Square of Area 10

This is another one from my PGCE; all those years ago now, but continues to be a favourite little investigation. I take no originality credit! I'm posting in response to @mathsatschool comment on #mathchat that students don't accept that "diamond"s are squares.

I say little, it can be anything from 1 lesson to a week or fortnight's worth.

Resources: Square spotty paper and lots of it.

1. Draw a square of area 1. Easy enough to start. I get peculiar looks at this stage...
2. Draw a sqaure of area 4. Have a look around, you get some 16's at this stage.
3. Draw a square of area 9. They are starting to think either "This is going to be easy" or "He's up to something" depending on how well they know me.
4. Draw a sqaure of ares ... wait for it (they'll be storming ahead with 16 if they're anything like my usual lot).... draw a square of area 10.

At this point you tend to get the stunned silence, followed by some frantic scribbling. Some clever so and so will root 10, and try to do that. "Exactly 10" usually stops them.

It's worth a play, but here's my offering - and indeed I usually have to show them the answer...

I find I need to show them the 'proof' of its area. My usual method is the Big Square around the outside, take away the triangles....

A = 16 - 4 * (0.5 * 3 * 1) = 10.

I find having a notation for labeling squares handy. This one I'd call 3_1. (To move long an edge you go along 3, down 1, etc).

So - what squares can you draw? Area 20? Area 14? Are there any you cannot do?

Whats great is that you can check their answers really quickly. You'll either see the answer really instantly, or you'll do the algebra with square x_y, and go "Oh - yeah".

Enjoy. Good for Shape / Area work, and for Algebra.