# Making mandalas – Stages 2 and 3

Stage 2 and 3 – A thinking mathematically targeted teaching opportunity focussed on developing number sense and early additive strategies

## 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
- MA2-2DS-02

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

## Collect resources

You will need:

- a piece of ribbon or string
- assorted objects to make the mandala (for example pegs, leaves, sticks, flower petals, rocks, small stones).

## Watch

Watch Making mandalas stage 2 video (10:01).

[White text over a navy-blue background: Making Mandalas. MathXplosion – Go Fly a Kite! In the lower right-hand corner is a red waratah logo of the NSW Government. In small font in the lower left-hand corner is the text: NSW Mathematics Strategy Professional Learning team (NSWMS PL team).]

### Speaker

Hey there, mathematicians, exploring symmetry in kites was fun, and it started to make me curious about symmetry in other geometric designs like mandalas. Let's get our brains sweaty and explore this further.

[New slide: Text: You will need…

- A piece of string or ribbon
- Assorted materials to make your mandala (for example pegs, small rocks, stones, leaves, flower petals etc)

Don’t forget to ask for permission before using natural materials around your home!]

### Speaker

You will need a piece of string or ribbon and assorted materials to make your mandala. For example, pegs, small rocks, stones, leaves, or flower petals. Don't forget to ask for permission before using natural materials from around your home.

[New slide. Text: Let’s play!]

### Speaker

Let's play.

[A pile of yellow, blue and red pegs, leaves, small rocks and a blue ribbon lie on the edges of a piece of butcher’s paper.]

### Speaker

As I start to think about my mandala design, I'm going to use what I know about symmetry to help make sure that it's a symmetrical design.

[The speaker takes a red peg and places it so that the open end of the peg is pointed towards the top.]

### Speaker

I'm going to start with a red peg. Red is my favourite colour. Here is the top part of my peg, and here's the bottom part of my peg. Now, I would like my design to have a line of symmetry that runs from side to side, meaning that the top half is a mirror image of the bottom half. And I got this blue piece of ribbon here today to help my mathematical imagination.

[She places the blue ribbon horizontally across the butchers paper just below the closed end of the peg.]

### Speaker

If I place it down like this. This is really going to help me.

[The speaker picks up another red peg and flips it, demonstrating that the peg is identical on both sides.]

### Speaker

Now, the great thing about pegs is that this side is the same as this side. And that's really helpful.

[The speaker places the second peg directly on top of the first peg. It is identical.]

### Speaker

If I place the peg here on top and overlay it, that's going to help me to see how I can make the top half become the mirror image of the bottom half.

[She places the second peg in the same position as the first peg below the blue ribbon with the open end pointing upwards.]

### Speaker

I could try translating my peg down like this. But that's not symmetrical.

[She places the second peg back on top of the first peg.]

### Speaker

Let's try a different approach. This time, if I reflect my peg along this bottom edge.

[She flips the second peg 180 degrees so that the closed part of the peg faces upwards. The second peg is below the blue ribbon, creating a mirror image of the peg above the line of the blue ribbon.]

### Speaker

Now, the top half of my design is a mirror image of the bottom part. Great. Let's use that strategy again.

[She takes two more red pegs and places one above the peg which is above the blue ribbon and one beneath the peg which is beneath the blue ribbon. There are now two pegs on either side of the blue ribbon, both facing the same way, to create a mirror image.]

### Speaker

That thinking as. I do another red peg. That is great.

[She removes the blue ribbon.]

### Speaker

I'm gonna move this out of the way as I think about the next part of my design.

[The speaker takes a blue peg and adds it to the right of the red pegs, on a 45 degree angle from the centre point so that the open end of the peg points towards the top right corner.]

### Speaker

I think I'm going to add in a blue peg on this side like this.

[She places the blue ribbon on top of the vertical line of four red pegs.]

### Speaker

Now I'm going to use my blue ribbon to help me think about the line of symmetry that I want to run top to bottom of my design.

[The speaker indicates to the left of the blue ribbon, where there is no blue peg.]

### Speaker

I know that I'll need a blue peg on this part. And I also know that it needs to be the mirror image.

[She places a second blue peg on top of the existing blue peg. It is identical.]

### Speaker

Let's try placing this blue peg on top.

[She places the second blue peg on the left of the blue ribbon, but below the centre line so that the blue pegs run diagonally pointing to the top right.]

### Speaker

If I were to translate the blue peg again, it's not quite in the right position, and it's not the mirror image that I'm after. So I need to use something else.

[The places the peg on the left of the blue ribbon so that it runs parallel with the red pegs.]

### Speaker

I could try reflecting the peg along this edge this time. Great. Almost.

[The speaker rotates the peg slightly to the left so that it is a mirror image of the blue peg on the right-hand side of the ribbon. The open part of the peg on the left-hand side of the ribbon is facing the top left corner.]

### Speaker

Good idea, now I just need to rotate it like this. You're right. And this side becomes the mirror image of that side.

[She takes another blue peg and places it on the right of the blue ribbon, but below the other blue peg. The open part of the peg points to the bottom right corner.]

### Speaker

Let's do that again down the bottom.

[The speaker tries to place a fourth blue peg on the left hand side of the ribbon where the open part of the peg also faces to the bottom right corner. It is not a mirror image.]

### Speaker

If I place my next blue peg here like this, we know that translating isn't going to be quite right. No, it doesn't quite work, does it?

[The speaker flips the peg from its position on top of the peg facing the bottom right corner. The fourth blue peg is now on the left side of the blue ribbon with the open part of the peg facing the bottom left corner.]

### Speaker

Let's try the thinking of reflecting along this edge, and then rotating it a little bit so that it's in the right position.

[The speaker places the blue ribbon along the horizontal line, demonstrating that two blue pegs lie above the line and two blue pegs below it. There is now an image of four red pegs and four blue pegs. The red pegs above the horizontal line are placed one above the other with open end pointing to the top edge and the blue pegs above the horizontal line are a on a 45 degree angle to the left and the right of the red pegs, with the open part of the pegs pointing to the top left and top right corners respectively. This layout is flipped for the four begs below the horizontal line creating a mirror image.]

### Speaker

Great. Now we can use my blue ribbon, and we can also check this line of symmetry. And the top part of my design is a mirror image of the bottom part of my design. That's great. I think I'm going to add in some leaves next. Now I'm going to use some direct comparison to check that my leaves are about the same size.

[The speaker places two leaves on top of each other. They are similar sizes.]

### Speaker

It's a little bit tricky with natural materials. I think these ones are a good match though.

[The speaker shows one side of the leaves is a bright green with dark stripes. She flips the leaves to show a darker, purplish hue with less defined stripes. She also shows that on one end of the leaves is a stem and one end is a pointed tip.]

### Speaker

Do you notice how my leaves have the stripy side and then they also have this purple side? That's going to be something I need to consider as I place them in the right position. Something else that's really important about these leaves is that they have this tip at one end and then they have their little stem at the other. This is really important as I start to place them in my design.

[The speaker places a leaf on the horizontal line between the two blue pegs on the right-hand side of the vertical line of red pegs. She has placed the leaf green side up with the stem pointing into the centre of the mandala. The pointed tip of the leaf points to the right edge.]

### Speaker

As I put my first leaf down, I can see that the stem is closest to the red and the blue pegs here and that this little tip of the leaf is facing outwards.

[The speaker places the second leaf in the same spot on the left hand side of the vertical line, but with the pointed tip facing the centre and the stem pointing to the left edge.]

### Speaker

Now, if I was to translate the leaf into position, it's a little bit tricky because it looks like it's symmetrical. I have stripes on this side. I have stripes on this side. But when you look more closely, the closest part to the pegs is not the stem side like on this side. So translating doesn't quite work.

[The speaker flips the leaf so that the stem points into the centre, but on the purplish side of the leaf.]

### Speaker

Let's try reflecting this one along its stem edge. Oh, no, look at that. Now, the stem part of the leaf is in the right spot, but now I have a purple side, and that's not symmetrical either. Now my design is asymmetrical. What else could we try?

[The rotates the leaf 180 degrees so that the leaf is still on the green side and the stem points inwards. The speaker uses the blue ribbon on the vertical line to demonstrate that the mandala is a mirror image.]

### Speaker

Great idea. Let's rotate a full half turn. And look at that. Now I can use my blue ribbon to check. We have this side is a mirror image of this side.

[She moves the blue ribbon to the horizontal line to demonstrate the symmetry.]

### Speaker

And if we move this in our mathematical imagination, thinks about this line of symmetry. I can see that this half of my design is a mirror image of this half of my design.

[She picks up a handful of yellow pegs.]

### Speaker

Great, I'm going to keep going. I think I'm going to use some bright yellow pegs this time.

[She places a yellow peg on the diagonal line above the blue peg with the open end facing the top right corner. The speaker takes the blue ribbon and places it on the vertical line and points to the right hand side, which includes a yellow peg pointing to the top right corner, and then the left hand side which does not have a yellow peg.]

### Speaker

Now, as I place the first peg down like this, I'm going to get my ribbon again. Can you think about how I can make sure that this side is a mirror image of that side?

[She picks up another yellow peg and places it on top of the existing yellow peg.]

### Speaker

I really liked the strategy I used before overlaying the peg.

[She translates the second yellow peg to the left-hand side of the vertical line so that the open ends of the pegs are facing in the same direction. It is not a mirror image.]

### Speaker

Now let's check if we translate. Not quite right. OK.

[She flips and rotates the yellow peg so that it is placed above the blue peg with the open end pointing towards the top left corner. It is now a mirror image along the vertical line.]

### Speaker

Let's see what happens if we reflect the peg along this edge. OK. And if I rotate it like we did last time. You're right. Now it's in the right spot.

[She places the blue ribbon on the horizontal line, showing the yellow pegs on the top half but not on the bottom half.]

### Speaker

I can do that along the bottom as well because good thing you've noticed if we were to think about the line of symmetry like this at the moment, the top half is not a mirror image of the bottom half.

[The speaker adds yellow pegs below the blue pegs in the bottom half so that the entire image is symmetrical on the vertical and horizontal line.]

### Speaker

That's right. We need those yellow pegs. So if I place this peg down in here and I reflect along this side. And then I can rotate it into where it goes. Now my design is symmetrical once more.

[She removes the blue ribbon.]

### Speaker

I'm almost done I think. I need to add a few more things in though. I'm going to add in some of my little stones.

[The speaker places a small brown stone in the open end of each of the yellow pegs.]

### Speaker

I'm gonna put one brown stone in each of the openings of the yellow pegs just like this. Great. Now I know that my design is symmetrical because each yellow peg has a brown stone.

[She adds two grey stones on either side of each of the brown stones, except for the top left quadrant where she only adds one grey stone.]

### Speaker

And to finish it off, I think I'm going to add in two grey stones just like this. OK. That's great.

[She places the blue ribbon on the horizontal line. She points out that the top left quadrant is missing a grey stone.]

### Speaker

Let's get that blue piece of ribbon to help our mathematical imaginations again. I place it down like this. Let's have a look at the line of symmetry. Running side to side is the top half and mirror in each of the bottom half? Oh, that's right. I've noticed that, too. I need to add one more grey stone over here. Oh, now I know that they are symmetrical because each yellow peg has two grey stones and one brown stone.

[She moves the blue ribbon to the vertical line, demonstrating that both sides are symmetrical.]

### Speaker

Great. And let's check this part. This line of symmetry. Is this side a mirror image of that side? You're right. They are. That's great. I had so much fun making my mathematical mandala. I hope you do, too.

[White text on a blue background: What’s (some of) the mathematics?]

### Speaker

Let's have a look at what some of the mathematics is in this activity.

[New slide. Text:

- We can use our mathematical imagination to help us explore lines of symmetry.
- We can make symmetrical designs by translating, reflecting and rotating materials.
- Mathematicians can describe two-dimensional designs as symmetrical, when they have one or more lines of symmetry, which means one side of the design is a mirror image of the other side.]

### Speaker

We can use our mathematical imagination to help us explore lines of symmetry. We can make symmetrical designs by translating, reflecting, and rotating materials. Mathematicians can describe two-dimensional designs as symmetrical when they have one or more lines of symmetry, which means one side of the design is a mirror image of the other side. Now it's over to you to create your own mandalas. I wonder how many lines of symmetry you'll be able to find in your design, or if you can create a design that won't have any lines of symmetry. It's time to explore.

[New slide. Text: Over to you!

New slide. The NSW Government logo flashes on screen. Text below reads: Copyright, State of New South Wales (Department of Education), 2021.]

[End of transcript]

## Instructions

- Collect assorted objects from around your house. You will need to collect a pair of each object (same colour, same size, same shape). Remember to ask permission before using natural materials.

- Select a pair of objects (two objects that are the same colour, size and shape) and place one object down as your starting point.
- Create your own mandala using the objects you have collected by rotating, translating and reflecting the objects.
- You can check the lines of symmetry using a ribbon to see if each side is a mirror image.

## Follow up

- Can you make a mandala that has no lines of symmetry or is asymmetrical? How can you prove this?
- What is a line of symmetry?
- How many can you see in your Mandala?
- Could you make more lines of symmetry if you moved or added and other objects?

## Collect resources

You will need:

- your mandala design
- 1 cm grid paper (PDF 91 KB)
- coloured markers
- a ruler.

## Watch

Watch Making mandalas stage 3 video (6:57).

[Blue text on a white background: Plot your mandala on the Cartesian plane! In the bottom right corner is the red waratah logo of the NSW government. In the top left corner is the small font text: NSW Department of Education

An image of a mandala appears below the text. The mandala is composed of a vertical line of 4 red pegs and 2 diagonal lines of a blue peg and a yellow peg, where the blue peg is closer to the centre. Two leaves form a horizontal line. There is a grey rock and 2 brown rocks in the open end of each of the 4 yellow pegs.

Dotted lines make a vertical and horizontal line in the mandala image.

Beside the mandala image, another image appears of a graph with an x and a y axis. The x axis runs horizontally from -6 to 6 and the y axis runs vertically from 6 to -6. The axes intersect at 0.]

### Speaker

Mathematicians, our lines of symmetry in our mandala design are beginning to remind me of the Cartesian plane. Look, our design has four quadrants, and so does the Cartesian plane.

[The cartesian plane image disappears. In its place, another copy of the mandala image appears with grid paper superimposed on it.]

### Speaker

We can use our mathematical imaginations like this, to help us plot the coordinates of our design into the full quadrants of the plane.

[New slide. Text: Plot your mandala on the Cartesian plane!

Both mandala images disappear. An image of blank grid paper appears and an image of a ruler and 6 coloured markers.]

### Speaker

To do this, you will need, 1cm grid paper like the one I am using, a ruler and markers to record the coordinates.

[The mandala image appears again with bold white lines running vertically and horizontally. An image of blank grid paper appears beside the mandala.]

### Speaker

Just like the two lines of symmetry that we found in our design, this can help us to draw our axes of the coordinate system. First, I need to find the middle of my grid paper, so that each of my four quadrants are equal.

[The right most column on the grid paper is outlined in yellow. It consists of 14 squares.

The yellow outline changes to be filled in and covering only the first 7 squares of the right column. A red arrow points from the mandala to the yellow line.]

### Speaker

I want to start with my horizontal line first. I know that I have 14cm squares vertically, so I am able to halve this and find the middle point where I can start drawing my horizontal axis.

[The yellow fill disappears, and a red line runs horizontally to divide the grid paper in half. There are 7 squares in the columns above the red line, and 7 squares in the columns below.]

### Speaker

Like this. Now I can use the same technique of halving to locate where I will begin my vertical axis.

[A red line appears, running vertically down the grid paper to divide the paper evenly into quadrants. Each quadrant is 7 squares by 7 squares.]

### Speaker

Like this. Now we have created two intersecting lines at right angles, which will form the axes of our coordinate system.

[The text: x-axis, appears beside the horizontal line.

The text: y-axis, appears beside the vertical line.]

### Speaker

Our plane is divided into four quadrants by these perpendicular axes. The horizontal line is called the x-axis, and the vertical line is the y-axis.

[The graph with the x and y axes is superimposed over the mandala.]

### Speaker

Now I can see the full quadrants of my mandala design and the four quadrants of my Cartesian plane. And now I'm thinking I could take one element of my design like the pebbles and plot their positions on the plane. I'll be able to use what I know about symmetry to help me do this.

[The images of the mandala and the Cartesian plane are separated again.

A black dot appears in the middle of the Cartesian plane, at the intersection of the 2 axes.]

### Speaker

The centre point of my Cartesian plane is where the two axes, meet. That's right here.

[On the image of the mandala, a yellow circle appears around the grey rock in the top right quardrant.

A yellow circle appears in the top right quadrant, at the intersection of the 5th line of the x-axis and the 4th line of the 7 axis.]

### Speaker

So now I can make a reasonable estimation that my pebble would be about here. But now I need to think about how I can describe the location of this rock. We've drawn it, but mathematicians also describe it. So I have my axes that I can use. I have my x-axis and my y-axis. And just like in a map, we use numbers to help us describe where something is located.

[A 0 appears beside the black dot at the intersection of the axes. Running to the right, the digits 1, 2, 3, 4, 5, 6 and 7 appear on the x-axis beside the column lines.]

### Speaker

So starting from our centre point which we label 0, now I can label each of these intersecting grid lines. This would be location 1, location 2, location 3, location 4, location 5, location 6 and location 7.

[On the y-axis and running upwards from 0, the digits 1, 2, 3, 4, 5, 6, 7 appear on the row lines]

### Speaker

And in fact, I can do the same along the y-axis as well. Starting at location 0, as I go up, it will increase like this, 1, 2, 3, 4, 5, 6, 7.

[The digit 5 on the x-axis and the digit 4 on the y-axis are highlighted in yellow. Lines form run from those 2 numbers to where the yellow circle is in the right quadrant.

The digits 5, 4 appear beside the yellow circle.]

### Speaker

So now what I can see in the top right quadrant, is that location 5 along the x-axis and location 4 along the y-axis, describe the position of our first rock. And that is 5,4. But now I have a bit of a predicament.

[A yellow circle appears around the grey rock in the top left quadrant of the mandala. A yellow arrow points to the circle.]

### Speaker

How can I describe the location of this rock, which is the same rock but in a different quadrant? I'll need to continue my number line.

[On the Cartesian plane, a yellow arrow appears below the x-axis pointing towards the number 7. A yellow dot appears over the axis intersection, point 0.

The arrow flips so that it points to the left of the graph on the x-axis.]

### Speaker

When we started at location 0 and travelled along the x-axis each move to the right, the numbers were increasing by 1. This time, when we start at location 0 and travel to the left, it means our numbers will decrease by 1 each time.

[To the left of the 0 on the x-axis, the following digits appear: -1, -2, -3, -4, -5, -6, -7.]

### Speaker

So 1 less than none is -1. 1 less than -1 is -2. 1 less than -2 is -3. 1 less than -3 is -4. 1 less than -4 is -5. 1 less than -5 is -6. 1 less than -6 is -7.

[A line appears from the -5 location on the x-axis and the 4 location on the y-axis. The two lines meet at the yellow circle in the left quadrant. The digits -5,4 appears beside the yellow circle.]

### Speaker

So now I can locate the coordinates for this rock in the top left quadrant, because I can see it's -5 on my x-axis and 4 on the y-axis. Which I can describe as -5,4. This rock is almost the same as our first rock, which was 5,4, except this rock is -5,4. And that makes sense because I was trying to make symmetrical mandala designs.

[A yellow circle appears around the grey rock in the bottom left quadrant of the mandala image. A yellow arrow points to the rock.

The digit -5 on the x-axis is highlight in yellow.]

### Speaker

Now I can locate this pebble down here in the bottom left quadrant. But while we already know the number along the x-axis, which will be -5, I need to label the rest of the y-axis to know what the second coordinate will be.

[A yellow arrow appears beside the y-axis, pointing upwards. The line flips so that it points downwards.]

### Speaker

When we travelled up the y-axis, each number increased by 1. So as we travel down the y-axis, each number will decrease by 1.

[The line disappears. From 0 and running down the y-axis, the following numbers appears: -1, -2 ,-3, -4, -5, -6, -7.]

We start at 0 and just like we worked out on our x-axis, 1 less than 0 is -1. 1 less than -1 is -2. 1 less than -2 is -3. 1 less than -3 is -4. 1 less than -4 is -5. 1 less than -5 is -6. 1 less than -6 is -7.

[A yellow circle appears in the bottom left quadrant of the Cartesian plane with the digits -5, -4 beside it.

A yellow circle appears in the bottom right quadrant with the digits 5, -4 beside it.]

### Speaker

So now I can easily see that the bottom will be in the location -5,-4. Now I can work out the pebble in the bottom right-hand quadrant. Because we have all the numbers on our axes, we can easily see that the last location we are after is 5,-4.

[New slide. Text: Over to you!]

### Speaker

Alright mathematicians, over to you to pick a piece of your mandala design to plot on your own Cartesian plane.

[The NSW Government logo flashes on screen. Text below reads: Copyright, State of New South Wales (Department of Education), 2021.]

[End of transcript]

## Follow up

Pick a piece of your mandala design to plot on your Cartesian plane.