Multiplicative Strategies progression, 4th step
The purpose of the activity
In this activity, the learners develop mental strategies for solving division problems with singledigit divisors. Materials are used to demonstrate strategies.
The teaching points

There are a variety of mental strategies available for solving division problems These include:

using tidy numbers with compensation (72 ÷ 4 can be solved as (80 ÷ 4) – 2)

deriving from known facts (72 ÷ 4 = can be solved by 72 ÷ 2 = 36 (known fact) and 36 ÷ 2 = 18 (known fact) because dividing by 4 is the same as dividing by 2 twice)

using reversibility (72 ÷ 4 can be solved by turning it into the multiplication 4 x ? = 72 and using multiplication strategies)

using equivalent expressions (72 ÷ 4 can be solved by 36 ÷ 2 (halving and halving))

360 ÷ 5 can be solved by 720 ÷ 10 (doubling and doubling)
Note: The answer to a division problem remains the same whenever you increase or decrease both the number to be divided and the divisor by the same factor (for example, multiply both by 10). This concept underpins the method commonly used, but poorly understood, for dividing with decimals. For example, 16 ÷ 0.4 is the same as 160 ÷ 4, where both numbers have been multiplied by 10.

Different problems lend themselves to different strategies. Competent learners have a range of strategies and choose the most appropriate in a given situation.

It is not intended that you name and ‘teach’ a range of strategies. This activity is designed for the learners to explore and share strategies in order to increase their range.

Using materials to demonstrate strategies helps develop understanding.

Discuss with the learners the strategies that are most familiar to them and how they can expand their repertoires.
Resources

Sticky notes.

Paper clips (lots).
The guided teaching and learning sequence
1. Tell the learners that you have 72 objects that need to be shared equally among 4 people and write 72 ÷ 4 on the board. Ask the learners to work out how many objects each person would get.
2. Ask the learners to share their strategies. If strategies only include using a calculator or algorithm, ask the learners if they can think of other ways of finding the answer. If the learners offer no strategies, prompt them by asking, for example, “What happens if I share 80 objects between 4 people … and can I use this to help solve sharing 72 objects between 4 people?”
Model the strategies discussed with paper clips or sticky notes.
Possible strategies include:
a) Adding 8 to the 72 gives 80 objects, and if you share 80 equally among 4 people, each gets 20. But I ‘shared’ out 8 too many, so each person has 2 too many. Each person should have only 18. (Using tidy numbers with compensation or deriving from known facts.)
Model this by showing with sticky notes that the 8 added to give 80 shared equally between 4 people results in each person having an extra 2.
b) Saying: “4 times what equals 72? I know 4 times 10 equals 40 and I have 32 still to share out. I know 4 times 8 equals 32. Each person gets 18.”
Another possibility is: “I know 4 times 15 equals 60, and I have 12 still to share out. I know 4 times 3 equals 12. Each person gets 18.” (Using reversibility and place value partitioning.)
c) “Dividing by 4 is the same as dividing by 2 twice so 72 ÷ 2 = 36 and 36 ÷ 2 = 18.” (Deriving from known facts.)
Model this, using sticky notes.
Is the same as:
d) 72 is twice 36, and 4 is twice 2. I know if 36 is shared out between 2 people each gets 18, so if 72 is shared out between 4 people each will get 18.
In division, if each number is increased or decreased by the same factor, the answer will be the same. 72 ÷ 4 = 36 ÷ 2 = 18.
3. Model this by asking the learners to share 72 paper clips equally between 4 people and count how many each person gets. Ask the learners to halve the number of paper clips (36) and halve the number of people (2). Ask the learners to share 36 paper clips between 2 people and count how many each person gets. Emphasise that when 72 paper clips are shared out between 4 people and 36 between 2 people, each person gets the same amount.
4. Reinforce this concept by posing further problems.
“Is 12 ÷ 6 the same as 6 ÷ 2?”
“Is 90 ÷ 6 the same as 45 ÷ 3?”
“Is 12 ÷ 6 the same as 6 ÷ 3?”
If necessary, encourage the learners to show with paper clips that the answer to the original problem and the halved problem(s) is the same.
5. Ask the learners to investigate, with paper clips if necessary, whether the answer is the same if both numbers are doubled.
“Is 15 ÷ 3 the same as 30 ÷ 6 and 60 ÷ 12?”
Ask how doubling both numbers could be used to solve problems such as 360 ÷ 5.
Followup activity
Write further problems on the board, for example 120 ÷ 8, 78 ÷ 6, 171 ÷ 9, 240 ÷ 5. Have the learners work in pairs and ask them to share their strategies for solving each problem. If they are unable to understand the strategy their partner is using, ask the partner to demonstrate it with materials.
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