3A · Chapter 3 · Lesson 3
Subtraction with Regrouping — Part 2
Regroup across a zero by borrowing from the nearest nonzero place.
Today's challenge
Calculate 2,020 − 770; explain how you cross the zero places. Try your own method, then compare it with a model.
What you'll learn
- Regroup across a zero by borrowing from the nearest nonzero place.
- Explain the connection between a diagram and a number sentence.
- Solve a fresh application without step-by-step hints.
Before this lesson
Subtraction with one exchange.
Prerequisite competency: 3A-C03-L02
Entry · 2 questions
Warm-up: prerequisite check
Try on paper before continuing. If the questions are hard, review the prerequisite concepts.
- E1
Subtract 1,022 − 838; show any place-value exchange.
Show answer and reasoning
Answer: 184
Exchange a larger unit into 10 smaller units when needed. Preserve the total and subtract. Difference = 184; add 838 to check.
- E2
Subtract 1,172 − 898; show any place-value exchange.
Show answer and reasoning
Answer: 274
Exchange a larger unit into 10 smaller units when needed. Preserve the total and subtract. Difference = 274; add 898 to check.
Learn the concept
Words to know
- regroup across zero
- exchange through one or more empty place-value columns
Regroup across a zero by borrowing from the nearest nonzero place.
A zero column cannot lend a unit until a larger unit is exchanged into it. Continue the exchange across each intermediate place, recording the new counts carefully.
Why this works
A zero in the tens place means there are no tens ready to exchange, not that subtraction must stop. To compute 3,000 − 1, rename one thousand as 10 hundreds, then one of those hundreds as 10 tens, then one ten as 10 ones. Remove one one to obtain 2,999. Record the exchanges so no units are lost or invented.
Check the meaning
Use expanded place-value notation to verify that your exchanges preserve the starting number.
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
Calculate 2,020 − 770; explain how you cross the zero places.
- Subtract 0 from 0 in the 1s place to obtain 0 units.
- Exchange from the 1,000s place through the empty places to the 10s place; there are now 12 units in this column.
- Subtract 7 from 12 in the 10s place to obtain 5 units.
- Subtract 7 from 9 in the 100s place to obtain 2 units.
- Subtract 0 from 1 in the 1,000s place to obtain 1 units.
- Check: 1,250 + 770 = 2,020.
Result: 1250
Understanding check 1 — your turn
Calculate 3,040 − 423; explain how you cross the zero places.
Reveal reasoning
Answer: 2617 — Rename a nonzero higher-place unit to supply ten of the next smaller units; continue exchanging across zeros. 3,040 − 423 = 2,617.
Worked example 2
Reason and verify
Calculate 3,040 − 843; explain how you cross the zero places.
- Exchange from the 10s place through the empty places to the 1s place; there are now 10 units in this column.
- Subtract 3 from 10 in the 1s place to obtain 7 units.
- Exchange from the 1,000s place through the empty places to the 10s place; there are now 13 units in this column.
- Subtract 4 from 13 in the 10s place to obtain 9 units.
- Subtract 8 from 9 in the 100s place to obtain 1 units.
- Subtract 0 from 2 in the 1,000s place to obtain 2 units.
- Check: 2,197 + 843 = 3,040.
Result: 2197
Understanding check 2 — your turn
Calculate 4,005 − 496; explain how you cross the zero places.
Reveal reasoning
Answer: 3509 — Rename a nonzero higher-place unit to supply ten of the next smaller units; continue exchanging across zeros. 4,005 − 496 = 3,509.
Worked example 3
Reason and verify
Calculate 4,005 − 336; explain how you cross the zero places.
- Exchange from the 1,000s place through the empty places to the 1s place; there are now 15 units in this column.
- Subtract 6 from 15 in the 1s place to obtain 9 units.
- Subtract 3 from 9 in the 10s place to obtain 6 units.
- Subtract 3 from 9 in the 100s place to obtain 6 units.
- Subtract 0 from 3 in the 1,000s place to obtain 3 units.
- Check: 3,669 + 336 = 4,005.
Result: 3669
Understanding check 3 — your turn
Calculate 1,000 − 569; explain how you cross the zero places.
Reveal reasoning
Answer: 431 — Rename a nonzero higher-place unit to supply ten of the next smaller units; continue exchanging across zeros. 1,000 − 569 = 431.
Guided · 4 questions
Guided practice
Practice with the model visible. Explain the units at each step.
- G1
Calculate 3,040 − 407; explain how you cross the zero places.
Show answer and reasoning
Answer: 2633
Rename a nonzero higher-place unit to supply ten of the next smaller units; continue exchanging across zeros. 3,040 − 407 = 2,633.
- G2
Calculate 4,005 − 480; explain how you cross the zero places.
Show answer and reasoning
Answer: 3525
Rename a nonzero higher-place unit to supply ten of the next smaller units; continue exchanging across zeros. 4,005 − 480 = 3,525.
- G3
Calculate 1,000 − 553; explain how you cross the zero places.
Show answer and reasoning
Answer: 447
Rename a nonzero higher-place unit to supply ten of the next smaller units; continue exchanging across zeros. 1,000 − 553 = 447.
- G4
Calculate 2,020 − 626; explain how you cross the zero places.
Show answer and reasoning
Answer: 1394
Rename a nonzero higher-place unit to supply ten of the next smaller units; continue exchanging across zeros. 2,020 − 626 = 1,394.
Independent · 8 questions
Independent practice
Work without relying on the model or checking answers prematurely.
- I1
Calculate 3,040 − 827; explain how you cross the zero places.
Show answer and reasoning
Answer: 2213
Rename a nonzero higher-place unit to supply ten of the next smaller units; continue exchanging across zeros. 3,040 − 827 = 2,213.
- I2
Calculate 4,005 − 320; explain how you cross the zero places.
Show answer and reasoning
Answer: 3685
Rename a nonzero higher-place unit to supply ten of the next smaller units; continue exchanging across zeros. 4,005 − 320 = 3,685.
- I3
Calculate 1,000 − 393; explain how you cross the zero places.
Show answer and reasoning
Answer: 607
Rename a nonzero higher-place unit to supply ten of the next smaller units; continue exchanging across zeros. 1,000 − 393 = 607.
- I4
Calculate 2,020 − 466; explain how you cross the zero places.
Show answer and reasoning
Answer: 1554
Rename a nonzero higher-place unit to supply ten of the next smaller units; continue exchanging across zeros. 2,020 − 466 = 1,554.
- I5
Calculate 3,040 − 539; explain how you cross the zero places.
Show answer and reasoning
Answer: 2501
Rename a nonzero higher-place unit to supply ten of the next smaller units; continue exchanging across zeros. 3,040 − 539 = 2,501.
- I6
Calculate 4,005 − 612; explain how you cross the zero places.
Show answer and reasoning
Answer: 3393
Rename a nonzero higher-place unit to supply ten of the next smaller units; continue exchanging across zeros. 4,005 − 612 = 3,393.
- I7
Calculate 1,000 − 685; explain how you cross the zero places.
Show answer and reasoning
Answer: 315
Rename a nonzero higher-place unit to supply ten of the next smaller units; continue exchanging across zeros. 1,000 − 685 = 315.
- I8
Calculate 2,020 − 758; explain how you cross the zero places.
Show answer and reasoning
Answer: 1262
Rename a nonzero higher-place unit to supply ten of the next smaller units; continue exchanging across zeros. 2,020 − 758 = 1,262.
Challenge · 2 questions
Challenge practice
Solve original application and explanation problems. Draw and defend a useful model.
- C1
Why does 3,000 − 1 equal 2,999? Explain the chain of exchanges.
Show answer and reasoning
Answer: 2,999
Exchange one thousand into 10 hundreds, one hundred into 10 tens, and one ten into 10 ones. Then remove one one.
- C2
A student writes 4,002 − 187 = 4,185 because zeros cannot lend. Show the correct exchanges and answer.
Show answer and reasoning
Answer: 3,815
Rename 4,002 as 3 thousands, 9 hundreds, 9 tens, 12 ones; subtract 0 thousands, 1 hundred, 8 tens, 7 ones, obtaining 3,815.
Exit · 3 questions
Exit check
Solve independently; revisit the skill if an important explanation is missing.
- E1
Calculate 3,040 − 507; explain how you cross the zero places.
Show answer and reasoning
Answer: 2533
Rename a nonzero higher-place unit to supply ten of the next smaller units; continue exchanging across zeros. 3,040 − 507 = 2,533.
- E2
Calculate 4,005 − 580; explain how you cross the zero places.
Show answer and reasoning
Answer: 3425
Rename a nonzero higher-place unit to supply ten of the next smaller units; continue exchanging across zeros. 4,005 − 580 = 3,425.
- E3
Calculate 1,000 − 653; explain how you cross the zero places.
Show answer and reasoning
Answer: 347
Rename a nonzero higher-place unit to supply ten of the next smaller units; continue exchanging across zeros. 1,000 − 653 = 347.