Opposite-Change Rule for Addition
Addition
The opposite-change rule says that if a number is added to one addend
and that same number is subtracted from the other addend, the sum will
be unaffected. And since it is arguably easier to add two addends when
one of them ends in one or more zeros, the goal is to adjust both addends
so that one of them is changed to the nearest ten (or hundred or thousand).
Students find this algorithm particularly useful when calculating mentally.
Build Understanding
Lead students in a quick, oral review of number pairs that add up to 10. Then
expand the review by asking students to identify number pairs that add up to
20, 30, 40, 50, and so on. Point out to students that these larger multiples-often number pairs are based on basic addition facts—for example, the 42 and 8
pair is based on 2 and 8; the 31 and 9 pair is based on 1 and 9; and the 25 and
5 pair is based on 5 and 5. Then reverse the review and test students on
multiples-of-ten number pairs based on basic subtraction facts: What is 40
minus 3? 70 minus 6? 90 minus 8?
Note: Some students may need to write the basic facts and fact extensions on
scratch paper. Others may need to see a demonstration: Display 10 counters
and have different students demonstrate how many different subgroup
pairings can be made with the ten counters while still maintaining the same
total number (10).
Using page 15, explain that with this method of adding, students will be
renaming the two addends (and rewriting the problem) one or two times before
they finally add—the goal being to adjust both addends so that one of them
ends in one or more zeros. Use questions like the following to guide students
through the examples:
• Which of the two addends is closer to an even ten (or hundred or
thousand)?
• What will you have to do to the other addend?
1. 1,480
• Do you need to adjust the addends again before you are ready to add
them together?
2. 912
Error Alert Watch for students who adjust one addend “up” or “down”
without also adjusting the other addend the opposite way. Explain that
students are taking the total value of the two numbers and shifting it around,
or redistributing it, between the two addends. To maintain the total value,
they cannot add a number to one addend without subtracting that same
number from the other addend.
3. 1,367
4. 4,227
5. 4,540
Check Understanding
Divide the class into groups of four, and assign a leader in each group to
explain which adjustments took place in each of the examples. Tell group
members to direct their questions to their group’s leader. When you are
reasonably certain that most of your students understand the algorithm,
assign the “Check Your Understanding ” exercises at the bottom of page 15.
(See answers in margin.)
6. 2,113
7. 5,964
8. 20,600
14
Copyright © Wright Group/McGraw-Hill
• How much will you have to add to (or subtract from) that addend to make
it an even ten (or hundred or thousand)?
Page 15
Answer Key
Teacher Notes
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Name
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Opposite-Change Rule for Addition
Addition
Decide which addend is closer to an even 10 (or 100 or 1,000).
Decide how to adjust that addend so that it ends in one or
more zeros.
Adjust the other addend in the opposite way.
Rename both addends until you reach your goal.
Then add the two addends together to find their sum.
364 (addend)
+ 278 (addend)
Example 1
First, adjust 364 down (by 2) to 362
and adjust 278 up (by 2) to 280.
362
+ 280
Then, adjust 362 down (by 20) to 342
and adjust 280 up (by 20) to 300.
Finally, add the two addends together.
342
+ 300
642 (sum)
→
5,261
+ 9,400
Copyright © Wright Group/McGraw-Hill
Example 2
First, adjust 9,400 down (by 400) to 9,000
and adjust 5,261 up (by 400) to 5,661.
Then add the two addends together.
5,661
+ 9,000
14,661
→
Check Your Understanding
Solve the following problems.
1. 504 + 976
2. 642 + 270
3. 823 + 544
4. 4,132 + 95
5. 972 + 3,568
6. 1,477 + 636
7. 2,675 + 3,289
8. 14,037 + 6,563
Write your answers on a separate sheet of paper.
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