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Objective
To introduce and provide practice with the trade-first
subtraction algorithm.
1
materials
Teaching the Lesson
Key Activities
Children learn the trade-first subtraction algorithm. They use the algorithm to solve subtraction
problems with and without trades. They make ballpark estimates to determine whether their
answers 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]
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 and
7 longs (optional)
grid paper (optional)
• 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.
2
Ongoing Learning & Practice
Children practice finding landmarks on a line plot.
Children practice and maintain skills through Math Boxes and Home Link activities.
3
Children solve subtraction problems using
base-10 blocks.
Math Journal 2, p. 270
My Reference Book, pp. 45 and 46
Home Link Masters (Math Masters,
pp. 353 and 354)
stick-on notes
materials
Differentiation Options
READINESS
materials
ENRICHMENT
Children make posterboards displaying
multidigit subtraction strategies.
Teaching Master (Math Masters,
p. 355)
base-10 blocks (1 flat, 20 longs, and
40 cubes)
Per group:
Teaching Aid Master (Math Masters,
p. 418; optional)
posterboard; markers
base-10 blocks (optional)
Technology
Assessment Management System
Mental Math and Reflexes
See the iTLG.
812
Unit 11 Whole-Number Operations Revisited
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Getting Started
Mental Math and Reflexes
Write problems like the following on the board.
Mix horizontal and vertical formats. Have children
solve the problems by using the number grid or mental
arithmetic. Have children record their solutions on an
Exit Slip (Math Masters, page 415).
50 35 ? 15
30 7 ? 23
50 43 ? 7
70 39 ? 31
160 14 ? 146
180 56 ? 124
Math Message
Make a ballpark estimate for each answer.
58 37 Sample answer: 60 – 40 20
143 65 Sample answer: 140 – 60 80
Unit
people
Home Link 11 2 Follow-Up
Review answers. Give children practice with estimation
skills by asking questions like the following:
• You buy a bottle of glitter. You pay with a $5 bill. Should you get
more or less than $3.00 in change? more
• You buy a box of crayons. You pay with a $1 bill. Should you get
more or less than 4 dimes in change? less
Ongoing Assessment:
Recognizing Student Achievement
Mental Math
and
Reflexes
Use Mental Math and Reflexes to assess children’s progress toward using
strategies for subtraction problems involving two digits. Children are making
adequate progress if they successfully complete the first two sets of problems
mentally or using the number grid. Some children may be able to solve the
third set.
[Operations and Computation Goal 2]
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
Remind children that one way to make a ballpark estimate is to
change the numbers in the problem to close but “easier” numbers
that can be subtracted mentally. For example:
58 37 is close to 60 40, so the exact answer to 58 37
should be close to 20.
143 65 is close to 140 60 or 140 70, so the exact answer
should be near 80 or 70. The exact answer is 143 65 78.
Point out that 65 is halfway between 60 and 70 on the number
line and the same number of spaces away from 60 and 70 on
the number grid. Therefore, 60 or 70 may be used as an
“easier” number substitute for 65.
Links to the Future
Today children will learn and practice one
paper-and-pencil strategy for subtraction.
Children will learn other paper-and-pencil
subtraction algorithms in third grade.
Encourage children to continue using other
favorite strategies (such as counting up) that
they may have learned or devised for
subtracting numbers.
Lesson 11 3
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Demonstrating the Trade-First
WHOLE-CLASS
ACTIVITY
Algorithm without Trades
Explain to children that they will be able to calculate the answers
to subtraction problems with larger numbers much more easily if
they learn to use the same strategy for all problems.
Show the paper-and-pencil method for problems that do not
require trading 1 ten for 10 ones.
Example: 45 – 22
longs
10s
4
2
2
cubes
1s
5
2
3
1. Write the problem on the board in vertical format. As a
reminder, write “longs/10s” and “cubes/1s” above the
columns. (See margin.)
2. Draw 4 longs and 5 cubes on the board to represent the
number 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 can
remove 2 longs. There are 5 cubes, so I can remove 2 cubes.
4. Subtract the tens. Erase 2 longs. Say, 4 longs minus 2 longs
results in 2 longs, or 4 tens minus 2 tens equals 2 tens. Write
2 in the tens place of the answer space.
Ongoing Assessment:
Informing Instruction
Watch for children who do not identify the
value of the digits correctly. For example, in
subtracting 79 34, children should say “7
tens minus 3 tens equals 4 tens” or “70 minus
30 equals 40,” not “7 minus 3 equals 4.”
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 do
not require trades, such as 79 34 and 87 15. Try to rely
less and less on the longs-and-cubes pictures to illustrate the
operation of subtraction.
Demonstrating the Trade-First
WHOLE-CLASS
ACTIVITY
Algorithm with Trades
longs
10s
4
5
3
1
cubes
1s
13
3
7
6
Example: 53 – 37
1. Write the problem on the board in vertical format. As a
reminder, write “longs/10s” and “cubes/1s” above the
columns. (See margin.)
2. Draw 5 longs and 3 cubes on the board to represent the
number 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’t
remove 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 up
53 now? 53 is now represented by 4 longs and 13 cubes.
Mark the problem to reflect this action: Draw a slash mark
through the 5 and write 4 above it; draw a slash mark
through the 3 and write 13 above it.
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Unit 11 Whole-Number Operations Revisited
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5. Subtract the tens. Erase 3 longs from the picture. Say, 4
longs minus 3 longs results in 1 long, or 4 tens minus 3 tens
equals 1 ten. Write 1 in the tens place of the answer space.
6. Subtract the ones. Erase 7 cubes from the picture. Say, 13
ones minus 7 ones equals 6 ones. Write 6 in the ones place of
the answer space.
Repeat this demonstration with several other problems that
require trades, such as 72 38 and 80 17. Try to rely less and
less on the longs-and-cubes pictures to illustrate the operation
of subtraction.
Summary: The method shown in these examples is called a
subtraction algorithm because it gives a step-by-step set of
instructions for solving a subtraction problem. It is called a tradefirst algorithm because the first step is to identify whether any
trade (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 tens
digits and the ones digits in either order. Use the term trade-first
with children, but it is not necessary to use the term algorithm. To
support English language learners, discuss and demonstrate the
meaning of trade.
Practicing the Trade-First
Date
Time
LESSON
Trade-First Subtraction
11 3
䉬
䉬
Make a ballpark estimate for each problem and write a number model
for your ballpark estimate.
䉬
Use the trade-first method of subtraction to solve each problem.
Example:
1. Ballpark estimate:
Ballpark estimate:
30 ⫺ 20 ⫽ 10
40 ⫺ 20 ⫽ 20
longs cubes
10s
1s
2
17
3
⫺1
7
9
1
8
Answer
18
2. Ballpark estimate:
3
⫺1
5
6
7
⫺5
Answer
17
6
9
5. Ballpark estimate:
40 ⫺ 30 ⫽ 10
longs cubes
10s
1s
9
80 ⫺ 60 ⫽ 20
14
1
7
Answer
8
9
longs cubes
10s
1s
Answer
4. Ballpark estimate:
3
⫺2
2
⫺1
3. Ballpark estimate:
30 ⫺ 20 ⫽ 10
longs cubes
10s
1s
longs cubes
10s
1s
40 ⫺ 30 ⫽ 10
Answer
9
longs cubes
10s
1s
4
⫺2
Answer
17
4
7
Math Journal 2, p. 269
INDEPENDENT
ACTIVITY
Algorithm
(Math Journal 2, p. 269)
10s
5
- 2
On the board, write several 2-digit subtraction problems that
require trading. Invite children to solve these problems at the
board. Ask children to describe exactly what they did. Correct
errors in calculation and in method. Make sure the numbers
children write are properly aligned in columns.
1s
1
9
Many students will find it helpful if they are reminded about place
value for the numbers they are subtracting. (See margin.)
Assign journal page 269. Children may continue to use base-10
blocks to model the problems if they wish. Check that children are
correctly marking the problems to show any trades. Remind
children to compare their ballpark estimates to their answers and
to look for mistakes if an answer is far away from an estimate.
ELL
Adjusting the Activity
Use base-10 blocks to model the problems. Children will soon discover
that 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
Encourage children to write “10s” and “1s” above
the tens and ones columns of a problem.
Ongoing Assessment:
Informing Instruction
Watch for children who have difficulty
aligning the digits when solving a vertical
problem. Draw lines to separate place-value
columns or have children use grid paper.
Children then can put each digit in a
separate box.
Lesson 11 3
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Date
Time
LESSON
11 3
䉬
1. The perimeter is about
13
2 Ongoing Learning & Practice
Math Boxes
2. Divide into:
cm.
halves
fourths
Describing a Data Set
Write ⬍, ⬎, or ⫽.
1
ᎏᎏ
2
1
ᎏᎏ
2
3. What is the value of the digit 4
⬎
⬍
1
ᎏᎏ
4
3
ᎏᎏ
4
⫽
2
ᎏᎏ
4
on a Line Plot
1
ᎏᎏ
2
(My Reference Book, pp. 45 and 46)
4.
in each number?
4
40
142
436 400
4,678 4,000
14
3
7
-by-
Use pages 45 and 46 in My Reference Book to review data
landmarks. Prepare a line plot on the board using stick-on notes
and 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, suggest
that the line plot tells the number of minutes children spent
doing homework.
array
21
How many in all?
10 11
5. I had a 10-dollar bill. I spent
6. Complete the Fact
$5.23. How much change did
I receive? Fill in the circle next
to the best answer.
A $3.80
C $5.00
B
D
Triangle. Write
the fact family.
18
⫻, ⫼
3
$4.77
3
6
18
18
$15.23
⫻
⫻
⫼
⫼
6
3
6
3
⫽
⫽
⫽
⫽
6
18
18
3
6
38
Math Journal 2, p. 270
Ask children to use the line-plot data to identify and label the
largest number of minutes (maximum) 20, the smallest number of
minutes (minimum) 5, the number of minutes that occurs most
often (mode) 5, and to calculate the difference between the
maximum and the minimum number of minutes (range) 15. Next,
have children remove the last stick-on note from each end of the
line plot over and over until one or two remain. This is the median
number of minutes. 10
Number
of
Children
0
HOME LINK
11 3
䉬
Family
Note
Date
Time
Trade-First Subtraction
Today your child learned about subtracting multidigit numbers using a procedure called the
trade-first method. Your child also used “ballpark estimates” to determine whether his or her
answers made sense.
The trade-first method is similar to the traditional subtraction method that you may be
familiar with. However, all the “regrouping” or “borrowing” is done before the problem is
solved—which gives the method its name, “trade-first.”
Example:
longs cubes
10s
1s
4
⫺3
6
9
䉬 Are there enough tens and ones to remove exactly 3 tens and 9 ones from 46? (No; there
are enough tens, but there aren’t enough ones.)
䉬 Trade 1 ten for 10 ones.
5
10 15 20
Number of Minutes
25
0
5
10 15 20 25
Number of Minutes
The median is 10 minutes.
Replace the stick-on notes on the line plot. Have children imagine
that you will put all the stick-on notes into a container and pick
one out at random (without looking). Ask children to compare the
chance of getting a note that has a 5 on it with the chance of
getting one with a 10 on it. The 5 has a greater chance because
there are more 5s than 10s. Ask children to compare the chances
of getting a 15 and a 20. There is a greater chance of getting 15
because there are more 15s. Ask children to give two numbers
that have the same chance. 10 and 15
longs cubes
10s
1s
3
4
⫺3
䉬 Solve. 3 tens minus 3 tens
leaves 0 tens. 16 ones minus
9 ones leaves 7 ones. The
answer is 7.
16
6
9
Math Boxes 11 3
longs cubes
10s
1s
3
4
⫺3
INDEPENDENT
ACTIVITY
(Math Journal 2, p. 270)
16
6
9
7
䉬 Make a ballpark estimate to see whether the answer makes sense: 46 is close to 50, and
39 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 child
may choose this method or a different procedure. What is most important is that your child
can 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.
34
Math Masters, p. 353
816
Number
of
Children
Line plot with stick-on notes
Home Link Master
Name
WHOLE-CLASS
ACTIVITY
Unit 11 Whole-Number Operations Revisited
Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lesson 11-1. The skill in Problem 6
previews Unit 12 content.
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Home Link Master
Home Link 11 3
INDEPENDENT
ACTIVITY
(Math Masters, pp. 353 and 354)
Name
Date
HOME LINK
11 3
Use the trade-first method of subtraction to solve each problem.
Example: Ballpark estimate:
1. Ballpark estimate:
70 ⫺ 40 ⫽ 30
30 ⫺ 20 ⫽ 10
longs cubes
10s
1s
1
8
16
2
⫺1
longs cubes
10s
1s
Answer
6
8
7
⫺4
8
2. Ballpark estimate:
longs cubes
10s
1s
4
⫺2
3 Differentiation Options
90 ⫺ 60 ⫽ 30
23
8
⫺5
4. Ballpark estimate:
PARTNER
ACTIVITY
3
⫺1
Answer
29
5
6
5. Ballpark estimate:
30 ⫺ 10 ⫽ 20
longs cubes
10s
1s
31
3
2
longs cubes
10s
1s
Answer
9
6
Answer
3. Ballpark estimate:
50 ⫺ 30 ⫽ 20
Modeling Subtraction with
cont.
Make a ballpark estimate for each problem and write a number model
for your estimate.
Home Connection Children use the trade-first algorithm
to solve subtraction problems. The trade-first algorithm is
similar to the subtraction method most parents learned,
but the term may not be familiar to them. The explanation in the
Family Note will help parents feel comfortable when participating
in these activities with their children.
READINESS
Time
Trade-First Subtraction
䉬
30 ⫺ 20 ⫽ 10
34 ⫺ 18
Answer
Answer
17
2
5
16
15–30 Min
Base-10 Blocks
Math Masters, p. 354
(Math Masters, p. 355)
Subtraction Strategies
To explore multidigit subtraction, have children model subtraction
with base-10 blocks. First have children build 23 using cubes only.
Remind children that there are different ways to build a number
with base-10 blocks. Ask them to build 23 using the smallest
number of base-10 blocks. Have them take away 12. Write the
number model 23 – 12 on the board. Have children share
their answers.
Ask children to build 23 again using the smallest number of
base-10 blocks. Have them take away 19. Write the number model
23 – 19 on the board. Have children share their answers and
strategies. Sample strategies: Rebuild 23 using more cubes;
physically trade a long for cubes; take away two longs and add one
cube back in.
372 - 157 = 215
Base-Ten Blocks
81 - 46 = 35
7 11
81
- 46
35
Trade First
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 mental
picture), base-10 blocks, counting up, trade-first algorithm, and
mental math. Children may use other methods as well. Display
the posters.
24
25
26
27
28
29
30
33
34
35
36
37
38
39
40
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
Date
LESSON
Time
Subtraction with Base-10 Blocks
11 3
Use base-10 blocks to help you subtract.
SMALL-GROUP
ACTIVITY
15–30 Min
23
32
41
Teaching Master
1.
ENRICHMENT
22
31
Counting Up Using My
Mental Picture of a
Number Grid
Name
䉬
Have children work together to complete Math Masters, page 355.
21
4.
longs cubes
10s
1s
3
⫺2
7
2
1
5
longs cubes
10s
1s
6
⫺3
2
9
2
3
2.
5.
3.
longs cubes
10s
1s
4
⫺3
3
1
1
2
2
⫺1
4
8
6
6.
longs cubes
10s
1s
5
⫺4
longs cubes
10s
1s
longs cubes
10s
1s
5
6
4
⫺2
7
9
9
1
8
7. Write a problem of your own. Record what you would do with
base-10 blocks to solve your problem.
Answers vary.
Math Masters, p. 355
Lesson 11 3
817