In summer, I had the pleasure of accompanying some students on a school trip to Europe.  Mr Mattock kindly stepped in to look after #mathschat while I was away. This is a blog that I’ve had on the back burner since then.

While we were waiting at a public travel hub in France (a place designed for the sole purpose of losing children, or at least scattering them so you can’t be certain they will all arrive in time for the departure), I was asked a number of times, by different children:

“Is this cheaper at home?”

Understanding the question

I answered in my best teacher voice: “I don’t know how much they cost at home, so I can’t tell”.

But that wasn’t what they were asking.

The students weren’t asking my advice.  They didn’t want me to tell them whether this set of pens was a bargain. They simply wanted to know whether €7.30 was more or less than £7.30.

A set of pens and a set of pencils with price labels in euros
If these pens were priced in pounds, would they be more or less expensive?

So I answered as a good mathematician should: “A euro is about 85p.” (It was a year ago.)

But that wasn’t the answer they wanted either. That a euro was 85p did not help them. More or less, that’s what they wanted to know.

To me it seemed obvious that €7.30 was less than £7.30. It just felt right. But not to them. And I didn’t want to just give them the answer, I wanted them to understand. I thought it could be an issue of multiplying and dividing by numbers close to one, but not quite one.

Since we had a 13-hour coach journey, and the book I’d brought with me was really rubbish, I dedicated the return journey to asking the students questions, to try to understand the issue.

Finding the right analogy

They had no problems working with New Zealand dollars (about two to the pound), because that was an easy conversion. But as we got away from whole multiples, and close to one, it went wobbly.

I needed an analogy to help them understand. I tried yards and metres, which came close, but didn’t quite manage it. I tried litres of petrol and pounds, but that blew up in my face because in that analogy, the petrol was the pounds and the pounds were the euros, which was far too confusing.

Then, after 13 hours on a coach in Europe, I had a break-through. Kilometres and miles. That was a great analogy, partially prompted by a Tweet I’d read:

Fibonacci numbers 0, 1, 1, 2, 3, 5, 8, 13 etc are useful in converting kilometres to miles. 3 miles equals 5 km, 5 miles is 8 km…

So I asked them: “If 1 mile is about 1.6 km, and the distance is 60 km, is this distance less or more than 60 miles?”

“Less,” they said.

Brilliant. That was the analogy sorted, now I just had to turn my attention to the language.

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