Magazine article Mathematics Teaching

Happy Lessons

Magazine article Mathematics Teaching

Happy Lessons

Article excerpt

My Y7 group are good but suffer from Friday afternoons like any class - like me too. So was this really the time to introduce a new idea? Would we run with it and would the rest of the double fly? Or would the lesson fall flat on its face?

"What makes a number happy?" I asked.

"Prime numbers aren't happy because they don't have any children." An interesting idea.

"666 is evil, but possibly happy."

So I told them what I thought made a number a 'happy number': square each digit and add to get a new number; eg, 23 [arrow right] 13 [arrow right] 10 [arrow right] 1. Clearly, 1 loops back onto itself (1^sup 2^ = 1). We say 23 is a happy number because the sequence ends up at 1. The same rule applies for 3-digit numbers, 4-digit numbers and so on: square and add the digits; eg, 103 [arrow right] 10 [arrow right] 1.

There are 20 happy numbers smaller than 100, including 1 itself. So my challenge to my Y7 group for the weekend was to find the 20 happy numbers.

"How do we do that? Do we have to check every number? That's long!"

I clearly hadn't enthused the troops yet! So we needed ideas about how to speed the task up.

"So 23 is a happy number. What else does our example tell us?" I asked.

They were pretty quick to realise that 13 and 10 are also happy numbers because they are in the same chain. So it's a matter of finding all the chains.

"That's obvious. It's still long!"

"OK. So 23 is a happy number. What else does that tell us?"

Silence for a moment. Then the answer I wanted to hear: "32 is a happy number too!"

We were fully armed for the weekend. My final piece of advice: show your working!

I suppose one measure of the success of a lesson is the teacher learning something from it. This, to my surprise, happened with happy numbers. Most of the class had found the 20 happy numbers: 1, 7, 10, 13, 19,23,28,31,32,44,49,68, 70, 79, 82, 86, 91, 94, 97, 100. …

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