3A · Chapter 3 · Lesson 2
Subtraction with Regrouping — Part 1
Exchange one ten for ten ones or one hundred for ten tens when needed.
Today's challenge
Subtract 2,982 − 428; show any place-value exchange. Try your own method, then compare it with a model.
What you'll learn
- Exchange one ten for ten ones or one hundred for ten tens when needed.
- Explain the connection between a diagram and a number sentence.
- Solve a fresh application without step-by-step hints.
Before this lesson
Place-value exchanges.
Prerequisite competency: 3A-C03-L01
Entry · 2 questions
Warm-up: prerequisite check
Try on paper before continuing. If the questions are hard, review the prerequisite concepts.
- E1
Add 2,118 and 1,997. Explain one exchange of ten units.
Show answer and reasoning
Answer: 4115
Combine ones, tens, hundreds, and thousands. Exchange 10 units for 1 in the next column whenever needed; 2,118 + 1,997 = 4,115.
- E2
Add 2,288 and 2,127. Explain one exchange of ten units.
Show answer and reasoning
Answer: 4415
Combine ones, tens, hundreds, and thousands. Exchange 10 units for 1 in the next column whenever needed; 2,288 + 2,127 = 4,415.
Learn the concept
Words to know
- exchange
- rename one higher-place unit as ten smaller units
- difference
- the amount remaining after subtraction
Exchange one ten for ten ones or one hundred for ten tens when needed.
When a smaller column has too few units, exchange one higher-place unit into ten of the next unit down. Subtract the units after you have renamed them.
Why this works
If the top number has too few ones to subtract, rename one of its tens as ten ones. In 62 − 35, the 62 becomes 5 tens and 12 ones. Now remove 5 ones and 3 tens to get 2 tens and 7 ones, or 27. The new grouping is equal to 62 before anything is subtracted.
Check the meaning
The represented total before and after the exchange is the same. Check by adding the difference and the amount subtracted.
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
Subtract 2,982 − 428; show any place-value exchange.
- When a smaller column has too few units, exchange one higher-place unit into ten of the next unit down. Subtract the units after you have renamed them.
- Exchange a larger unit into 10 smaller units when needed. Preserve the total and subtract. Difference = 2,554; add 428 to check.
- The represented total before and after the exchange is the same. Check by adding the difference and the amount subtracted.
Result: 2554
Understanding check 1 — your turn
Subtract 2,042 − 958; show any place-value exchange.
Reveal reasoning
Answer: 1084 — Exchange a larger unit into 10 smaller units when needed. Preserve the total and subtract. Difference = 1,084; add 958 to check.
Worked example 2
Reason and verify
Subtract 3,142 − 498; show any place-value exchange.
- When a smaller column has too few units, exchange one higher-place unit into ten of the next unit down. Subtract the units after you have renamed them.
- Exchange a larger unit into 10 smaller units when needed. Preserve the total and subtract. Difference = 2,644; add 498 to check.
- The represented total before and after the exchange is the same. Check by adding the difference and the amount subtracted.
Result: 2644
Understanding check 2 — your turn
Subtract 2,202 − 228; show any place-value exchange.
Reveal reasoning
Answer: 1974 — Exchange a larger unit into 10 smaller units when needed. Preserve the total and subtract. Difference = 1,974; add 228 to check.
Worked example 3
Reason and verify
Subtract 3,302 − 568; show any place-value exchange.
- When a smaller column has too few units, exchange one higher-place unit into ten of the next unit down. Subtract the units after you have renamed them.
- Exchange a larger unit into 10 smaller units when needed. Preserve the total and subtract. Difference = 2,734; add 568 to check.
- The represented total before and after the exchange is the same. Check by adding the difference and the amount subtracted.
Result: 2734
Understanding check 3 — your turn
Subtract 2,362 − 288; show any place-value exchange.
Reveal reasoning
Answer: 2074 — Exchange a larger unit into 10 smaller units when needed. Preserve the total and subtract. Difference = 2,074; add 288 to check.
Guided · 4 questions
Guided practice
Practice with the model visible. Explain the units at each step.
- G1
Subtract 2,012 − 508; show any place-value exchange.
Show answer and reasoning
Answer: 1504
Exchange a larger unit into 10 smaller units when needed. Preserve the total and subtract. Difference = 1,504; add 508 to check.
- G2
Subtract 2,162 − 568; show any place-value exchange.
Show answer and reasoning
Answer: 1594
Exchange a larger unit into 10 smaller units when needed. Preserve the total and subtract. Difference = 1,594; add 568 to check.
- G3
Subtract 2,322 − 638; show any place-value exchange.
Show answer and reasoning
Answer: 1684
Exchange a larger unit into 10 smaller units when needed. Preserve the total and subtract. Difference = 1,684; add 638 to check.
- G4
Subtract 2,482 − 708; show any place-value exchange.
Show answer and reasoning
Answer: 1774
Exchange a larger unit into 10 smaller units when needed. Preserve the total and subtract. Difference = 1,774; add 708 to check.
Independent · 8 questions
Independent practice
Work without relying on the model or checking answers prematurely.
- I1
Subtract 3,612 − 908; show any place-value exchange.
Show answer and reasoning
Answer: 2704
Exchange a larger unit into 10 smaller units when needed. Preserve the total and subtract. Difference = 2,704; add 908 to check.
- I2
Subtract 3,762 − 968; show any place-value exchange.
Show answer and reasoning
Answer: 2794
Exchange a larger unit into 10 smaller units when needed. Preserve the total and subtract. Difference = 2,794; add 968 to check.
- I3
Subtract 1,122 − 238; show any place-value exchange.
Show answer and reasoning
Answer: 884
Exchange a larger unit into 10 smaller units when needed. Preserve the total and subtract. Difference = 884; add 238 to check.
- I4
Subtract 1,282 − 308; show any place-value exchange.
Show answer and reasoning
Answer: 974
Exchange a larger unit into 10 smaller units when needed. Preserve the total and subtract. Difference = 974; add 308 to check.
- I5
Subtract 1,442 − 368; show any place-value exchange.
Show answer and reasoning
Answer: 1074
Exchange a larger unit into 10 smaller units when needed. Preserve the total and subtract. Difference = 1,074; add 368 to check.
- I6
Subtract 1,602 − 438; show any place-value exchange.
Show answer and reasoning
Answer: 1164
Exchange a larger unit into 10 smaller units when needed. Preserve the total and subtract. Difference = 1,164; add 438 to check.
- I7
Subtract 1,762 − 508; show any place-value exchange.
Show answer and reasoning
Answer: 1254
Exchange a larger unit into 10 smaller units when needed. Preserve the total and subtract. Difference = 1,254; add 508 to check.
- I8
Subtract 1,922 − 568; show any place-value exchange.
Show answer and reasoning
Answer: 1354
Exchange a larger unit into 10 smaller units when needed. Preserve the total and subtract. Difference = 1,354; add 568 to check.
Challenge · 2 questions
Challenge practice
Solve original application and explanation problems. Draw and defend a useful model.
- C1
You need to subtract 8 ones but have only 2 ones and 6 tens. How can you rename the number?
Show answer and reasoning
Answer: 5 tens and 12 ones
Exchange one of the 6 tens for 10 ones: 6 tens + 2 ones equals 5 tens + 12 ones.
- C2
Explain how 6 hundreds 2 tens 1 one can be renamed before subtracting 35.
Show answer and reasoning
Answer: 5 hundreds, 11 tens, 11 ones
621 = 5 hundreds + 12 tens + 1 one = 5 hundreds + 11 tens + 11 ones. Now 35 can be subtracted from ones/tens.
Exit · 3 questions
Exit check
Solve independently; revisit the skill if an important explanation is missing.
- E1
Subtract 1,212 − 908; show any place-value exchange.
Show answer and reasoning
Answer: 304
Exchange a larger unit into 10 smaller units when needed. Preserve the total and subtract. Difference = 304; add 908 to check.
- E2
Subtract 1,362 − 968; show any place-value exchange.
Show answer and reasoning
Answer: 394
Exchange a larger unit into 10 smaller units when needed. Preserve the total and subtract. Difference = 394; add 968 to check.
- E3
Subtract 1,522 − 238; show any place-value exchange.
Show answer and reasoning
Answer: 1284
Exchange a larger unit into 10 smaller units when needed. Preserve the total and subtract. Difference = 1,284; add 238 to check.