Showing posts with label patterns. Show all posts
Showing posts with label patterns. Show all posts

Sunday, 12 July 2015

#tiling again

Well, John Golden has posted about his pattern block explorations, so I will too. Along with Dani Ruiz Aguilera he should accept a little of the "blame" for me spending so much time shuffling small shapes around. Dani has been doing some industrial-scale pattern block workshops himself:
I don't know whether I should put this post here or over on seekecho.blogspot.fr. I put my own things over there usually, including some of the things I've done with pattern blocks. But, although I've kind of "got the bug" with this, hopefully there are spin-offs for the classroom, like Teflon in space travel and the kitchen.

I've posted about some of the work I did in the class with a particular set of pattern blocks, the square, the hexagon and the thin white rhombus, already.

We've used Math Toybox's great Pattern Block tool - and a great thing about this is that you can save your creations onto the gallery, and even edit them again later on. John used it to respond to what we'd done in class:
I liked it that there were some interesting rules for these line patterns:
We looked at that and one or two other things, and then I asked them to work in bigger groups to extend one of them:
Meanwhile, something was intriguing me: a non-periodic, almost free-form, tiling using these shapes:
It has a really interesting balance between being forced to put the tiles in certain places, and having some freedom. To start with, if you begin with a hexagon, there are a lot of different ways of surrounding it. Here are some of them:
Similarly, there are a lot of ways of surrounding a square (finding them all might be an interesting task). You can see I've restricted myself. I never put two rhombuses against each other. Or two squares. I've got a lot of questions about this, and here are some of them:
  1. It feels like it would carry on forever in all sorts of ways. Can we prove it does?
  2. Once there's a "line" of squares and rhombuses, the line won't go away. What are the rules for it's behaviour?
  3. I've got an intuitive feel for an algorithm for making it carry on. Can we write down an algorithm for making sure it continues to grow without getting into any "impossible" situations?
  4. It feels like this could make a good game. What would the rules be?
  5. You can get areas of different regular tessellation - how many kinds of these are possible? 
  6. Can we - it feels like we can - create writing or pictures with this?
  7. Can I have a large public space, St Mark's Square in Venice for instance, to tile in this way? Or failing that, a playground?
Meanwhile, I was exploring other patterns:
Another thing that caught my attention was dodecagons.
Those holes can hold a dodecagon:
These can "point" in one of twelve directions. The idea of embedding these pointers and maybe other dodecagons, in the tessellatin grabbed me, and for a while I thought about how numbers can be patterned in this kind of space.
Here are some rotating dodecagons embedded in a simpler matrix:
John meanwhile had created his own beauty:
(At the same time, I've been reading Roger Penrose on, among other things, his work on non-periodic tiling. I guess I'm thinking about those 17 wallpaper groups, and trying to push other symmetries embedded within a simple translation.)

On a more practical note, I saw that you can get pattern block stickers!
These would really help the students recording a creation directly into their books. And something they could do this for - why hadn't I thought of it before? - popped into my head: pattern block equations:
 As Mary Pardoe tweeted:
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Saturday, 17 January 2015

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:
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:
And he tweeted back with this wonderful visual proof:

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?

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Later edit: interesting developents
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