From TES.

Two years ago I had a decision to make. Should I teach kids the identity symbol? I discussed it with my head of department, who wasn’t really in favour of the idea, though was happy for me to try. The argument against is that it would introduce an unnecessary layer of complexity. It’s a valid argument, and I wasn’t sure I was doing the right thing. However, I went ahead, and two years on, I’m glad I did.

First, what *exactly* is an identity? It took me quite a while to figure it out, and I spent a long time asking fellow teachers and tutors, none of whom had an answer. From A Levels all I was ever told was that it means ‘exactly equal to…’ but, doesn’t the equals sign also mean ‘exactly equal to’?!

Here are two pairs of expressions:

Do the expressions in pair 1 have the same value? It should be apparent that they do, since one is the expansion of the other. So, you don’t need me to tell you anything more to know that those two are equal; me telling you they’re equal adds nothing, that was already obvious.

What about pair 2? We can’t know whether or not they have equal values. So, if I *now* tell you that they’re equal, by including an equals sign, I’ve added additional information. Those two expressions aren’t going to be equal for all values of x… in fact, with a little analytic algebra, we can see that they are only equal if x = 6. So, if they are equal, the unknown quantity x must be 6.

By contrast…

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## About Kris Boulton

Teach First 2011 maths teacher, focussed on curriculum design.

I’ve started to teach the identity symbol more recently. I often explain it with the terms “always true” and “sometimes true”. Your first example is always true, no matter the value of ‘x’, so is an identity. The second is sometimes true (in this case when x=6), so = is needed. Twinned with an “Always, Sometimes, Never” sorting activity, the symbols can be clarified quite well.

That is such a good idea… Always, Sometimes, Never… Though, might it risk some confusion? I guess it depends on how we think about the letter. If we treat it as a variable, then ASN True is a valid question. On the other hand, if we consider the equality to be irrevocably true, then the letter can represent only one number. Which is the correct conceptualisation?

I mean, if you read the statement 2x + 5 = 5x – 1 out loud – and it *is* a statement, then it reads ‘Two x plus five

is equal tofive x subtract 1.’ In which case, it is stating that the first expression is equal to the first; I suppose the word ‘always’ injects the time dimension into the statement… 2x + 5 = 5x – 1 means the two expressions 2x + 5 and 5x – 1 happen to be equal at the moment; perhaps they won’t be in the future… hmm… needs some more thinking.Sorry Kris, I totally missed your reply to this back in September. I see your point, I think the time element conveys the idea well, but it is still a tricky concept for students to get their heads around.

Here’s another question: is it only reserved for algebraic statements? Why, for instance, do we not use the identity symbol in 2+2=4, where we could in x+x=2x?

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