Add four partial products: 1 1 1 8

NOTE Make sure the digits students write are properly aligned in columns. It will also help if they write place-value reminders (such as 100s, 10s, and 1s) above thecolumns.

Ongoing Assessment: Informing InstructionWatch for students who do not recognize the value of the digits in a number.Have them write the factors in expanded notation.

� Reviewing Multiplication of Decimals(Student Reference Book, pp. 38 and 39)

Ask students to solve the following problem: 1.3 � 5. After a coupleof minutes, have them share their solution strategies. Expect thatthey may have difficulty because one of the factors is a decimal.

Explain that one way to solve multiplication problems containingdecimal factors is to multiply as though both factors were wholenumbers and then adjust the product. Specifically:

1. First make a magnitude estimate of the product.

2. Multiply the numbers as though they were whole numbers.

3. Then use the magnitude estimate as a guide to inserting thedecimal point at the correct location in the answer.

NOTE At this time, do not teach the shortcut of counting decimal places in thefactors as a way of locating the decimal in the product. Students will benefit long-term from becoming proficient at estimating decimal answers.

Example: 1.3 � 5 � ?

1. Make a magnitude estimate: Ask students to justify how theywould round 1.3. Answers vary. Students may use the nearestmultiple of powers of ten (1.3 is closer to 1 (100) than to0.01(10-2); the number line model for rounding; or the stepmethod for rounding. 1.3 rounds to 1; because 1 � 5 � 5, theproduct will be in the ones.

WHOLE-CLASS ACTIVITY


2. Ignore the decimal point and multiply 13 � 5 as though bothfactors were whole numbers: 13 � 5 � 65.

3. Since the magnitude estimate is in the ones, the product mustbe in the ones. The answer must be 6.5. So, 1.3 � 5 � 6.5.

Ask partners to solve several multiplication problems in which one of the factors is a decimal. Suggestions: 25 � 0.6 15; 400 � 1.7 680.

Next use examples like those on pages 38 and 39 of the StudentReference Book to demonstrate how to find the product of two decimals.

Example: 3.4 � 4.6 � ?

1. Round 3.4 to 3 and 4.6 to 5. Since 3 � 5 � 15, the product willbe in the tens.

2. Ignore the decimal points and multiply 34 � 46 as though bothfactors were whole numbers: 34 � 46 � 1,564.

3. Since the magnitude estimate is in the tens, the product mustbe in the tens. The answer must be 15.64. Thus, 3.4 � 4.6 � 15.64.

Ask partners to solve several multiplication problems in whichboth factors are decimals. Suggestions: 6.3 � 1.8 11.34; 0.71 � 3.2 2.272.

NOTE There are borderline cases where a magnitude estimate is not accurateenough to guide the correct placement of the decimal point in a product. For example, 3.4 � 3.4 ➝ 3 � 3 � 9. Place the decimal point to make the product asclose to 9 as possible: 34 � 34 � 1,156; 3.4 � 3.4 � 11.56. Remind students that the placement of the decimal point should result in a product that is reasonable.

� Practicing Multiplication of Whole Numbers and Decimals(Math Journal 1, pp. 50 and 51)

Students estimate the answers to Problems 1–6 on journal page 50 and Problems 1–6 on journal page 51. They will find theexact answer only for some, not all, of these problems. Studentsmay use whatever method they prefer to make a ballpark estimate. They should write the number sentence they used tomake their estimate on the line and then circle the magnitude oftheir estimate.

Have students complete both pages. Then have partners checkeach other’s answers and share solutions with the class.

NOTE Some students will use friendly numbers to make an estimate rather thanrounding. Making the appropriate magnitude estimate is the important concept, notwhether the student uses rounding to make an estimate.

PARTNER ACTIVITY

52

Solving Number Stories LESSON

2 � 8

Date Time

For each problem, fill in the blanks and solve the problem.

1. Linell and Ben pooled their money to buy a video game. Linell had $12.40 and Benhad $15.88. How much money did they have in all?

a. List the numbers needed to solve the problem.

b. Describe what you want to find.

c. Open sentence:

d. Solution: e. Answer:

2. If the video game cost $22.65, how much money did they have left?



c. Open sentence:


3. Linell and Ben borrowed money so they could also buy a CD for $13.79. How much did they have to borrow so they would have enough money to buy the CD?



c. Open sentence:


4. How much more did the video game cost than the CD?


b. Describe what you want to find out.

c. Open sentence:

d. Solution: e. Answer: $8.86m � 8.8622.65 � 13.79 � m

game costsHow much more the video

22.65 and 13.79

$8.16m � 8.165.63 � m � 13.79

borrowedThe amount of money

13.79 and 5.63

$5.63m � 5.6328.28 � 22.65 � m

The amount of money left22.65 and 28.28

$28.28m � 28.2812.40 � 15.88 � m

The total amount of money12.40 and 15.88


Student Page

53

Math Boxes LESSON

2 � 8

Date Time

4. Acute angles measure greater than0 degrees and less than 90 degrees.Circle all the acute angles below.

2. a. Make up a data set of at least 12 numbers that have the followinglandmarks.

maximum: 18 mode: 7

range: 13 median: 12

b. Make a bar graph of the data.

7, 12, 14, 15, 16, 16, 18, 18Sample answer: 5, 6, 7, 7, 7,

1. Use the map on page 339 of your Student

Reference Book to answer the followingquestions.

Choose the best answer.

a. About how many miles is it from theequator to the Arctic Circle?

2,000 mi 6,000 mi

4,000 mi 8,000 mi

b. About how many miles long is SouthAmerica?

5,000 mi

6,000 mi

7,000 mi 211

119 122

119 139

05 6 7 8 9 10 11 12 13 14 15 16 17 18

1234

Sample answer:

3. Find the missing numbers and landmarksfor the set of numbers below:

18, 20, 20, 24, 27, 27, , 30, 33, 34,

36, 36,

a. range: 22

b. mode: 27

c. minimum:

d. maximum: 4018

4027


Student Page

Lesson 2�8 123

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LESSON

2�8

Name Date Time

Materials � array grid (Math Masters, pp. 416 and 417)

� base-10 blocks

Directions

� Draw a line around rows and columns on the grid to model each problem.� Cover the array you made using as few base-10 blocks as possible.� Solve using the partial-products method.� Then match each part of the array with a partial product.� Record the solution, filling in the sentences to match the blocks you used.

1. 6 º 23 � 138In each of6 rowsthere are…

longs, so there are cubes.

cubes, so there are cubes.

There are cubes in all.138183

1202 Write the problem showingthe partial products.

120 � 18 � 138

In eachof 20 rowsthere are…

In eachof 6 rowsthere are…





There are cubes in all.468488601

16082001 Write the problem showing

the partial products.

200 � 160 � 60� 48 � 468

2. 26 º 18 � 468

Model the Partial-Products Method

Math Masters, p. 56

Teaching Master

STUDY LINK

2�8

Name Date Time

� For each problem, make a magnitude estimate.� Circle the appropriate box. Do not solve the problem.� Then choose 3 problems to solve. Show your work on the grid.

1. 8 º 19

How I estimated

2. 155 º 6

How I estimated

3. 37 º 58

How I estimated

4. 5 º 4.2

How I estimated

5. 9.3 º 2.8

How I estimated9 º 3 � 27

26.04

5 º 4 � 2010s 100s 1,000s 10,000s

21

40 º 60 � 2,40010s 100s 1,000s 10,000s

2,146

150 º 6 � 90010s 100s 1,000s 10,000s

930

8 º 20 � 16010s 100s 1,000s 10,000s

152

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

Estimating and Multiplying

247

Math Masters, p. 55

Study Link Master

Ongoing Assessment:Recognizing Student Achievement

Use journal page 50, Problems 1–6 to assess students’ understanding of howto make magnitude estimates. Students are making adequate progress if theymake reasonable magnitude estimates based on their number sentences.

[Operations and Computation Goal 6]

� Solving Number Stories(Math Journal 1, p. 52)

Students solve addition and subtraction number stories. Theywrite an open number sentence for each problem and solve theopen sentence to find the answer to the problem.

� Math Boxes 2�8(Math Journal 1, p. 53)

Mixed Practice Math Boxes in this lesson are paired withMath Boxes in Lesson 2-10. The skill in Problem 4previews Unit 3 content.

� Study Link 2�8(Math Masters, p. 55)

Home Connection Students make magnitude estimatesfor multiplication problems in which the factors are wholenumbers and/or decimals. They then select 3 problems tosolve for exact answers.

� Modeling the Partial-Products Method(Math Masters, pp. 56, 416, and 417)

To provide experience with multiplication using a concrete model,have students solve multidigit multiplication problems with base-10 blocks. Use transparencies of Math Masters, pages 416

5–15 Min

SMALL-GROUP ACTIVITYREADINESS

3 Differentiation Options

INDEPENDENTACTIVITY

INDEPENDENTACTIVITY

INDEPENDENTACTIVITY

2 Ongoing Learning & Practice

Journal 0 �Problems 1–6

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and 417. (See Advance Preparation.) Place the assembled grids ona table. Do not use an overhead projector. Gather a small group ofstudents around the table. Use an overhead marker to show a 4-by-28 array. (See Figure 1.)

Have students cover the array using as few base-10 blocks aspossible.

Guide students’ use of the partial-products method to solve 4 � 28.Match each part of the 4-by-28 array with a partial product. (See Figure 2.)

1. There are 2 longs in each of 4 rows, so there are 80 cubes.

2. There are 8 cubes in each of 4 rows, so there are 32 cubes.

3. There are 80 � 32, or 112 cubes in all.

Clear the masters. Now use an overhead marker to mark off a 17-by-32 array.

Ask students to cover the array using as few base-10 blocks (flats,rods, and cubes) as possible.

Guide students’ use of the partial-products method to solve 17 � 32.Now match each part of the 17-by-32 array with a partial product.

1. There are 10 rows with 30 cubes in each row (3 flats).

2. There are 7 rows with 30 cubes in each row (21 longs).

3. There are 10 rows with 2 cubes in each row (2 longs).

4. There are 7 rows with 2 cubes in each row (14 cubes).

5. There are 300 � 210 � 20 � 14, or 544 cubes in all.

Ask students to work with partners using base-10 blocks to solvethe multiplication problems on Math Masters, p. 56.

� Multiplying Numbers That End in 9(Math Masters, p. 57)

To further explore multiplication strategies, have students solveproblems using a mental multiplication strategy. Students readMath Masters, page 57 and use the mental math strategy given toanswer the questions on the page. If necessary, read and discussExample 1 as a class.

INDEPENDENTACTIVITYENRICHMENT

pyg

gp

LESSON

2�8

Name Date Time

A Mental Calculation Strategy

When you multiply a number that ends in 9, you can simplify the calculation by changing it into an easier problem. Then adjust the result.Example 1: 2 º 99 � ?� Change 2 º 99 into 2 º 100.

� Find the answer: 2 º 100 � 200

� Ask: How is the answer to 2 º 100 different from the answer to 2 º 99?100 is 1 more than 99, and you multiplied by 2.So 200 is 2 more than the answer to 2 º 99.

� Adjust the answer to 2 º 100 to find the answer to 2 º 99:200 � 2 � 198. So 2 º 99 � 198.

Example 2: 3 º 149 � ?� Change 3 º 149 into 3 º 150.

� Find the answer: 3 º 150 � (3 º 100) � (3 º 50) � 450.

� Ask: How is the answer to 3 º 150 different from the answer to 3 º 149?150 is 1 more than 149, and you multiplied by 3.So 450 is 3 more than the answer to 3 º 149.

� Adjust: 450 � 3 � 447. So 3 º 149 � 447.

Use this strategy to calculate these products mentally.

1. 5 º 49 2. 5 º 99

3. 8 º 99 4. 4 º 199

5. 2 º 119 6. 3 º 98 294238796792495245

Math Masters, p. 57

Teaching Master

Lesson 2�8 125

Figure 1: Array model of 4 � 28

Figure 2: Base-10 block model of 4 � 28


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