# One Week’s Photographs

When I was about ten I’d be given a roll of film to take on holiday. Deciding which scenes to use my exposures on was a tricky business. Using them too early meant I lost any opportunity to take a special souvenir late in the week, but what if it rained solidly in the last few days? In practice it didn’t matter much, because the resulting pictures were always pathetic – blurred, dull grey, with the carefully composed subject barely visible in the middle of the landscape.

Now of course, we carry with us the chance to capture almost anything, instantly, and of good quality. Here are some pictures from the last seven days.

I went to a meeting at the Royal Society on Friday, and have the badge to prove it:

Now don’t imagine the Royal Society and I are closely connected, but once a year they kindly host a meeting of the Joint Primary Group of the Mathematical Association and the Association of Teachers of Mathematics, and give us a splendid buffet lunch.

Considerably more attractive to my mind than that offered to the Fellows themselves – if you’re an FRS you clearly take very seriously indeed the idea of fish as food for the brain:

A couple of days earlier I was in Wetherspoon’s. Can anyone possibly explain why an Americano – which Wetherspoon’s defines as black without milk or cream – contains six times as many calories as black coffee, and more than filter coffee with milk? (The staff certainly couldn’t, and I don’t blame them):

Here’s another I don’t understand (particularly as by my count, the bus only has 28 seats):

On one of the cooler, damper days last week, this one made me smile:

And on one of the sunnier, warmer days here’s the canal. One of the delightful things about canals is that you get these lovely tranquil scenes when you’re in a shopping street with dozens of cars trundling past, and with the supermarket 100m away to your left, and the railway station 100m away on your right.

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# Farewell Amy

I’d be grateful if you’d help me in a little experiment.

In the photo the letters A to J stand, in some order, for the digits 0 to 9. (As you’ll no doubt guess, a two-letter item stands for a number such as 23 rather than 2×3.)

The first question I’d ask you is roughly how long you needed to solve the whole set of statements and discover the unique solution. Did you see everything straight away? Five minutes? Ten? Were there any blind alleys, and fresh starts?

Secondly, you’ll want to reflect on the mathematical and reasoning skills you needed to call upon.

Thirdly, who might you see as suitable pupils for the problem?

For what it’s worth, I think it probably took me about 15 minutes, perhaps 20; I found four possibilities and had to explore each of them. It wasn’t we were working through it together that I realised there was a much better approach that avoided multiple possibilities and allows you to home in smoothly on the unambiguous solution.

And, as you’ve realised, “we” means the remarkable Amy and her partner Paddy; I’ve written before a couple of times (March 2017) about her unusually highly developed reasoning abilities. Now 55 years of teaching give me a pretty solid feeling that this isn’t a problem you dish out to your average 11-year-old (actually Amy was ten when we first me, but like me she’s had a birthday since then).

But I’ve had plenty of pupils who’ve been able to tackle this problem, and the biggest reason they can handle it is motivation. It’s one of many challenges they meet in Anita Straker’s “Martello Tower” adventure game, and by the time they’ve invested several weeks of effort they’re not going to let one more problem put them off.

Usually, however, I offer a clue or two, and my contribution with Amy and Paddy was much more limited. I did write out each of the statements onto card so they could sequence them as they wished and write on them to keep track, but otherwise

my contributions were restricted to comments like “What does that tell you?” and “What could you do next?” On completion I congratulated them, and Paddy said something to the effect that he didn’t know what I was making a fuss about, it had all seemed pretty easy!

This was our final session together, and eight weeks of working with them has reminded me yet again just how localised children’s abilities can be. Their performance in some other arithmetical problems was nothing like as advanced. In the adventure they need to identify a four-digit number using ‘more than’ / ‘less than’ clues and neither of them were great at that, and Amy was worse than Paddy. In Martello Tower they repeatedly need to use triangular numbers and neither of them ever really reached the stage where they could find TN16 without working from and earlier one like TN10 or TN12. My other pupils have almost invariably called upon the streamlined method long before the end of the adventure.

It’s not simply that some children are good at number and less so at spatial stuff, and vice versa. Amy and Paddy are able to operate a very high level in some number work, and much more mundanely at other activities even in related areas, and the difference in maturity can be quite dramatic.

I’m reminded that one year I was asked to lead the national evaluation of pupil performance in the Key Stage 2 national tests. One thing jumped out at me: for virtually every one of the hardest (level 5) questions something like 10% of the correct answers were given by children whose overall achievement was graded at below average level 3. And of course it was a different 10% each time; clearly there are a lot of Amys around, with a very jagged profile of skills across different areas of the curriculum.

Potentially this has huge consequences for the way we group and teach children, and I thought it was so important that we should be shouting it from the rooftops. But no-one else seemed at all interested, and rather to my relief the curriculum authority decided to keep the process in-house and never invited me to do the job again.

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# More Envelopes, More Amy

I spent most of last week telling everyone I could think of about Amy’s insight into the Envelope puzzle, and I couldn’t wait to throw some of my more difficult puzzles at her and her partner. I gave them a ten-envelope set where each contains two cards from a 1 to 20 set, and the displayed products are 10, 24, 26, 45, 55, 63, 136, 168, 320, 342.

They dealt with this quite happily, so I gave them a smaller set, with just four envelopes and a set of 1 to 12 cards – but three cards to an envelope and their products shown:

Once again Amy did something I didn’t expect. Padraig, much as I would have done, targeted the 14 envelope and deduced it contained the 1, 2, and 7. But Amy zoomed straight in on the biggest number, treated it as 96 x 10, recalled that 96 is 12×8, and was home and dry.

What I can’t get my head around is that she’s got brilliant things about how numbers work going on in her head and yet she’s someone who hasn’t found much success in maths. I’m going to have to devise something really special for next week.

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# Stars (v) Some Work With Children

*This is an account of some sessions I did with children aged six and seven, exploring some aspects of different shapes using paths traced out by a beanbag being slid from person to person.*

In the Hall, ask three people to sit on the floor. Give them a beanbag and tell them to slide it from one to another, and keep sliding it, so it goes from Sasha to Luke to Rufina, to Sasha to Luke to Rufina, to Sasha ….

Pretend the beanbag is covered in paint and marks out a trail. What shape is the path it traces out?

The beanbag marks out a triangle. If three more volunteers take their place do they produce a triangle as well? Does it look the same as the first one? Can they design a triangle to order? Can they make a big triangle? A small one? One where all the sides are equal? (Do we know a name for such a triangle?)

Looking at triangles in this sort of way has several advantages. It gives you a new insight; instead of being pictures on paper, or rigid shapes of card or plastic, we’re working with mental images, where we see a triangle as a path dependent upon the positions of the three points which are its corners. Change one – or more – of the corners and you change the triangle. We’re starting to see geometry not just as a static process but a dynamic one.

There are more questions waiting to be asked, some of them distinctly tricky. Can we make a triangle with a right-angled corner? Can we make one with two right-angled corners? (If not, why not?) Can we make a triangle to enclose as much floor space as possible? Can we make a triangle whose sides are big but which encloses a very small area? And we ought to ask whether it’s possible for three people to position themselves so that they __don’t__ make a triangle.

Of course, neither you nor the children will want to stop with triangles. Four people will make quadrilaterals. What instructions do they need to make squares, oblongs, rhombuses, ….? If we fix the positions of two people where could two more sit so the four of them make a square?

*In my next session we took the beanbag work into the Stars arena.*

A very small modification moves the investigation into new and fascinating territory. Start with five people; ask them to sit down and slide the beanbag from one their next-door neighbour.

The path is a pentagon, and if the children spread themselves out equally – not necessarily an easy matter – the pentagon is regular.

Now ask what would happen if instead of sliding to the next person each child misses out their neighbour and slides to the __next-but-one person__ instead. Can they visualise what shape will be traced out?

Ben’s diagram shows a five-pointed star. It’s quite a nice way of drawing a five-pointed star, in fact.

What will happen if we have __seven__ people in the group and miss out one? A seven-pointed star is perhaps the natural prediction, and that’s what Craig gets. And if we miss out two people each time, Alice finds we get a __different__ star, and a much more elegant one.

Any predictions for a six-person group, missing out one each time? Hands up, all those voting for a six-point star! The first time I did this, with Y2 pupils, was the only occasion in my life when I’ve actually seen someone’s jaw literally drop. A boy sat there with his hands open, waiting for the bag to reach him, and it went round and round and round, passing him each time. “I’m __never__ going to get it”, he wailed! (There are some quite deep ideas in this statement.) Eden’s diagram (1) shows why.

What happens if you miss out two people rather than one? This time the path is even more limited; Eden’s diagram (2) shows it’s simply a straight line as the bag travels backwards and forwards between just two people.

There are lots of questions waiting to be asked. Why are things completely different with six people? What happens with other numbers?

The children were full of suggestions and ideas they wanted to explore. Nearly all used abstract diagrams from the start, but a couple preferred to draw people, while two others wanted to set their examples in a concrete setting (rabbits in a field and fish in an “Ekweriam”). I could see who chose to use a ruler, and who put explanations and descriptions on their diagrams.

Alexander was particularly keen to see what happened if he used a composite rule (miss out 1, then miss 2, then miss 1, miss 2, miss 1, …) with seven people. I certainly didn’t know what would happen, and our joint diagram delighted us when we found the final pattern included both the stars.

Someone explored all the patterns you can make with a group of nine people. Jo showed tremendous skill at spotting the group sizes and rules which generated triangles, squares, and even hexagons rather than producing stars. How much insight and understanding did this 7-yearold display to be able to create this diagram? “I’ve made a Hexagon!” she recorded with justifiable pride.

The Beanbag is one of my favourite themes. It’s instantly accessible and it’s a lovely example of the fact that mathematics does not consist of blocks of learning which are isolated from each other. Rather, they are linked so closely it can be impossible to disentangle them. Surprisingly, this investigation is at least as much about numbers as it is about geometry. It can be used with almost any group – I’ve used it with Classroom Assistants, B.Ed. students, teachers in disadvantaged schools in South Africa, and of course the 60 Y3 children whose recordings I’ve used throughout this article.

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# Stars – part (iv) Some Reflections

I’ve no intention of giving away too much. I had to explore everything for myself and enjoyed it immensely. Most of my first hazy guesses didn’t work out but it was well worth it in the end. It was wonderfully satisfying to find that the investigation brought together so many topics in mathematics.

We started with what was quite obviously an enquiry into geometrical shapes, but gradually it dawned on me that it was just as much an arithmetical topic as a spatial one. You – or your pupils – may follow the same path as I did, or you may take a different route, but by the time you’ve found out everything you want to you’ll have used factors, multiples, and prime numbers. Whoever would have thought that the key to drawing stars lie in prime numbers?

And a real enquiry it was. I had to hypothesise, explore, evaluate, improve, amend, conjecture – all the thinking skills you want your pupils to explore. And yet it’s pretty well untouched in any school programme of study. Yes, __any__ school. What age-group can you see yourself using these ideas with? After all, it worked with me at my level.

So would you use it with secondary children?

Or could you use the theme with primary pupils? In my final part I’ll give some examples what happened when I used some of these ideas with 6 and 7 year-olds.

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# Stars – part (iii) Some Glimpses of Daylight

*So far I’ve mentioned drawing stars with 5, 6, 7, and 8 points. Sometimes you can compose a stars by overlapping shapes; other times you draw a single continuous zigzagging line.*

Most of this was pretty new to me, and I figured that if I was going to get any insight into things I’d have to collect as much data as possible, and do so in a systematic manner.

So for the 9-point situation I sketched out 9 points equally spaced round a circle. Joining each to the next one gave me a regular nonagon, which isn’t a star at all. But when I joined each point to the next-but-one point, i.e. missing out one point each time, I got a genuine star. (A)

What if I miss two points each time? I get an equilateral triangle, and if I make all the three possible equilateral triangles I’ve got a new star – an overlap. (B)

And if I miss three points each time there’s a second zig-zag star. (C)

There aren’t going to be any more, because I can see that missing out four points each time (or five, or six, …) simply gives me shapes I’ve already discovered.

So my table of result now looks like this:

Points | Zig-zag | Overlap | Total | |

5 | 1 | — | 1 | |

6 | — | 1 | 1 | |

7 | 2 | — | 2 | |

8 | 1 | 1 | 2 | |

9 | 2 | 1 | 3 |

It had been a long time coming, but I was now indeed beginning to get some insights into what was happening. I still had plenty of questions –

are all stars either overlaps or zig-zags, or is there another type?

are there any numbers which will give no stars at all?

can I predict which types of stars, and how many, we’ll get for any number?

does every number after 6 give at least one zig-zag star?

can I predict how many zig-zags there’ll be for each number?

are there any numbers which will give more overlaps than zig-zags?

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If you want some practical help, you can find circles with every number of dots up to 24 at __http://nrich.maths.org/8506__

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# Stars – part (ii)

*The previous post was built on the observation that to draw a five-pointed star you use a single zig-zag line, and to draw a six-pointed star you must overlap two equilateral triangles. But there’s much more to find out.*

The first time I explored these ideas I found some delightful surprises lying in wait, and the first one comes with seven-pointed stars. Actually there’s not one, but two seven-pointed stars you can draw, both by the single zig-zag line method. One version is – to my mind – elegant, and the other is fat and rather bloated

And what about eight? You may be beginning to expect that the superposition method will give an eight-pointed star, and if you use two squares that’s what you get.

But, wait a moment, there’s a second eight-pointed star – and this one is drawn by the single-line zig-zag method!

So we’ve already got a rich collection of results. With 5 points you get a zig-zag star; with 6 you get a star formed by overlap. 7 points gives you not one but two zig-zag stars, and 8 points produces both one zig-zag and one overlapping star. I’ll be very disappointed if you’re not already wondering about 9 pointed stars – zig-zags (and how many?), or overlap – or perhaps both?

Points | Zig-zag | Overlap | Total | |

5 | 1 | — | 1 | |

6 | — | 1 | 1 | |

7 | 2 | — | 2 | |

8 | 1 | 1 | 2 | |

9 | ? | ? | ? |

It looks as if there are some rather seductive patterns to explore, doesn’t it?

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More next time.

# Stars – An Introduction

*Every now and again you come across an idea that’s immensely rich in all the good things you want a topic to offer – accessibility, scope for exploration, coverage of a wide range of mathematical ideas, available to a wide range of ages and abilities, …. – but which is totally ignored in syllabuses and schemes of work.*

*Just what was wanted, in fact, when I was asked to do a series of sessions to be televised to a selection of schools in England and Pakistan. There was no chance of finding what each different group had covered, so I needed to use topics which were visual, accessible, exploratory – and which I could be pretty sure were not covered in their schemes of work.*

*I was working with secondary schools, but in different circumstances I’ve used most of the ideas equally well with primary children.*

*(There will be a short pause while you think of the topics you might have chosen in similar circumstances.)*

*Very high up on my list was the theme of Stars, and here’s where I got my starting point from.*

Service was rather slow in Pizza Express and we discovered my wife and I were both doodling stars. Jill was drawing six-pointed stars, and all mine had five points. Jill drew hers by drawing an equilateral triangle and then another on top of it, turned through 180°.

Unlike hers, my pencil never left the paper. I drew one continuous line which turned at four points and eventually got back where it started.

Our sketches looked something like this:

A couple of questions immediately arose. Could our methods be interchanged? Could we produce a five-pointed star by overlapping two shapes, and could we draw a six-pointed star from one continuous line?

Now that gave us a lot of fun, and you may like to think about it for a while and try a few sketches.

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Probably before long you’ve convinced yourself that the two methods are __not__ interchangeable. But that only generates further questions, and the very first is to wonder what happens if you try stars with a different number of points.

So how are you going to draw a seven-pointed star? Overlap? Or zig-zag? And eight? And are any hypotheses developing?

I’ll post a second part soon.

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# Slicing Up Squares With My Paper Trimmer

*My wife would probably allege that if the house were on fire my first priority would be to make sure the paper trimmer was safe. Not quite true perhaps, but it’s certainly true I do have a bit of an addiction problem. My cosy office is less than about 3 metres square, but it does hold five paper trimmers.*

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I make a lot of cards for number and sorting games for my pupils, and the other day I needed to convert a 24cm x 24cm card square from a cereal packet into 36 4cm x 4cm cards.

It seems intuitively obvious that some ways of cutting up the square will be more efficient than others, but in spite of my long experience I wasn’t clear what the best strategy would be.

Would it be best to (i) first slice the 24cm x 24cm square into six strips, or (ii) first cut the big square into quarters, or (iii) perhaps cut the big square into six 8cm x 8cm squares and then cut these up? Or perhaps another strategy is better than any of these.

Over to you. If you’re going to put yourself into my shoes I need to emphasise that I’m cutting card rather than paper, and my trimmer cannot handle more than one thickness. Do let me know how you get on with this exploration, which I’ve never seen offered anywhere else.

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# An Update

There’s no reason why anyone should have noticed, but this is the first post for quite a while. There’s nothing sinister about that; it was basically just work pressures and hobby stuff had to take a back seat.

But something funny’s been happening – some people have clearly decided they quite like the idea of a silent blog, so that new readers have appeared, and earlier this week the biggest ever number of readers gave a totally new look to the statistics.

So a big welcome to one and all. The About page tells you a bit more about me, but this is basically a blog about the teaching of mathematics to children of primary age. There are oodles of terrific maths blogs by secondary teachers, and a good few about general primary education, but there very seem to be very few whose focus is primary maths – so if you know of more I’d love to hear about them.

And my idea of a blog is that it’s an individual and personal thing, perhaps quirky, and as unique as you can get it. I asked someone who I greatly respect to take a look at it. “What an engaging blog”, she said. That’ll do for me; “Engaging” suits me just fine, and I hope you find it engaging as well.

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