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.
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.
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.)
It was a great pleasure to be working with a couple of pupils I knew would accept any challenge I offered, so you won’t be surprised to know that we spent two or three sessions exploring all the ideas around Stars that I wrote about in several recent posts.
I may well have been the only teacher in the country disappointed that the end of the summer term was coming up fast, but there was still time for one further session. I really don’t think there’s any exploration more accessible and productive than the Tower of Hanoi. It’s intensely practical and visual and you need just two simple rules. I was using it with two very bright nine-yearolds, but I’ve used it both with teachers and with much younger children – one teacher used it with her Reception class “Baby Teddy can sit on Mummy Teddy’s lap or Daddy Teddy’s lap ….” and it worked a treat.
There’s so much to find that even now I’m still discovering new aspects, but it won’t take long to start wondering how many moves it takes to move a stack of 3, a stack of 4, a stack of 5, …., or to observe a dazzling array of patterns and movement rules.
If you need refreshing on the rules and background there must be hundreds of websites devoted to the problem, with diagrams, formulae, and animations. Many of them spoil the fun, but you’ll easily find all the information you could possibly want and much more besides.
In the spring I used it with a Masterclass group of Y6 children and we dealt with numbers up to quintillions, and derived a procedure to allow them to solve the puzzle for a stack of any size. We used boxes gleaned from the supermarket, and I was struck that for these children it’s probably rather rare that they get they chance to manipulate apparatus. It seems a little sad, but I suspect that one reason they enjoyed the session so much was that there was a strong element of play involved. There were 30 people in the group and next year the organiser has decided she wants to invite 90. Collecting enough boxes will be a massive task, and we’re hoping we can persuade IKEA to sponsor us with a few dozen sets of their toddlers’ stacking cups at £1.50 a set.
The previous post used an activity where cards have different numbers on each side, and the possible totals are found.
I followed that by using another set of numbered cards. There were six in the set, each with a different number on each side, i.e. there were twelve different numbers in all. I gave Jenna and Den a free choice of which number should be face up on each of the cards, and gave them an opportunity to revise their choice. I asked them to add the six numbers on display – and then produced a sealed envelope which they opened to find I’d predicted the correct total in advance!
You don’t have to work very hard on presentation for your audience to be wholly baffled. I performed the trick with a set of eight cards in South Africa, and the audience included Toni Beardon, who’s the founder of NRICH and a very clever person indeed. I treasure the look of complete amazement on her face when the sealed envelope was opened and the prediction displayed.
I encouraged Jenna and Den to inspect the cards carefully. Their first observation was that each had an odd number one side and an even number the other. Secondly, on each card the even number was the lower one. Thirdly, on each, the odd number was 17 more than the even number.
So the total of the numbers on display would always be the sum of the six even numbers, plus 17 for however many odds were visible.
From the work earlier in the session the children told me there would be 64 possible arrangements. A significant number of these – twenty – show three odds and three evens, and that’s the situation I need to see for the trick to work. It doesn’t matter which three odds /three evens they are, and on about one occasion in three this will happen anyway, but Jenna and Den’s original selection showed four odds and two evens. So I invited them to “do a further randomisation” and turn over one of the six, and not surprisingly they turned one of the four odds. So we now had my desired situation of three odds and three evens, and the total had to be 51 more than the total of the six even numbers and it was safe to open the envelope.
Of course, it’s possible the “further randomisation” doesn’t do what you want, and you’re now looking at five odds and one even, at which point you need to request a final randomisation of two cards – but whenever I’ve done it one randomisation has been sufficient, and frequently the initial arrangement does the trick and you can open the envelope immediately.
Performing has always been part of teaching, and hamming up the amount of choice you’re giving the children not only disguises the fact that you’re actually controlling the situation, but should make the opening of the envelope both dramatic and amazing.
For my second session with Jenna and Den I used another of my long-time favourite number activities. It another one that’s very accessible but can make people think quite hard.
On each side of a card square write a number. You don’t actually have to use a different number on each side, and they don’t actually have to be whole numbers, but that’s what most people do. And on another piece of card do things similarly – again, you don’t have to use whole numbers, and they don’t actually have to be different from the ones on the first card, but that’s what usually happens.
Now toss the two cards as if they were coins, and add the two numbers you see. Do it again, and record the totals you see; do this until you’re satisfied you’re not going to get any new totals.
If you do this with a class some groups are likely to find they’ve made three different totals, and some will find they’ve got four. If they have two new pieces of card and number these, do they still get three (or four) totals? Can they discover how you ensure you always get three different totals, or four different totals?
With only a small number of children I may steer it in a different direction. Here’s the account I wrote up for school of what happened with Jenna and Den, including some false starts and blind alleys:
Today I asked them to devise two double-sided cards with different numbers on each face, so that the four possible totals they could display would give a set of consecutive numbers. Before long they found – not quite by accident, but not completely by design (Den had first decided that one card should be 0/1, and suggested 3/4 for the other) – the cards 0/1 and 2/4, which generate 0+2, 1+2, 0+4, 1+4 (i.e. 2,3,4,5).
I asked them to find a second set and Jenna offered 2/3 and 4/6, making 6,7,8,9.
I asked them to generalise from this and they suggested one card had to be even / odd and the other even / even, but it didn’t take long to find a counter-example, before Den came up with the correct suggestion that the numbers on one card should have a difference of 1, and on the other a difference of 2.
I asked them to explore the situation with three cards. They thought there would be six combinations, and Jenna suggested the cards would need differences of 1, 2, and 3. They used a logical process to derive each combination in turn, and both contributed equally. Having reached six they realised there would be eight possibilities, and they observed that each number appeared in four combinations and were able to use this to check they had a complete set. However, one of the totals in their set was repeated, and Jenna then suggested the cards needed to show differences of 1, 2, and 4 (rather than 1, 2, and 3).
They wanted to explore four cards, which Jenna suggested would need to display differences of 1, 2, 4, and 8. They quickly devised the set 2/3, 7/5, 4/8, and 9/1. Den thought there would be 12 combinations, but they again used their logical strategy for generating every combination, and so decided there would be 16. They found these with no slips, and found the 16 showed every total from 12 to 27, once each.
It was fun working with these two All I Can Throwers, but in some ways I prefer using Two Cards with students whose thinking isn’t so streamlined. Jenna and Den did offer a couple of suggestions which didn’t work out, but they immediately corrected them and got back on track, so they missed out on a lot of the red herrings that most people might experience. Incidentally, I was intrigued that in both the explorations they’d done so far neither of them had shown the slightest inclination to make notes or do any recording on paper.
Actually, this exploration was only half of what we did in the session, and I’ll tell you about the other activity in my next post.
Enough people have noticed the Consecutive Numbers piece to justify a follow-up.
Discovering just which numbers can’t be made by summing a set of consecutive positive whole numbers is such an elegant and surprising result it brings a smile to the face. On the way – and worthy of being an enquiry in its own right – is the key observation that while any set of three consecutive numbers sums to a multiple of 3, four consecutive numbers do not sum to a multiple of 4.
There are other spinoffs as well.
* Those sets which start with 1, e.g. 1, 2, 3, 4 sum to give the triangular numbers.
* What if you do the Consecutive Numbers enquiry with just the consecutive odd numbers? In this case, of course, those sets which start with 1 (e.g. 1, 3, 5) sum to give the square numbers.
* Which rather suggests that if you use numbers from the series 1, 4, 7, 10, 13, … you ought to find something interesting.
And that leads me to Roof Numbers, which you probably won’t have heard of. I was asked to work with some B.Ed. students, and in every respect except one I was given a totally free hand. So I was able to create a course built around exploratory maths, and a quite wonderful term it was too (I was asked to give them a second course the following year, and they insisted I was a guest to their graduation).
The fly in the ointment was the university’s requirement that I set a timed unseen written examination. But at least I got to set it, and Roof Numbers were my response. If the Consecutive Numbers question is a highly open problem, then Roof Numbers are hyper-open.
Here’s the problem:
Start with a bottom row of dots.
Above it, add a row which is three shorter than the bottom row.
Keep going till you feel like stopping, or until it’s impossible to carry on.
You have made a Roof Number.
o o o o o
o o o o o o o o
o o o o o o o o o o o
So 24 is a (level 3) Roof Number.
11 is a level 2 Roof Number:
o o o o
o o o o o o o
What can you find out about Roof Numbers?
I promised anyone who was desperate could buy a hint, but I knew perfectly well that anyone who’d spent a term doing problem-solving investigative mathematics would be able to spend their hour finding out interesting things, such as:
* are there numbers which are roof numbers in more than one way?
* are there numbers which cannot be made as roof numbers?
* what can you find out about level 3 (for example) roof numbers?
And since the step size of 3 is wholly arbitrary, you could just as well have roof numbers where the step size is 4, or 2 – and if the step size is 1, then you get the original Consecutive Numbers enquiry as simply a special example of Roof 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).
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.)
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.
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.
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:
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?
If you want some practical help, you can find circles with every number of dots up to 24 at http://nrich.maths.org/8506