Blog: Ian Davies , Director of Secondary | Posted: 1/09/15

Happy new school year! Let’s start with a calculation: think of a few different ways to mentally work out the answer to 13 × 12.

Some possible approaches are:

- I know 12 × 12 = 144, so 13 × 12 is 12 more. 144 + 12 = 156. You could call this “using known facts”.

- 13 × 12 = (10 × 12) + (3 × 12) = 120 + 36 = 156. You could call this “using partitioning”, but it also involves “using known facts”.

- 13 × 12 = (13 × 10) + (13 × 2) = 130 + 26 = 156, which is a variant of (2), although perhaps (13 × 10) and (13 × 2) are more likely to be worked out/derived than known.

There are of course many other ways and a great way of starting a lesson is to share strategies like these and to compare and contrast different approaches. As well as possibly learning a few new strategies for future use, students get the chance to demonstrate their reasoning, explain and structure a solution which helps build the skills they need to develop more complex arguments and justifications later. I’ve written before about the meaning of mastery and this is great example of what it means to “know your tables”; being able to recite 1 × 1 up to 12 × 12 is very useful, but if you don’t have strategies to then easily find 13 × 12 you can hardly claim to have “mastered” them!

Equally interesting is looking at errors that might arise here; some students may think 13 × 12 = (10 × 10) + (3 × 2) and there are many ways to illustrate why this is not the case using concrete manipulatives or pictorial representations e.g.

“Maths talks” – in particular here “number talks” – can then be extended as far as appropriate to continue to work on number sense and/or to practice and develop mathematical vocabulary. We place a great emphasis on “talk tasks” like these in the materials we share with partner schools in the Mathematics Mastery partnership. I might continue the above discussion by asking “Now we know 13 × 12 is equal to 156, how can we find 13 × 24?” Double 156 (and there are quite a few ways doing that!) is, to me at least, the most obvious and leads to my next question “Why does that work?”

Well, 13 × 24 = 13 × (12 × 2) = (13 × 12) × 2 using both factorising and the associative law and I think it’s important to use and revisit these terms often so that later on (in the same lesson perhaps) when looking at a more complex examples students understand what they are doing. I’ve heard “explanations” for 3*ab* × 6*a*^{2}*b* = 18*a*^{3}*b*^{3} along the lines of you “do the numbers, then do the *a*s and then the *b*s”. Compare that to because we’re 3 × *a *× *b* × 6 ×*a*^{2 }× *b* = 3 × 6 × *a *× *a*^{2} × *b* × *b* using the associative law, like we do for numbers and consider which students will have a deeper conceptual understanding and be able to simplify similar expressions in a few weeks’ time?

Incidentally, did you notice that all the methods I suggested for 13 × 12 were examples of the distributive law?

13 × 12 = (12 + 1) × 12

13 × 12 = (10 + 3) × 12

13 × 12 = 13 × (10 + 2)

…maybe a bit of cheeky commutativity thrown in for the last one too. In my experience, students love using the “posh words” and, reinforced often enough, don’t have a problem learning them.

There’s a lot more to say about talk and how to develop productive talk in the maths classroom and I’ll return to this topic with some more examples and ideas very soon – if you have any ideas to share as well, please let us know. In the meantime, all best wishes to you all for the new academic year.

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