The Greeks, Geometry and Proof

As we begin the PYP, we're trying things out. We've got some initial ideas of our own, but we're looking for ideas from the students that we can follow up on. We want them to be genuinely following an inquiry that they're interested in, or better still, that they have originated. But what follows are some initial ideas that could serve as a series of six or so lessons.

Rosy and Rachel, my colleagues in Year 5, are planning a unit on myths, focussing on the Greeks. I'm so pleased that they're using Hugh Lupton and Daniel Morden's retelling of The Odyssey as a starting point. A few years back my class got a lot out of that book (see some products on that year's blog). If I did it again, I'd definitely use Peter Worley's The If Odyssey to get to some great philosophical themes in the Odyssey too.

But Rosy was thinking whether it might be a thing to link in with the geometry that the Greeks did in ancient times. She asked me about it, and here  are my thoughts about where I might start. I've not done these lessons before, but now I want to!

As I say, I haven't done this sequence before. What I'm trying to lead to is the idea of proof. I've pared down Euclid, to try and find the simplest possible starting points. But maybe you can think of an easier or better way of doing it? Maybe you've had some experience with this? Please post a comment!

I think I might introduce the topic with this image, asking, of course, What do you notice? What do you wonder? I might also tell the story of the Delian Problem.

An English illustration of a translation of an
Arabic translation of the ancient Greek Elements by Euclid
Source 

Lesson 1

I think I would introduce students to the tools the Greeks used for geometry – get the compasses and rulers out and make something artistic with circles.

Lesson 2

I would make things in GeoGebra too. Let’s do a little exploring of this brilliant tool!

(I like this too: http://sciencevsmagic.net/geo/ )

Lesson 3

I might start by showing this image:



And asking “What do you notice?” + “What do you wonder?”

Will someone say something to the effect that some of those angles are the same? Perhaps…

You could then write this up as eg. “Maria’s Claim”.

Lesson 4

Return to Maria’s proof. Could anyone say how her claim might be proved?

Show them Euclid’s book (I’ll lend it to you).

The Greeks developed the idea of proving something, of being completely sure.

Each proof in the book rests on other ones, like bricks in a house, going back down to the things Euclid said were obvious. (We worked on proving a little last year - example)

Here’s an example.



Look at it for a while together & see if anyone can see what’s going on in this proof.

(You could tell them that ∴ means “therefore”.)

Basically, the same thing is added to the two opposite angles giving the same sum = the angles on a straight line, 180°, so the two opposite angles must be the same. Hopefully some of the children will be able to see this and explain it. It really is best if most of the thinking can come from them.

It might be best for them to actually play with physical angles, arranging them – maybe they could do a video of how they know it’s true? Fraction circles would do the job well.

Lesson 5

What do children notice / wonder about this one?

 Hopefully, eg, that the two red angles are equal.

Perhaps they could have a big copy on paper to cut up & see if they can show that this is so.

(Could make a poster, or video, or just demo to class)

Lesson 6

Look at this image:

What do you notice? What do you wonder?

Here, there’s the idea that the angles inside a triangle add up to the angles on a straight line, 180°.

How might the students go towards proving this? What they’ve done with the previous two images might help. Another route would be to get everyone to make a different triangle on A4 paper. Colour the three corner angles different colours. Photocopy it so there are three copies altogether.

They could then cut the three out and arrange the triangles next to each other. If they put the three different-colour corners together, they should line up on a straight line. 

Odds and Evens

We had the idea that, it being Christmas, it might be the time to do some mathematical stars. And it was. (Click link to see details about the lesson.) We made them on paper, and we made them in Geogebra. And people noticed things! The same pattern was repeating with different numbers. And the pattern of repeats was interesting.

To give them another way of looking at the pattern of repeats, I showed them a multiplication square. We looked at the last digits in the multiples of 4 and how they are the same as the ones of 6 but backwards.

But then people started noticing other kinds of unrelated things.

How the second row is double the first row. How the bottom left is the same as the top right. How the square numbers in the red diagonal go odd, even, odd, even. Someone said that there are more even numbers than odd ones. Then T said that only odd numbers times odd numbers give odd numbers; the rest are even. I perhaps showed some sign of approval at this point; in any case T suggested that should go up on the mathematical claims board. Yes, it would, I said.

There seemed to be so much noticing going on, that the next day I gave them an image of what we'd discovered with the stars and also a blank multiplication square, and asked them to write about, and illustrate, what they'd noticed about one or both of these. I wanted to do this because I thought I would get a variety of responses, and get away from the idea that there was one thing I expected, towards the idea of their own individual directions being important. And they were able to describe and illustrate lots of noticings:

Today T's claim went up. I wondered whether they might be able to justify it - especially if they'd had some concrete experience fresh in their minds. So we borrowed the Numicon, which because it's lined up in two rows is great for odds and evens. We represented some multiplications and saw whether T's claim held true. 

.

They documented T's claim and our examples of it, but justifying it was generally a step too far. I got the sense that they intuitively sensed it from our practical work, but articulating the intuition was just too demanding. It's a class which demands more time in terms of dealing with emotions, behaviour and relationships, and perhaps the level of focus was just not there, but in any case perhaps I should have saved T's claim up for justification later, after we'd look at simpler cases, like the addition of odds and evens. I think it was worthwhile, even if we weren't able to round it off with a justification.

It was interesting that there were more things observed when we looked at T's claim together:

These could be followed up, but I sense it's time to move on, and perhaps return to the whole thing later, before anyone gets boggled.

Same difference again

Last school year I blogged about how children in year 4 were proving a generalisation about subtraction. It was time to try the same thing again with my new class. They took a little longer to get the task, but after a while all got to the stage of creating a set with a difference of three:

When I asked what they noticed, a few people saw that the set would continue forever:

But they didn't see mention how the pairs increment. So we left it at that.

When we'd earlier made equations about the number fifteen,

T had written this:

30 - 15 = 15

29 - 14 = 15

28 - 13 = 15

So I took it as my second chance, and a couple of days later showed the class the pattern. They got the idea quickly. What do you notice?

and

Everyone agreed with these generalisations and they went up on the wall.

I asked them if they were able to explain why these claims were true, using words and pictures. This year however they found it much harder to explain themselves, and I don't know why. Sure, the class has a different character. It's earlier in the year. But I'm puzzled that they found it harder to explain what they thought. Most managed to get something on paper, but it didn't seem to convey the general nature of the situation like it did last year.

Proof

I've been thinking lots about children articulating claims in maths lessons. Luckily, I'm not thinking alone.A claim came up,

S claim: U can add the same number to both #s in a sub prob and get the same difference.His representa. @MikeFlynn55 pic.twitter.com/GRmDUqG4f5

— Kristin Gray (@MathMinds) May 13, 2015

and naturally I turned to Cuisenaire rods:

@MathMinds @kassiaowedekind @ekazemi or Cuisenaire rods pic.twitter.com/KeiQlICv5L

— Simon Gregg (@Simon_Gregg) May 13, 2015

A lot more Twitter conversations happened. Tracy Zager was tweeting out lots of pointers. For instance:

@kassiaowedekind @Simon_Gregg @MathMinds @Zakchamp @MikeFlynn55 From Mason & Burton: pic.twitter.com/iIACvLMFok

— Tracy Johnston Zager (@TracyZager) May 21, 2015

Blogs were helping me think about it too. Kristin Gray's post The Meaning of Subtraction Kassia Omohundro Wedekind's blog post Counting, Conjectures, and Claims

On Twitter I got a great tip off that I've been following up:

@Simon_Gregg I like the practical, intuitive approach this enables. Caroline Ainsworth has done some great work on this - on NCETM website

— WellingtonPrepHead (@WellingtonPHead) May 17, 2015

So I went to Caroline Ainsworth's work on using Cuisenaire rods.

Of course I tried this approach with the Year 4s (I had the whole year group working in the two adjoining classrooms because my colleague was away) and you can see the results here on the Year 4 blog.

And Marie and Samyak made a claim, I'm pleased to say.

The next day when I reminded them about this claim (which was by then up on the new Claims Board) almost everyone agreed with it. So I asked them to show why it's true on paper, using pictures of the rods and words that would explain it to someone that didn't know about it. (I mentioned proof, but didn't stress this word.)

There were a few that didn't get very far. But there were a lot of good explanations, all different in their presentation:

Some of these explanations seem to amount to a visual proof for me. OK, they are not showing a generalised case for every difference, but taking Angiolina's example just above, she's adding the same thing to the two same left hand ends of the rods. Euclid had it as an axiom, and we know without articulating it that "if equals be added to equals the sums will be equal."

To me, the quality and individuality of the children's proofs justify this activity. They're returning to something very basic, but they're looking at it through more algebraic and logical eyes. And they're taking possession of their knowledge by experiencing it physically and articulating it in words and pictures.

Tracy pointed me towards Avery Pickford's great posts about proof. (I've been playing a 2-digit version of Mastermind with the class since reading them.)

 I've got lots of questions. Like:

Are these 'proofs'? Where does explanation end and proof begin? Which is the most valuable?

The understanding is the key thing; what does proof add?

How does the social dimension enhance this learning?

What is the place of un-articulated experience and learning in all this?

What other activities will lend themselves to this kind of thinking?

Developments

Since my last post about Cuisenaire Squares there have been developments.

One is that I did the lesson. I managed to resist temptation, and allow the students to explore on their own without any eye-glazing attempt to explain a proof or whatever. The only major intervention was to say that the ill-starred attempt of some of the boys to find a mathematical pattern in Minecraft Creeper faces with Cuisenaire rods wasn't going to work.

And it was worthwhile. There was all sorts of initiative (which is what I was wanting) and reflection following the square-filling, which led off in all sorts of directions.

The second development is that I've been thinking about Miles Berry's proof, and backsliding a little.
The problem that was nagging at me, was that his way of filling the hundred-square so that it can be (almost) filled with 4--rods with four different colours

or suchlike

isn't the only way.
There's going to be a kind of repeating latin square. For instance, taking this square:

and repeating it will make e a 10 x 10 square like the one below. Like Miles' square it has more of one colour than the other colours, so the 4-rods wouldn't work with it.

 But you could find a latin square like this:

which would give this satisfying 10 x 10 square, with twenty five of each:

So, such a colouring wouldn't disprove that the square can be filled by 4-rods.

I'm really not sure I've got to the bottom of this though, and would value someone else's thought.

The proof still stands for the position of the lone 1-rod in a hundred-square filled with 3-rods though:

and that's because there's only basically one way of doing the 3 x 3 latin square.

And the last development is that Helen J Williams has had a go at the same activity with her maths club too, with good results:

@Simon_Gregg @CuisenaireCo "see this gap it's because 30x3=90 &3x3=9 so you'll still have 1 left over." pic.twitter.com/xU7hS0VXWb
— Helen J Williams (@helenjwc) January 25, 2015

Fantastic!

Cuisenaire squares

I was wanting my Year 4 class to think and talk about the way odd and even numbers add, so I borrowed the numicon from Year 2.

When I was about to give it back, I noticed some square trays for Cuisenaire rods still in their bag.

Great! So we used them for exploring number patterns.

But I liked the frames a lot; I felt there must  be more to them. So, when Justus and Anabel had finished their work, I asked them to fill the hundred-frame with threes. Unfortunately it wasn't possible, so they added a one to try to cover up the nasty gap.

I, being the kind of teacher that assesses students' efforts carefully, was not fooled. "There's a one in the corner," I said. They must have a growth mindset, because they admitted it was true. "Can't you move the threes round so you don't have that white one?" I asked.

"No, there'll always be a one," said Justus. 

"How do you know?" I asked.

"Because a hundred isn't in the three times table," he said.

So, naturally, I tweeted about it later, and Miles Berry kindly replied:

@Simon_Gregg how do they know there'll always be a gap?
— Miles Berry (@mberry) January 15, 2015

I was beginning to feel a lesson idea coming on, so at home I played with nrich's Cuisenaire environment, this time with fours instead of threes:

Will there always be a gap with fours, @mberry? A hundred is divisible by four... pic.twitter.com/22bASoKkdK
— Simon Gregg (@Simon_Gregg) January 16, 2015

And he tweeted back with this wonderful visual proof:

@Simon_Gregg Yes. Nice question. pic.twitter.com/vGQfvfcIUB
— Miles Berry (@mberry) January 17, 2015

This proves that it's impossible: if you count up the four colours there aren't 25 of each. And yet every 4-rod has to cover one of each colour.

so there would need to be equal numbers of each colour for the square to be filled with them.

It's like the mutilated chess board problem. And a bit like the solution to the impossible peg solitaire problem.

So, all sorts of thoughts come now. Is there enough here for the students themselves to be presented with the rods and frames and asked to come up with their own lines of inquiry?

What about if there were two colours? Which combinations could fill and which couldn't?

What abut the question of where that last white one goes? Can  it go anywhere?

I check with Miles Berry's method... 

In the diagram below, each rod must cover an orange, a yellow and a pale green square. But there are 34 orange squares and only 33 of the yellows and pale greens. So the spare white one must be on a red squares, which can either be in the position on the left or in a position in the middle. Superimposing the two, it must be in one of the positions on the right:

This is so neat. I want to share it with the kids, or at least a simpler case. But then, no, my real hope is that there's enough in the materials that they can explore themselves, not follow. The kids could explore this from another entry point: trying with simple cases:

possible on the side?

Finding which combinations work and how, then documenting it would give plenty of arithmetic in a meaningful context, and hopefully develop those exploration muscles, more important.

Here's another question: if there are ones left over, how many could there be? That is, how many so that for example the ones are isolated?

I do this thing, you see, that perhaps we're not meant to do. I start with the materials and think, where could that go? Usually we're meant to start with the curriculum and work down to materials. But as I have a little more leeway than some, I can kind of make it up as I go along. Find wild places to explore. And for me, learning to explore is a lot of what the curriculum should be.

What do you think? Would it work as a lesson? Can you see other avenues that might be explored?

_______________________________________________________________________________

Later edit: interesting developents