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126 Unit 2 Estimation and Computation

Advance PreparationFor Part 1, draw a 1-by-3 and a 2-by-2 lattice on the board. You might find it helpful to copy the lattice computation

grids on Math Masters, page 57 on a transparency. Each student will need 2 copies of Math Masters, page 418.

Teacher’s Reference Manual, Grades 4–6 pp. 126–132

Key Concepts and Skills• Apply multiplication facts.  

[Operations and Computation Goal 2]

• Use the lattice method for multiplying

whole numbers and decimals.  

[Operations and Computation Goal 3]

• Use magnitude estimates to verify lattice

method solutions.  

[Operations and Computation Goal 6]

Key ActivitiesStudents review and practice the lattice

method for multiplying multidigit whole

numbers. They practice the lattice method

for multiplying decimals.

Ongoing Assessment: Informing Instruction See page 129.

Ongoing Assessment: Recognizing Student Achievement Use an Exit Slip (Math Masters, page 414). [Operations and Computation Goal 3]

Key Vocabularylattice method � lattice

MaterialsMath Journal 1, pp. 54 and 55

Student Reference Book, pp. 20 and 40

Study Link 2�8

Math Masters, pp. 58, 414, and 418

Class Data Pad

Playing Factor BingoMath Masters, p. 452

per group: 4 each of number cards

2–9 (from the Everything Math Deck,

if available) � 12 counters

Students practice applying

multiplication facts and number

properties, recognizing factors, and

factoring numbers.

Multiplying and Dividing by Powers of 10Math Journal 1, pp. 56A and 56B

Students practice multiplying and

dividing by powers of 10.

Math Boxes 2�9Math Journal 1, p. 56

Students practice and maintain skills

through Math Box problems.

Study Link 2�9Math Masters, p. 59

Students practice and maintain skills

through Study Link activities.

ENRICHMENTExploring an Ancient Multiplication MethodMath Masters,p. 60

Students analyze an ancient multiplication

method and use that method to solve a

multiplication problem.

EXTRA PRACTICE

5-Minute Math5-Minute Math™, p. 186

Students practice multiplying decimals.

EXTRA PRACTICE

Multiplying DecimalsMath Masters, p. 414

Students practice multiplying decimals and

writing about their strategies.

Teaching the Lesson Ongoing Learning & Practice

132

4

Differentiation Options

�������

The Lattice Methodof Multiplication

Objective To review and practice the lattice method for

multiplication of whole numbers and 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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Date Time

Study the problems and solutions in Column A. Then use lattice multiplication

to solve the problems in Column B.

Column A

Example 1: 5 ∗ 486 = 2,430

Example 2: 87 ∗ 59 = 5,133

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

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

5

4 8

202

4 3

40

30

0

6

8

7

5

5

13 3

9

Column B

1. 8 ∗ 274 = 2,192

2. 67 ∗ 45 = 3,015

40

72

35 6 3

2 7 41 5 3

82 6 6 21 9 2

1

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

6 7

1 5

2 24 8

3 30 5

4

53

0

1 1

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

1 1

Lattice PracticeLESSON

2�9

EM3cuG5MJ1_U02_029-059.indd 54 1/11/11 11:32 AM

Math Journal 1, p. 54

Student Page

Lesson 2�9 127

Getting Started

Mental Math and Reflexes Write problems on the board or the Class Data Pad so students can visually recognize the patterns of decimal point placement in the products. Ask students to describe the patterns they notice in each set.

Math MessageStudy the problems in Column A on journal page 54. Then use lattice multiplication to solve the problems in Column B.

Study Link 2�8 Follow-Up Allow students five minutes to work together to compare their answers and correct any errors.

6 ∗ 3 18

6 ∗ 0.3 1.8

6 ∗ 0.03 0.18

8 ∗ 7 56

8 ∗ 0.7 5.6

0.8 ∗ 0.7 0.56

8 ∗ 5 40

8 ∗ 0.5 4

0.8 ∗ 0.5 0.4

1 Teaching the Lesson

▶ Math Message Follow-Up WHOLE-CLASSDISCUSSION

(Math Journal 1, p. 54)

Ask volunteers to demonstrate on the board how they solved the problems. Note students’ explanations of their processes, including their vocabulary, for use in the following review.

▶ Reviewing the Lattice

WHOLE-CLASS ACTIVITY

Method of Multiplication(Student Reference Book, pp. 20 and 40;

Math Journal 1, p. 54; Math Masters, p. 418)

The lattice method for multiplying has been used since Third Grade Everyday Mathematics. This method is very easy to use because it relies almost entirely on the recall of the basic multiplication facts.

Use the student examples on the board from the Math Message Follow-Up to emphasize the vocabulary of the lattice method.

� The box with squares and diagonals is called a lattice. Ask students how they know what size lattice to use and where to write the factors. The factors tell which size lattice to use. Write the second factor above the lattice and the first factor on the right side. Some students might think of the rectangular array model for multiplication: The number of digits in the first factor is the number of rows, and the number of digits in the second factor is the number of columns.

� Multiply the digits. Write the answers. Ask students how they know where to write these partial products. Write the answers in the square where the multiplied digits intersect on the lattice, 1 digit per diagonal. Some students may note that each partial product is never more than 2 digits: Tens go in the left diagonal and ones go in the right diagonal.

Algorithm Project The focus of this

lesson is the lattice method for multiplying

whole numbers and decimals. To teach U.S.

traditional multiplication with whole numbers

and with decimals, see Algorithm Project 5

on page A22 and Algorithm Project 6 on

page A27.

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Lattice MethodThe lattice method for multiplying has been used forhundreds of years. It is very easy to use if you know the basicmultiplication facts.

Whole Numbers

Check your answers on pages 433 and 434.

Draw a lattice for each problem. Then multiply.1. 6 * 78 2. 55 * 25 3. 77 * 89 4. 8 * 444 5. 357 * 6

6 * 815 = ?

The box with cells and diagonals is called a lattice.Write 815 above the lattice.Write 6 on the right side of the lattice.

Multiply 6 * 5. Then multiply 6 * 1. Then multiply 6 * 8.Write the answers as shown.

Add the numbers along each diagonal, starting at the right.

Read the answer. 6 * 815 = 4,890

42 * 37 = ?

Write 37 above the lattice.Write 42 on the right side of the lattice.

Multiply 4 * 7. Then multiply 4 * 3.Multiply 2 * 7. Then multiply 2 * 3.Write the answers as shown.

Add the numbers along each diagonal, starting at the right.

When the numbers along a diagonal add up to 10 or more:• record the ones digit in the sum.• add the tens digit to the sum along the next diagonal above.

Read the answer. 42 * 37 = 1,554

6

8 1 5

6

8 14

80

63

0

5

6

8 14

88 9

06

30

0

5

4

4

3 71

21

5 4

21

80

61

452

4

3 71

22

80

61

42

Student Reference Book, p. 20

Student Page

Check your answers on page 434.

Draw a lattice for each problem and multiply.

1. 32.5 * 2.5 2. 4.02 * 17 3. 8.1 * 23.4

The lattice method ofmultiplication was usedby Persian scholars aslong ago as the year1010. It was often calledthe “grating” method.

2

3 40

8

0

51

00

60

00

00

0

51

52

02

5

Decimals and Percents

Lattice Multiplication with Decimals

Find 34.5 * 2.05 using lattice multiplication.

Step 1: Make a magnitude estimate. 34.5 * 2.05 � 35 * 2 � 70The product will be in the tens. (The symbol � means is about equal to.)

Step 2: Draw the lattice and write the factors, including the decimalpoints, at the top and right side. In the factor above the grid, thedecimal point should be above a column line. In the factor on the rightside of the grid, the decimal point should be to the right of a row line.

Step 3: Find the products inside the lattice.

Step 4: Add along the diagonals, moving from right to left.

Step 5: Locate the decimal point in the answer as follows. Slide thedecimal point in the factor above the grid down along the columnline. Slide the decimal point in the factor on the right side of the gridacross the row line. When the decimal points meet, slide the decimalpoint down along the diagonal line. Write a decimal point at the endof the diagonal line.

Step 6: Compare the result with the estimate.

The product, 70.725, is very close to the estimate of 70.

2

3 4

001

8

70

51

00

60

00

00

0

7 20

5

5

15

20

25

Find 73.4 * 10.5 usinglattice multiplication.

A good magnitude estimate is 73.4 * 10.5 � 73 * 10 � 730. (Thesymbol � means is about equal to.)

The product, 770.70, is close to the estimate of 730.

1

7 3

00

3

70

401

40

70

00

00

0

0 77

5

0

35

15

20

Steps 1–3

Steps 4–6

Student Reference Book, p. 40

Student Page

128 Unit 2 Estimation and Computation

� Add the numbers along each diagonal. Ask students how they know where to begin the addition. Begin with the diagonal in the bottom right-hand corner. Explain that the extended diagonal lines outside of the lattice squares are the answer spaces. Ask students how the sum is written when the sum on a diagonal is 10 or more. Write the ones digit in the answer space and the tens digit at the top of the next diagonal.

Review these steps by demonstrating with the following problem.

The Lattice Method for Multiplying Whole Numbers

Example: 42 ∗ 37 = ?

1. Write 37 above the lattice. Write 42 on the right side.

2. Multiply 4 ∗ 7. Then multiply 4 ∗ 3. Multiply 2 ∗ 7. Then multiply 2 ∗ 3. Write the answers as shown.

3. Add the numbers along each diagonal. Begin with the diagonal in the bottom right-hand corner. The sum of the numbers in the second diagonal is 15, so write 1 above the 2 in the third diagonal. The sum of the numbers in that diagonal is 1 + 2 + 2 = 5.

4. Stress that the answer is shown, starting on the upper left side of the lattice and continuing below the lattice.42 ∗ 37 = 1,554

2

4

3 7

214

06

4

3 7

28

12

1

55 4

214

06

4

3 7

21

812

Pass out two lattice-computation grids (Math Masters, page 418) to each student. Students work with partners to use the lattice method to solve several multiplication problems in which both factors are whole numbers. Remind students to make a magnitude estimate for the problem before beginning the lattice. Suggestions: 7 ∗ 37; 43 ∗ 96; 32 ∗ 146; 513 ∗ 658.

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LESSON

2�9

Name Date Time

Lattice Multiplication Table

9 8 7 6 5 4 3 2 1 0

0

1

2

3

4

5

6

7

8

9

0 0 0 0 0 0 0 0 00

0

1

2

3

4

5

6

7 6 5 4 4 3 2 1 0

78 6 5 4 3 2 1 0 0

5 4 4 3 2 2 1 0

4 4 3 3 2 1 1 0

4 3 3 2 2 1 1 0

3 2 2 2 1 1 0 0

2 2 1 1 1 0 0 0 0

0

0

0

0

0

1 1 1 1 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 00

9

8

7

6

5

4

3

2 4 6 8 0 2 4 6 8

21 3 4 5 6 7 8 9 0

6 9 2 5 8 1 4 7

8 2 6 0 4 8 2 6

0 5 0 5 0 5 0 5

2 8 4 0 6 2 8 4

4 1 8 5 2 9 6 3 0

0

0

0

0

0

6 4 2 0 8 6 4 2 0

8 7 6 5 4 3 2 1 0

Math Masters, p. 58

Teaching Master

Date Time

Multiplication by the Lattice MethodLESSON

2 � 9For each problem: � Make a magnitude estimate. Circle the appropriate box.

� Solve the problem using the lattice method. Show your work below.

1. 7 � 349 � 2. 48 � 72 �

3. 384 � 256 � 4. 6.15 � 8.3 �

5. 1.7 � 5.6 � 6. 82 � 4.9 � 401.89.52

51.04598,304

3,4562,44310s 100s 1,000s 10,000s 10s 100s 1,000s 10,000s

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

10s1s 100s 1,000s 10s 100s 1,000s 10,000s

Math Journal 1, p. 55

Student Page

Lesson 2�9 129

Ongoing Assessment: Informing Instruction

Watch for students who are still having difficulty with the automatic recall of

multiplication facts. Provide them with copies of the lattice multiplication facts

table (Math Masters, page 58).

The Lattice Method for Multiplying Decimals

An advantage of the lattice method is that products of decimals are as easy to calculate as products of whole numbers, and the grid automatically locates the placement of the decimal point in the answer.

� When writing the factors above and on the right side of the lattice, include the decimal points. In the factor above the grid, the decimal point should be above a column line. In the factor on the right side of the grid, the decimal point should be to the right of a row line.

� Locate the decimal point in the answer as follows: Slide the decimal point in the factor above the grid down. Slide the decimal point in the factor on the right side of the grid across. The decimal points will intersect on a diagonal line. Slide that decimal point down along the diagonal line. Place a decimal point at the end of the diagonal line.

Model how to use the lattice to locate the decimal point in problems with factors such as 0.5 and in problems where one factor is a whole number such as 3.7 ∗ 42.

Have students work with partners to use the lattice method to solve several multiplication problems in which at least one factor is a decimal. Remind students to make a magnitude estimate for the problem before beginning the lattice. Suggestions: 7 ∗ 3.7 25.9; 4.3 ∗ 9.6 41.28; 32 ∗ 14.6 467.2; 5.13 ∗ 6.58 33.7554.

▶ Multiplying Whole Numbers and

INDEPENDENT ACTIVITY

Decimals by the Lattice Method(Math Journal 1, p. 55)

Students solve the problems independently and then check each other’s work. Circulate and assist. Bring the class together to share magnitude estimates and solutions and to explain strategies and the reasoning used.

4.2 ∗ 3.7 = 15.54

1

55 4

214

06

4

3 • 7

21

812

•

•

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Date Time

Math Boxes LESSON

2�9

5. Measure angle BOP to the nearest degree.

1. Give the value of the boldface digits ineach numeral.

a. 390.81

b. 8,092,741

c. 4,350.47

d. 32,768.9 2 thousand4 tenths8 million

1 hundredth2. Solve.

a. n � 45 � 190

n �

b. 360 � n � 270

n �

c. 23.14 � p � 30.59

p � 7.45

90

145

3. Write the prime factorization of 72.2 � 2 � 2 � 3 � 3

4. Solve.

a. 8 � 400 �

b. 36,000 � � 60

c. 420,000 � 700 �

d. 9,000 � � 72,000

e. 5,000 � 8,000 � 40,000,0008

6006003,200

4

12 18 219

138204 205 142 143

219

B

O P

6. Cross out the shapes below that are not polygons.

�BOP: 125�

Math Journal 1, p. 56

Student Page

STUDY LINK

2�9 Multiply with the Lattice Method

Name Date Time

For each problem:� Make a magnitude estimate. Circle the appropriate box.

� Solve using the lattice method. Show your work in the grids.

1. 94 º 73 �

3. 5.4 º 6.18 �

5. 17.7 º 19.3 �

2. 24 º 3.7 �

4. 384 º 261 �

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

100,224

0.1s 1s 10s 100s

88.8

0.1s 1s 10s 100s

341.61

0.1s 1s 10s 100s

33.372

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

6,862

6. 7,402 � 2,587 �

7. 37 � 7 ∑

8. 328 � 237 �

9. $15.75 � $3.25 � $19.0091

5 R29,989

Practice

20

Math Masters, p. 59

Study Link Master

130 Unit 2 Estimation and Computation

Ongoing Assessment: Exit Slip

Recognizing Student Achievement

Use an Exit Slip (Math Masters, page 414) to assess students’ ability

to use a multiplication method of their choice. Have students write a

response to the following: Explain how to use the multiplication method of

your choice to solve 82 ∗ 75 =? and 8.2 ∗ 7.5 =? Students are making

adequate progress if their explanations support their multiplication strategy and

appropriate placement of the decimal.

[Operations and Computation Goal 3]

2 Ongoing Learning & Practice

▶ Playing Factor Bingo

SMALL-GROUP ACTIVITY

(Math Masters, p. 452)

Students practice applying multiplication facts and number properties, recognizing factors, and factoring numbers by playing Factor Bingo. See this lesson guide, page 50, for game directions. It can also be played as a whole-class activity.

▶ Multiplying and Dividing by

INDEPENDENT ACTIVITY

Powers of 10(Math Journal 1, pp. 56A and 56B)

Students practice multiplying and dividing by powers of 10.

▶ Math Boxes 2�9

INDEPENDENT ACTIVITY

(Math Journal 1, p. 56)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 2-6. The skills in Problems 4 and 5 preview Unit 3 content.

▶ Study Link 2�9

INDEPENDENT ACTIVITY

(Math Masters, p. 59)

Home Connection Students make estimates and then solve multiplication problems by the lattice method.

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Date Time

Dividing by Powers of 10LESSON

2�9

Use the strategy described in Extended Division Facts on page 21 of the Student

Reference Book to solve Problems 1– 6.

Example: 45,000 / 100 = ? Divide a whole number that ends in zero by a power of 10.

45,000 / 100 = 45,000 Count the number of zeros in the divisor (power of 10).

45,000 / 100 = 45,000 Cross out the same number of zeros in the answer (quotient)

as there are zeros in the divisor (power of 10), starting in the

ones place.

45,000 / 100 = 450 The numbers not crossed out give the answer.

1. 3,200 / 10 = 320

2. 78,000 / 10 = 7,800

3. 902,000 / 100 = 9,020

4. 35,000 / 1,000 = 35

5. 450,000 / 5,000 = 90

6. 28,000 / 40 = 700

Solve the following problems. Show your work.

7. Every year, Mrs. Connick asked her students to

bring in bottle caps for charity. It took Mrs. Connick

10 years to collect one million bottle caps. About

how many bottle caps did each class collect every

year?

100,000 bottle caps

8. The entire fifth grade collected about 8,000 buttons

for their Math Fun Fair estimation jar. There are

200 fifth-grade students. If each student collected

the same number of buttons, about how many

buttons did each student bring in?

40 buttons

9. Elves love to bake cookies. Last year, they had a

contest to see who could bake the most cookies

in a week. Effie, the fastest elf, baked 70 cookies

an hour. She baked for 10 hours per day. About

how many cookies did she bake in a week?

4,900 cookies

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Math Journal 1, p. 56A

Student Page

Date Time

Multiplying and Dividing Decimals by Powers of 10 LESSON

2�9

Study the problems below.

1. Explain a strategy for finding a product when you multiply a number by a power

of 10 greater than 1. Do the same for a power of 10 less than 1.

Sample answer: When multiplying by 10,100, or 1,000, you move the decimal point tothe right; when multiplying by 0.1, 0.01, or 0.001, you move the decimal point to the left.

Study the problems below.

327÷ =10 03 2 7 8 ÷ =1004 3 4380

1,000967 4 7694÷ = 0 10,00042 2462÷ = 06 2

2. Explain a strategy for finding the quotient when you divide a number by a power

of 10 greater than 1.

Sample answer: When dividing by 10, 100, or1,000, you move the decimal point one place

to the left for each 0 in the power of 10.

3. Use your strategies to solve the problems below.

a. 59.87 ∗ 10 = 598.7

b. 20.477 ∗ 1,000 = 20,477

c. 6.32 ∗ 1 _ 10 =

0.632 d. 959.8 ∗ 1

_ 100 =

9.598

e. 56.02 ÷ 100 = 0.5602

f. 49,034 ÷ 1,000 = 49.034

7º =10 323 2 7 8 8º =100 430 4 3

67 9 4 694º = 71

10 642 2 462º = 21

100

EM3cuG5MJ1_U02_029-059.indd 56B 1/22/11 9:52 AM

Math Journal 1, p. 56B

Student Page

Lesson 2�9 131

3 Differentiation Options

ENRICHMENT PARTNER ACTIVITY

▶ Exploring an Ancient 15–30 Min

Multiplication Method(Math Masters, p. 60)

To apply students’ understanding of multiplication, have them explore and analyze how to use an ancient Egyptian method of multiplying. When students have finished,

have them describe the pattern that is used in the Egyptian method.

NOTE This method is also used in Project 3: An Ancient Multiplication Algorithm.

See page 446 in this lesson guide.

EXTRA PRACTICE SMALL-GROUP

ACTIVITY

▶ 5-Minute Math 5–10 Min

To offer students more experience with multiplying decimals, see 5-Minute Math, page 186.

EXTRA PRACTICE INDEPENDENT ACTIVITY

▶ Decimal Computation 5–10 Min

(Math Masters, p. 414)

To offer students more experience with decimal multiplication, have them record the following problems onto an Exit Slip, Math Masters, page 414. Ask students to make magnitude estimates for each product and then solve the problems. Students should use their estimates to ensure that their answers are reasonable. Have them write an explanation of their reasoning in solving Problem a.

a. 3.35 ∗ 8.4 = 28.14 c. 82.6 ∗ 6.5 = 536.9

b. 16.05 ∗ 400 = 6,420 d. 61.04 ∗ 3.9 = 238.056

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