Tangrams 2

Alright mathematicians. Welcome back. I started thinking about we could now do a bit of a challenge to play around with making special kinds of rectangles, but as I started playing with this, it got me thinking about something else.

So the first thing I do want to do is think about what are some rectangles I could make that are actually the same size as this one. Using some or all, if I wanted, of my tangram pieces so this one I have the square. And I have two triangles that are formed into the same size of the square. Look, because if I cover that up you can see they are forming the same area.

So what I wondered is what other rectangles can I make of exactly those same dimensions using just my tangram pieces?

Over to you for minute mathematicians. And then of course, record your thinking. Ok.

Alright mathematicians, welcome back. Did you find any others?

Ah ha, so you, well we knew this one but you might have found this one here using the parallelogram. Oops, I'll just turn it around. Yeah. And yes, if you're sceptical you could use a measure, a ruler to measure, to check. Oh, you'd like me to prove it? I can do that ready? Look, if I actually, that's a very good idea. If I do this here like that.

And what I'll do is this, very carefully trace the corners. And I'm just doing them a little bit outside. So that you can see, where they fit. Ok, so they're the corners of my rectangle and if I use my parallelogram, mhm you can see that, that sits inside those same corners. Yes, so it's got the same dimensions. Nice, nice wanting proof mathematicians. I like this a lot.

Yes, and the other one involved the medium triangle. If I move these ones around and bring this guy down it goes into here like this, look. Yeah, so I can draw this one also.

Yes, so this now started to get me really interested. Yes, so it really got me wondering about if my rectangle here around the outside. If that's defined as one or one whole. Then what's the value of my square? My small triangle? The parallelogram? And the medium triangle?

Over to you mathematicians to work that out.

Alright mathematicians, welcome back. How did you go?

Yeah, I agree with you. Some of this was easier to workout than other parts and I think I found these two triangles the easiest to go, yeah I know what they're worth. Yes, and some of you guys were thinking the same thing too. Let's go back to our first one here, because if this is the boundary of our rectangle, then the square here, yes is half its area, so it's a fraction. The fraction is one half and I can prove that by, if I actually just flip it over very carefully.

Yes, I can see this and if I wanted to I could draw a little line here. To prove that both sides, so what that means is that when I put this triangle here. Yes, it's now half of half and a half of a half is equivalent to a quarter. So we know that our square is worth one half. And the small rectangles are worth one quarter.

But then we're wondering about, well, what's the value of the parallelogram and the triangle? Yeah, and you, you're right, we can use reasoning because if we know if this is worth one half. And that's worth one half. They are the same triangles. So when I have my triangles like this. Yes, and I move this out. I slide that triangle across and in comes my parallelogram.

They are still the same triangles, so they have to be worth, oh sorry, one quarter. One quarter. Nice pick up. One quarter and one quarter. Yes, and so this has to be worth a half because this square is worth one half, so that must be worth a half as well, because there's no space is left and it's not taking over the rectangle. Ah ha, and the same then for the triangle.

Yes, 'cause all that happened. Yeah, was I rotated and moved the small triangle and the big triangle came in. So that's worth one quarter. That's worth one quarter and that must be worth one half. But now what I'm curious about is, these shapes look very different, but they have the same area, they cover the same surface of our larger rectangle. But how could we prove that they have the same area? That they're all in fact half of our larger rectangle?

That sounds like an investigation to me mathematicians. Over to you. How can we prove that the medium triangle, the parallelogram and the square are all one half of the larger rectangle? How can we prove they all cover the same surface area?

Off you go. Have fun.