There were other things I would have done had time allowed. One is to create similar representations in pattern blocks, like those of Daniel Ruiz Aguilera, but simpler:

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Tres vistes (3) pic.twitter.com/B00TAg45mj
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— Daniel Ruiz Aguilera (@druizaguilera) May 9, 2014

I'm hoping that other people can also see that there's a cuboid outlined in the middle of the fold. It's a 3cm x 2cm x 6cm cuboid (albeit with top and bottom faces made of air). Here it is, made with the NCTM isometric drawing tool:

(I particularly like this cuboid, because it's central diagonal is a whole number. But that's not for my students.) |

I'll show another example with a

*cube*. Ask students to:

- make a smallish (< 8) cube out of interconnecting cm cubes;
- ask them to fold squared paper along a central line (scoring with scissors first might help);
- then make cuts away from that fold the size of the cube;
- score the new folds they need to make;
- fold in the cube;
- make annotations about how many squares there are on each face, what the total number of squares covering the cube would be and how many little cubes are needed to make this cube (lots of just-right calculations here).

*cuboid*like this and ask them whether it's possible to make a popup version of it. This is a little tricky because if all the dimensions are different you can't use that first central fold. Some experimentation would be essential. If someone finds the way - great! We'll go ahead and do roughly the same list of things I wrote for the cube.

The way of working that you are describing, which allows the students to travel back and forth between 2- and 3- dimensions, not once but multiple times has the same feel to it as going back and forth between equations and real objects The way the students are building with blocks, then transforming what they've built on to paper, then using tiles or cut paper shapes to "fill In" the drawing, then also creating a pop-up from the same shape, making sure to attend to concepts like area, well the whole sequence feels so exciting to me. I can't help but think that the students are absorbing some deep understanding of geometric thinking.

ReplyDeleteThe idea of also asking students, to reproduce, in pop-up form, a cuboid, is something that I've never even thought about. It's non-symmetric qualities does not lend itself to intuitive thinking. Just sitting down, making mistakes and keeping at it is the only way to go. It's the sort of thing that making mistakes is the only road to success. I hope you get a chance to try this out. I hope I get the chance to try this out.

So you made pop-up chicks too? You found my template for little chicks then? When I've done this with 6 year-olds I spend some time talking to students about making their chicks unique, giving them personality. I'll show them different, cartoon type, ways of drawing eyes which convey personality, ask them to choose feather colors, consider if they want to adorn their chicks with accessories. Each chick becomes a playful expression of th e student.. BTW, I've created a page on the side bar of my blog that lists all the PDFs I've happened to make, to make this sort of thing easier to find.

Thanks for writing about this way of continuing to work with shapes!

Thank you so much Paula, both for inspiration, and for this thoughtful and encouraging comment!

DeleteI did go with some of what I've mentioned here:

http://followinglearning.blogspot.fr/2015/04/popup-cuboids.html

I haven't printed out your PDF - but I like the idea of making the whole thing a lot more artistic! I would maybe go with multilink cubes and make bigger cuboids and then recreate them as 2D images with your coloured triangles.

The travelling back and forth seems to be a worthwhile model in teaching. It's a good model isn't it, and one that lots of us follow, to climb up and down the ladder of abstraction. I mentioned it here:

http://followinglearning.blogspot.fr/2014/09/the-ladder-of-abstraction.html

The only possible wrong step in all this is when I got them to annotate their cuboid fruit carton frames with both volume and surface area. I think this will be new terminology for some of them, and to learn two terms at once can be confusing ("oh, which one is it now?"). Even to be discussing volume and surface area without the terminology, as I did today ("How many little cubes?", "How many squares to cover your big cube?") is pushing it a bit, and something I generally avoid at this age. (Questions like what's the maximum area you can get for a given perimeter for instance seem to work better when the students are a year older. At eight and nine it seems to work better at first to deal with the ideas in separate months.)