Other Ways to Add and Subtract — Part 1
4A · Chapter 2 · Lesson 3 · MATH-4A-C02-L03
Learning goal. Choose decompositions and compensation strategies and justify equivalent sums.
Before you begin
Recall: 4A Ch1: million-range place value; Grade 3: column algorithms and comparison bars; Subtraction. Try these three warm-up problems. If they are difficult, revisit the prerequisite lesson before advancing.
Calculate 482,740 − 92,763.
Show answer & reasoning
Subtract like place values, exchanging one higher unit for ten lower units when necessary. Check: 389,977 + 92,763 = 482,740.
Calculate 315,793 − 202,788.
Show answer & reasoning
Subtract like place values, exchanging one higher unit for ten lower units when necessary. Check: 113,005 + 202,788 = 315,793.
Calculate 402,964 − 34,225.
Show answer & reasoning
Subtract like place values, exchanging one higher unit for ten lower units when necessary. Check: 368,739 + 34,225 = 402,964.
Key vocabulary
- regroup
- exchange equal-value units
- compensation
- temporary adjustment corrected afterward
- sum
- result of addition
- difference
- result of subtraction
Understand the concept
Compensation makes awkward addition friendlier. For 3,998 + 246, temporarily raise 3,998 to 4,000, add 246, and remove the extra two: 4,244. You can also move two from one addend to the other, leaving the sum unchanged. State which adjustment was made so you do not forget the correction.
Model it: number bonds and balancing equal sums. Explain what every part of the model represents, then write a matching equation.
Fully worked examples
Read the problem first and predict the answer. Then study how each quantity is represented and calculated.
Worked example 1
Use compensation to add 470,851 and 65,950. What is the exact sum?
Increase 470,851 by 49 to a friendly multiple of 100, then subtract 49 from the sum to preserve the total.
Check your understanding 1
Use compensation to add 90,630 and 195,452. What is the exact sum?
Show answer & reasoning
Increase 90,630 by 70 to a friendly multiple of 100, then subtract 70 from the sum to preserve the total.
Worked example 2
Use compensation to add 122,266 and 206,363. What is the exact sum?
Increase 122,266 by 734 to a friendly multiple of 1,000, then subtract 734 from the sum to preserve the total.
Check your understanding 2
Use compensation to add 54,742 and 11,182. What is the exact sum?
Show answer & reasoning
Increase 54,742 by 8 to a friendly multiple of 10, then subtract 8 from the sum to preserve the total.
Worked example 3
Use compensation to add 288,984 and 216,133. What is the exact sum?
Increase 288,984 by 6 to a friendly multiple of 10, then subtract 6 from the sum to preserve the total.
Check your understanding 3
Use compensation to add 571,751 and 19,804. What is the exact sum?
Show answer & reasoning
Increase 571,751 by 9 to a friendly multiple of 10, then subtract 9 from the sum to preserve the total.
Guided practice
Use compensation to add 409,040 and 170,790. What is the exact sum?
Show answer & reasoning
Increase 409,040 by 60 to a friendly multiple of 100, then subtract 60 from the sum to preserve the total.
Use compensation to add 66,159 and 80,906. What is the exact sum?
Show answer & reasoning
Increase 66,159 by 41 to a friendly multiple of 100, then subtract 41 from the sum to preserve the total.
Use compensation to add 343,604 and 172,800. What is the exact sum?
Show answer & reasoning
Increase 343,604 by 96 to a friendly multiple of 100, then subtract 96 from the sum to preserve the total.
Use compensation to add 433,238 and 201,677. What is the exact sum?
Show answer & reasoning
Increase 433,238 by 2 to a friendly multiple of 10, then subtract 2 from the sum to preserve the total.
Use compensation to add 542,070 and 155,201. What is the exact sum?
Show answer & reasoning
Increase 542,070 by 930 to a friendly multiple of 1,000, then subtract 930 from the sum to preserve the total.
Independent practice
Use compensation to add 529,837 and 183,627. What is the exact sum?
Show answer & reasoning
Increase 529,837 by 3 to a friendly multiple of 10, then subtract 3 from the sum to preserve the total.
Use compensation to add 336,133 and 4,147. What is the exact sum?
Show answer & reasoning
Increase 336,133 by 67 to a friendly multiple of 100, then subtract 67 from the sum to preserve the total.
Use compensation to add 490,287 and 136,693. What is the exact sum?
Show answer & reasoning
Increase 490,287 by 713 to a friendly multiple of 1,000, then subtract 713 from the sum to preserve the total.
Use compensation to add 631,397 and 202,485. What is the exact sum?
Show answer & reasoning
Increase 631,397 by 3 to a friendly multiple of 100, then subtract 3 from the sum to preserve the total.
Use compensation to add 488,127 and 120,004. What is the exact sum?
Show answer & reasoning
Increase 488,127 by 73 to a friendly multiple of 100, then subtract 73 from the sum to preserve the total.
Use compensation to add 562,342 and 64,303. What is the exact sum?
Show answer & reasoning
Increase 562,342 by 658 to a friendly multiple of 1,000, then subtract 658 from the sum to preserve the total.
Use compensation to add 349,787 and 148,721. What is the exact sum?
Show answer & reasoning
Increase 349,787 by 13 to a friendly multiple of 100, then subtract 13 from the sum to preserve the total.
Use compensation to add 320,590 and 25,541. What is the exact sum?
Show answer & reasoning
Increase 320,590 by 410 to a friendly multiple of 1,000, then subtract 410 from the sum to preserve the total.
Challenge and transfer
Explain the relationships and justify your method. Multiple correct approaches may be accepted if the reasoning is valid.
A student says 299 + 21 = 300 + 21. Identify the compensation error and find the actual sum.
Show answer & reasoning
Increasing 299 to 300 adds one extra, which must be removed: 300 + 21 − 1 = 320.
Use compensation to add 693,927 and 80,752. What is the exact sum?
Show answer & reasoning
Increase 693,927 by 73 to a friendly multiple of 100, then subtract 73 from the sum to preserve the total.
Exit check
Use compensation to add 10,202 and 204,475. What is the exact sum?
Show answer & reasoning
Increase 10,202 by 98 to a friendly multiple of 100, then subtract 98 from the sum to preserve the total.
Use compensation to add 255,799 and 166,748. What is the exact sum?
Show answer & reasoning
Increase 255,799 by 1 to a friendly multiple of 100, then subtract 1 from the sum to preserve the total.
Use compensation to add 16,800 and 136,447. What is the exact sum?
Show answer & reasoning
Increase 16,800 by 100 to a friendly multiple of 100, then subtract 100 from the sum to preserve the total.