ALG 08

Week 8: In Week 1 we looked at the distributive law, which allowed us to perform certain calculations more efficiently. Here we look at the effect of changing one or both elements of a division. The tasks are relatively simple but give pupils the opportunity to think formally, and thus to perform quite complex seeming calculations.
In each task, we start with the division 21 ÷ 7 = 3 and then alter the numbers in some way. You might want to create your own variants, for example by starting with a different division. And it is worth taking this work to the next stage by starting off with a challenging division like Monday's 21÷3½, and asking pupils to find a simpler equivalent division.
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Monday: Here we halve the divisor. What does this do the the result (the quotient)?

Some pupils might baulk at being asked to divide by 3½. Others might initially think that halving the divisor halves the result. However, it is likely that many pupils will successively solve the task by arguing that halving the divisor halves the result, without having to think specifically about what dividing by 3½ might mean. Thus the task gives pupils the opportunity to adopt a quite formal approach, albeit involving the relatively simple and familiar notion of halving.
It is nonetheless worth asking pupils what sense they can make of dividing by 3½. The notion of division as 'quotition' or 'measurement' is relatively straightforward here ("How many 3½ litre jugs can I fill from a barrel containing 21 litres of water?"). However, the more familiar view of division as 'sharing' is more awkward here ("21 walnuts shared equally between 3½ people, what does each person get?"). Nonetheless, with a bit of flexible thinking one can make 'sharing' (or partitive division) work: imagine dealing out 21 cards, say, one at a time into 3½ piles, so each pile gets one card in turn, while the 'half pile' gets half a card in turn. After 6 rounds, all the cards are dealt, with each pile having received 6 cards and the half-pile having received 3 cards!
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Tuesday: A simple variant on Monday's task. Here we operate on both dividend and divisor, by multiplying them both by the same amount. What does this do to the result?

84 ÷ 28 looks to be much more challenging that 21 ÷ 7, but we have simply scaled up the expression, so it gives the same result.
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Wednesday: Another variant. This time the dividend and divisor have been scaled by different amounts.

We have mutliplied the dividend by 10 and the divisor by 5, ie by only half as much. How does this affect the result?
The task can again be solved by a relatively simple formal argument, albeit involving two steps rather than one. Is this a step too far for some pupils?
It might help to make the two steps explicit, as here: 21÷7 = 3, so 210÷7 = 10×3 = 30, so 210÷35 = 30÷5 = 6. It might also help to couch each step in a story.
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Thursday: Wow, this variant looks horrible! The divisor is larger than the dividend and the dividend involves a fraction. Help!

If pupils are happy to work at a purely formal level, the task involves quite simple steps: the dividend has been halved (so we need to halve the result); the divisor has been doubled (so we need to halve the result again); half of half of 3 is ¾!
We could check this, for example by using a bar model, as below.

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Friday: Another variant! Thursday's task provides a clue to solving this....

We can solve this in a series of formal steps, as here: 21÷7 = 3, so 10½ ÷ 7 = 1½, so 31½÷7 = 3×1½ = 4½. Or we could short circuit the steps like this: 21 + half of 21 = 31½, 3 + half of 3 = 4½.
We could check the work by performing the actual division: 31½÷7 = 4 + 3½/7 = 4 + ½ = 4½.