3B · Chapter 8 · Lesson 4
Dividing by 6 and 7
Apply inverse operations to solve equal-sharing and number-of-groups problems.
Today's mathematical question
Each tray holds 7 counters. How many full trays are required to arrange 49 counters?
Learning target
Apply inverse operations to solve equal-sharing and number-of-groups problems.
Before starting: Prior lesson: Multiplying by 6 and 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
Explain how splitting 7 equal groups of 3 helps you find their total.
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
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. - P3
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.
Learn and understand
Division reverses multiplication. When 42 objects form six equal groups, you look for the number in each group: 6 × ? = 42. In grouping division, the group size is known and you find how many groups fit. Both interpretations should give a matching multiplication equation. Suppose 42 counters are placed into groups of seven. There are six groups, because 6 × 7 = 42. What changes if the story gives six groups instead of seven counters per group?
Represent it three ways
- Concrete: Build equal pieces, measures, counters or shapes with real objects when useful.
- Pictorial: Represent a known total using equal boxes. Count groups or objects in each group, then verify by multiplying quotient and divisor.
- Abstract: Write the matching number sentence, calculation, comparison or mathematical description. Explain each number's unit.
Important vocabulary
- quotient
- the answer to division
- divisor
- the number used to divide
- inverse operations
- operations that undo one another
Fully worked examples and checks
Try each new understanding check before revealing the explanation.
Worked example 1
Explore, calculate, verify
Each tray holds 7 counters. How many full trays are required to arrange 49 counters?
- Identify the whole, the units, and what the question asks.
- Find how many groups of 7 fit into 49: 49 ÷ 7 = 7.
- Check that the answer uses the right units and makes sense.
Answer: 7
Check your understanding 1
Each tray holds 7 counters. How many full trays are required to arrange 21 counters?
Reveal answer and explanation
3 — Find how many groups of 7 fit into 21: 21 ÷ 7 = 3.
Worked example 2
Explore, calculate, verify
35 counters are shared equally into 7 groups. How many counters are in each group?
- Identify the whole, the units, and what the question asks.
- Use inverse multiplication: 7 × 5 = 35; therefore 35 ÷ 7 = 5.
- Check that the answer uses the right units and makes sense.
Answer: 5
Check your understanding 2
54 counters are shared equally into 6 groups. How many counters are in each group?
Reveal answer and explanation
9 — Use inverse multiplication: 6 × 9 = 54; therefore 54 ÷ 6 = 9.
Worked example 3
Explore, calculate, verify
Each tray holds 7 counters. How many full trays are required to arrange 35 counters?
- Identify the whole, the units, and what the question asks.
- Find how many groups of 7 fit into 35: 35 ÷ 7 = 5.
- Check that the answer uses the right units and makes sense.
Answer: 5
Check your understanding 3
Each tray holds 7 counters. How many full trays are required to arrange 56 counters?
Reveal answer and explanation
8 — Find how many groups of 7 fit into 56: 56 ÷ 7 = 8.
Guided · 5 questions
Guided practice
Keep the model and earlier examples visible while practising.
- G1
70 counters are shared equally into 7 groups. How many counters are in each group?
Reveal explanation
Answer: 10
Use inverse multiplication: 7 × 10 = 70; therefore 70 ÷ 7 = 10.
7 rows of 10 dots = 70 dots. - G2
21 counters are shared equally into 7 groups. How many counters are in each group?
Reveal explanation
Answer: 3
Use inverse multiplication: 7 × 3 = 21; therefore 21 ÷ 7 = 3.
7 rows of 3 dots = 21 dots. - G3
60 counters are shared equally into 6 groups. How many counters are in each group?
Reveal explanation
Answer: 10
Use inverse multiplication: 6 × 10 = 60; therefore 60 ÷ 6 = 10.
6 rows of 10 dots = 60 dots. - G4
Each tray holds 7 counters. How many full trays are required to arrange 42 counters?
Reveal explanation
Answer: 6
Find how many groups of 7 fit into 42: 42 ÷ 7 = 6.
7 rows of 6 dots = 42 dots. - G5
49 counters are shared equally into 7 groups. How many counters are in each group?
Reveal explanation
Answer: 7
Use inverse multiplication: 7 × 7 = 49; therefore 49 ÷ 7 = 7.
7 rows of 7 dots = 49 dots.
Independent · 8 questions
Independent practice
Work these without hints and explain the method used. Write answers on paper.
- I1
18 counters are shared equally into 6 groups. How many counters are in each group?
Reveal explanation
Answer: 3
Use inverse multiplication: 6 × 3 = 18; therefore 18 ÷ 6 = 3.
6 rows of 3 dots = 18 dots. - I2
63 counters are shared equally into 7 groups. How many counters are in each group?
Reveal explanation
Answer: 9
Use inverse multiplication: 7 × 9 = 63; therefore 63 ÷ 7 = 9.
7 rows of 9 dots = 63 dots. - I3
Each tray holds 7 counters. How many full trays are required to arrange 28 counters?
Reveal explanation
Answer: 4
Find how many groups of 7 fit into 28: 28 ÷ 7 = 4.
7 rows of 4 dots = 28 dots. - I4
42 counters are shared equally into 6 groups. How many counters are in each group?
Reveal explanation
Answer: 7
Use inverse multiplication: 6 × 7 = 42; therefore 42 ÷ 6 = 7.
6 rows of 7 dots = 42 dots. - I5
Each tray holds 6 counters. How many full trays are required to arrange 30 counters?
Reveal explanation
Answer: 5
Find how many groups of 6 fit into 30: 30 ÷ 6 = 5.
6 rows of 5 dots = 30 dots. - I6
28 counters are shared equally into 7 groups. How many counters are in each group?
Reveal explanation
Answer: 4
Use inverse multiplication: 7 × 4 = 28; therefore 28 ÷ 7 = 4.
7 rows of 4 dots = 28 dots. - I7
Each tray holds 6 counters. How many full trays are required to arrange 18 counters?
Reveal explanation
Answer: 3
Find how many groups of 6 fit into 18: 18 ÷ 6 = 3.
6 rows of 3 dots = 18 dots. - I8
Each tray holds 6 counters. How many full trays are required to arrange 36 counters?
Reveal explanation
Answer: 6
Find how many groups of 6 fit into 36: 36 ÷ 6 = 6.
6 rows of 6 dots = 36 dots.
Challenge · 2 questions
Challenge and error analysis
Go beyond arithmetic: justify a claim, correct reasoning, or connect two representations.
- C1
35 counters are shared among 7 students. Someone claims each receives 6. Explain why this cannot be correct and find the correct share.
Reveal explanation
Answer: 5
7 × 6 = 42 is more than 35. The correct share is 35 ÷ 7 = 5.
- C2
37 counters are put into full bags of 7, leaving extras. How many full bags and extras are there, and why?
Reveal explanation
Answer: 5 bags and 2 extras
37 = 7 × 5 + 2, with 2 less than the group size 7.
Exit · 3 questions
Exit check
Solve independently, then return on another day to check retention.
- E1
48 counters are shared equally into 6 groups. How many counters are in each group?
Reveal explanation
Answer: 8
Use inverse multiplication: 6 × 8 = 48; therefore 48 ÷ 6 = 8.
6 rows of 8 dots = 48 dots. - E2
Each tray holds 6 counters. How many full trays are required to arrange 42 counters?
Reveal explanation
Answer: 7
Find how many groups of 6 fit into 42: 42 ÷ 6 = 7.
6 rows of 7 dots = 42 dots. - E3
Each tray holds 7 counters. How many full trays are required to arrange 70 counters?
Reveal explanation
Answer: 10
Find how many groups of 7 fit into 70: 70 ÷ 7 = 10.
7 rows of 10 dots = 70 dots.