It was only supposed to be a one-off activity. When the students had settled into the room, I asked them if they could spot the envelope. I told them that inside the envelope was a number and that they would all - hopefully - finish on that number after following the instructions I gave them. Once they had spotted the envelope, I started on the instructions.
Write down any positive whole number. (With some groups I say the number should be less than 10 to make the calculations simpler.) Add 3 and double your answer. Add 1 and double your answer. Add on the number you started with and double your answer. Cross off the 8 on the end (technically, you subtract 8 and divide by 10, but that creates less interest).
At this point, the students will look at each other and ask if the other person also has an 8 on the end. They will be quite bemused when it becomes apparent that everyone does.
Finally, take away the number you started with.
Then the person who first spotted the envelope is allowed to open it and hold up the 2. Again, the children all start asking each other: "Have you finished on 2?"
As a volunteer who works with small groups in my local primary school, I have learned to keep an eye on the students who need support. I focus on those who might go wrong at some point, so that they can share the experience of getting the right end number.
As I said at the beginning, this was supposed to be just a one-off activity, but virtually every group I have tried it with instinctively looks around for the envelope when they arrive at my next lesson.
With the brighter groups, after a few simple examples, I have started to throw in a final twist. I say to the class, "I bet you won't get the answer the way it's written in the envelope", and then deliver a final instruction: "Divide by 3." This leads us into recurring decimals. You could also divide by 0 and provoke a discussion on infinity.
Whenever we discuss why it works, I always give credit to this thing called algebra, which can answer lots of questions at once, while arithmetic answers just one at a time. I claim that if the local secondary schools all tried this exercise together, they would all get the answer 2 using the above instructions. The children usually like that.
Michael Rath is a retired secondary maths teacher. He now volunteers two mornings a week with key stage 2 students at his local primary school.
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