3A · Chapter 2 · Lesson 4
Mental Subtraction — Part 2
Change both numbers by the same amount to keep their difference unchanged.
Today's challenge
Use equal adjustment to find 675 − 198. Try your own method, then compare it with a model.
What you'll learn
- Change both numbers by the same amount to keep their difference unchanged.
- Explain the connection between a diagram and a number sentence.
- Solve a fresh application without step-by-step hints.
Before this lesson
Mental subtraction using place values.
Prerequisite competency: 3A-C02-L03
Entry · 2 questions
Warm-up: prerequisite check
Try on paper before continuing. If the questions are hard, review the prerequisite concepts.
- E1
Find 444 − 81 by breaking apart the amount subtracted.
Show answer and reasoning
Answer: 363
Decompose 81 into tens and ones, subtract each part from 444, and check the difference 363.
- E2
Find 504 − 104 by breaking apart the amount subtracted.
Show answer and reasoning
Answer: 400
Decompose 104 into tens and ones, subtract each part from 504, and check the difference 400.
Learn the concept
Words to know
- equal adjustment
- change both subtraction terms equally to preserve the difference
Change both numbers by the same amount to keep their difference unchanged.
Imagine both numbers as points on a number line. Moving both to the right by the same distance does not change the gap. Shift the smaller number to a friendly hundred.
Why this works
In subtraction, the distance between two numbers stays the same when you add the same amount to both. For 503 − 198, add two to each number: 505 − 200 = 305. Changing only one number would change the difference, so always adjust both.
Check the meaning
The new pair must differ by the same amount as the original pair; adjust BOTH numbers, not just one.
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
Use equal adjustment to find 675 − 198.
- Imagine both numbers as points on a number line. Moving both to the right by the same distance does not change the gap. Shift the smaller number to a friendly hundred.
- Add 2 to BOTH numbers: 677 − 200 = 477. Their difference is unchanged.
- The new pair must differ by the same amount as the original pair; adjust BOTH numbers, not just one.
Result: 477
Understanding check 1 — your turn
Use equal adjustment to find 371 − 198.
Reveal reasoning
Answer: 173 — Add 2 to BOTH numbers: 373 − 200 = 173. Their difference is unchanged.
Worked example 2
Reason and verify
Use equal adjustment to find 471 − 297.
- Imagine both numbers as points on a number line. Moving both to the right by the same distance does not change the gap. Shift the smaller number to a friendly hundred.
- Add 3 to BOTH numbers: 474 − 300 = 174. Their difference is unchanged.
- The new pair must differ by the same amount as the original pair; adjust BOTH numbers, not just one.
Result: 174
Understanding check 2 — your turn
Use equal adjustment to find 517 − 297.
Reveal reasoning
Answer: 220 — Add 3 to BOTH numbers: 520 − 300 = 220. Their difference is unchanged.
Worked example 3
Reason and verify
Use equal adjustment to find 617 − 396.
- Imagine both numbers as points on a number line. Moving both to the right by the same distance does not change the gap. Shift the smaller number to a friendly hundred.
- Add 4 to BOTH numbers: 621 − 400 = 221. Their difference is unchanged.
- The new pair must differ by the same amount as the original pair; adjust BOTH numbers, not just one.
Result: 221
Understanding check 3 — your turn
Use equal adjustment to find 663 − 396.
Reveal reasoning
Answer: 267 — Add 4 to BOTH numbers: 667 − 400 = 267. Their difference is unchanged.
Guided · 4 questions
Guided practice
Practice with the model visible. Explain the units at each step.
- G1
Use equal adjustment to find 509 − 99.
Show answer and reasoning
Answer: 410
Add 1 to BOTH numbers: 510 − 100 = 410. Their difference is unchanged.
- G2
Use equal adjustment to find 655 − 198.
Show answer and reasoning
Answer: 457
Add 2 to BOTH numbers: 657 − 200 = 457. Their difference is unchanged.
- G3
Use equal adjustment to find 451 − 297.
Show answer and reasoning
Answer: 154
Add 3 to BOTH numbers: 454 − 300 = 154. Their difference is unchanged.
- G4
Use equal adjustment to find 597 − 396.
Show answer and reasoning
Answer: 201
Add 4 to BOTH numbers: 601 − 400 = 201. Their difference is unchanged.
Independent · 8 questions
Independent practice
Work without relying on the model or checking answers prematurely.
- I1
Use equal adjustment to find 359 − 99.
Show answer and reasoning
Answer: 260
Add 1 to BOTH numbers: 360 − 100 = 260. Their difference is unchanged.
- I2
Use equal adjustment to find 505 − 198.
Show answer and reasoning
Answer: 307
Add 2 to BOTH numbers: 507 − 200 = 307. Their difference is unchanged.
- I3
Use equal adjustment to find 651 − 297.
Show answer and reasoning
Answer: 354
Add 3 to BOTH numbers: 654 − 300 = 354. Their difference is unchanged.
- I4
Use equal adjustment to find 797 − 396.
Show answer and reasoning
Answer: 401
Add 4 to BOTH numbers: 801 − 400 = 401. Their difference is unchanged.
- I5
Use equal adjustment to find 547 − 99.
Show answer and reasoning
Answer: 448
Add 1 to BOTH numbers: 548 − 100 = 448. Their difference is unchanged.
- I6
Use equal adjustment to find 693 − 198.
Show answer and reasoning
Answer: 495
Add 2 to BOTH numbers: 695 − 200 = 495. Their difference is unchanged.
- I7
Use equal adjustment to find 489 − 297.
Show answer and reasoning
Answer: 192
Add 3 to BOTH numbers: 492 − 300 = 192. Their difference is unchanged.
- I8
Use equal adjustment to find 635 − 396.
Show answer and reasoning
Answer: 239
Add 4 to BOTH numbers: 639 − 400 = 239. Their difference is unchanged.
Challenge · 2 questions
Challenge practice
Solve original application and explanation problems. Draw and defend a useful model.
- C1
Does changing only the first number in 503 − 198 to 505 keep the difference the same? Explain.
Show answer and reasoning
Answer: No
505−198=307, not 305. To preserve the difference also change 198 to 200.
- C2
Make 691 − 298 easier without rounding the answer. Write an equivalent subtraction and solve.
Show answer and reasoning
Answer: 693 − 300 = 393
Add two to both numbers; the distance between them does not change.
Exit · 3 questions
Exit check
Solve independently; revisit the skill if an important explanation is missing.
- E1
Use equal adjustment to find 409 − 99.
Show answer and reasoning
Answer: 310
Add 1 to BOTH numbers: 410 − 100 = 310. Their difference is unchanged.
- E2
Use equal adjustment to find 555 − 198.
Show answer and reasoning
Answer: 357
Add 2 to BOTH numbers: 557 − 200 = 357. Their difference is unchanged.
- E3
Use equal adjustment to find 701 − 297.
Show answer and reasoning
Answer: 404
Add 3 to BOTH numbers: 704 − 300 = 404. Their difference is unchanged.