Tangrams (exploring trapeziums)

Stage 2 and 3 – A thinking mathematically targeted teaching opportunity focussed on classifying and creating trapeziums using 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

• MAO-WM-01
• MA3-2DS-01
• MA3-2DS-03

You will need:

Watch

Watch Tangrams (exploring trapeziums) part 1 video (0:39) to get started.

Create examples of trapeziums using tangrams.

Transcript of Tangrams (exploring trapeziums) part 1

(Duration: 39 seconds)

[White text on a navy-blue background reads ‘Tangrams 1 – 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-outs 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.]

Speaker

Hi there, mathematicians. Welcome back. We hope you're having a really delightful day today. We are going to use our tangram pieces here to explore trapeziums. So before we get started, can you please write down in your notebook what you think a trapezium is? So we've got a Frayer model for you to fill out.

[A white form has blue text that reads ‘Adapted Frayer model’ at the top. Small text above it reads ‘NSW Department of Education’. Small black text below reads ‘A graphic organiser for building understanding’. A table has a black header that reads ‘TRAPEZIUMS’ and is divided into three sections ‘Examples’, ‘Non-examples’ and ‘Definition and features’.]

Speaker

What do you think are some examples of a trapezium, some non-examples of a trapezium, and what you think a definition might be. So 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

Use a Frayer chart to record your thinking.

Watch

When you're ready, watch Tangrams (exploring trapeziums) part 2 video (5:19).

Explore trapezium properties using tangrams.

Transcript of Tangrams (exploring trapeziums) part 2

[White text on a navy-blue background reads ‘Tangrams 1 – 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. On the right, a pen sits on the lined page of a notebook folded back on itself.]

Female speaker

Let's talk about what a trapezium is defined as.

[The speaker places a square of pale pink paper that has black handwritten text on it that reads ‘Trapezium: quadrilateral with at least one pair of parallel sides.’ The speaker starts to arrange the green paper shapes on the notebook page when mentioned.]

Female speaker

So a trapezium has this definition, that it's a quadrilateral with at least one pair of parallel sides. So, if I take this square, for example, it actually fits the definition of a trapezium, it's a quadrilateral, meaning it has 4 angles and 4 sides, and at least one pair is parallel because this side is parallel to this side, and actually this side is parallel to this side. So, a square is a special kind of trapezium.

And yes, that means I could also think a rectangle or an oblong is also a kind of trapezium.

[The speaker places 2 small triangles together to form a square. She then places the 2 shapes together to form a rectangle.]

Female speaker

Ah-ha because it's a quadrilateral that I've just made. It has 4 angles and 4 sides. And at least one pair is parallel. These 2 lines here run in parallel. And these lines here are also parallel. Yes. So your challenge today, mathematicians, is to think about how many trapeziums, using this definition here, can you make using 2 tangram pieces. So, it could be possible that our tangrams look different. So, let's try to make our, trapezium sorry, look different.

Let's try to think about making one. What are you thinking?

[The speaker arranges the parallelogram and a triangle together on the blue paper.]

Female speaker

OK, if I take the parallelogram and the triangle. Oh, yes, that forms a trapezium, doesn't it?

[The speaker uses her pen to draw and write on the blank notebook page. Further steps explained by speaker.]

Female speaker

OK, so here's one that I can make. I've got my triangle and my parallelogram. So a triangle and a parallelogram... is equivalent to or makes a trapezium. Yeah. That's actually a pretty cool idea.

What's another way I could make one?

[The speaker arranges further green shapes on the blue page (steps explained by speaker).]

Female speaker

Oh yeah, If I keep the parallelogram and use a triangle. What about this? Like that. Is it a quadrilateral? Let's check. 1, 2, 3, 4 angles. And 1, 2, 3, 4 sides. Is at least one pair of the lines parallel? Well, these ones aren't parallel, because if we continued them, they would intersect. But these ones. Yeah, so this is also a trapezium.

[The speaker uses her pen to draw and write on the notebook page.]

Female speaker

Yeah, so I have a triangle combined with my parallelogram. So a triangle... and a parallelogram is an equivalent shape to a trapezium or equivalent in definition. Yeah, actually look, they both use the triangle and the parallelogram. I wonder if I could make it another way without using the parallelogram.

What are you thinking?

[The speaker places a square on the left of a triangle to form a different quadrilateral shape.]

Female speaker

OK, like that. Oh, yes. Let's check. Has it got 4 angles? Yes. 1, 2, 3, 4. Does it have 4 sides? Yes. 1, 2, 3, 4. At least one pair of sides parallel? These ones are parallel, aren't they? Because if we continued them, they wouldn't intersect with these ones. We can imagine they would cross over about here. So this was another way.

[The speaker uses her pen to draw and write on the notebook page.]

Female speaker

I have a square. Yes, you're right, a trapezium plus a triangle is a trapezium. That's a really nice way of thinking about it. I'm going to call it a square because it's a special kind plus a triangle equals a trapezium.

Alright, mathematicians, it's now over to you. Your challenge is, we made trapeziums using just 2 of the shapes. Yeah, what I wonder is, can we make some trapeziums using 3 of our 7 shapes from our tangram? And then, of course, what about using 4 of them? 5 of them? 6 of them, and 7 of them?

[White text on a blue background reads ‘Make trapeziums using:

• 2 tangram pieces
• 3 tangram pieces
• 4 tangram pieces
• 5 tangram pieces
• 6 tangram pieces
• all 7 tangram pieces’ . ‘Record your creations.’]

Female speaker

Yes and there could be lots of different ways, as you're seeing. Alright, mathematicians, it's now over to you.

Have fun.

[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]

Instructions

How many different trapeziums can you make using your tangram pieces?

Make a trapezium using:

• 2 tangram pieces
• 3 tangram pieces
• 4 tangram pieces
• 5 tangram pieces
• 6 tangram pieces
• 7 tangram pieces

Watch

Watch Tangrams (exploring trapeziums) part 3 video (4:15).

Explore trapeziums created by different amounts of tangram pieces.

Transcript of Tangrams (exploring trapeziums) part 3

[White text on a navy-blue background reads ‘Tangrams 1 – 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 positioned to form a larger quadrilateral shape. On the right, a lined notebook folded back on itself has a handwritten table on it with 3 columns. From the left, the first column has a header that reads ‘no. of pieces’ with ‘2’ at the top to ‘7’ at the bottom; the second column ‘Trapezium’ has different hand-drawn trapezoid shapes made up of 2 to 7 different tangram pieces; and the third column ‘Composed of…’ has a handwritten outline of what shapes have been used for each trapezium.]

Female speaker

OK then, mathematicians, how did you go? Some of them really made me have to think hard, too, and I had to keep coming back to this idea of what a trapezium is defined as.

[The speaker places a square of pale pink paper that has black handwritten text on it that reads ‘Trapezium: quadrilateral with at least one pair of parallel sides.’]

Female speaker

That it's a quadrilateral, meaning it has the four angles, which also means it has four sides, and that at least one pair had to be parallel. Yeah. So, here is just one for each that I found. I found some others, but I put one in a table for you, mostly just because, you know, I love organising my information into a table 'cause it helps me as a mathematician to see my thinking and to share it with other people as well. And I kept my seventh one here just because I really liked it, and it took me a while to figure that out. But what I did start thinking about was keeping something consistent.

So, I don't know if you can see this, how I had the square and a triangle and then I kept that and added something to the side to see if it worked. And then I did the same here where I kept my square and two triangles and tried adding another shape to see if I could just build on to keep the trapezium, yeah.

And then sometimes that didn't work. So, I had to, like, move things around and rotate. But then in these other parts, you can see the square and the triangle and the parallelogram here stay the same. Yeah. And that triangle, there's a little core there that remains. And down on the seventh, it was just the square and the triangle. And did you notice this also with the shapes?

[White text on a blue background (read by speaker).]

Female speaker

That you can combine two dimensional shapes to form other shapes. Yeah. And that we can partition or break apart shapes to make other shapes. Just like how we can partition numbers to make other numbers. I thought this was a really cool thing that you can see a trapezium being made of a square and a triangle or a trapezium being made of a square and two triangles. Or that you could partition a trapezium to make a square into triangles. So cool.

So, mathematicians, back to you to your Frayer chart now to think about now that we've played around with this, what would you change?

[A white form has blue text that reads ‘Adapted Frayer model’ at the top. Small text above it reads ‘NSW Department of Education’. Small black text below reads ‘A graphic organiser for building understanding’. A table has a black header that reads ‘TRAPEZIUMS’ and is divided into three sections ‘Examples’, ‘Non-examples’ and ‘Definition and features’.]

Female speaker

So, what are some other examples or non-examples of trapeziums that you could include, and what would you refine or add on to your definition and characteristics?

Over to you. Oh, and before you go, mathematicians, you're right. I just took this away really quickly.

[The speaker replaces the pink handwritten note from earlier back onto the blue paper.]

Female speaker

But if I kept it there, actually, we could add another row into this table, you're right, because technically, we can make a trapezium with just one. We could have made it with a square. Aha. And we could also have made it with a parallelogram. Yes, because they fit this definition of a trapezium. So, let's quickly squeeze that in together.

[The speaker writes on the notebook above the ‘2 pieces’ row.]

Female speaker

Which one would you prefer to choose? The square or the parallelogram? The square, OK. And that is a square. Plus nothing else. Yes. Alright. Over to you, mathematicians. Nice pick up.

[White text on a blue background reads ‘Over to you!’. Further white text read by speaker.]

Female speaker

So, what's some of the mathematics we're seeing here today?

[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.]

Female speaker

Yeah, that you can combine two dimensional shapes to form other shapes, which means that you can also decompose or partition or break apart two dimensional shapes to form other shapes. Yes. And this reminds us of how numbers work, that inside bigger numbers are smaller numbers. Just like inside bigger shapes are smaller shapes.

[Below the 2 trapeziums, further black text (read by speaker) and a hand drawn petal shape that has 6 green circles and text above that reads ‘6 3 twos’.]

Female speaker

Yes. It reminds us of that task we looked at with the youcubed number visuals, where we could see things like inside of 6 are 3 x 2, inside the trapezium are 2 triangles, a parallelogram, and a square.

[Black text and a bullet point on a white background (as read by speaker). Below, 7 tangram shapes filled in black.]

Female speaker

We also saw that shapes can look different but have some important characteristics that allow them to be classified in the same way.

[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]