Week 7: This week we look at 'function machines' and consider what happens when we change the order of the operations or change the numbers slightly. It is likely that many pupils will explore the change of order in an empirical way. This is a useful and important first step, since the fact that changing the order of operations can change the output of the machine, may well come as a surprise. Some pupils may grasp the opportunity to take this further, by thinking about why a change in order has a particular effect.
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Monday: It will be interesting to observe how pupils approach this task. Do they simply try various changes to see what happens, or do they try to analyse and predict the effect of various changes?
Some changes have no effect on the output, for example changing –2, +9, ×4 to +9, –2, ×4. Performing ×4 first will reduce the output, while starting with –2, followed by ×4, will reduce it further. The largest output is given by +9, ×4, –2. Why?
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Tuesday: This involves four operations rather than three, but in some ways the task is simpler than Monday's task. We revisit the key underlying idea that multiplication is distributive over addition (and subtraction): if a multiplication follows an addition (or subtraction), it 'amplifies'* the addition (or subtraction); however, if the multiplication comes first, it has no effect on the addition (or subtraction) operation.
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*When the multiplier is smaller than 1, it would be more accurate to say it 'diminishes' rather than 'amplifies' the addition (or subtraction) that precedes it.]
With an input of 5, if we multiply first (×2, ×10 or ×10, ×2) the machine has an intermediate output of 100 (ie 20×5) and a final output of 107. It doesn't matter whether we follow the multipliers with +3 then +4, or +4 then +3. So there are 4 ways (2×2) of getting this (smallest) output.
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Wednesday: Here we look at division instead of multiplication. It is likely that pupils will find it more challenging to envisage the effect of division rather than multiplication on other operations.
Dividing by 3 will 'lessen' the effect of adding 18 or subtracting 6. If we want an output that is as large as possible, then we want to 'apply' ÷3 to –6 and not to +18, so the order should be –6, ÷3, +18.
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Thursday: This task is relatively easy to solve, but it should allow pupils to discover, and perhaps gain insight into, the fact that for an input of 5, if we increase the multiplier by 1, then the (intermediate) result increases by 5 - and so we need successively to reduce the additive operation by 5 to get the constant output of 100.
You might want to check whether pupils do indeed spot the fact that as the multiplier increases by 1, the additive operation decreases by 5. And you might want to tease out what sense pupils make of this. Is it just a consequence of some empirical number facts (eg "5×8 happens to be 40 so I get to 100 by adding 60") or do they sense that the overall pattern arises from the way multiplication 'works' (put formally: if we increase the multiplier by 1, we add the multiplicand to the result)?
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Friday: Here we vary Thursday's task slightly, by changing the input from 5 to 6. This should give pupils a fresh look at the nature of the resulting pattern and at the underlying structure that brought it about.
Just as 5×7 is 5 more than 5×6 and 5×8 is 5 more than 5×7, here we find that 6×7 is 6 more than 6×6 and 6×8 is 6 more than 6×7. You might want to illuminate this by drawing diagrams (eg arrays of dots), or by inventing stories (8 boxes of 6 eggs compared to 7 boxes of 6 eggs...), or by examining expressions like 6×8 = 6×(7+1) = 6×7 + 6×1.
You could also push this further by asking questons like, "How much bigger than 34×97 is 34×98?"
