# The All I Can Throwers – Sessions with Den and Jenna. #1 – Consecutive Numbers

I think most of us are pretty uncomfortable with labels like “slow learners”, “less able” etc. And terms like “gifted”, “quick”, “high ability” aren’t much better. Apart from anything else, many are highly pejorative and they all suggest that everyone has a fixed level of learning.

All the same, I do from time to time meet children who do have the experience and background that lets me know they can take all I can throw at them, and recently I had the chance to work with a couple of them for a few weeks. The challenge was not to accelerate them through the syllabus but to give them the opportunity to explore some ideas at greater depth than is normally possible.

So what did I do with my All I Can Throwers? In looking for ideas I had many of the same criteria I’d use for any group, but basically I wanted themes that were accessible, intriguing, offered scope for asking questions, and lots of things to find out.

My first almost chose itself. I was asked to go to Cambridge a while back to talk to NRICH about my favourite activities and we both agreed that number 1 on our list would be the Consecutive Numbers question.

- I can write 12 as the sum of three consecutive numbers; 12 = 3 + 4 + 5
- Another example: 9 = 4 + 5
- Another: 14 = 2 + 3 + 4 + 5
- (and some numbers can be made in more than one way: 21 = 10 + 11, and also 21 = 5 + 6 + 7).

So the question is whether __all__ numbers can be made in this way.

In truth, it’s the perfect enquiry for almost anyone. It’s immediately accessible using the simplest arithmetic, and offers scope to explore in your own way. There’s lots of scope for formulating questions, making observations, and reasoning and generalising. Perhaps best of all, it gives up its secrets gradually. It won’t take long to make the first observation, which explains half of all numbers, but others may take a little longer. There are further generalisations on the way, each adding a little more to the understanding.

Some of these discoveries will be made by any pupil who tackles the question, but there’s a final gorgeous climax in store for those who, like my pupils Den and Jenna, are able to dig deep and make generalisations. It had been a great way to spend my first hour with them, and with just five minutes to go they realised, and were able to explain, just which numbers __cannot__ be made as the sum of consecutive numbers. But you don’t have to have a Den or Jenna in your group – it’s a great topic to explore (and makes for a fine display for the first parents’ evening of the year).

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# An Afternoon With Keena

I enjoy working with nearly all my pupils, but this year Keena was right at the top of the list. It’s fair to say that many of her successes are hard won, but she’ll tackle everything, and when things go wrong she’d far rather have another go to sort things out than ask for my help, and she does it all with a big smile on her face.

One day last term she left with an even bigger smile.

A nice accessible starter challenge is to ask children to fold a piece of paper so that *when you make a single straight cut* you’re left with a square hole in your piece of paper. It probably takes a couple of tries, but nobody minds having another go:

So if everyone can do a square, then creating an oblong (just one single straight cut, remember) can’t be hard, can it?

It was so obviously an easy challenge that I hadn’t bothered checking it out completely myself, and I contributed fully to the increasing number of rejects on the table. I think it took a little over five minutes and it was a deadheat between Keena and myself, though I do claim my expression of satisfaction was a little more restrained than Keena’s warwhoop.

Could she do it again? Yes indeed, and now they knew it was possible, everyone else tackled the challenge with renewed intensity. Soon everyone, with perhaps a little advice from Keena, could do it. Keena, meanwhile, was busy exploring more shapes. An equilateral triangle? Certainly. A regular hexagon? Why not?

Comfortingly, there’s a wonderful lesson plan from Joel Hamkins on his website at http://jdh.hamkins.org/math-for-nine-year-olds-fold-punch-cut/ He mentions that – incredibly – any shape whatsoever that’s made up of straight lines can be made in this way, and we ended the lesson trying to make a five-pointed star.

Keena wasn’t the only one to leave with a smile on her face. I think we all did. I was delighted that we’d found something that had given her so much success, but more widely it had proved yet again how powerful a good Low Threshold High Ceiling activity can be. No matter what may be included in your list of requirements for a good starter I reckon this one ticks all the boxes.

(It probably helps if you’re on good terms with the caretaker, but that’s another story. Oh Heck, let’s include it here. I’ve known John since he joined the school around 1980. At that time I used to run the six miles into school once a week, and he does a lot of running as well – i.e. unlike me, he still does. We sponsored him to run in the Ottawa Marathon recently and when I coughed up the money he looked a bit frazzled. He explained that before committing himself he’d checked that the run is normally in a nice comfortable 17°. A week before he checked the weather forecast and was a little worried at the predicted temperature of 27°. On the day itself it was a quite ludicrous 37° ! Yes, he completed the full 26 miles, but not surprisingly he’s decided he won’t do another.)

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# Two Olympic Questions

I’ve long believed that asking someone to measure something is to give them an instant challenge in problem-solving. So I suppose I’m particularly vulnerable to a couple of questions thrown up in the Olympics.

In the pool Michael Phelps and two others tied for a silver medals with each given a time of 51.14. Now, today’s systems would allow times to be recorded to far greater accuracy – in cycling for example times are reported to thousands of a second, so surely there’s no problem in giving times in swimming to three decimal places?

No there’s not, but in a thousandth of a second Phelps will travel a bit under 3mm, and when a 50m pool is constructed the tolerance allowed is more than ten times that – as much as 3cm in a lane. In the extreme case in the pool one contestant could be swimming 49.97m and another 50.03m. (That seems quite a lot; however, it’s pointless trying to build to a greater accuracy – the length will vary not only with the temperature, but even with the number and position of people in the pool). In a sport where everyone uses the same track – motor-racing for example – it doesn’t matter, but whenever you have lanes then you’d have a problem.

So the swimming authorities seem to have got things right. If you use technology to give you times more accurately you run the risk that the technology will also demonstrate that the reason someone finishes before someone else, and hence records a shorter time, is because they’ve had to complete a shorter course than their opponent. This might be terrific news for the manufacturers of measuring equipment and for a host of lawyers but the extra accuracy would cause more problems than it solved.

Incidentally, some back of the envelope calculations tell me that if there is just 2cm (well under the allowed tolerance) difference between two lanes that equates to around 0.01 second – in other words, even giving timings to hundredths of the a second is bit dubious. Good news for the lawyers after all!

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There are plenty of mathematical expressions that are simple and elegant, but they don’t include the following, which gives you the number of points won in the high jump in the women’s heptathlon: 1.84523(M-75)^{1.348} where M = height in cm.

And just in case you were interested, here are the other formulae. They’ve been in use since 1984 and are scaled with the intention that a good performance should gain 1000 points and a minimal performance (e.g. 0.75m in the high jump) gets 0. They’re designed so that regular increments in performance gain fair increments in points scored:

4.99087(42.5-S)^{1.81} 200 metres

0.11193(254-M)^{1.88} 800 metres

9.23076(26.7-X)^{1.835} 100 metres hurdles

0.188807(Z-210)^{1.41} long jump

56.0211(D-1.50)^{1.05} shot put

15.9803(W-3.80)^{1.04} javelin

I’ve never seen it mentioned as an issue, but the range of accuracies implied varies from three to six. In particular, the performances themselves are measured to an accuracy of three figures in the jumps, and to five figures in the 800m. They’re then plugged into the values in the brackets, which are to three figures.

Now I was going to say that conditions are the same for all competitors in all seven events, so none of them suffers from anything corresponding to the variable lane situation. But I’m not sure that’s true. Aren’t the jumps weak links here? Aren’t there problems at both ends of the long jump? Can we be certain the plasticine is identically positioned for each competitor (Greg Rutherford certainly felt he was hard done by in the men’s event)? And how you determine the exact point of the depression produced on landing seems very uncertain.

What about the high jump? Each time anyone knocks the bar off it has to be replaced. Can the bar be repositioned, even automatically, so that the next competitor faces an identical height to the last? And can we be sure the bar hasn’t been minutely distorted by the previous failure?

Can contestants really be certain that when the bar is set at 1.92m this is actually is 1.92m, and not a few millimetres either way taking it closer to 1.93m or 1.91m? A measured high jump of 1.92m is worth 1132 points, but someone at the very limit of her range who attempts what is actually nearer to 1.93m is likely to have to settle for her previously attained height of 1.89m, worth just 1093. That’s a difference of 39 points, and if Jessica Ennis-Hill had scored another 39 points she’d have taken the gold medal. Indeed, no fewer than four of the top eight in Rio would have been placed higher if they’d scored another 39 points.

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The ideal heptathlon and decathlon scoring systems (or perhaps the ideal heptathlon and decathlon performances, which is of course a totally different matter) would generate approximately equal points in each event. That was the aim of the 1984 scoring systems, which were based on the then current world record performance at each of the separate events, but in 2016 we can all think of several different reasons why that might not be such a good idea today.

Whenever we watch British women in the heptathlon they seem to get off to a good start in the 100m hurdles and the high jump, then do less well in the shot and javelin, but actually they’re simply modelling the performance of heptathletes as a whole. In heptathlons generally, the 100m hurdles and high jump produce scores around 1000 points, while in the throws typical scores are closer to 800. In other words, the scoring systems and indeed the very events in both heptathlon and decathlon favour tall slim athletes who do well in the running and jumping events.

There’s all sorts of scope for improving the scoring system in the heptathlon and decathlon – among the many suggestions is one by John Barrow to relate performances to kinetic energy – but one look at the heptathlon performers themselves makes it clear that an event where five of the seven events depend largely on speed and running is heavily biased in favour of tall competitors with low body mass.

But if we really do want a more balanced event then how about a hexathlon, with a flat race, a hurdles, one of the jumps, and three throwing events? This would be a much better way of finding a genuinely all-round athlete, but I’m pretty sure she wouldn’t look much like Katarina Johnson-Thompson.

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There’s one question I’ve tried to find an answer to for many years. Shorter track events in imperial units converted very well to metric measurements – 110 yards is 100.58 metres, and 220 yards is just over 201m. The one-lap quarter of a mile and the two-lap half-mile easily became translated to the 400m and 800m. So it seems the most natural thing in the world to replace the classic blue-riband track event, the mile, by the four-lap 1600m.

What I’ve never understood is why the so-called “metric mile” was instead taken as 1500 metres. This is more than 100m less than a mile (a mile is a little over 1609 metres) so there’s about 15 seconds difference in performance. Taking 1500m as the equivalent event to the mile not only discarded the close link in distance and time, but it also made life far harder for officials, with the need to start the race on a bend the other side of the track.

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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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# Plus Ça Change

*These extracts are taken from “A Social History of Education in England” by John Lawson and Harold Silver. The authors are writing about the “Payment By Results” system introduced in 1862. Grants to schools were determined by the proficiency of children from six years old as tested individually by the inspectors.*

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“…. As T H Huxley later put it: ‘the Revised Code did not compel any schoolmaster to leave off teaching anything; but, by the very simple process of refusing to pay for many kinds of teaching, it has practically put an end to them.’ Matthew Arnold, poet and HMI, was the inspector most responsive to the effects of the code. Under the old system a good inspector heard selected children read and questioned whole classes on all their subjects: ‘the whole life and power of a class, the fitness of its composition, its handling by the teacher, were well tested.’ Under the new system, however, he was unable to test any of these: ‘he hears every child in the group before him read, and so far his examination is more complete than the old inspection. But he does not question them; he does not … go beyond the three matters, reading, writing, and arithmetic.’

“The result was generally an increase in rote learning, and even inspectors not opposed to the principle of the revised code reported on its deadening and disheartening effects. The need to drill the children to meet the inspection requirements was reflected in the schools’ activities throughout the year. Frequent testing became common. Some of the improvements of the 1850s in curriculum and method in many schools were cut short. Even religious instruction was sometimes dropped as the inspection approached.

“…. The possibility of new thinking about educational methods and about the curriculum became paralysed by the operation of the code.

“…. Payment by results was a view of the nature of elementary education from which it took the system generations to recover. Edward Thring looked back at the experience of payment by results and the inspection of minds like ‘specimens on a board with a pin stuck through them like beetles’, and appealed to teachers to get rid of the vestiges of the system: ‘strive for liberty to teach, have mercy on the slow, the ignorant, the weak’. A former inspector looked back in 1911 at the thirty or more years of ‘Code despotism’ in which he had been involved, and thought the efforts were still being felt of ‘that deadly system … which seems to have been devised for the express purpose of arresting growth and strangling life, which bound all of us, myself included, with links of iron’.”

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