Week 2: This week is all about matchsticks (but please don't light up on a plane).
In this week's tasks we find the number of matchsticks in a pattern by seeing how the pattern differs from one with a known number of matchsticks. Can pupils focus on the
difference between the number of elements in two sets while leaving aside, for a moment, the actual number of elements in the sets.
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MONDAY: We start with a fairly simple task: there are 2 extra yellow matches, making 14+2 = 16 matches in all.
It is possible that some pupils will find the answer by counting all the yellow matches. Can they be persuaded that this is not necessary!
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TUESDAY: If we pair-off blue and yellow matches, some blue matches are left over. How many?
If we move the yellow matches up or to the right, until they cover the neighbouring blue matches, 2 blue matches are left over. So there are 13–2 yellow matches.
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WEDNESDAY: Here again, the focus is on the
difference in the number of blue and yellow matches, not on the total numbers. Can pupils adopt this focus or are they tied to working only with the total numbers?
To get pupils to focus on the difference, you might want to ask, "If we had a pattern like this, but with 99 blue matches, how many yellow matches would there be?" Or, "How many fewer yellow matches are there than blue matches?". Or present this version of the task:
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Thursday: The temptation to count here is strong, but doing so is also quite tedious, so can we find the number of yellow matches by matching (sic!) the blue and yellow matches?
A nice result, which is likely to come as a surprise the first few times one meets it. "Can we make other 'staircases' like this, without changing the number of yellow matches?"
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Friday: Here the relation between the number of blue and yellow matches is no longer just a simple 'difference': there are 4 times as many yellow matches, plus another 3 matches.
Here we are close to working with a general rule, ie one that doesn't depend on the specific number of blue matchsticks. For example, if the line of blue matches were 100 matches 'long', it would be reasonable to assume that we would have 4 lines of yellow matches, also 100 matches 'long', plus another 3 matches. We could put the question like this:
