# Ways with Boxes

I’m a big fan of “Boxes”; I came across this through Don Steward, Chris McGrane at #mathsconf19 and then it is in Visible Maths as Ratio tables through NCETM.

Recently I had been training my new Year 10 class up on these and I asked them this question:

You’ll have seen this question at the end of the thread about Boxes. However, until I did it with a class, I didn’t see how far this could go. It opened up a really interesting answer and has opened my eyes to even more possibilities with boxes.

You may want to find some answers before you go on!

Soon I am going onto teaching factors and multiples. These students already know factors and multiples. I am sure when I ask them to find a HCF or a LCM, or product of prime factors, I won’t be met with blank faces – they see this in their KS3 lessons.

However seeing these concepts, in particular prime factorisation, applied to work we’ve already done should make the maths come alive. Jonny Hall opened my eyes to prime factor tiles at #mathsconf22 in Manchester in March and I would plan to use them to look at equivalent fractions, fraction multiplication, fraction division, types of numbers and other things we have covered this year. But if we look at boxes, there are lots of beautiful patterns to spot.

This is the introduction to the task that Don Steward gives. The right hand side is my version with prime factor tiles. Here the relationship is easier to see (than with just numbers) as we replace a 3 with a 2 going down, and a 2 with a 3 going across. We unpick the bones of the relationship.

Here is another. 3 replaced by 7 going down – 5 replaced by 2 going right.

So here’s an interesting relationship. If we take two numbers; say 30 and 50 and put them at opposite corners of the boxes.

What happens if we put the highest common factor, 10, in the bottom left hand corner?

What is the answer? Why is the answer interesting? Does this always happen?

Prime factor tiles give us a way of *seeing* this in action, but also help us to specialise so that we can generalise. This also strikes me as a lot nicer and more visual than the Venn diagram method (but I suspect the Venn diagram method is enhanced by the prime factor tiles).

Before we go back to the original problem, let us look at a completed set of boxes with prime factor tiles.

How does this help us? Well through this act of specialising we can *feel the maths* and note this:

Both diagonals contain the same prime factors and therefore multiply to 210. This helps to explain the patterns we see in the original problem.

So we could have 3 and 45, 9 and 5, 1 and 225 because this is how we can pair off 3, 3, 3 and 5.

Try a similar approach for this one:

Going back to this:

urr… so you can apply this to factorising non-monic quadratics? Well, yes actually.

Take this:

We are now looking for a diagonal that sums to 41. (There is nothing to stop us putting x squared by the 28, but actually I would endorse playing around with the numbers before introducing any algebra here)

Now we try and find pairs of numbers that we can put into these boxes to make 41.

This provides us with a neat way of splitting the middle term as so:

I appreciate there is an awful lot of work to do to get students to this point, including with negatives/positives. I would certainly go about exploring what types of numbers we can get as sums across those diagonals and relate the boxes diagrams to multiplying out via the grid method first.