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.
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