3A · Chapter 5 · Lesson 5
Multiplication with Regrouping Ones and Tens
Use successive place-value exchanges during multiplication.
Today's challenge
Calculate 65 × 3 and show the exchanges. Try your own method, then compare it with a model.
What you'll learn
- Use successive place-value exchanges during multiplication.
- Explain the connection between a diagram and a number sentence.
- Solve a fresh application without step-by-step hints.
Before this lesson
Both one-position regrouping methods.
Prerequisite competency: 3A-C05-L04
Entry · 2 questions
Warm-up: prerequisite check
Try on paper before continuing. If the questions are hard, review the prerequisite concepts.
- E1
Calculate 15 × 2 and show the exchanges.
Show answer and reasoning
Answer: 30
15×2 = 0 + 20 + 10 = 30; combine partial products and exchange groups of ten.
- E2
Calculate 25 × 2 and show the exchanges.
Show answer and reasoning
Answer: 50
25×2 = 0 + 40 + 10 = 50; combine partial products and exchange groups of ten.
Learn the concept
Words to know
- successive regrouping
- more than one place-value exchange in an algorithm
Use successive place-value exchanges during multiplication.
Multiply ones first and record the tens exchanged. Multiply the tens and include the carried amount, then exchange into hundreds if necessary.
Why this works
Some products need an exchange in both the ones and the tens. For 78 × 4, 8 × 4 = 32 ones, or three tens and two ones. Then 7 tens × 4 plus three carried tens makes 31 tens, or three hundreds and one ten. The final product is 312.
Check the meaning
Estimate before applying the algorithm and re-expand the final product into place-value parts.
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 65 × 3 and show the exchanges.
- 5 1-unit groups × 3, plus 0 carried 1-unit groups, gives 15 groups. Write 5 in the 1s place and exchange 1 groups into the next place.
- 6 10-unit groups × 3, plus 1 carried 10-unit groups, gives 19 groups. Write 9 in the 10s place and exchange 1 groups into the next place.
- Record the final 1 group(s) in the next place.
- Check by partial products: 180 + 15 = 195.
Result: 195
Understanding check 1 — your turn
Calculate 67 × 3 and show the exchanges.
Reveal reasoning
Answer: 201 — 67×3 = 0 + 180 + 21 = 201; combine partial products and exchange groups of ten.
Worked example 2
Reason and verify
Calculate 75 × 3 and show the exchanges.
- 5 1-unit groups × 3, plus 0 carried 1-unit groups, gives 15 groups. Write 5 in the 1s place and exchange 1 groups into the next place.
- 7 10-unit groups × 3, plus 1 carried 10-unit groups, gives 22 groups. Write 2 in the 10s place and exchange 2 groups into the next place.
- Record the final 2 group(s) in the next place.
- Check by partial products: 210 + 15 = 225.
Result: 225
Understanding check 2 — your turn
Calculate 77 × 3 and show the exchanges.
Reveal reasoning
Answer: 231 — 77×3 = 0 + 210 + 21 = 231; combine partial products and exchange groups of ten.
Worked example 3
Reason and verify
Calculate 85 × 3 and show the exchanges.
- 5 1-unit groups × 3, plus 0 carried 1-unit groups, gives 15 groups. Write 5 in the 1s place and exchange 1 groups into the next place.
- 8 10-unit groups × 3, plus 1 carried 10-unit groups, gives 25 groups. Write 5 in the 10s place and exchange 2 groups into the next place.
- Record the final 2 group(s) in the next place.
- Check by partial products: 240 + 15 = 255.
Result: 255
Understanding check 3 — your turn
Calculate 87 × 3 and show the exchanges.
Reveal reasoning
Answer: 261 — 87×3 = 0 + 240 + 21 = 261; 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 55 × 3 and show the exchanges.
Show answer and reasoning
Answer: 165
55×3 = 0 + 150 + 15 = 165; combine partial products and exchange groups of ten.
- G2
Calculate 65 × 3 and show the exchanges.
Show answer and reasoning
Answer: 195
65×3 = 0 + 180 + 15 = 195; combine partial products and exchange groups of ten.
- G3
Calculate 75 × 3 and show the exchanges.
Show answer and reasoning
Answer: 225
75×3 = 0 + 210 + 15 = 225; combine partial products and exchange groups of ten.
- G4
Calculate 85 × 3 and show the exchanges.
Show answer and reasoning
Answer: 255
85×3 = 0 + 240 + 15 = 255; 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 95 × 3 and show the exchanges.
Show answer and reasoning
Answer: 285
95×3 = 0 + 270 + 15 = 285; combine partial products and exchange groups of ten.
- I2
Calculate 56 × 3 and show the exchanges.
Show answer and reasoning
Answer: 168
56×3 = 0 + 150 + 18 = 168; combine partial products and exchange groups of ten.
- I3
Calculate 66 × 3 and show the exchanges.
Show answer and reasoning
Answer: 198
66×3 = 0 + 180 + 18 = 198; combine partial products and exchange groups of ten.
- I4
Calculate 76 × 3 and show the exchanges.
Show answer and reasoning
Answer: 228
76×3 = 0 + 210 + 18 = 228; combine partial products and exchange groups of ten.
- I5
Calculate 86 × 3 and show the exchanges.
Show answer and reasoning
Answer: 258
86×3 = 0 + 240 + 18 = 258; combine partial products and exchange groups of ten.
- I6
Calculate 96 × 3 and show the exchanges.
Show answer and reasoning
Answer: 288
96×3 = 0 + 270 + 18 = 288; combine partial products and exchange groups of ten.
- I7
Calculate 57 × 3 and show the exchanges.
Show answer and reasoning
Answer: 171
57×3 = 0 + 150 + 21 = 171; combine partial products and exchange groups of ten.
- I8
Calculate 67 × 3 and show the exchanges.
Show answer and reasoning
Answer: 201
67×3 = 0 + 180 + 21 = 201; 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
Calculate 78 × 4 by partial products and explain both exchanges.
Show answer and reasoning
Answer: 312
70×4=280 and 8×4=32. Together 312; 32 ones create 3 tens, and 31 tens create 3 hundreds plus 1 ten.
- C2
Calculate 69×4 and identify the two exchanges that are needed.
Show answer and reasoning
Answer: 276
9×4=36 ones=3 tens+6 ones. Then 6×4 tens +3 tens=27 tens=2 hundreds+7 tens. Product 276.
Exit · 3 questions
Exit check
Solve independently; revisit the skill if an important explanation is missing.
- E1
Calculate 77 × 3 and show the exchanges.
Show answer and reasoning
Answer: 231
77×3 = 0 + 210 + 21 = 231; combine partial products and exchange groups of ten.
- E2
Calculate 87 × 3 and show the exchanges.
Show answer and reasoning
Answer: 261
87×3 = 0 + 240 + 21 = 261; combine partial products and exchange groups of ten.
- E3
Calculate 97 × 3 and show the exchanges.
Show answer and reasoning
Answer: 291
97×3 = 0 + 270 + 21 = 291; combine partial products and exchange groups of ten.