3A · Chapter 2 · Lesson 2
Mental Addition — Part 2
Use compensation: make an addend easier and undo the adjustment.
Today's challenge
Find 198 + 140 using compensation. Try your own method, then compare it with a model.
What you'll learn
- Use compensation: make an addend easier and undo the adjustment.
- Explain the connection between a diagram and a number sentence.
- Solve a fresh application without step-by-step hints.
Before this lesson
Mental addition by decomposition.
Prerequisite competency: 3A-C02-L01
Entry · 2 questions
Warm-up: prerequisite check
Try on paper before continuing. If the questions are hard, review the prerequisite concepts.
- E1
Add 273 + 155. Show a place-value breakdown.
Show answer and reasoning
Answer: 428
273 is 200 + 73; add the two parts to 155. The sum is 428.
- E2
Add 310 + 178. Show a place-value breakdown.
Show answer and reasoning
Answer: 488
310 is 300 + 10; add the two parts to 178. The sum is 488.
Learn the concept
Words to know
- compensation
- an adjustment that is later undone
- sum
- the result of addition
Use compensation: make an addend easier and undo the adjustment.
Raise one addend to a nearby hundred. The total changes by exactly the same amount, so undo that increase. Write the adjusted and original sums side by side.
Why this works
Friendly numbers can make mental addition easier. For 398 + 57, temporarily replace 398 with 400. Now 400 + 57 = 457, but you added two extra. Remove those two: 457 − 2 = 455. Compensation means you undo exactly the change you made.
Check the meaning
Try compensation only when the adjusted number makes the arithmetic simpler. Decomposing by place value is equally valid.
Concrete → Pictorial → Abstract
- Build: Use counters, grid paper, or labeled bundles to represent the story or quantity.
- Draw: Sketch an array, place-value exchange, or equal-unit bar; label what is known.
- Calculate: Write an equation matching the representation and solve.
- Verify: Use an inverse operation or a second strategy to check.
For advanced students: if you can already explain this concept, start with the independent or challenge set.
Worked examples and immediate checks
Explain each step aloud or in writing. Work the fresh question before revealing its solution.
Worked example 1
Reason and verify
Find 198 + 140 using compensation.
- Raise one addend to a nearby hundred. The total changes by exactly the same amount, so undo that increase. Write the adjusted and original sums side by side.
- Increase 198 by 2 to get a multiple of 100; then subtract that same adjustment from the total. Result: 338.
- Try compensation only when the adjusted number makes the arithmetic simpler. Decomposing by place value is equally valid.
Result: 338
Understanding check 1 — your turn
Find 99 + 72 using compensation.
Reveal reasoning
Answer: 171 — Increase 99 by 1 to get a multiple of 100; then subtract that same adjustment from the total. Result: 171.
Worked example 2
Reason and verify
Find 297 + 163 using compensation.
- Raise one addend to a nearby hundred. The total changes by exactly the same amount, so undo that increase. Write the adjusted and original sums side by side.
- Increase 297 by 3 to get a multiple of 100; then subtract that same adjustment from the total. Result: 460.
- Try compensation only when the adjusted number makes the arithmetic simpler. Decomposing by place value is equally valid.
Result: 460
Understanding check 2 — your turn
Find 198 + 95 using compensation.
Reveal reasoning
Answer: 293 — Increase 198 by 2 to get a multiple of 100; then subtract that same adjustment from the total. Result: 293.
Worked example 3
Reason and verify
Find 396 + 186 using compensation.
- Raise one addend to a nearby hundred. The total changes by exactly the same amount, so undo that increase. Write the adjusted and original sums side by side.
- Increase 396 by 4 to get a multiple of 100; then subtract that same adjustment from the total. Result: 582.
- Try compensation only when the adjusted number makes the arithmetic simpler. Decomposing by place value is equally valid.
Result: 582
Understanding check 3 — your turn
Find 297 + 118 using compensation.
Reveal reasoning
Answer: 415 — Increase 297 by 3 to get a multiple of 100; then subtract that same adjustment from the total. Result: 415.
Guided · 4 questions
Guided practice
Practice with the model visible. Explain the units at each step.
- G1
Find 99 + 119 using compensation.
Show answer and reasoning
Answer: 218
Increase 99 by 1 to get a multiple of 100; then subtract that same adjustment from the total. Result: 218.
- G2
Find 198 + 142 using compensation.
Show answer and reasoning
Answer: 340
Increase 198 by 2 to get a multiple of 100; then subtract that same adjustment from the total. Result: 340.
- G3
Find 297 + 165 using compensation.
Show answer and reasoning
Answer: 462
Increase 297 by 3 to get a multiple of 100; then subtract that same adjustment from the total. Result: 462.
- G4
Find 396 + 188 using compensation.
Show answer and reasoning
Answer: 584
Increase 396 by 4 to get a multiple of 100; then subtract that same adjustment from the total. Result: 584.
Independent · 8 questions
Independent practice
Work without relying on the model or checking answers prematurely.
- I1
Find 99 + 705 using compensation.
Show answer and reasoning
Answer: 804
Increase 99 by 1 to get a multiple of 100; then subtract that same adjustment from the total. Result: 804.
- I2
Find 198 + 579 using compensation.
Show answer and reasoning
Answer: 777
Increase 198 by 2 to get a multiple of 100; then subtract that same adjustment from the total. Result: 777.
- I3
Find 297 + 631 using compensation.
Show answer and reasoning
Answer: 928
Increase 297 by 3 to get a multiple of 100; then subtract that same adjustment from the total. Result: 928.
- I4
Find 396 + 683 using compensation.
Show answer and reasoning
Answer: 1079
Increase 396 by 4 to get a multiple of 100; then subtract that same adjustment from the total. Result: 1,079.
- I5
Find 491 + 285 using compensation.
Show answer and reasoning
Answer: 776
Increase 491 by 9 to get a multiple of 100; then subtract that same adjustment from the total. Result: 776.
- I6
Find 99 + 337 using compensation.
Show answer and reasoning
Answer: 436
Increase 99 by 1 to get a multiple of 100; then subtract that same adjustment from the total. Result: 436.
- I7
Find 198 + 389 using compensation.
Show answer and reasoning
Answer: 587
Increase 198 by 2 to get a multiple of 100; then subtract that same adjustment from the total. Result: 587.
- I8
Find 297 + 441 using compensation.
Show answer and reasoning
Answer: 738
Increase 297 by 3 to get a multiple of 100; then subtract that same adjustment from the total. Result: 738.
Challenge · 2 questions
Challenge practice
Solve original application and explanation problems. Draw and defend a useful model.
- C1
A student says 398 + 57 = 400 + 57 + 2. Identify the error and correct it.
Show answer and reasoning
Answer: 455
Raising 398 to 400 adds 2 extra, so 400+57−2=455.
- C2
Without calculating the whole sum, predict whether 497 + 186 is greater or less than 500 + 186, and by exactly how much.
Show answer and reasoning
Answer: Less by 3
Replacing 497 by 500 adds three, so the original sum is exactly three less.
Exit · 3 questions
Exit check
Solve independently; revisit the skill if an important explanation is missing.
- E1
Find 99 + 579 using compensation.
Show answer and reasoning
Answer: 678
Increase 99 by 1 to get a multiple of 100; then subtract that same adjustment from the total. Result: 678.
- E2
Find 198 + 631 using compensation.
Show answer and reasoning
Answer: 829
Increase 198 by 2 to get a multiple of 100; then subtract that same adjustment from the total. Result: 829.
- E3
Find 297 + 505 using compensation.
Show answer and reasoning
Answer: 802
Increase 297 by 3 to get a multiple of 100; then subtract that same adjustment from the total. Result: 802.