Expanding brackets

Last week after teaching an algebra topic to a low ability year 8 class I tweeted:

Used coloured counters in cups to introduce expanding brackets: 3(2b+5g). Worked really well! #mathchat

The tweet seemed to provoke a reasonable amount of interest – it received a few retweets, several replies, and a few direct messages. All of the direct messages were along the lines of: sounds interesting – I’d like to hear more about that.

So here goes.

Aside from mini whiteboards (a post for another day), I think cups and counters are possible some of the most useful pieces of equipment for teaching mathematical concepts – particularly for algebra.

At its simplest level, algebra is usually introduced in terms of items in a cafe or shop – 2c standing for 2 cups of tea, or 2b standing for 2 buns. As well as being simplistic, this introduces a conceptual error right from the start; mathematically 2c represents c + c. One cannot add a cup of tea to another cup of tea – one can, however, add the price of a cup of tea, to the price of a cup of tea.

Tal Greengard expands on this – his blog is worth a read.

I’ve now moved on to introducing algebra using cups. Cheap plastic cups – easily written on using a whiteboard marker. Instead of a standing for apple, instead a represents the cup with a written on it. I then use counters inside the cup to show what value a represents.

So how do you expand brackets?

Last week I used the cup to represent the bracket. My example was 3(2b + 5g). I had 3 cups, each containing 2 black counters and 5 green counters. Previously when teaching brackets without visual aids, pupils have struggled to understand why we’re multiplying the contents of the bracket by the number on the outside. Doing it this way only took three practical demonstrations, backed up with the maths on the board, before the pupils were confident with what they were doing and ready to move on to practising on mini whiteboards.


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