Tangrams 3

Ok mathematicians, how did you go proving that question? So I wrote it down here for us. How can I prove the triangle, the square, oh and the parallelogram, not the rectangle.

I know it's really great that mathematicians revise and edit their work all the time. All have the same area. Ah, I, I use that too, a strategy of folding and cutting. So to help you guys see it, because what I was thinking about is when I lay my square over the top of my triangle, I can see that it overlaps. Yes, so there's these bits here. These two small triangles here overlap, so I can't just use direct comparison and here, I've got these two small triangles that don't have anything.

Ah ha, yes, and actually. I'm going to use a different coloured triangle. You can see it's the same size. So that you can see that more clearly, look. But yes, if I turn that over, I know some of you like me too, you can visualise that this portion of this triangle it's hanging over looks like it might fit into there.

Ah ha, yes or yes, some of you are thinking well hold on a second, you could lay the the square over one half of the triangle. Yeah, and if I fold it down over. Ah ha. If it covers half of the area, the surface area of the triangle on this side, and half of the surface area of the triangle on this side, it must cover the whole surface area of one face, one side.

Yes, 'cause look if I put that section there and then move it around and that section there. It has the same area, uh-huh. Would you like me to cut it to show you? Let's see. So now if I lay this down here, I can prove that. It's the same area.

Yeah, and look I can now put it back on to here and it's the same area and if I had sticky tape I could reform it back into my square.

Ok, let's deal with the parallelogram and I'm going to use the triangle for the same base and this time when I lay it over, look, I have this portion, ahhh, what are you thinking with this part? Well, if I fold that over. Look what happens. I still have a bit here. And that triangle doesn't look like, it looks too big, doesn't it? To fit there and it looks too big to fit there.

Ah, so what? Do you think I should go this way? Ok, and then what? Ah ha, and then fold it? And what are you seeing now? Ah, that this portion covers half but this, oh, slide it. Oh yeah, do you want to see that again? Look, if we turn it over, let me fold it so it's a little bit better.

So this triangle covers half of the red triangle. This green triangle covers half of the red triangle 'cause we folded our parallelogram, in half so it looks like it makes like a capital M and this half of the red triangle is covered by this triangle. And if I slide it across. Yes, this half of the triangle, red triangle is covered by the green triangle.

So you're saying that because this half is covered and this back half is covered. That would cover the whole of the surface area. You'd like to cut to see? Let's check. Let's have a look.

So one half of my parallelogram and another half of my parallelogram and voi-la. Isn't that amazing?

Alright, mathematicians we're about to give you another challenge. Get ready.

So what some of the mathematics here? Yes, so remember we saw that you can combine two-dimensional shapes to form other shapes and that you can also decompose two-dimensional shapes into other shapes. Yeah, and so inside the biggest shapes there's small shapes. So we can use this knowledge to help us prove that even though two shapes look different they have the same area. 'Cause inside a parallelogram there's two smaller triangles, the same as the medium triangle and in fact the same as the square. They are just orientated differently in space. Yes, this is a great strategy to help us prove.

Ok mathematicians, here's your challenge. What are all the different ways you can show half using this rectangle? Could look like this, could look like this. And don't forget you're going to have to be able to prove your thinking so some of those things that we learned today could be really useful. Ok, have fun making mathematicians.