# Tangrams (investigating fractions) Stage 2

A thinking mathematically targeted teaching opportunity, focussed on creating and finding the fractional value of rectangles made from tangram pieces

## Syllabus

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

• MAO-WM-01
• MA2-2DS-01
• MA2-2DS-02

You will need:

## Watch

Watch Tangrams 2 part 1 video (1:18) to start exploring fractions with tangrams.

Use tangrams to create equivalent rectangles.

### Transcript for Tangrams 2 part 1 video

[White text on a navy-blue background reads ‘Tangrams 2 – part 1’. 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 are 5 triangles of various sizes, a square and a parallelogram. On the right, a pen sits on the lined page of a notebook folded back on itself.]

### Female speaker

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.

[The speaker arranges the square and 2 of the smaller triangles to form a rectangle.]

### Female speaker

But as I started playing with this, it got me thinking about something else. So, the first thing I do wanna do is think about what are some rectangles that I could make that are actually the same size as this one, using some, or all if I wanted, of my tangram pieces.

[The speaker uses the pen to draw the shapes on the blank notebook page.]

### Female speaker

So this one I have the square, and I have 2 triangles that are formed into the same size of the square, look, because if I cover that up, you can see they’re 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 a minute, mathematicians. And then, of course, record your thinking. OK.

[White text on a blue background reads ‘Over to you!’.]

[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]

## Instructions

• What other rectangles can you make of exactly the same dimensions?

## Watch

When you're ready, watch Tangrams 2 part 2 video (2:49).

Explore ways to create and record equivalent rectangles.

### Transcript for Tangrams 2 part 2 video

[White text on a navy-blue background reads ‘Tangrams 2 – part 2’. 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 are 5 triangles of various sizes, a square and a parallelogram. The square and 2 of the small triangles form a rectangle. On the right, a pen sits on the lined page of a notebook folded back on itself. It has a hand drawn square alongside 2 small triangles that form the rectangle on it.]

### Female speaker

Alright, mathematicians welcome back, did you find any others? Aha! So, well we knew this one, but you might have found this one here using the parallelogram. Oops! Had to turn it around.

[The speaker uses the parallelogram and a triangle on each end to form another rectangle shape. She then draws it on the notebook with the pen.]

### Female speaker

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 just very carefully trace the corners.

[The speaker uses a blue felt tip pen to outline the corners of the first rectangle shape. Further steps explained by speaker.]

### Female speaker

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, you can see that sits inside those same corners. Yes so, it's got the same dimensions. Nice, nice. Wanting proof, mathematicians, I like this a lot.

[The speaker uses 2 small triangles and a medium sized triangle to form a third rectangle.]

### Female speaker

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

[The speaker draws the third rectangle shape onto the notebook page with her pen.]

### Female speaker

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.

[The speaker uses a blue felt tip pen to draw the rectangle outline at the top of the notebook page. Further steps explained by speaker.]

### Female speaker

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?

[White text on a blue background reads ‘If the rectangle is defined as the whole, what’s the value of:

• the square ?
• the small triangle?
• the parallelogram?
• the medium triangle?

### Female speaker

Over to you mathematicians to work that out!

[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]

## Instructions

• If the rectangle is the whole, what’s the value of the square, the small triangle, the medium triangle and the parallelogram?

## Watch

Finally watch Tangrams 2 part 3 video (3:22).

Explore fractional representations of tangram pieces.

### Transcript for Tangrams 2 part 3 video

[White text on a navy-blue background reads ‘Tangrams 2 – part 3’. 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 are 5 triangles of various sizes, a square and a parallelogram. On the right, a pen sits on the lined page of a notebook folded back on itself. It has 3 hand drawn tangram shapes that form 3 different rectangles of the same size on it.]

### Female speaker

Alright, mathematicians, welcome back. How did you go? Yeah, I agree with you. Some of this was easier to work out than other parts.

[On the left, 2 of the small green triangles are positioned either side of a medium triangle to form a rectangle.]

### Female speaker

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 that same thing too.

[The speaker moves aside the shapes and places the square inside an area marked out by the corners of the earlier rectangle shape in blue felt tip pen.]

### Female speaker

Let's get 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.

[The speaker uses a pencil to mark a line down the middle of the rectangle outline. She shuffles the green square piece from one side to the other. Further steps explained by speaker.]

### Female speaker

So what that means is that when I put this triangle here, yes, it's now half of a 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'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 triangle. So when I have my triangles like this, yes, and I move this out, I slide that triangle across, and in come's my parallelogram, they are still the same triangles so they still have to be worth, oh, sorry, one quarter. 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 spaces left, and it's not taking over the rectangle.

[The speaker moves the parallelogram out and replaces it with the medium triangle alongside the 2 small triangles.]

### Female speaker

A-ha, and the same then for the triangle. Yes, because 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 1-quarter, and that must be worth 1-half.

But now what I'm curious about is how 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?

[Black text on a white background reads ‘Over to you mathematicians!’ Below, blue text (as read by speaker) and 4 labelled shapes – a blue rectangle, a black square, a medium triangle and a parallelogram.]

### Female speaker

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 1-half of the larger rectangle? How can we prove they all cover the same surface area? Off you go, have fun.

[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]

## Share/submit

• How can we prove that the medium triangle, the parallelogram and the square are all equal in area?