812 Unit 11 Whole-Number Operations Revisited

Teaching the Lesson materials

Key ActivitiesChildren learn the trade-first subtraction algorithm. They use the algorithm to solve subtractionproblems with and without trades. They make ballpark estimates to determine whether theiranswers to subtraction problems are reasonable.

Key Concepts and Skills• Make equivalent numbers using base-10 blocks. [Number and Numeration Goal 5]• Use and explain strategies for solving subtraction problems.

[Operations and Computation Goal 2]• Make ballpark estimates for subtraction problems. [Operations and Computation Goal 3]

Key Vocabulary algorithm • trade-first (subtraction)Ongoing Assessment: Recognizing Student Achievement Use Mental Math and Reflexes. [Operations and Computation Goal 2]

Ongoing Assessment: Informing Instruction See pages 814 and 815.

Ongoing Learning & Practice materialsChildren practice finding landmarks on a line plot.

Children practice and maintain skills through Math Boxes and Home Link activities.

Differentiation Options materials

Children solve subtraction problems usingbase-10 blocks.

Children make posterboards displaying multidigit subtraction strategies.

� Teaching Master (Math Masters, p. 355)

� base-10 blocks (1 flat, 20 longs, and40 cubes)

Per group:� Teaching Aid Master (Math Masters,

p. 418; optional)� posterboard; markers� base-10 blocks (optional)

ENRICHMENTREADINESS

3

� Math Journal 2, p. 270� My Reference Book, pp. 45 and 46� Home Link Masters (Math Masters,

pp. 353 and 354)� stick-on notes

2

� Math Journal 2, p. 269� Home Link 11�2� Teaching Aid Masters (Math

Masters, p. 415; and p. 418,optional)

� base-10 blocks: 18 cubes and7 longs (optional)

� grid paper (optional)

1

Objective To introduce and provide practice with the trade-first subtraction algorithm.

Technology Assessment Management SystemMental Math and ReflexesSee the iTLG.

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� Math Message Follow-UpRemind children that one way to make a ballpark estimate is tochange the numbers in the problem to close but “easier” numbersthat can be subtracted mentally. For example:

� 58 � 37 is close to 60 � 40, so the exact answer to 58 � 37should be close to 20.

� 143 � 65 is close to 140 � 60 or 140 � 70, so the exact answershould be near 80 or 70. The exact answer is 143 � 65 � 78.Point out that 65 is halfway between 60 and 70 on the numberline and the same number of spaces away from 60 and 70 onthe number grid. Therefore, 60 or 70 may be used as an“easier” number substitute for 65.

WHOLE-CLASSDISCUSSION

1 Teaching the Lesson

Lesson 11�3 813

Getting Started

Math MessageMake a ballpark estimate for each answer.

58 � 37 �Sample answer: 60 – 40 � 20

143 � 65 �Sample answer: 140 – 60 � 80

Mental Math and ReflexesWrite problems like the following on the board.Mix horizontal and vertical formats. Have childrensolve the problems by using the number grid or mentalarithmetic. Have children record their solutions on anExit Slip (Math Masters, page 415).

�

50 � 35 � ? 1530 � 7 � ? 2350 � 43 � ? 770 � 39 � ? 31160 � 14 � ? 146180 � 56 � ? 124

Unit

people

Home Link 11�2 Follow-UpReview answers. Give children practice with estimationskills by asking questions like the following:

• You buy a bottle of glitter. You pay with a $5 bill. Should you getmore or less than $3.00 in change? more

• You buy a box of crayons. You pay with a $1 bill. Should you getmore or less than 4 dimes in change? less

Links to the FutureToday children will learn and practice onepaper-and-pencil strategy for subtraction.Children will learn other paper-and-pencilsubtraction algorithms in third grade.Encourage children to continue using otherfavorite strategies (such as counting up) thatthey may have learned or devised for subtracting numbers.

Ongoing Assessment:Recognizing Student Achievement

Use Mental Math and Reflexes to assess children’s progress toward usingstrategies for subtraction problems involving two digits. Children are makingadequate progress if they successfully complete the first two sets of problemsmentally or using the number grid. Some children may be able to solve thethird set.

[Operations and Computation Goal 2]

Mental Math and Reflexes �

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814 Unit 11 Whole-Number Operations Revisited

longs10s

4� 2

2

cubes1s

523

� Demonstrating the Trade-FirstAlgorithm without TradesExplain to children that they will be able to calculate the answersto subtraction problems with larger numbers much more easily ifthey learn to use the same strategy for all problems.

Show the paper-and-pencil method for problems that do notrequire trading 1 ten for 10 ones.

Example: 45 – 22

1. Write the problem on the board in vertical format. As areminder, write “longs/10s” and “cubes/1s” above thecolumns. (See margin.)

2. Draw 4 longs and 5 cubes on the board to represent thenumber 45.

3. Ask: Are there enough longs and cubes so I can remove 22 (2 longs and 2 cubes)? Yes. There are 4 longs, so I canremove 2 longs. There are 5 cubes, so I can remove 2 cubes.

4. Subtract the tens. Erase 2 longs. Say, 4 longs minus 2 longsresults in 2 longs, or 4 tens minus 2 tens equals 2 tens. Write2 in the tens place of the answer space.

5. Subtract the ones. Erase 2 cubes. Say, 5 cubes minus 2 cubes results in 3 cubes, or 5 ones minus 2 ones equals 3 ones. Write 3 in the ones place of the answer space.

Repeat this demonstration with several other problems that donot require trades, such as 79 � 34 and 87 � 15. Try to relyless and less on the longs-and-cubes pictures to illustrate theoperation of subtraction.

� Demonstrating the Trade-First Algorithm with TradesExample: 53 – 37

1. Write the problem on the board in vertical format. As areminder, write “longs/10s” and “cubes/1s” above thecolumns. (See margin.)

2. Draw 5 longs and 3 cubes on the board to represent thenumber 53.

3. Ask: Are there enough longs and cubes so I can remove 37 (3 longs and 7 cubes)? No. There are only 3 cubes, so I can’tremove 7 cubes.

4. Trade a long for cubes. Erase 1 long and replace it by drawing 10 cubes. Ask: How many longs and cubes make up53 now? 53 is now represented by 4 longs and 13 cubes.Mark the problem to reflect this action: Draw a slash markthrough the 5 and write 4 above it; draw a slash markthrough the 3 and write 13 above it.

WHOLE-CLASS ACTIVITY

WHOLE-CLASS ACTIVITY

longs10s

45

� 31

cubes1s

13376

Ongoing Assessment:Informing Instruction

Watch for children who do not identify thevalue of the digits correctly. For example, insubtracting 79 � 34, children should say “7tens minus 3 tens equals 4 tens” or “70 minus30 equals 40,” not “7 minus 3 equals 4.”

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Adjusting the Activity

5. Subtract the tens. Erase 3 longs from the picture. Say, 4longs minus 3 longs results in 1 long, or 4 tens minus 3 tensequals 1 ten. Write 1 in the tens place of the answer space.

6. Subtract the ones. Erase 7 cubes from the picture. Say, 13ones minus 7 ones equals 6 ones. Write 6 in the ones place ofthe answer space.

Repeat this demonstration with several other problems thatrequire trades, such as 72 � 38 and 80 � 17. Try to rely less andless on the longs-and-cubes pictures to illustrate the operation of subtraction.

Summary: The method shown in these examples is called asubtraction algorithm because it gives a step-by-step set ofinstructions for solving a subtraction problem. It is called a trade-first algorithm because the first step is to identify whether anytrade (such as 1 ten for 10 ones) is required. If a trade is required,the trade is carried out first, followed by subtraction of the tensdigits and the ones digits in either order. Use the term trade-firstwith children, but it is not necessary to use the term algorithm. Tosupport English language learners, discuss and demonstrate themeaning of trade.

� Practicing the Trade-First Algorithm(Math Journal 2, p. 269)

On the board, write several 2-digit subtraction problems thatrequire trading. Invite children to solve these problems at theboard. Ask children to describe exactly what they did. Correcterrors in calculation and in method. Make sure the numberschildren write are properly aligned in columns.

Many students will find it helpful if they are reminded about placevalue for the numbers they are subtracting. (See margin.)

Assign journal page 269. Children may continue to use base-10blocks to model the problems if they wish. Check that children arecorrectly marking the problems to show any trades. Remindchildren to compare their ballpark estimates to their answers andto look for mistakes if an answer is far away from an estimate.

ELL

Use base-10 blocks to model the problems. Children will soon discoverthat they can begin to solve problems without the blocks.

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

INDEPENDENTACTIVITY

Lesson 11�3 815

Trade-First SubtractionLESSON

11�3

Date Time

longs cubes10s 1s

7 6� 5 9

longs cubes10s 1s

3 5� 2 6

longs cubes10s 1s

4 4� 2 7

longs cubes10s 1s2 173 7

� 1 91 8

� Make a ballpark estimate for each problem and write a number modelfor your ballpark estimate.

� Use the trade-first method of subtraction to solve each problem.

longs cubes10s 1s

2 8� 1 9

longs cubes10s 1s

3 1� 1 7

Answer

9

Answer

14

Answer

9

Answer

17

Answer

17

Answer

18

Example:

Ballpark estimate:1. Ballpark estimate:

2. Ballpark estimate: 3. Ballpark estimate:

4. Ballpark estimate: 5. Ballpark estimate:40 � 30 � 1040 � 30 � 10

80 � 60 � 2030 � 20 � 10

30 � 20 � 1040 � 20 � 20

Math Journal 2, p. 269

Student Page

10s 1s5 1

- 2 9

Encourage children to write “10s” and “1s” abovethe tens and ones columns of a problem.

Ongoing Assessment:Informing Instruction

Watch for children who have difficulty aligning the digits when solving a verticalproblem. Draw lines to separate place-valuecolumns or have children use grid paper.Children then can put each digit in aseparate box.

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Math BoxesLESSON

11�3

Date Time

1. The perimeter is about

cm.13

3. What is the value of the digit 4in each number?

14

142

436

4,678 4,000400404

6. Complete the FactTriangle. Writethe fact family.

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� �

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� � 6318361818361863

2. Divide into:

halves fourths

Write �, �, or �.

�12� �

14� �

24� �

12�

�12� �

34��

��

4.

-by- array

How many in all? 2173

5. I had a 10-dollar bill. I spent$5.23. How much change did I receive? Fill in the circle nextto the best answer.

A $3.80 B $4.77

C $5.00 D $15.23

3 6

18�,�

10 11

38

Math Journal 2, p. 270

Student Page

Name Date Time

Today your child learned about subtracting multidigit numbers using a procedure called thetrade-first method. Your child also used “ballpark estimates” to determine whether his or heranswers made sense.

The trade-first method is similar to the traditional subtraction method that you may befamiliar with. However, all the “regrouping” or “borrowing” is done before the problem issolved—which gives the method its name, “trade-first.”

Example:

� Are there enough tens and ones to remove exactly 3 tens and 9 ones from 46? (No; thereare enough tens, but there aren’t enough ones.)

� Trade 1 ten for 10 ones.

� Solve. 3 tens minus 3 tens leaves 0 tens. 16 ones minus 9 ones leaves 7 ones. The answer is 7.

� Make a ballpark estimate to see whether the answer makes sense: 46 is close to 50, and39 is close to 40. 50 � 40 � 10. 10 is close to the answer of 7, so 7 is a reasonable answer.

The trade-first method is one of many ways people solve subtraction problems. Your childmay choose this method or a different procedure. What is most important is that your childcan successfully solve subtraction problems using a method that makes sense to him or her.

Please return the second page of this Home Link to school tomorrow.

FamilyNote

HOME LINK

11� 3 Trade-First Subtraction

longs cubes10s 1s3 164 6

� 3 9

longs cubes10s 1s

4 6� 3 9

longs cubes10s 1s3 164 6

� 3 97

34

Math Masters, p. 353

Home Link Master

816 Unit 11 Whole-Number Operations Revisited

� Describing a Data Set on a Line Plot(My Reference Book, pp. 45 and 46)

Use pages 45 and 46 in My Reference Book to review datalandmarks. Prepare a line plot on the board using stick-on notesand this data: 5, 5, 5, 5, 10, 10, 15, 15, 20. (See the routine in Part 2 of Lesson 9-8.) To provide a context for the children, suggestthat the line plot tells the number of minutes children spent doing homework.

Ask children to use the line-plot data to identify and label thelargest number of minutes (maximum) 20, the smallest number ofminutes (minimum) 5, the number of minutes that occurs mostoften (mode) 5, and to calculate the difference between themaximum and the minimum number of minutes (range) 15. Next,have children remove the last stick-on note from each end of theline plot over and over until one or two remain. This is the mediannumber of minutes. 10

Line plot with stick-on notes The median is 10 minutes.

Replace the stick-on notes on the line plot. Have children imaginethat you will put all the stick-on notes into a container and pickone out at random (without looking). Ask children to compare thechance of getting a note that has a 5 on it with the chance ofgetting one with a 10 on it. The 5 has a greater chance becausethere are more 5s than 10s. Ask children to compare the chancesof getting a 15 and a 20. There is a greater chance of getting 15because there are more 15s. Ask children to give two numbersthat have the same chance. 10 and 15

� Math Boxes 11�3(Math Journal 2, p. 270)

Mixed Practice Math Boxes in this lesson are paired withMath Boxes in Lesson 11-1. The skill in Problem 6 previews Unit 12 content.

INDEPENDENTACTIVITY

0 5 10 15 20 25Number of Minutes

Numberof

Children

0 5 10 15 20 25Number of Minutes

Numberof

Children

WHOLE-CLASS ACTIVITY

2 Ongoing Learning & Practice

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� Home Link 11�3(Math Masters, pp. 353 and 354)

Home Connection Children use the trade-first algorithmto solve subtraction problems. The trade-first algorithm issimilar to the subtraction method most parents learned,

but the term may not be familiar to them. The explanation in theFamily Note will help parents feel comfortable when participatingin these activities with their children.

� Modeling Subtraction with Base-10 Blocks(Math Masters, p. 355)

To explore multidigit subtraction, have children model subtractionwith base-10 blocks. First have children build 23 using cubes only.Remind children that there are different ways to build a numberwith base-10 blocks. Ask them to build 23 using the smallestnumber of base-10 blocks. Have them take away 12. Write thenumber model 23 – 12 on the board. Have children share their answers.

Ask children to build 23 again using the smallest number ofbase-10 blocks. Have them take away 19. Write the number model23 – 19 on the board. Have children share their answers andstrategies. Sample strategies: Rebuild 23 using more cubes;physically trade a long for cubes; take away two longs and add onecube back in.

Have children work together to complete Math Masters, page 355.

� Exploring Subtraction Strategies(Math Masters, p. 418)

To further explore strategies for solving subtraction problems,children work together to write a 2- or 3-digit subtraction problem.They use as many methods as they can to solve the problem.Children create posters showing their methods. (See margin.)Possible methods include using a number grid (real or a mentalpicture), base-10 blocks, counting up, trade-first algorithm, andmental math. Children may use other methods as well. Displaythe posters.

15–30 Min

SMALL-GROUP ACTIVITY

ENRICHMENT

15–30 Min

PARTNER ACTIVITYREADINESS

3 Differentiation Options

INDEPENDENTACTIVITY

Lesson 11�3 817

Use base-10 blocks to help you subtract.

1. 2. 3.

4. 5. 6.

7. Write a problem of your own. Record what you would do withbase-10 blocks to solve your problem.Answers vary.

LESSON

11�3

Name Date Time

Subtraction with Base-10 Blocks

longs cubes10s 1s

3 7� 2 2

1 5

longs cubes10s 1s

4 3� 3 1

1 2

longs cubes10s 1s

2 4� 1 8

6longs cubes10s 1s

6 2� 3 9

2 3

longs cubes10s 1s

5 5� 4 6

9

longs cubes10s 1s

4 7� 2 9

1 8

Math Masters, p. 355

Teaching Master

Name Date Time

Make a ballpark estimate for each problem and write a number modelfor your estimate.

Use the trade-first method of subtraction to solve each problem.

HOME LINK

11� 3 Trade-First Subtraction cont.

longs cubes10s 1s

3 2� 1 5

Example: Ballpark estimate: 1. Ballpark estimate:

2. Ballpark estimate: 3. Ballpark estimate:

4. Ballpark estimate: 5. Ballpark estimate:

34 � 18

30 � 20 � 1030 � 10 � 20

90 � 60 � 3050 � 30 � 20

70 � 40 � 3030 � 20 � 10

Answer

17Answer

16

longs cubes10s 1s1 162 6

� 1 88

Answer

8

longs cubes10s 1s

7 3� 4 2

Answer

31

longs cubes10s 1s

4 9� 2 6

Answer

23longs cubes10s 1s

8 5� 5 6

Answer

29

Math Masters, p. 354

Home Link Master

Subtraction Strategies372 - 157 = 215

Base-Ten Blocks

81 - 46 = 35

81

35- 46

7 11

Trade First

Counting Up Using MyMental Picture of a

Number Grid

30292827262524232221

40393837363534333231

50494847464544434241

60595857565554535251

70696867666564636261

80797877767574737271

90898887868584838281

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