I’m guessing, but I imagine that if you’re reading this there’s a good chance that you see a multiplication square as a beautifully structured array, logical and full of intriguing patterns, some known and some waiting to be discovered. For most of the children I work with the situation is very different. For them a tables square can be an intimidating affair, rather like one of those medieval maps of a largely unexplored continent where only the outer fringes are known and the inner areas are largely unfamiliar – “Here be dragons” indeed. In the last couple of years I’ve made much less use of a multiplication square; instead I’ve very often used a set of multiplication strips – a tables square cut into ten separate strips, one for each multiplication table. The strips are much less daunting and far more accessible, and they introduce a tactile and exploratory aspect. Give a child a multiplication square and they’ll look at it – and that’s about it. Give them a set of multiplication strips and they’ll pick them up and start to do something. Perhaps they’ll put them together and reassemble the tables square:
or often they’ll start to pair them up or sort them into groups. Sometimes they’ll sort them into two groups, the odd-numbered tables, and the even-numbered. Straight away there are interesting discoveries to be made. Choose one of the even-table strips. Can you find some odd numbers on it? No, they’re all even. So what do you expect if we look at the odd-table strips? Here’s an observation that tends to come as a complete surprise – three times as many of the numbers within a tables square are even as odd, and there are no odd numbers to be found anywhere within any even table.
More often, the initial sort puts strips into doubling pairs, x3 with x6, and x5 with x10. Immediately all sorts of questions arise. If you pair x2 with x4, then what do you do with the x8, and the x1? So in fact some of your sorting puts the strips into families rather than pairs. Once you’ve got the x1, x2, x4, x8 family isn’t it tempting to wonder if there’s not a further member waiting to be included – so you can incorporate the x16, and … ? And when you sort into pairs, what about those strips which don’t have a partner, like the x7 ? Perhaps we really ought to have a x14 strip? Or could we have a x1½ strip to provide an alternative partner for the x3 ? Of course none of these questions arise every time, but all of them are questions which have arisen with my pupils, and I find it useful to have some empty strips on hand. There’s no need actually to write out the full x14 table onto the strip, simply labelling it x14 makes the point perfectly well.
Of course, I didn’t really have all those ideas about groupings and fractions in mind when I made my first set of strips. I simply saw them as a means of developing tables mastery. It’s easy to concentrate on a single multiplication table, or, when you want to focus upon the links between two multiplication tables you can look at just these two strips and not get distracted by all the other numbers in a tables square.
A natural thing to do is hold a strip so the first number is obscured. Can the child identify it? If I cover one number in the middle of the strip what is the missing number? Or I can do both.
This can get progressively harder. Cover two numbers, or cover three adjacent numbers – what is the middle one? Take it further still, and cover the majority of numbers; if the only uncovered numbers are 27 and 63 which strip is it? Are there any other possibilities? What if the numbers are 18 and 30?
One pupil took things further than I’d expected. I chose a strip and covered all the numbers except 40. I’d rather expected he’d suggest it must be the x10 strip, but he immediately rejected that possibility. “What’s wrong with the x10 ?”, I asked. “The 40’s in the wrong place”, he said, “if it was the x10 strip it would be in the fourth square, but it’s in the fifth square instead”. “Wow!”, I thought. Likewise, because the 40 was near the middle of the strip he knew it couldn’t be the x4 strip or the x5, and there was no point in considering strips like x3 or x7; it was very satisfying to observe him rule out all the impossibilities and home in on the correct answer. He’s nowhere near the school’s top group, and has a very incomplete knowledge of his tables, but he consistently used the same reasoning no matter what I threw at him.
On another occasion I found something else I could do with the strips that wouldn’t have worked nearly as well with a tables square. We were exploring the question of which numbers appear most often on strips (and for that matter which don’t appear at all). I was using a set of twelve strips rather than ten and I asked which tables included 24. With the help of a quickly improvised x24 strip the pupil located all eight strips. I then asked if 24 ever appeared in the same position on different strips. He put them together and found the pattern in the photograph to show the answer in the most elegant way.
So many ideas can crop up that blank strips are always useful. What about the x14 strip we needed as a doubled partner for x7? Can we fill it in completely? To fill it in it might be helpful to use the x7 (the x4 might also be useful).
(I find the Number Dials ITP [ http://webarchive.nationalarchives.gov.uk/20110809101133/http://www.nsonline.org.uk/node/47776 ] very useful – the ITPs are very robust, and this one’s very flexible.
Don’t get me wrong. I’m not silly enough to claim that the strips give all my pupils immediate and permanent mastery of their multiplication tables. Of course they don’t – but I have found them a useful addition to the armoury in the never-ending quest to make children a little more comfortable and secure in their number knowledge.
(This piece first appeared in the Mathematical Association’s journal Primary Mathematics.)