Gordon Hamilton's Squaring and Subtracting is a great activity, and just right for 8 and 9 year olds. It gets them subtracting, and adding, but more than that reasoning about number relationships in a geometric context that means minimal explanation is necessary, language or symbols have a very small part.

I know that we want children to articulate their mathematical thinking, but it seems to me that part of building up number sense, or intuition about numbers is also working without symbols or language for a while (obviously the numbers themselves are symbols, but you know what I mean).

Another nice thing about these rectangles is they give their own feedback - a bit like with a jigsaw puzzle, where if you squeeze the wrong piece in, the rest doesn't work.

I used the worksheet PDF last time, which was great for giving different difficulties to different people.

The only problem I found was that it was over a bit too quickly! It felt too much like a race, and I want to slow it down.

So this time I think we should get out the Cuisenaire rods. Make squares that fit together first of all:

They could do this in small groups or pairs. Just keep on growing your squares. Can you limit the sizes?

Then, perhaps, can you make a rectangle out of squares?

Then I think it's time to come together, and look at one example. Say,

Adapted from here. |

And it's going to be, "What do you notice?' Think on your own. Tell a partner. Someone tell all of us.

And together we can fill it out, step by step.

Then, in pairs, have a go at solving lots, one of these:

(I think also, that making one of these in Cuisenaire rods would be worthwhile too. Just to be sure everyone has a really solid feel for what's going on.)

Then, I think is the time to use the PDF - with perhaps a choice of how hard you go?

And to finish, coming together for the story of finding squares packed in a

*square*. This one is the smallest possible with all the squares different:
As mathematicians were so pleased with this, it's been made in all sorts of materials:

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