# Tangrams (investigating fractions) Stage 3

A thinking mathematically targeted teaching opportunity, focussed on finding the fractional value of tangram pieces used to create a square

## 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

## Outcomes

- MAO-WM-01
- MA3-RQF-01
- MA3-RQF-02
- MA3-2DS-01
- MA3-2DS-03

## Collect resources

You will need:

- pencils or markers
- your mathematics workbook
- your tangram pieces (how to make a tangram).

## Watch

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

[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?
- Record your thinking in your workbook.

## Watch

When you're ready, watch Tangrams 3 part 1 video (1:35).

[White text on a navy-blue background reads ‘Tangrams 3 – 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 positioned to form a large square. There are 5 triangles of various sizes, a square and a parallelogram. On the right there is a lined page of a notebook folded back on itself.]

### Speaker

Hello there, mathematicians, welcome back. We have a tangram challenge for you. We were wondering something about our tangram, because as we cut it up and created it, we realised that we were partitioning our tangram, and making them into parts. And some of the parts are equal sized, like these 2 triangles are the same size. But some of the parts are smaller, but it all came from the same square originally.

[The speaker uses a pen to draw the tangram square onto the notebook page.]

So what we were wondering is if the square of the rectangle, the square of the tangram, which all the pieces inside of it partitioned, like about this. Oops, it's a bit wonky. If this whole thing is worth one, what are all of the other pieces worth? A-ha, this represents one whole. So what that means is, what's the value of the 2 large triangles?

[Black text on a blue background reads ‘Your challenge…’ Below, blue text reads ‘If all 7 tangram pieces formed into a square has a value of 1, what’s the value of the individual pieces?

- the large triangle?
- the medium triangle?
- the parallelogram?
- the square?
- the small triangle?]

What's the value of the medium triangle? What's the value of the parallelogram? What's the value of the two small triangles? And what's the value of the square? Over to you, mathematicians.

[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

If all the 7 tangram pieces formed into a square has a value of 1 whole, what’s the value of the individual pieces?

- The large triangle?
- The medium triangle
- The parallelogram?
- The square?
- The small triangle?

## Watch

Watch Tangrams 3 part 2 video (12:38).

[A title over a navy-blue background: Tangrams 3 – part 2. Small font text in the upper left-hand corner reads: NSW Department of Education. In the lower left-hand corner is the white waratah of the NSW Government logo.

On a blue paper is a lined notebook with a red square paper on it and some text. On the left of the notebook is a cut-up green square paper.]

### Speaker

Welcome back, mathematicians. How did you go? I agree with you, some of them made my brain sweat a little bit more than some of the others. Shall we talk about it together? OK, good. So what I found really useful was to think about how I built my tangram…

[She holds the red paper.]

…to help me think about, if this is the whole…

[She traces the shape of the red paper. Then she places it over the green paper.]

…then what is the value of each of my tangram puzzle pieces?

[She points to the shapes on the green paper.]

And so I thought about what did I do to my whole to make the puzzle pieces to start with? So can you remember? Yeah, the first move was that we halved our whole…

[She aligns the top right corner of the red paper to the bottom left corner. She folds the paper.]

…that's right. So we now have two large triangles…

[She opens the paper.]

…and each of these is one half of the whole.

[She takes a scissor and cuts at the fold.]

And I'll cut it out so that we can keep manipulating it. And as I'm cutting it out, what was the next move that we made when we made our tangrams?

[She holds up the triangle on the right.]

Yeah, we partitioned this triangle into two equal triangles, our large triangles. So if this is a half…

[She aligns the bottom right corner of the triangle to the top left corner. She folds the paper.]

…and when I half my half, I end up with quarters, yes.

[She opens the paper.]

So a half of a half is a quarter.

[She takes a scissor and cuts at the fold.]

### Speaker

I know, and that seems weird because I go from writing something that looks like this…

[On top of the notebook page, she writes: ½. Next to it, she writes: ¼.]

…to something that looks like this.

And usually in whole numbers, four means more than two. And in this case it does too, it means I have more pieces, but the pieces are getting smaller. Look, here's a half…

[She holds up a larger triangle.]

…and here's a quarter…

[She holds up a smaller triangle. She moves the larger triangle up and down.]

…and this is definitely bigger. Whereas in whole numbers two is smaller than four, and so sometimes yeah, we can get ourselves muddled mathematically if we use whole number thinking when we're working with fractions. So, this then means it's a quarter, so let's just check…

[She brings a yellow square paper.]

…I've got another square of paper and I've got a different colour, so it's easier to see. And this was our half…

[She places the larger triangle on the yellow square.]

…that we halved again, see?

[She places the smaller triangle on the left side of the larger one.]

And so, if that was there like this…

[She places the triangles over the left side of the yellow square.]

…I can now say I've halved it to make quarters…

[She removes the larger triangle, leaving the smaller triangle.]

…and I would need four of them. Look one, if I rotate it…

[She turns the triangle clockwise until its side is aligned with the square.]

…two…

[She turns the triangle clockwise until its side is aligned with the square two more times as she counts.]

… three and four, and it's the same size over here.

[She places the triangle over the top shape in the green square.]

### Speaker

So the large triangle is worth one-quarter.

[On the triangle, she writes: 1 quarter.]

And symbolically, we would write that like this.

[On the triangle, she writes: 1/4. She gets the other smaller triangle.]

So this is also worth one-quarter.

[On the triangle, she writes: 1 quarter 1/4.]

Yeah, so the large triangle, we have worked that out, I can put this over here actually, if I like.

[She places one triangle over the top shape in the green square and the other triangle over the left shape.]

So now, I still have this other half left.

[She takes the larger triangle. She places it over the left side of the yellow square.]

And actually this whole half is partitioned into…

[She points to the shapes of the green square.]

…one, two, three triangles, a square and a parallelogram. And can you remember how we made this?

[She points to the red triangle.]

So let's see if this will help us work it out.

[She folds the triangle in half.]

So, we did a little mark down here to show halfway and…

[She opens it. She takes the bottom point to the foldmark and folds.]

…we folded this down…

[She opens it up. She cuts at the fold.]

…and we cut this out like this. Now, this one isn't so obvious, I don't think. Whereas, this…

[She points to the quarter triangle.]

…was easy because when you, not easy but less brain sweaty, because when you halve a half, you end up with quarters. But we're not entirely sure what we did…

[She picks up the larger piece of the cut-up triangle shapes.]

…to this piece now, except that we made a triangle into a trapezium and a triangle.

[She places the smaller piece of the cut-up triangle above the larger piece.]

### Speaker

And so if I lay this on here…

[She places the smaller piece of the cut-up triangle over the left side of the larger piece.]

…though, can you see this? I'll just use one of another colour…

[She places a green triangle over the red one.]

…so it's the same size triangle…

[She moves the red triangle offscreen.]

…so that piece fits there. And, yeah, like you, I'm imagining now this spinning around in my brain, and I'm using my imagination. And if I spin this around, I think another triangle would fit there.

[She points to the middle section of the larger red piece.]

And then I think there'd be a third one here.

[She points to the right section of the larger red piece.]

Should we try it? So let's spin that around like that.

[She turns the green triangle to the right until it’s bottom side aligns with the larger red piece.]

### Speaker

Yeah, and actually now we can see for sure because these are the same size and area and so is this one. So what happened actually when I cut that tip off? Yes, I've now quartered my half, which means I made eighths. Yes, because for each half I cut it into four pieces. And so one half has four pieces and the other half has four pieces, which is eight pieces all together. So this is now one-eighth…

[She takes back the smaller red triangle and on it she writes: 1 eighth.]

…and that's this piece over here.

[She places the triangle over the bottom left of the green shapes.]

And you're right, I could check that by laying it over this quarter, let's check.

[She takes the top red quarter and places it over the yellow square.]

Because if I have two eighths, that's the same as one quarter. So, let's use this one…

[She picks up a green eighth triangle.]

…'cause it's a different colour.

[She places it over the right side of the quarter triangle.]

So, that's one eighth over the top of a quarter. And if I doubled it…

[She flips it over to the left.]

…I'd have two eighths, which is a quarter.

[She places the quarter triangle back over the green shapes.]

So that gives me confidence that my claim is accurate, I've got some nice proof.

[She lays the eighth red triangle over the bottom left of the green shapes.]

And now what I'm going to do with this other part…

[She picks up the larger piece of the red triangle.]

…which at the moment is three eighths worth, isn't it? Yes, 'cause we had the triangle fit there, another triangle here and another one there, see?

[She places a green triangle over the middle section of the larger piece.]

### Speaker

So one eighth, two eighths and three eighths more. But now with my three eighths, can you remember what we did? We partitioned it into half…

[She folds it in half.]

…so I've now halved my three eighths.

[She opens the paper and cuts at the fold.]

Yes, this part does make your brain sweaty, it's good, our brains are growing. And we now halved…

[She places the cut-up pieces over the yellow square.]

…our trapezium to end up with two smaller trapeziums that are now both half of three eighths, I know. But then we kept partitioning it, didn't we? And we folded this little edge over here…

[She aligns the bottom right point of the trapezium to the bottom left point. She folds the paper.]

…to make a little triangle and that gave us a triangle and a square.

[She opens the paper and cuts at the fold.]

So, yes, I was thinking the same thing as you, I used my eighth as a measure to help me work these out. Look, here's my eighth here…

[She takes the red and the green eighth triangles and brings them together.]

…and I'll use the green one, I'll just prove to you they're the same size to help me show that if I take this piece…

[She picks up the small triangle.]

…and lay it on here, like this..

[She lays it over the left side of the green triangle.]

…what do you notice? It's half of the eighth. And if I halve an eighth, I get a sixteenth.

[She places the pieces down. On the red triangle, she write: 1 sixteenth 1/16.]

### Speaker

So this is one sixteenth now. And I can use that actually to work out this…

[She places the triangle over the left side of the square.]

…look, because if I lay this over here that covers half the surface area of my square.

And if I flip it over or rotate it, it covers the other half. So that means that this…

[She points to the square.]

…is two sixteenths, which is an eighth, one eighth.

[On the square she writes: 1 eighth 1/8. She moves the square and triangle to the top of the yellow square.]

So now we know actually that half of three eighths is one eighth and one sixteenth more.

[She picks up the trapezium.]

Yes, you're right.

[She folds in the bottom right side of the trapezium.]

So if I do this action now to make my parallelogram and my triangles, I'm decomposing my trapezium now into a parallelogram and a triangle.

[She opens the paper and cuts at the fold.]

And these were the same size to start with, remember?

[She picks up the square and triangle above. She puts down the triangle and places the square on its right.]

This looked like this, so they are the exact same shape. So this triangle…

[She points to the cut triangle from the trapezium then the 1 sixteenth triangle.]

…is the same area as that triangle, which means it's one sixteenth, one sixteenth.

[On the triangle, she writes: 1 sixteenth 1/16.]

### Speaker

Which means that this parallelogram must also be one eighth, I agree with you.

[On the parallelogram, she writes: 1 eighth 1/8.]

Because they occupy the exact same area or same portion of our original square, even though they look really weird. So, that means that over here…

[She places the 1 sixteenth triangle over the green triangle in the middle of the green shapes.]

…this is a sixteenth, this is an eighth…

[She places the parallelogram over the green parallelogram.]

…this is a sixteenth…

[She places the other 1 sixteenth triangle over the green triangle on the left of the green shapes. She places the 1 eighth square over the green square.]

…and this is an eighth. Wow, isn't that really cool? Yeah, and now mathematicians, after all those sweaty brains, I really want to draw your attention to these guys.

[She moves the eighth triangle, the parallelogram and square back onto the yellow square.]

So here's my triangle, here's the parallelogram, and here is the square. Now, these guys look really different as shapes, don't they? But they, we think, are all the same portion of our large square, meaning they all cover the same surface area in this instance. And now what I'm wondering is over to you, mathematicians. Can you think of a way to prove that these are, in fact, all exactly the same or cover the exact same surface area of our original square? Over to you mathematicians.

[Text over a blue background: What’s (some of) the mathematics?]

Before we get to your challenge, let's talk about some of the mathematics today.

[A title on a white background reads: What’s (some of) the mathematics?

Below the title is an image of the cut-up green squares. Next to the image is text: When you have a half…]

### Speaker

So here's our tangram and it revealed some really cool things for us.

[The large triangle on the right is replaced by a blue triangle with text: 1-half.]

That when you halve a half…

[The blue triangle is replaced by red and yellow triangles with text: 1-half of 1-half 1 quarter 1/4.

On the right side, text with bullet points appear:

When you halve a half, you create quarters

- 1-half of 1-half is 1 quarter
- ½ of ½ = ¼

This helps us realise 2 important fractional ideas:

- As the number of parts we break something into gets bigger, the size of the parts get smaller
- Fractional numbers don’t always work the same as whole numbers.]

…you create quarters. So one-half of one half is one quarter. And this helps us realise two really important fractional ideas. One, that as the number of parts we break something into gets bigger, the size of the parts get smaller. And that fractional numbers don't always work the same as whole numbers. And these are two really, critical ideas.

Yes, for us to have discovered today, so thanks to the tangram for that.

[The previous shapes and text are cleared. The bottom left triangle is replaced by a pink triangle with text: 1-quarter of 1-half is 1-eighth.

On the right side, text with a bullet point appears:

We can use other fractional parts as measures. We worked out this section with 1-quarter of 1-half, which is equivalent to 1-eighth.

- ¼ of ½ is 1/8.]

### Speaker

We also realised that we can use other fractional parts as measures. So we worked out this section with one quarter of one half, which is equivalent to an eighth.

[The middle triangle has been replaced by a yellow triangle with text: ?

On the right side, text appears: To work out the yellow section:

And we then used it to work out the yellow section…

[The yellow triangle moves to next to the pink triangle.

On the right side, text appears: we can compare it to the pink section (the 1-eighth). Since it is 1-half of 1-eighth, the yellow part is 1-sixteenth.]

…we laid it over the top and since a half of an eighth is a sixteenth, the yellow part must be equivalent to…

[The middle triangle has been replaced by a yellow triangle with text: 1-sixteenth.]

…one sixteenth of the whole area of our original square.

[Text over a blue background: Now, for your challenge…]

So now for your challenge, mathematicians.

[A title on a white background reads: Over to you, mathematicians!

Text below reads: How can we prove the medium triangle, the parallelogram and the square are all equal in area (they are all 1-eight of the original square)?

Below the text is a row of 3 shapes: on the left is a square with text below that reads: square; in the middle is an upside-down triangle with text below that reads: medium triangle; on the right is a parallelogram with text below that reads: parallelogram.]

How can we prove the medium triangle, the parallelogram and the square are all equal in area? And in fact, they're all an eighth of our original tangram puzzle. Over to you.

[Text over a blue background: Over to you!

Over a grey background, the red waratah of the NSW Government logo appears amongst red, white and blue circles. Text: Copyright State of New South Wales (Department of Education), 2021.]

[End of transcript]

## Discussion

- How can we prove that the medium triangle, the parallelogram and the square are all equal in area?
- Record your thinking in your workbook.

## Watch

Finally, watch this Tangrams 3 part 3 video (5:25) to continue exploring.

[A title over a navy-blue background: Tangrams 3 – part 3. Small font text in the upper left-hand corner reads: NSW Department of Education. In the lower left-hand corner is the white waratah of the NSW Government logo.

On a sheet of butcher’s paper are some green shapes on the left and a blank sheet of paper on the right.]

### Speaker

Hello there mathematicians, welcome back. How did you go with our challenge? Yes, there are, in fact, a few strategies that you could have used. One would have been to use another piece…

[The speaker brings in a red triangle.]

…of our Tangram as a measure. So, remember, we made another one in red.

[She brings other red shapes in.]

Yes, and that was my triangle. There was at the top that I can not even remember how to put back together. There it is.

[She places two red triangles with writing: 1 quarter ¼ on them together.]

And then from there, yes, I know this.

[She puts the shapes together to form a red square.]

It turns out to have a parallelogram, and the portion, and the triangle here, and a square, and the other triangle. And we had worked out that these two smaller triangles…

[She points to 2 triangles with writing: 1 sixteenth on them.]

…were equivalent in area to one-sixteenth of the entirety of our square. So, I could actually use this…

[She takes 1 of the 1-sixteenth triangles and sets aside the rest of the shapes.]

…as a measure with these shapes over here…

[She moves the green shapes.]

…because what I can see in my mind is that this square…

[She places the green square on the paper and lays the red triangle on it.]

…is half the area of this triangle. So, if this is one-sixteenth, and if I spin it…

[She turns the triangle over to the other side.]

### Speaker

…and this is one-sixteenth, then I have two one-sixteenths.

[She draws a line from the bottom left point to the top right point of the square, forming 2 triangles. In each of the triangles, she writes: 1/16.]

So, that's one-sixteenth and one-sixteenth, and together…

[She turns the square over and writes: 2/16 = 1/8.]

…that's one-eighth or two sixteenths.

So, that's one way I can think about it.

[She picks up the 1/16 triangle.]

And then I could use the same unit of measure - my one-sixteenth. So, if I take my parallelogram now…

[She places the parallelogram onto the paper and lays the triangle on top.]

…and if I line it up, I can see the same thing then. That would be one-sixteenth, and that would be…

[She turns the triangle until it aligned with the other side of the parallelogram.]

…the second sixteenth. And so that would also be two sixteenths, which is equivalent to one-eighth.

[She turns the parallelogram over and writes: 2/16 = 1/8.]

So, even though they look really different, they have the same area, they're both one-eighth of our original square. And I could use the same unit of measure for my triangle.

[She places the green triangle onto the paper and lays the red triangle on top.]

So, that's one-sixteenth, and if I spin it around, another sixteenth.

[She turns the red triangle until it aligned with the other side of the green triangle.]

And so, two-sixteenths is equivalent to one-eighth.

[She turns the triangle over and writes: 2/16 = 1/8.]

So, that's one strategy I could have used.

[She sets the red triangle aside. She pulls down the green shapes closer to her.]

### Speaker

Another strategy I could have used is to use my square as a measure, for example, or in fact, any of my other shapes. But with the square, what I could do is decompose my other shapes and recompose them as a square to prove that they're equivalent. So, I think I have a square. Yes, I do. I'll use this one…

[She picks up a red 1-eighth square.]

…as my measure just because it's easier for you to see.

And I'm gonna think about my parallelogram…

[She picks up the parallelogram.]

…because if I lay that over the top of my square, what I can actually imagine here in my mind's eye is this portion…

[She points to the area of the parallelogram over the side of the square.]

…that's overhanging here. If that was cut off and rotated around, then that would cover the same surface area. And I could use a similar strategy for the triangle, where if I covered it like this…

[She places the green triangle over the square.]

…I can also imagine this portion being cut off, and flicking over. And that's another way I could prove. But let's do it. Let's chop this guy in half.

[She cuts at the fold.]

### Speaker

So, now my one-eighth becomes two-sixteenths.

[She places a triangle on the bottom side of the square.]

And if I place one-sixteenth here, and then this part that I imagined…

[She places the other triangle on the top side of the square.]

…yes, it's mandatory to make those sounds. Goes over here, I can see, huh, that is equivalent to the area of the square. And I could do the same with the parallelogram. Look, if I do this…

[She places the parallelogram on the left side of the square.]

…and then this was the portion we imagined chopping, so let's chop it.

[She cuts at the fold.]

Let's decompose our parallelogram into two triangles.

[She places a triangle on the both sides of the square.]

One goes there, and this section here that I imagined sliding up to there.

And so that's how we can prove, another way we can prove, that they all have the same area. They're all one-eighth of a portion of our large Tangram puzzle. Alright, let's get ready for your next investigation.

[Text over a blue background: What’s (some of) the mathematics?]

So, what's some of the mathematics here?

[A title on a white background reads: What’s (some of) the mathematics?

Below the title is an image of the cut-up green squares with the middle triangle replaced with a yellow one with text: 1-sixteenth. Next to the image is text: We can use other fractional parts as measures. We used 1-sixteenth to prove the square, parallelogram and medium triangle are all 2-sixteenths (1-eighth), the area of the original square.]

### Speaker

So, we realised that we can use other fractional parts as measures. So, we use the one-sixteenth to prove the square, parallelogram and medium triangle are all two sixteenths which is equivalent to one-eighth, the area of the original square.

[The 2 bottom triangles are outlined yellow.]

So, there we see two of the sixteenths…

[The 2 bottom triangles turn white with text: 2-sixteenths = 1-eighth.

The square turns into 2 triangles outlined yellow.]

…to be one-eighth, another two of the sixteenths in the square - one-eighth.

[Text appears in the square: 2-sixteenths = 1-eighth.

The medium triangle turns into 2 triangles outlined yellow.]

And another two sixteenths to be the area of the medium triangle.

[The 2 triangle turn blue and text appears: 2-sixteenths = 1-eighth.

Nice work today mathematicians.

[Over a grey background, the red waratah of the NSW Government logo appears amongst red, white and blue circles. Text: Copyright State of New South Wales (Department of Education), 2021.]

[End of transcript.]