Tangrams exploring halves
Stage 2 – A thinking mathematically targeted teaching opportunity, focussed on understanding fractions whilst exploring area and classifying 2D shapes
Syllabus outcomes and content descriptors from Mathematics K–10 Syllabus © NSW Education Standards Authority (NESA) for and on behalf of the Crown in right of the State of New South Wales, 2023
You will need:
- pencils or markers
- your mathematics workbook
- your tangram pieces (how to make a tangram).
Watch Tangrams exploring halves video to start thinking (5:50).
[White text on a navy-blue background reads ‘Tangrams 2 – part 4’. On the right, a blue half circle at the top and a red half circle at the bottom. In the middle bottom, a line of red dots forms another half circle. In the bottom left corner, a white NSW Government ‘waratah’ logo.]
[On a white desk, a sheet of pale blue paper on the left has green paper cut-out shapes spread across it. There is a square, a parallelogram and a medium triangle positioned near the corner outlines of a rectangle. On the right, a notebook folded back on itself has hand drawn tangram shapes on it and a handwritten question (as read by speaker).]
OK, mathematicians, how did you go in 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 used 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. Yeah, 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.
[The speaker brings in a red medium sized triangle paper cut-out. She places it on top of the green square.]
A-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 are like me too, you can visualise that this portion of this triangle that's hanging over looks like it might fit into there. Aha! Yes, or some of you are thinking, well, hold on a second, you could lay the square over one half of the triangle. Yeah, and if I fold it down over, aha!
[The speaker folds the green square diagonally over the red triangle.]
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, because look, if I put that section there and then move it around and that section there, it has the same area. Aha! Would you like me to cut it to show you? Let's see.
[The speaker uses some blue-handled scissors to cut the square in half diagonally. The cut pieces are arranged on top of the red triangle.]
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 onto 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 I'm gonna lay it over, look.
[The speaker lays the parallelogram on top of the red triangle and folds it in various ways (as explained by speaker).]
I have this portion here. Ah, 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. Oh, so you think I should go this way? OK, and then what? Aha, and then fold it, and what are you seeing now? 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 and we fold it, so it's a little bit better. Mm-hmm. So, this triangle covers half of the red triangle. This green triangle covers half of the red triangle because we folded our parallelogram in half so it looks like it makes a capital M. Mm-hmm. 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. Mm-hm. 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.
[The speaker uses the scissors again to cut the parallelogram in half. She positions the cut pieces onto the red triangle.]
Let's have a look. So, one half of my parallelogram and another half of my parallelogram, and voila! Isn't that amazing? Alright, mathematicians, we're about to give you another challenge. Get ready.
[White text on a blue background reads ‘What's (some of) the mathematics?’.]
So, what's some of the mathematics here?
[Black text and bullet points on a white background (as read by speaker). Below, 2 trapeziums of the same shape. The left trapezium is filled in black and the right trapezium is white with black outlines of a parallelogram, triangle, square and another triangle that form the shape of it.]
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.
[An additional bullet point on a white background (as read by speaker). Below, 3 black shapes – a square, triangle and parallelogram. Shapes within the shapes are highlighted in yellow and orange as mentioned by speaker.]
Yeah, and so, inside the bigger shapes there's smaller shapes. So, we can use this knowledge to help us prove that even though two shapes look different, they have the same area. Because inside a parallelogram there are 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.
[Black text on a white background reads ‘Your challenge…’ Below, a blue text question reads ‘What are the different ways you can show half using this rectangle?’. At the bottom, a solid blue rectangle on a grid sheet. Shapes within the rectangle are highlighted in yellow as mentioned by speaker.]
What are all the different ways you can show a 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 learnt today could be really useful. OK, have fun making, mathematicians.
[White text on a blue background reads ‘Have fun making’.]
[The NSW Government waratah logo turns briefly in the middle of various circles coloured blue, red, white and black. A copyright symbol and small blue text below it reads ‘State of New South Wales (Department of Education), 2021.’]
[End of transcript]
- What are all the different ways you can show half using this rectangle?
- Record your thinking in your workbook.