*Every now and then a child says something that really makes you sit up and go Wow! See what you think about this Wow! moment. *

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I’ve borrowed a vast number of ideas from other people, but I have had one or two good ones of my own, and Envelope puzzles are up there with the best of them. I’ve written about them before (April 2015) but I’ve no hesitation in doing so again. They do give a hugely accessible way for children to develop a chain of rigorously justified reasoning.

I gave Amy and her partner this set of envelopes. They knew each envelope contained two cards from a 0 to 9 set of digits and that the product of the two digits was displayed on each envelope. Their job of course was to identify the cards in each envelope.

Amy’s partner and I agreed it would be sensible to leave the 0 envelope till last, since though we could be sure it contained the 0 we wouldn’t know which the other digit was until we’d eliminated all the other possibilities.

“No”, said Amy, “you can say immediately that the 0 envelope must have the 0 and the 1”.

“Why’s that?” I said. I rather assumed Amy was a bit unclear about the multiplicative properties of 0 and 1.

“Well”, she said, “if the 1 is in any other envelope then it must have a single-digit number as its partner. That would mean that one envelope would have a single-digit number written on it, but none does. So 1 cannot be in any other envelope, and so it must be in the 0 envelope.”

Wow! indeed. What a terrific and totally water-tight chain of reasoning that had never occurred to me when I devised the set. With a National Curriculum which aims that we focus upon problem solving, reasoning and fluency I reckon Amy’s pretty much on the right lines.

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A footnote: I was almost as flabbergasted at the end of the afternoon when I eagerly buttonholed a couple of teachers. “Can I tell you about Amy?”, I said. “Ah, Amy”, they said ruefully, “she’s always had problems with maths!”

(Don’t get me wrong – I’m not saying this to show how brilliant I am; these are experienced and committed expert teachers who spend every moment every day devoted to thirty pupils, many very challenging. I, on the other hand, merely swan in for the afternoon and have no other responsibility than to work with two or three children on aspects of their mathematics. My point is rather that locked away in Amy’s head was potential and insight and I was lucky enough to find the right key to bring some of this out into the light of day.)

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