Tangrams (exploring quadrilaterals)
A thinking mathematically targeted teaching opportunity focussed on classifying and creating quadrilaterals using tangram pieces
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, 2021
You will need:
- pencil or marker
- adapted frayer chart (PDF 119 KB)
- your tangram pieces (how to make a tangram).
Watch Tangrams: exploring quadrilaterals part 1 video (0:58).
Hello there mathematicians. We hope you're having a really nice day today.
[Screens shows 6 green shapes. There are 2 large triangles of the same size, one medium triangle, one small triangle, one square and one parallelogram. Presenter moves the shapes around and places the 2 large triangles together making a square. They move the medium triangle and the small triangle to the right of the 2 large triangles. Presenter moves the square and parallelogram beneath the other shapes.]
We were playing around with our tangram pieces, and we started to wonder about something, because we noticed that there's 4 triangles. 2 of them are the same size and then there's a medium triangle and a small triangle.
And then there's a square and a parallelogram. There's actually two quadrilaterals.
And this made us start thinking about well what actually is a quadrilateral and how many of them can we make?
So, before we get started, your job mathematicians, is to fill in your Frayer chart, which is in your mathematics workbook, and define what you think a quadrilateral is.
[Presenter places a sticky note next to the shapes with the word ‘quadrilateral’ written on it and points to the word ‘quadrilateral’.]
So, what are some examples? What are some non examples? And then what are the characteristics?
[Screen shows a document called ‘Adapted Frayer model’, with subheading ‘A graphic organiser for building understanding’. Below the subheading is the word ‘Trapeziums’. The graphic organiser is divided into 3 sections for writing information, with two sections on the top and one section below. The top left section has the text ‘Examples (draw, write or describe some examples)’. The top right section has the text ‘Non-examples (draw, write or describe some non-examples. Ask yourself: “What isn’t it?”)’ The bottom section has the text ‘Definition and features (draw, write or describe a definition and some really important features.)’]
Over to you.
[End of transcript]
Use the adapted Frayer chart to define a quadrilateral.
Watch Tangrams: exploring quadrilaterals part 2 video (4:35).
Okay mathematicians, welcome back.
So, we're going to use this definition of a quadrilateral.
[Screen shows a workbook open at the first page, and a sticky note which reads: a shape with 4 straight sides and 4 vertices/corners. There are also 6 shapes: 2 large triangles, one medium triangle, one small triangle, one small square and a parallelogram.]
That it's a shape with 4 straight sides and 4 vertices or corners.
So, so a square fits this definition because it has 4 sides.
[Presenter picks up the square and points to the writing].
Look, 1, 2, 3, 4, they're all straight and it has 4 corners and 4 internal angles.
[Presenter traces the outline of the square, points to its 4 corners and its 4 internal angles.]
So, that's actually a quadrilateral.
[Presenter places it onto the right page of the workbook.]
And so is a parallelogram.
Look, 1, 2, 3, 4 sides, 1, 2, 3, 4 corners or vertices and 4 internal angles.
[Presenter points to the parallelogram and traces the 4 sides, the 4 corners/vertices and the 4 internal angles and places it underneath the square.]
So, but now I started wondering well what other quadrilaterals can I make with my tangram?
[Presenter moves the parallelogram and square back over to the left.]
And I thought well, is this a quadrilateral?
[Presenter joins the square and the smaller triangle together by putting the right angle of the triangle against the right side of the square.]
Aha, look, let's check, 1, 2, 3, 3, 4 sides. 1, 2, 3, 4 vertices or corners.
[The presenter now traces around the 4 sides and then traces the vertices or corners.]
So, then I started to wonder, mathematicians, about what are all the different shapes I could make?
So, if I have, yes, using some or all of my tangram pieces.
[Presenter creates a new heading in the notebook which says ‘places’. Underneath her heading she writes the number 1, and them draws an image of a square. She then draws an image of a parallelogram. She labels each shape with their names on the right of the drawings.]
So, using one tangram piece I can make a square and I could also make a parallelogram.
Oops, well they were already made, mm-hmm, a square and a parallelogram.
But what about if I have 2 pieces of my tangram? What would we call this quadrilateral?
Mm-hmm, yeah, it's a trapezium.
So, I could make a trapezium and I will draw it like this so I can see the square and the triangle or look I could even write it like this square plus triangle equals trapezium.
[Presenter draws a line underneath the 2 drawings and writes the number 2 under the heading ‘pieces’. On the left, she pushes the square and the triangle together to create a trapezium. She draws an image of the trapezium, ensuring that the square and triangle shapes can be identified, and next to it writes: square plus triangle equals trapezium.]
I know or I could just label it as a trapezium. Mm-hmm, but it's still also a quadrilateral because it fits our definition.
Uh-huh. And then, what if I did this?
[Presenter now joins a parallelogram and the small triangle together. She traces along the 4 sides and the 4 corners or vertices.]
Does it have 4 sides? Yeah, 1, 2, 3, 4.
Does it have 4 vertices? 4 corners? Mm, 1, 2, 3, 4.
So, this is also a quadrilateral.
[Presenter draws a picture of the quadrilateral that she’s made into her table and labels it as a trapezium.]
Yeah, and it's actually also a trapezium, and I could label it like that if I wanted to. Is there another 4-sided shape I could make with 2 pieces? Ahh, yes.
Look, I could use 2 of my triangles to make, aha, a square.
[Presenter joins the two larger triangles together, making a square. She draws a representation of this underneath her previous drawing, making sure that the triangle shapes can be seen. She labels this image as a square.]
Oh, this is cool.
So now I have a square.
Oh, and if I turn it, if I rotate things, oh, no, oh that makes another triangle.
Oh, I see, like this, oh, now it has, aha, 1, 2, 3, 4 sides and 1, 2, 3, 4 vertices so I've made another parallelogram using my triangles.
[Presenter rotates the triangles and forms a parallelogram, which she draws into her table and labels.]
But it's also using just 2 pieces.
Mm-hmm, so mathematicians here is your challenge, parallelogram, is how many different quadrilaterals can you make using the pieces of your tangram?
And can you actually make at least 1 for 1 piece, 2 pieces, 3 pieces, 4 pieces, 5 pieces, 6 pieces and all 7 pieces?
Once you've had a go at that, then come back to your Freya chart and think about revisiting it.
Would you add any more information or revise your thinking?
Over to you.
[End of transcript]
- How many different quadrilaterals can you make using your tangram pieces?
- Try making at least 1 quadrilateral with:
- 1 tangram piece
- 2 tangram pieces
- 3 tangram pieces
- 4 tangram pieces
- 5 tangram pieces
- 6 tangram pieces
- 7 tangram pieces
- Record your thinking in your workbook.