3A · Chapter 5 · Lesson 7
Multiplication with Regrouping More Than Once
Coordinate two or more exchanges in a three-digit product.
Today's challenge
Calculate 256 × 3 and show the exchanges. Try your own method, then compare it with a model.
What you'll learn
- Coordinate two or more exchanges in a three-digit product.
- Explain the connection between a diagram and a number sentence.
- Solve a fresh application without step-by-step hints.
Before this lesson
Three-digit products with one exchange.
Prerequisite competency: 3A-C05-L06
Entry · 2 questions
Warm-up: prerequisite check
Try on paper before continuing. If the questions are hard, review the prerequisite concepts.
- E1
Calculate 136 × 2 and show the exchanges.
Show answer and reasoning
Answer: 272
136×2 = 200 + 60 + 12 = 272; combine partial products and exchange groups of ten.
- E2
Calculate 236 × 2 and show the exchanges.
Show answer and reasoning
Answer: 472
236×2 = 400 + 60 + 12 = 472; combine partial products and exchange groups of ten.
Learn the concept
Words to know
- partial products
- products of place-value parts
- regrouping
- place-value exchanges without changing a total
Coordinate two or more exchanges in a three-digit product.
Keep a place-value chart visible throughout. Each exchange changes the representation but never changes the total number of ones in the product.
Why this works
For 286 × 4, begin at the ones: 6 × 4 = 24, leaving four ones and two carried tens. Eight tens × 4 plus two tens makes 34 tens, or four tens and three carried hundreds. Two hundreds × 4 plus three hundreds makes eleven hundreds. The total is 1,144.
Check the meaning
Verify with distributive partial products or division by the factor.
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 256 × 3 and show the exchanges.
- 6 1-unit groups × 3, plus 0 carried 1-unit groups, gives 18 groups. Write 8 in the 1s place and exchange 1 groups into the next place.
- 5 10-unit groups × 3, plus 1 carried 10-unit groups, gives 16 groups. Write 6 in the 10s place and exchange 1 groups into the next place.
- 2 100-unit groups × 3, plus 1 carried 100-unit groups, gives 7 groups. Write 7 in the 100s place.
- Check by partial products: 600 + 150 + 18 = 768.
Result: 768
Understanding check 1 — your turn
Calculate 379 × 3 and show the exchanges.
Reveal reasoning
Answer: 1137 — 379×3 = 900 + 210 + 27 = 1137; combine partial products and exchange groups of ten.
Worked example 2
Reason and verify
Calculate 356 × 3 and show the exchanges.
- 6 1-unit groups × 3, plus 0 carried 1-unit groups, gives 18 groups. Write 8 in the 1s place and exchange 1 groups into the next place.
- 5 10-unit groups × 3, plus 1 carried 10-unit groups, gives 16 groups. Write 6 in the 10s place and exchange 1 groups into the next place.
- 3 100-unit groups × 3, plus 1 carried 100-unit groups, gives 10 groups. Write 0 in the 100s place and exchange 1 groups into the next place.
- Record the final 1 group(s) in the next place.
- Check by partial products: 900 + 150 + 18 = 1,068.
Result: 1068
Understanding check 2 — your turn
Calculate 479 × 3 and show the exchanges.
Reveal reasoning
Answer: 1437 — 479×3 = 1200 + 210 + 27 = 1437; combine partial products and exchange groups of ten.
Worked example 3
Reason and verify
Calculate 456 × 3 and show the exchanges.
- 6 1-unit groups × 3, plus 0 carried 1-unit groups, gives 18 groups. Write 8 in the 1s place and exchange 1 groups into the next place.
- 5 10-unit groups × 3, plus 1 carried 10-unit groups, gives 16 groups. Write 6 in the 10s place and exchange 1 groups into the next place.
- 4 100-unit groups × 3, plus 1 carried 100-unit groups, gives 13 groups. Write 3 in the 100s place and exchange 1 groups into the next place.
- Record the final 1 group(s) in the next place.
- Check by partial products: 1200 + 150 + 18 = 1,368.
Result: 1368
Understanding check 3 — your turn
Calculate 189 × 3 and show the exchanges.
Reveal reasoning
Answer: 567 — 189×3 = 300 + 240 + 27 = 567; combine partial products and exchange groups of ten.
Guided · 4 questions
Guided practice
Practice with the model visible. Explain the units at each step.
- G1
Calculate 379 × 3 and show the exchanges.
Show answer and reasoning
Answer: 1137
379×3 = 900 + 210 + 27 = 1137; combine partial products and exchange groups of ten.
- G2
Calculate 479 × 3 and show the exchanges.
Show answer and reasoning
Answer: 1437
479×3 = 1200 + 210 + 27 = 1437; combine partial products and exchange groups of ten.
- G3
Calculate 189 × 3 and show the exchanges.
Show answer and reasoning
Answer: 567
189×3 = 300 + 240 + 27 = 567; combine partial products and exchange groups of ten.
- G4
Calculate 289 × 3 and show the exchanges.
Show answer and reasoning
Answer: 867
289×3 = 600 + 240 + 27 = 867; combine partial products and exchange groups of ten.
Independent · 8 questions
Independent practice
Work without relying on the model or checking answers prematurely.
- I1
Calculate 389 × 3 and show the exchanges.
Show answer and reasoning
Answer: 1167
389×3 = 900 + 240 + 27 = 1167; combine partial products and exchange groups of ten.
- I2
Calculate 489 × 3 and show the exchanges.
Show answer and reasoning
Answer: 1467
489×3 = 1200 + 240 + 27 = 1467; combine partial products and exchange groups of ten.
- I3
Calculate 199 × 3 and show the exchanges.
Show answer and reasoning
Answer: 597
199×3 = 300 + 270 + 27 = 597; combine partial products and exchange groups of ten.
- I4
Calculate 299 × 3 and show the exchanges.
Show answer and reasoning
Answer: 897
299×3 = 600 + 270 + 27 = 897; combine partial products and exchange groups of ten.
- I5
Calculate 399 × 3 and show the exchanges.
Show answer and reasoning
Answer: 1197
399×3 = 900 + 270 + 27 = 1197; combine partial products and exchange groups of ten.
- I6
Calculate 499 × 3 and show the exchanges.
Show answer and reasoning
Answer: 1497
499×3 = 1200 + 270 + 27 = 1497; combine partial products and exchange groups of ten.
- I7
Calculate 155 × 3 and show the exchanges.
Show answer and reasoning
Answer: 465
155×3 = 300 + 150 + 15 = 465; combine partial products and exchange groups of ten.
- I8
Calculate 255 × 3 and show the exchanges.
Show answer and reasoning
Answer: 765
255×3 = 600 + 150 + 15 = 765; combine partial products and exchange groups of ten.
Challenge · 2 questions
Challenge practice
Solve original application and explanation problems. Draw and defend a useful model.
- C1
Find 278 × 4 using expanded form rather than only a compact algorithm.
Show answer and reasoning
Answer: 1,112
200×4 +70×4 +8×4 = 800+280+32 =1,112.
- C2
A child finds 286×4=824 by multiplying each digit and writing the final ones digits. What happened to the exchanges?
Show answer and reasoning
Answer: 1,144
6×4=24 gives 2 carried tens; 8 tens×4+2=34 tens gives 3 carried hundreds; 2 hundreds×4+3=11 hundreds; answer 1,144.
Exit · 3 questions
Exit check
Solve independently; revisit the skill if an important explanation is missing.
- E1
Calculate 355 × 3 and show the exchanges.
Show answer and reasoning
Answer: 1065
355×3 = 900 + 150 + 15 = 1065; combine partial products and exchange groups of ten.
- E2
Calculate 455 × 3 and show the exchanges.
Show answer and reasoning
Answer: 1365
455×3 = 1200 + 150 + 15 = 1365; combine partial products and exchange groups of ten.
- E3
Calculate 165 × 3 and show the exchanges.
Show answer and reasoning
Answer: 495
165×3 = 300 + 180 + 15 = 495; combine partial products and exchange groups of ten.