3B · Chapter 8 · Lesson 8
Dividing by 8 and 9
Find quotients using inverse multiplication and interpret remainders in context.
Today's mathematical question
Each tray holds 9 counters. How many full trays are required to arrange 90 counters?
Learning target
Find quotients using inverse multiplication and interpret remainders in context.
Before starting: Prior lesson: Multiplying by 8 and 9. 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
An array has 8 rows, with 7 objects in each row. Split it into smaller arrays and find the total.
Reveal explanation
Answer: 56
8 groups of 7 make 8 × 7 = 56. Or find 7 × 7 and add 7.
8 rows of 7 dots = 56 dots. - P2
Explain how splitting 9 equal groups of 10 helps you find their total.
Reveal explanation
Answer: 90
9 groups of 10 make 9 × 10 = 90. Or find 8 × 10 and add 10.
9 rows of 10 dots = 90 dots. - P3
Use two useful multiplication facts to work out 8 × 4. What is the product?
Reveal explanation
Answer: 32
8 groups of 4 make 8 × 4 = 32. Or find 7 × 4 and add 4.
8 rows of 4 dots = 32 dots.
Learn and understand
Use inverse multiplication for dividing by eight or nine. If a total does not divide into complete groups, the remainder tells you how many objects are left. For example, 50 ÷ 8 gives six complete groups and two left, because 8 × 6 + 2 = 50. The leftover must be smaller than eight. For 58 ÷ 8, seven complete groups use 56, leaving 2. Check 8 × 7 + 2 = 58. If the remainder were 8, another group would fit.
Represent it three ways
- Concrete: Build equal pieces, measures, counters or shapes with real objects when useful.
- Pictorial: Draw equal boxes for the full groups, then place leftover counters outside them. Write dividend = divisor × quotient + remainder.
- Abstract: Write the matching number sentence, calculation, comparison or mathematical description. Explain each number's unit.
Important vocabulary
- remainder
- items left after all complete groups are formed
- quotient
- number of full groups or items in a group
- inverse
- operation that undoes another
Fully worked examples and checks
Try each new understanding check before revealing the explanation.
Worked example 1
Explore, calculate, verify
Each tray holds 9 counters. How many full trays are required to arrange 90 counters?
- Identify the whole, the units, and what the question asks.
- Find how many groups of 9 fit into 90: 90 ÷ 9 = 10.
- Check that the answer uses the right units and makes sense.
Answer: 10
Check your understanding 1
Each tray holds 9 counters. How many full trays are required to arrange 36 counters?
Reveal answer and explanation
4 — Find how many groups of 9 fit into 36: 36 ÷ 9 = 4.
Worked example 2
Explore, calculate, verify
25 stickers are arranged in complete groups of 8. How many complete groups and how many stickers remain?
- Identify the whole, the units, and what the question asks.
- 8 × 3 = 24, leaving 1. The remainder is smaller than 8.
- Check that the answer uses the right units and makes sense.
Answer: 3 groups, 1 left
Check your understanding 2
89 stickers are arranged in complete groups of 9. How many complete groups and how many stickers remain?
Reveal answer and explanation
9 groups, 8 left — 9 × 9 = 81, leaving 8. The remainder is smaller than 9.
Worked example 3
Explore, calculate, verify
Each tray holds 8 counters. How many full trays are required to arrange 64 counters?
- Identify the whole, the units, and what the question asks.
- Find how many groups of 8 fit into 64: 64 ÷ 8 = 8.
- Check that the answer uses the right units and makes sense.
Answer: 8
Check your understanding 3
Each tray holds 8 counters. How many full trays are required to arrange 24 counters?
Reveal answer and explanation
3 — Find how many groups of 8 fit into 24: 24 ÷ 8 = 3.
Guided · 5 questions
Guided practice
Keep the model and earlier examples visible while practising.
- G1
36 counters are shared equally into 9 groups. How many counters are in each group?
Reveal explanation
Answer: 4
Use inverse multiplication: 9 × 4 = 36; therefore 36 ÷ 9 = 4.
9 rows of 4 dots = 36 dots. - G2
Each tray holds 8 counters. How many full trays are required to arrange 80 counters?
Reveal explanation
Answer: 10
Find how many groups of 8 fit into 80: 80 ÷ 8 = 10.
8 rows of 10 dots = 80 dots. - G3
67 stickers are arranged in complete groups of 8. How many complete groups and how many stickers remain?
Reveal explanation
Answer: 8 groups, 3 left
8 × 8 = 64, leaving 3. The remainder is smaller than 8.
- G4
Each tray holds 8 counters. How many full trays are required to arrange 40 counters?
Reveal explanation
Answer: 5
Find how many groups of 8 fit into 40: 40 ÷ 8 = 5.
8 rows of 5 dots = 40 dots. - G5
27 counters are shared equally into 9 groups. How many counters are in each group?
Reveal explanation
Answer: 3
Use inverse multiplication: 9 × 3 = 27; therefore 27 ÷ 9 = 3.
9 rows of 3 dots = 27 dots.
Independent · 8 questions
Independent practice
Work these without hints and explain the method used. Write answers on paper.
- I1
40 counters are shared equally into 8 groups. How many counters are in each group?
Reveal explanation
Answer: 5
Use inverse multiplication: 8 × 5 = 40; therefore 40 ÷ 8 = 5.
8 rows of 5 dots = 40 dots. - I2
73 stickers are arranged in complete groups of 8. How many complete groups and how many stickers remain?
Reveal explanation
Answer: 9 groups, 1 left
8 × 9 = 72, leaving 1. The remainder is smaller than 8.
- I3
Each tray holds 8 counters. How many full trays are required to arrange 48 counters?
Reveal explanation
Answer: 6
Find how many groups of 8 fit into 48: 48 ÷ 8 = 6.
8 rows of 6 dots = 48 dots. - I4
81 counters are shared equally into 9 groups. How many counters are in each group?
Reveal explanation
Answer: 9
Use inverse multiplication: 9 × 9 = 81; therefore 81 ÷ 9 = 9.
9 rows of 9 dots = 81 dots. - I5
58 stickers are arranged in complete groups of 8. How many complete groups and how many stickers remain?
Reveal explanation
Answer: 7 groups, 2 left
8 × 7 = 56, leaving 2. The remainder is smaller than 8.
- I6
63 counters are shared equally into 9 groups. How many counters are in each group?
Reveal explanation
Answer: 7
Use inverse multiplication: 9 × 7 = 63; therefore 63 ÷ 9 = 7.
9 rows of 7 dots = 63 dots. - I7
65 stickers are arranged in complete groups of 8. How many complete groups and how many stickers remain?
Reveal explanation
Answer: 8 groups, 1 left
8 × 8 = 64, leaving 1. The remainder is smaller than 8.
- I8
48 counters are shared equally into 8 groups. How many counters are in each group?
Reveal explanation
Answer: 6
Use inverse multiplication: 8 × 6 = 48; therefore 48 ÷ 8 = 6.
8 rows of 6 dots = 48 dots.
Challenge · 2 questions
Challenge and error analysis
Go beyond arithmetic: justify a claim, correct reasoning, or connect two representations.
- C1
36 counters are shared among 9 students. Someone claims each receives 5. Explain why this cannot be correct and find the correct share.
Reveal explanation
Answer: 4
9 × 5 = 45 is more than 36. The correct share is 36 ÷ 9 = 4.
- C2
38 counters are put into full bags of 9, leaving extras. How many full bags and extras are there, and why?
Reveal explanation
Answer: 4 bags and 2 extras
38 = 9 × 4 + 2, with 2 less than the group size 9.
Exit · 3 questions
Exit check
Solve independently, then return on another day to check retention.
- E1
52 stickers are arranged in complete groups of 8. How many complete groups and how many stickers remain?
Reveal explanation
Answer: 6 groups, 4 left
8 × 6 = 48, leaving 4. The remainder is smaller than 8.
- E2
Each tray holds 8 counters. How many full trays are required to arrange 32 counters?
Reveal explanation
Answer: 4
Find how many groups of 8 fit into 32: 32 ÷ 8 = 4.
8 rows of 4 dots = 32 dots. - E3
64 counters are shared equally into 8 groups. How many counters are in each group?
Reveal explanation
Answer: 8
Use inverse multiplication: 8 × 8 = 64; therefore 64 ÷ 8 = 8.
8 rows of 8 dots = 64 dots.