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Why do this problem?
requires learners to apply knowledge of multiplication facts, and stresses the importance of knowing the squares of numbers, particularly the squares of primes. It will increase their familarity with representing multiplication facts as a grid or matrix.
You could start by giving the whole group some numbers such as $18$, $33$, $35$, $48$ and $56$ and asking for the multiplication facts that generate them. Include some squares such as $25$, and even $49$, if you feel the group might have a problem identifying it.
You could then introduce the first matrix given in the top example and get the group to help fill it in. Point out the $4$ which is generated by $2 \times 2$ and therefore a square and the only number which has been used twice. As you go round, ask the learners how they know which number should go in the various square spaces. You could make a list of all the numbers from $2$ - $9$ and cross
them off as they are used. As an alternative, you could give them the completed example and ask them to talk about it in pairs so that in a whole-group discussion, you come to an agreement about how the matrix represents these multiplication facts. All this should help them to be able to tackle the main part of the problem with confidence.
After this learners could work in pairs on the actual problem so that they are able to talk through their ideas with a partner. They could use this sheet which has two copies of the main puzzle matrix on it.
At the end of the lesson the whole group could come together again to discuss, not only the answers, but how they worked them out and which multiplication tables they needed to know well in order to do a problem such as this.
Where might you start? Why?
How can you decide which number is used twice?
Which multiplication table does that number come into? Is that the only table it is in?
How can you keep a check on which numbers you have used?
Learners could make a similar puzzle with the least amount of information given so that it can still be done. They could then swap puzzles with someone else. Missing Multipliers
, a Stage 3 task, would make a great follow-up to this one.
Some learners might find it helpful to start by looking for the number which is used twice. You could, perhaps, suggest they find all the numbers given on a multiplication square and write down the factors that are less than $13$. Get them to a make a list of all the numbers from $2 - 12$ and cross the numbers off as they use them.