3A · Chapter 6 · Lesson 6
Dividing a 3-digit Number by 3, 4, and 5
Use known multiples and place-value exchanges for three-digit exact division.
Today's challenge
Calculate 464 ÷ 4. Explain how you check your quotient. Try your own method, then compare it with a model.
What you'll learn
- Use known multiples and place-value exchanges for three-digit exact division.
- Explain the connection between a diagram and a number sentence.
- Solve a fresh application without step-by-step hints.
Before this lesson
Two-digit division by 3–5.
Prerequisite competency: 3A-C06-L04
Entry · 2 questions
Warm-up: prerequisite check
Try on paper before continuing. If the questions are hard, review the prerequisite concepts.
- E1
Calculate 52 ÷ 4. Explain how you check your quotient.
Show answer and reasoning
Answer: 13
Group 52 into 4 equal parts using place-value exchanges. Each group has 13. Verify 13×4=52.
- E2
Calculate 90 ÷ 5. Explain how you check your quotient.
Show answer and reasoning
Answer: 18
Group 90 into 5 equal parts using place-value exchanges. Each group has 18. Verify 18×5=90.
Learn the concept
Words to know
- partial quotient
- a useful part of the final quotient
- divisor
- the number used to divide
Use known multiples and place-value exchanges for three-digit exact division.
Use a multiple of ten or a suitable hundred when the dividend is large. Decompose the dividend into pieces that divide by the same divisor.
Why this works
Use place-value sharing or partial quotients. For 624 ÷ 4, share 400 ÷ 4 = 100 per group, then 200 ÷ 4 = 50, then 24 ÷ 4 = 6. Combine 100 + 50 + 6 = 156, and check that 156 × 4 = 624.
Check the meaning
Verify every place-value contribution in the quotient.
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 464 ÷ 4. Explain how you check your quotient.
- At the 100s place, make groups of 4: record 1 quotient digit; no units remain to exchange.
- At the 10s place, make groups of 4: record 1 quotient digit and exchange remainder 2 to smaller units.
- At the 1s place, make groups of 4: record 6 quotient digit; no units remain to exchange.
- Read all quotient digits in their correct columns, including zero placeholders: 116. Check: 116 × 4 = 464.
Result: 116
Understanding check 1 — your turn
Calculate 414 ÷ 3. Explain how you check your quotient.
Reveal reasoning
Answer: 138 — Group 414 into 3 equal parts using place-value exchanges. Each group has 138. Verify 138×3=414.
Worked example 2
Reason and verify
Calculate 635 ÷ 5. Explain how you check your quotient.
- At the 100s place, make groups of 5: record 1 quotient digit and exchange remainder 1 to smaller units.
- At the 10s place, make groups of 5: record 2 quotient digit and exchange remainder 3 to smaller units.
- At the 1s place, make groups of 5: record 7 quotient digit; no units remain to exchange.
- Read all quotient digits in their correct columns, including zero placeholders: 127. Check: 127 × 5 = 635.
Result: 127
Understanding check 2 — your turn
Calculate 596 ÷ 4. Explain how you check your quotient.
Reveal reasoning
Answer: 149 — Group 596 into 4 equal parts using place-value exchanges. Each group has 149. Verify 149×4=596.
Worked example 3
Reason and verify
Calculate 414 ÷ 3. Explain how you check your quotient.
- At the 100s place, make groups of 3: record 1 quotient digit and exchange remainder 1 to smaller units.
- At the 10s place, make groups of 3: record 3 quotient digit and exchange remainder 2 to smaller units.
- At the 1s place, make groups of 3: record 8 quotient digit; no units remain to exchange.
- Read all quotient digits in their correct columns, including zero placeholders: 138. Check: 138 × 3 = 414.
Result: 138
Understanding check 3 — your turn
Calculate 800 ÷ 5. Explain how you check your quotient.
Reveal reasoning
Answer: 160 — Group 800 into 5 equal parts using place-value exchanges. Each group has 160. Verify 160×5=800.
Guided · 4 questions
Guided practice
Practice with the model visible. Explain the units at each step.
- G1
Calculate 575 ÷ 5. Explain how you check your quotient.
Show answer and reasoning
Answer: 115
Group 575 into 5 equal parts using place-value exchanges. Each group has 115. Verify 115×5=575.
- G2
Calculate 378 ÷ 3. Explain how you check your quotient.
Show answer and reasoning
Answer: 126
Group 378 into 3 equal parts using place-value exchanges. Each group has 126. Verify 126×3=378.
- G3
Calculate 548 ÷ 4. Explain how you check your quotient.
Show answer and reasoning
Answer: 137
Group 548 into 4 equal parts using place-value exchanges. Each group has 137. Verify 137×4=548.
- G4
Calculate 740 ÷ 5. Explain how you check your quotient.
Show answer and reasoning
Answer: 148
Group 740 into 5 equal parts using place-value exchanges. Each group has 148. Verify 148×5=740.
Independent · 8 questions
Independent practice
Work without relying on the model or checking answers prematurely.
- I1
Calculate 260 ÷ 4. Explain how you check your quotient.
Show answer and reasoning
Answer: 65
Group 260 into 4 equal parts using place-value exchanges. Each group has 65. Verify 65×4=260.
- I2
Calculate 380 ÷ 5. Explain how you check your quotient.
Show answer and reasoning
Answer: 76
Group 380 into 5 equal parts using place-value exchanges. Each group has 76. Verify 76×5=380.
- I3
Calculate 261 ÷ 3. Explain how you check your quotient.
Show answer and reasoning
Answer: 87
Group 261 into 3 equal parts using place-value exchanges. Each group has 87. Verify 87×3=261.
- I4
Calculate 392 ÷ 4. Explain how you check your quotient.
Show answer and reasoning
Answer: 98
Group 392 into 4 equal parts using place-value exchanges. Each group has 98. Verify 98×4=392.
- I5
Calculate 545 ÷ 5. Explain how you check your quotient.
Show answer and reasoning
Answer: 109
Group 545 into 5 equal parts using place-value exchanges. Each group has 109. Verify 109×5=545.
- I6
Calculate 360 ÷ 3. Explain how you check your quotient.
Show answer and reasoning
Answer: 120
Group 360 into 3 equal parts using place-value exchanges. Each group has 120. Verify 120×3=360.
- I7
Calculate 524 ÷ 4. Explain how you check your quotient.
Show answer and reasoning
Answer: 131
Group 524 into 4 equal parts using place-value exchanges. Each group has 131. Verify 131×4=524.
- I8
Calculate 710 ÷ 5. Explain how you check your quotient.
Show answer and reasoning
Answer: 142
Group 710 into 5 equal parts using place-value exchanges. Each group has 142. Verify 142×5=710.
Challenge · 2 questions
Challenge practice
Solve original application and explanation problems. Draw and defend a useful model.
- C1
Split 468 ÷ 4 into two convenient divisions.
Show answer and reasoning
Answer: 117
400÷4=100 and 68÷4=17. Their sum is 117, checked by 117×4=468.
- C2
Use place-value partial quotients to find 624÷4 without long division.
Show answer and reasoning
Answer: 156
400÷4=100; 200÷4=50; 24÷4=6. Add 100+50+6=156.
Exit · 3 questions
Exit check
Solve independently; revisit the skill if an important explanation is missing.
- E1
Calculate 477 ÷ 3. Explain how you check your quotient.
Show answer and reasoning
Answer: 159
Group 477 into 3 equal parts using place-value exchanges. Each group has 159. Verify 159×3=477.
- E2
Calculate 680 ÷ 4. Explain how you check your quotient.
Show answer and reasoning
Answer: 170
Group 680 into 4 equal parts using place-value exchanges. Each group has 170. Verify 170×4=680.
- E3
Calculate 905 ÷ 5. Explain how you check your quotient.
Show answer and reasoning
Answer: 181
Group 905 into 5 equal parts using place-value exchanges. Each group has 181. Verify 181×5=905.