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Maths - Learn from your mistakes

What's it all about?

I was marking Amy's algebra assignment when I saw this (see picture). I paused. As an attempt at simplification it was pretty bad a few weeks before the exam, writes Jonny Griffiths.

I think Amy has said to herself, "If you do the same thing to the top and bottom of a fraction, its value is unchanged", before taking the square root of top and bottom. Then there is the sub-mistake of saying:

(square root)a(2)(minus)b(2)=a(minus)b

All A-level maths teachers will have seen this a million times. The sub- mistake is easily revealed to be an error by squaring both sides.

It is partially true that "If you do the same thing to the top and bottom of a fraction, its value is unchanged". If you are doing multiplication or division, no problem. But if you are adding, subtracting, squaring or, as in this case, square rooting, there is a problem. For example, is the following correct?

a(divided by)b=(square root)a(divided by)(square root)b

This would mean that a(square root)b = b(square root)a, and so a = b or a = 0. We can now be even more charitable towards Amy's mistake. She effectively said this:

a(2)(divided by)b(2)(minus)c(2)(identically equal)a(divided by)b(minus)c

We will call this Amy's rule, which has now generated healthy discussion about the difference between an identity and an equation. Amy's rule is wrong, but there are cases when you can "get away with it".

What else?

Need help identifying the causes of misconceptions? Try Secondary Maths NatStrats' module on learning with mistakes.

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