3B · Chapter 8 · Lesson 3
Multiplying by 6 and 7
Choose a strategy for 6s and 7s, verify with arrays or distributive reasoning.
Today's mathematical question
An array has 7 rows, with 9 objects in each row. Split it into smaller arrays and find the total.
Learning target
Choose a strategy for 6s and 7s, verify with arrays or distributive reasoning.
Before starting: Prior lesson: The Multiplication Table of 7. Try the warm-up independently.
Prerequisite · 3 questions
Check earlier skills first
These are earlier ideas you need for today; revisit them if they are difficult.
- P1
Compute 7 × 3. Explain a strategy using a known fact.
Reveal explanation
Answer: 21
7 groups of 3 make 7 × 3 = 21. Or find 6 × 3 and add 3.
7 rows of 3 dots = 21 dots. - P2
There are 7 equal trays with 7 counters on each tray. How many counters are there?
Reveal explanation
Answer: 49
7 groups of 7 make 7 × 7 = 49. Or find 6 × 7 and add 7.
7 rows of 7 dots = 49 dots. - P3
Find 7 groups of 8. Can you use 6 groups plus one more?
Reveal explanation
Answer: 56
7 groups of 8 make 7 × 8 = 56. Or find 6 × 8 and add 8.
7 rows of 8 dots = 56 dots.
Learn and understand
The six- and seven-times tables become easier when you use facts you know. A useful strategy is 6 × n = 5 × n + n. For 7 × n, use 5 × n + 2 × n. Remember that each subtotal is a number of objects, not a number of groups. For 7 × 8, splitting into five eights and two eights gives 40 + 16 = 56. Test your mental strategy on another factor before choosing the written calculation.
Represent it three ways
- Concrete: Build equal pieces, measures, counters or shapes with real objects when useful.
- Pictorial: Choose a factor that you can build from known multiples. Sketch the array, separate it into two rectangles, and add their counts.
- Abstract: Write the matching number sentence, calculation, comparison or mathematical description. Explain each number's unit.
Important vocabulary
- distribute
- split one group count into parts
- partial product
- one of the smaller products
- factor
- number multiplied
Fully worked examples and checks
Try each new understanding check before revealing the explanation.
Worked example 1
Explore, calculate, verify
An array has 7 rows, with 9 objects in each row. Split it into smaller arrays and find the total.
- Identify the whole, the units, and what the question asks.
- 7 groups of 9 make 7 × 9 = 63. Or find 6 × 9 and add 9.
- Check that the answer uses the right units and makes sense.
Answer: 63
Check your understanding 1
Explain how splitting 7 equal groups of 4 helps you find their total.
Reveal answer and explanation
28 — 7 groups of 4 make 7 × 4 = 28. Or find 6 × 4 and add 4.
Worked example 2
Explore, calculate, verify
Explain how splitting 6 equal groups of 7 helps you find their total.
- Identify the whole, the units, and what the question asks.
- 6 groups of 7 make 6 × 7 = 42. Or find 5 × 7 and add 7.
- Check that the answer uses the right units and makes sense.
Answer: 42
Check your understanding 2
Use two useful multiplication facts to work out 6 × 4. What is the product?
Reveal answer and explanation
24 — 6 groups of 4 make 6 × 4 = 24. Or find 5 × 4 and add 4.
Worked example 3
Explore, calculate, verify
Use two useful multiplication facts to work out 6 × 6. What is the product?
- Identify the whole, the units, and what the question asks.
- 6 groups of 6 make 6 × 6 = 36. Or find 5 × 6 and add 6.
- Check that the answer uses the right units and makes sense.
Answer: 36
Check your understanding 3
An array has 7 rows, with 6 objects in each row. Split it into smaller arrays and find the total.
Reveal answer and explanation
42 — 7 groups of 6 make 7 × 6 = 42. Or find 6 × 6 and add 6.
Guided · 5 questions
Guided practice
Keep the model and earlier examples visible while practising.
- G1
Explain how splitting 7 equal groups of 6 helps you find their total.
Reveal explanation
Answer: 42
7 groups of 6 make 7 × 6 = 42. Or find 6 × 6 and add 6.
7 rows of 6 dots = 42 dots. - G2
Use two useful multiplication facts to work out 7 × 10. What is the product?
Reveal explanation
Answer: 70
7 groups of 10 make 7 × 10 = 70. Or find 6 × 10 and add 10.
7 rows of 10 dots = 70 dots. - G3
An array has 7 rows, with 4 objects in each row. Split it into smaller arrays and find the total.
Reveal explanation
Answer: 28
7 groups of 4 make 7 × 4 = 28. Or find 6 × 4 and add 4.
7 rows of 4 dots = 28 dots. - G4
Explain how splitting 7 equal groups of 5 helps you find their total.
Reveal explanation
Answer: 35
7 groups of 5 make 7 × 5 = 35. Or find 6 × 5 and add 5.
7 rows of 5 dots = 35 dots. - G5
Use two useful multiplication facts to work out 7 × 9. What is the product?
Reveal explanation
Answer: 63
7 groups of 9 make 7 × 9 = 63. Or find 6 × 9 and add 9.
7 rows of 9 dots = 63 dots.
Independent · 8 questions
Independent practice
Work these without hints and explain the method used. Write answers on paper.
- I1
Explain how splitting 6 equal groups of 8 helps you find their total.
Reveal explanation
Answer: 48
6 groups of 8 make 6 × 8 = 48. Or find 5 × 8 and add 8.
6 rows of 8 dots = 48 dots. - I2
Use two useful multiplication facts to work out 6 × 9. What is the product?
Reveal explanation
Answer: 54
6 groups of 9 make 6 × 9 = 54. Or find 5 × 9 and add 9.
6 rows of 9 dots = 54 dots. - I3
Explain how splitting 7 equal groups of 9 helps you find their total.
Reveal explanation
Answer: 63
7 groups of 9 make 7 × 9 = 63. Or find 6 × 9 and add 9.
7 rows of 9 dots = 63 dots. - I4
Use two useful multiplication facts to work out 6 × 3. What is the product?
Reveal explanation
Answer: 18
6 groups of 3 make 6 × 3 = 18. Or find 5 × 3 and add 3.
6 rows of 3 dots = 18 dots. - I5
An array has 7 rows, with 8 objects in each row. Split it into smaller arrays and find the total.
Reveal explanation
Answer: 56
7 groups of 8 make 7 × 8 = 56. Or find 6 × 8 and add 8.
7 rows of 8 dots = 56 dots. - I6
Explain how splitting 6 equal groups of 5 helps you find their total.
Reveal explanation
Answer: 30
6 groups of 5 make 6 × 5 = 30. Or find 5 × 5 and add 5.
6 rows of 5 dots = 30 dots. - I7
Use two useful multiplication facts to work out 6 × 10. What is the product?
Reveal explanation
Answer: 60
6 groups of 10 make 6 × 10 = 60. Or find 5 × 10 and add 10.
6 rows of 10 dots = 60 dots. - I8
An array has 7 rows, with 7 objects in each row. Split it into smaller arrays and find the total.
Reveal explanation
Answer: 49
7 groups of 7 make 7 × 7 = 49. Or find 6 × 7 and add 7.
7 rows of 7 dots = 49 dots.
Challenge · 2 questions
Challenge and error analysis
Go beyond arithmetic: justify a claim, correct reasoning, or connect two representations.
- C1
Without counting individual counters, show two different decompositions for 7 × 4, and explain why both work.
Reveal explanation
Answer: 28
One decomposition is (6 × 4) + (1 × 4) = 28. Another is (3 × 4) + (4 × 4) = 28. The partitions use different group splits but preserve the same product.
- C2
An array has 7 rows of 5 dots. A student says turning it creates 5 more dots. Explain the error and find the total.
Reveal explanation
Answer: 35
Turning an array swaps rows and columns but adds no dots. Both 7 × 5 and 5 × 7 equal 35.
Exit · 3 questions
Exit check
Solve independently, then return on another day to check retention.
- E1
Use two useful multiplication facts to work out 7 × 8. What is the product?
Reveal explanation
Answer: 56
7 groups of 8 make 7 × 8 = 56. Or find 6 × 8 and add 8.
7 rows of 8 dots = 56 dots. - E2
An array has 7 rows, with 10 objects in each row. Split it into smaller arrays and find the total.
Reveal explanation
Answer: 70
7 groups of 10 make 7 × 10 = 70. Or find 6 × 10 and add 10.
7 rows of 10 dots = 70 dots. - E3
An array has 6 rows, with 4 objects in each row. Split it into smaller arrays and find the total.
Reveal explanation
Answer: 24
6 groups of 4 make 6 × 4 = 24. Or find 5 × 4 and add 4.
6 rows of 4 dots = 24 dots.