# On Estimating

An exam question was recently brought to my attention: students were asked to estimate $31.28^2$.

### Whoosh

“$31^2 = 961$ and $31.5^2 = 992.25$, so it’s about 978.”

“Very good, sensei, but no marks. Uhuhuh, put that down - you need to take it up with the exam board, not with me.”

A narrowing of the Mathematical Ninja’s eyes.

“The ‘correct’ answer, the one that would get you the point, is… wait for it… 900.”

Usually, the Mathematical Ninja makes other people explode. It was nice to be on the other side of the exchange for once.

### Why 900?

In my opinion, 900 is a perfectly reasonable estimate: round the number to one significant figure and do the sums from there. $31.28^2$ is about $30^2$ which is 900.

However, it’s not an especially accurate estimate - it’s not correct even to one significant figure.

### Why I dislike this question

That’s not fair, actually. I don’t dislike the question at all: estimating is a really valuable skill and the ‘zequals’ approach Rob Eastaway promotes is a fine method for getting a ballpark answer to a complicated questions.

My problem is the answer: 900 is OK, as I said. But what about students who have put the effort in to learn their squares? They may know (like the Ninja) that $31^2 = 961$, so 1000 is a better guess. They may know that $\pi^2 \approx 10$, and reason that 1000 is a good guess. They might know that 32 is $2^5$ and $2^{10} = 1024$. They may have several other strategies for estimating that aren’t precisely what the exam board had in mind.

### “Estimate.”

You keep using that word. I do not think it means what you think it means.

I think exams have a responsibility to be accessible. The word ‘estimate’ does not mean - in everyday parlance or in any specific technical sense that I’m aware of - “round everything to 1sf and do the appropriate calculation”. It means “come up with a reasonable, rough answer” and to pretend otherwise does a disservice to students and educators.

In my opinion “900 or better (978.4384)” would be a fine thing to have in the mark scheme. Would some candidates work it out by hand? Sure. They would lose time as a result. That’s a consequence of doing things the hard way.

But if you ask students to estimate a number and they estimate it correctly - more correctly than the ‘correct’ answer - I do not believe they should be penalised for it.

## Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

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