# Problems with Gabriel’s Problem

*Here’s an activity I’d like some help with. It seemed it would chime pretty well with my Envelope puzzles (see for example, https://established1962.wordpress.com/2015/04/ and also https://established1962.wordpress.com/2017/03/03/a-wow-conversation-with-amy/ ) Things started well enough, but as the session went on I couldn’t find the right way to take it further.*

We started with me sketching out a 2×2 grid. I asked Chaz and Jo to put the digit cards 1 to 4 into the squares. I then wrote in the products of the rows and columns.

The next stage was to give them another grid, this time showing the products only. Could they place the digit cards?

Of course they were happy to do this, and we were quickly ready to move on.

I think I missed a good line of development here. Chaz and Jo have good knowledge of the multiplication tables, and could have handled a situation where I used other combinations of digit cards. They would have been able to work out the cards used to make this grid:

Instead, I moved to a 2×3 grid, with the numbers 1 to 6. This worked fine. There were lots of possibilities, some decent practice in handling two- and three-factor multiplications, and plenty of decisions to make.

So, after several of these, and with twenty minutes to go, we simply had to take things a bit further forward. It was pretty obvious that we had to tackle 3×3 grids, using cards from 1 to 9. They were very happy to kick me out of the classroom while they composed puzzles for me to solve. Well, I’m glad I tried it this way round instead of me setting them problems, because the 3×3 grid turned out to be a different beast entirely.

Chaz’ problem made me think for a while, but would have been too much of a challenge for my Y6s, though I’m pretty sure some of their more confident classmates could have wrestled it out.

But I found Jo’s to be a complete brute. I gave it 15 minutes of effort, but there were too many Sudoku-type multiplicities and I had no clue which one to follow. Later I did have another look and found a more successful line of attack, but it was definitely more difficult than Chaz’ puzzle.

I was really disappointed at the way this had developed. It had begun with a nice gentle starting point accessible to all, and when I’d thought I’d turn up the challenge dial one just more notch I found it had jumped from strength 3 to about strength 10.

Afterwards I spent some time trying to figure out how I could move from the rather successful 2×3 grid and introduce a new challenge without making it so intimidating. In fact I spent most of the weekend trying, and rejecting, ideas. I’ll offer a couple of the better ones next time, but in the meantime I’d welcome any suggestions you might have.

And who’s Gabriel? This was my attempt at using a simpler version of the very similar NRICH puzzle called Gabriel’s Problem. The NRICH version is targeted at secondary pupils, and is located at https://nrich.maths.org/11750 The challenges set by Jo and Chaz fit very well with the NRICH puzzle.

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# A Wow! Conversation with Amy

*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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# Envelopes

*This is a revised and expanded version of a piece I published a little while ago which my pupils found gave a very accessible route into some very meaty challenges. They enjoyed them so much that I kept on thinking of new things to do and which gave them some high-quality experiences in the National Curriculum’s requirements for Fluency, Reasoning, and Problem-solving.*

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Here’s a screenshot from what you’ll see is a very old – and incidentally my very favourite – computer game. The letters A to J, in no particular order, stand for the digits 0 to 9. It struck me recently how neatly this 30-year-old task met today’s demands for Fluency, Reasoning, and Problem-solving. It’s one of the best activities I know for generating high-quality discussion, and it requires the use of number facts in an situation that’s both challenging and purposeful.

However, I’m pretty sure you’ll agree that as it stands it does look rather bleak and even threatening, so I wanted to find some form of presentation that enhanced its accessibility across the ability range.

So I show children a set of 0 to 9 digit cards and – hiding the numbers – put two in each envelope. On the front of each envelope I put the sum of the two cards.

Here’s a snatch from a recent discussion between two Y6 girls (they’d begun by deciding that the 1 envelope must contain the 0 and the 1):

“Well, 6 could be 3 and 3, or 5 and 2. No, that has to be 5 and 1. Or 4 add 2. Oh, and 6 and 0. But we’ve used the 0 and the 1, and there aren’t two 3s – so it must be 4 and 2.”

Now these two are bang in the middle of the attainment range, and it seems to me you’ll very rarely hear such a long chain of watertight reasoning.

And here’s how they (spontaneously) made a record of their progress through the problem.

I’ve used my envelopes with lots of children and I can’t recall any group not wanting to tackle another set. Once they’ve done two or three it’s pretty seamless to move onto a set where the envelope shows the *product* of the two numbers inside (which is the sort of problem presented by the computer game).

Pupils discover that the envelopes with bigger numbers tend to be a good place to start (and the 0 is the very worst).

For a long time that’s all I’d do with the envelope idea, but this term’s pupils have been so enthusiastic that I’ve made lots more sets.

This set uses 0 to 15 cards and hence needs eight envelopes:

These weren’t enough, so we went to bigger sets still. I discarded the 0, so here we have a set of ten envelopes, using the cards from 1 to 20 – why settle for tables up to 12 when you can use your 17, 18, and 19 times tables as well?

And of course, just as it’s usually nice to work with bigger numbers, it’s normally a good idea to work with small ones, so these use a set of twelve cards using ¼, ½, ¾, ….., 2½, 2¾, 3.

These envelopes carry the sum of the two cards, but my most adventurous pupils have even been happy to use this set of cards and explore what might appear on envelopes which show the *product* of the cards inside.

Working with the envelopes has been enormously successful with my pupils, none of whom are of higher than average attainment. I’ve only called a halt because I didn’t want anyone to say “Oh no, not envelopes again!” (though I’m pleased to say there haven’t been any signs of that).

We certainly went out on a high.

One group made their own sets with 1 to 16 cards and used them to challenge the pupils in the group taking the Level 6 paper – yes, they managed to do them, but they certainly didn’t find the exercise trivial.

Another group – of average ability, remember, were happy to solve this multiplication set and do so mentally, where each envelope contains *three* cards from a 1 to 12 set. The challenge to fluency, problem-solving, and reasoning seems gratifyingly high.

I’m sure others will be able to come up with other interesting envelope activities, and I’d love to hear of further suggestions.

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# A Coincidence

I guess that by the time you’ve notched up several dozen years it’s pretty likely that some distinctly unlikely things have happened to you.

I went to a lecture at Logan Hall in London. The Logan Hall is a huge lecture theatre and holds nearly a thousand people. Speaking in public has never been much of a problem for me, and as I looked around I wondered what speaking to such a large audience might feel like. The only venue I could think of that was more prestigious was the Lecture Theatre at the Royal Institution.

When I got home I switched on the computer to check my emails. At the top of the list – and I still have it today – I read:

*“[We] were wondering whether you would be able to do a Primary *

*Maths lecture for us at the Royal Institution in London …. *

*“We would be very keen for you to do something based on your “Take Ten *

*Cards” activities which you introduced to Jenni during the ATM conference *

*(although Jenni has already used the envelope problem at a lecture last *

*week!).”*

I really don’t think anything has topped the excitement and pride of speaking to 300 children at the Royal Institution, standing at the very desk where Michael Faraday and some of the world’s most famous scientists stood. Evening discourses at the RI were one of the great events in the London calendar, so much so that the resultant congestion meant that Albemarle Street became London’s first-ever one-way street.

I did indeed include the envelope problem, and that’s what I want to write about next time.

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