I was marking Amy's algebra assignment when I saw this (see picture, above). I paused. As an attempt at simplification, how bad a mistake is it? Pretty bad, perhaps, for an A2 student 12 weeks from her exams. I wondered how to unravel it.
What mistake has Amy actually made? I think she 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 [s49]a2 - b2 = a - b, which all A-level maths teachers will have seen a million times. The sub-mistake is easily revealed to be an error by squaring both sides.
The next lesson, with Amy's permission, I put her mistake on the board. Can I show simply that this is indeed wrong? Well, put x = 4 into Amy's mistake. The left-hand side becomes 167, while the right-hand side is 4. We can now be sure that what Amy has written is not generally true.
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?
This would mean that a[s49]b = b[s49]a, and so a = b or a = 0. We can now be even more charitable towards Amy's mistake. She effectively said this:
We will call this Amy's Rule, which has now generated a very healthy discussion about the difference between an identity and an equation. If we take Amy's Rule as an equation, what then? When is it correct? A little manipulation gives a = b + c. So if, for example, c = 5, b = 6 and a = 11, we have the following, which is true:
Conclusion? Amy's Rule is wrong, but there are special cases when you can "get away with it". As with most mistakes, there was something right about it. And I must confess that marking mountains of perfect work is far less interesting than tackling the occasional mysterious error.
Jonny Griffiths teaches at a sixth-form college in Norfolk
Need help identifying the causes of common misconceptions? Try Secondary Maths NatStrats' module on learning from mistakes.
Get creative with differential equations in phildb's pendulum lesson.
Turn pupils into square root stalkers as they find and follow answers in mrbuckton4maths's loop card activity.
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