# Super shapes (NRICH)

Stage 3 – A thinking mathematically targeted teaching opportunity focussed on reasoning and using additive and multiplicative strategies to find the value of shapes.

## 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, 2021

• MAO-WM-01
• MA3-AR-01
• MA3-RQF-01
• MA3-RQF-02

You will need:

## Watch

Watch Super shapes video (5:08).

Find the value of shapes using mathematical reasoning.

### Transcript of Super shapes

[Text over a navy-blue background: Super Shapes. Beneath this, text reads: From NRICH maths https://nrich.maths.org/1056. Small font text in the top left-hand corner reads: NSW Department of Education. In the lower left-hand corner is the red waratah of the NSW Government logo.]

Hi Barbara.

### Barbara

Hello Michelle. How are you today?

### Barbara

I'm very well.

[Two pieces of paper sit side by side in portrait orientation, the piece of paper on the right is light-blue and blank, the piece of paper on the left has the following text in a blue banner at the top of the page: Super Shapes. To the left of this, on the other end of the banner, is a sequence of progressively smaller blue squares forming a spiral, with the text:’ nrich’ to it’s right.

Beneath this is the following text: Each of the following shapes has a value: Beneath this text is a green triangle with the text: ‘= 7’ to it’s right, and an orange rectangle, with the text: ‘= 17’ to it’s right.

Beneath this is the following text: The value of the red shapes changes in each of the following problems. Can you discover its value in each problem below, if the values of the shapes are being added together?

Beneath this are 5 rows of assorted shapes labelled (a), (b), (c), (d) and (e). Row A contains a green triangle followed by a red rounded shape and an orange rectangle. To the right of these shapes is the text: = 25.

Row B contains an orange rectangle followed by two green triangles and a red rounded shape. To the right of these shapes is the text: = 51.

Row C contains 2 green triangles followed by 2 red pentagons and 2 orange rectangle. To the right of these shapes is the text: = 136.

Row D contains 3 red triangular shapes. To the right of these shapes is the text: = 48.

Row E contains a green triangle followed by a red circular shape, a green triangle, an orange rectangle, a green triangle, a red rounded shape and another green triangle. To the right of these shapes is the text: ‘= 100’.]

### Michelle

I have got a brain sweating problem for us that I've borrowed from NRICH maths.

Bring it on.

### Michelle

And I really liked it because it's got, instead of numbers, it's got shapes and colours and things and I thought it looked a bit interesting and intriguing.

(LAUGHTER)

### Barbara

Two things I like.

### Michelle

So do work together?

### Barbara

[Michelle points to the green triangle and then the orange rectangle at the top of the page.]

### Michelle

Well, I think what this is saying is that this particular shape, the green triangle, is worth seven and the orange rectangle is worth 17. That's its value.

### Barbara

OK, and then, that's interesting because we can use that information here.

[Barbara points at the row labelled (a).]

### Michelle

Yes, and I was thinking A could be a good place to start, but do you agree?

### Michelle

Alright, so this is how I was thinking was that I would start there because we know this is seven and that's 17. And so we just have to work out this one here.

[In row A, Michelle points to the green triangle then the orange rectangle then the red round shape.]

### Barbara

And do you think they're being added? Is that what's happening?

### Michelle

Yeah, I think so, because seven and 17, if you multiply it, it would be heaps more than 25.

[Michelle points to the green triangle then the orange square in row A, then to the text reading: = 25.]

### Barbara

Oh, you're right, of course.

### Michelle

But if you joined seven and 17, so from 17, three more is 20 and then four more would get to 24.

### Barbara

So this red shape must be one?

### Michelle

I think so. So I think A would be 7+1+17, which is 25, so that...

[Using a black texter, Michele writes: a. 7 + 1 + 17 = 25 on the blue piece of paper.]

That works.

### Michelle

Almost like a red pebble...

(LAUGHTER)

..red alien blob is 1.

[Using a red texter, Michele draws a round red shape above the number ‘1’.]

### Barbara

So is red always 1?

I think so...

OK, so...

### Michelle

Let's check with D. Oh, no.

[Michelle points to the three triangular red shapes making up row D and then to the text to the right of the row reading: = 48]

No, it can't be.

### Michelle

It can't because that would be 1+1+1=48.

### Barbara

That's crazy. OK, no... so it must be a combination of shape and colour. It gives you the value. So not just the shape and not just the colour, but together.

### Michelle

And look, like something like this one here where there's four green triangles.

[Michele points to the 4 green triangles and then the orange rectangle in row E.]

### Michelle

We know that's four sevens.

### Barbara

We do and then another 17.

### Michelle

One 17. So it's 28 and 17 more.

### Barbara

OK, so 28 and two more gives me 30 and then I need to add another 15. So it's 45.

[Barbara writes the following text on the blue page: e. 28 + 2 = 30. Beneath this she writes: 30 + 15 = 45.]

Is that correct?

### Michelle

Yes, I think so, because seven, that's the four triangles and then the two and the 15, yes...

[Michelle draws a line to a triangle from the number ‘28’.]

Yeah, OK.

### Michelle

The rectangle is 45 and so then subtract that from 100, makes 55.

[Michelle outlines the number 2 and the number 15, which are aligned vertically, and draws a rectangle above them. To the right, she then writes: = 55.]

That's right.

### Michelle

And then we'd have to half twice of those.

[Michelle points to the two round red shapes in row E.]

### Barbara

So it's actually got halves in it?

Must do.

### Barbara

OK, because I know that half of 50 is 25 and I know that half of five is two and a half.

### Michelle

Yes. So that would make it 27.5.

[Michelle draws two lines out from the bottom of the number ‘55’ and at the end of each of these lines writes: 27 ½.]

OK.

### Michelle

And 27.5.

[ On the blue piece of paper, beneath the numbers for row A, Michelle writes: e. 7+25 ½ + 7 + 17 + 7 + 27 ½ . ]

### Michelle

So that very messy scribbling would mean E is 7+25.5 +7+17+7+27.5.

[Michelle crosses the 5 in the first 25 ½ and writes 7 above it.]

### Michelle

Thanks for helping me revise, and plus seven more.

[Michelle adds ‘+ 7’ to the end of the equation.]

### Barbara

Wow, and that equals 100.

### Michelle

And that equals 100. Actually, even though this is very messy, I liked this better when we had four sevens.

Yeah.

Plus one 17 22

(CROSSTALK)

Two 27.5.

### Michelle

Like that.

[Beneath the previous equation, Michelle writes: 4 sevens + 1 seventeen + 2 x 27 ½.]

### Michelle

I got lazy like a mathematician then.

### Barbara

And that makes 100, equals to 100. There's actually quite a lot we can work out now, I think, because we now know what the triangle is, we know what the orange rectangle is, and we know what the skinny oval is. So we know quite a lot.

Oh, yeah.

### Barbara

Oh, and the blob.

And the blob.

### Barbara

And the blob, so we have got a few more things to work out.

### Michelle

And so this one here would be about dividing 48 by three.

[Michelle points to row D.]

### Barbara

OK, of course, because there's three equal parts.

### Michelle

Oh, I really like this.

Me too.

### Michelle

OK, over to you mathematicians.

[Text on a blue background reads: Over to you, mathematicians!]

### Michelle

OK, so what is the mathematics?

[Text on a white background reads:

* This task requires a LOT of mathematical reasoning. You have to analyse problems so you can think about using what you already know to solve what you don’t know yet.

* There is also a lot of computation as you use additive and multiplicative strategies to help you crack the codes!]

### Michelle

So in this task, it requires a lot of mathematical reasoning. You have to analyse problems, so you can think about using what you already know to solve what you don't know yet. And there's a lot of computation as you use your additive and multiplicative strategies with all of the operations. So addition, subtraction, multiplication, and division to help you crack each one of those codes. So we hope you enjoyed this challenge, Mathematicians. Have a great day.

## Instructions

• Can you discover the value of each of the shapes in each of the problems?