# Spirolaterals

Stage 2 and 3 – a Thinking Mathematically targeted teaching opportunity focussed on using mathematical reasoning to develop and test conjectures involving repeated visual patterns and sequences

## 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-GM-02
• MA2-GM-03
• MA2-2DS-02
• MA2-2DS-03

• MAO-WM-01
• MA3-GM-02
• MA3-GM-03
• MA3-2DS-03

You will need:

## Watch

Watch the Spirolaterals part 1 video (4:57).

Use predetermined sequences of movement to create spiral patterns.

### Transcript of Spirolaterals part 1

[Text over a blue background: ‘Spirolaterals’ From youcubed.

In the lower right-hand corner of the screen is the waratah of the NSW Government logo. Small font text in the lower left-hand corner of the screen reads: NSW Mathematics Strategy Professional Learning team (NSWMS PL team).

Black text on a white background: You will need…

• Grid paper
• Four coloured markers

Below the text is an image of mathematical grid paper and four different coloured highlighters.]

### Speaker

Before we explore spirolaterals from youcubed, today you will need grid paper and four coloured markers.

[White text over a blue background: Let’s play!

A sheet of mathematical grid paper sits on a table. There are 4 uncapped coloured highlighters on the paper; one orange, one blue, one green, and one pink.]

### Speaker

Today we are going to represent numbers creatively by making spirolaterals. Mathematics can be creative and even elegant, and you will see this today when you make a spirolateral. To make a spirolateral, you first need to select three to eight single digit numbers.

[Using a blue marker, the speaker writes the numbers 4, 3, 2 and 7 at the top of the grid paper.]

### Speaker

You can pick randomly like me. Today, I will use the numbers four, three, two, and seven. Your coloured markers will be used to draw your spirolateral. You need to pick a colour for each line direction.

[The speaker picks up the orange highlighter and draws an arrow facing right on top of the grid paper. They use the blue highlighter to draw a downward-facing arrow, the green highlighter to draw a left-facing arrow, and the pink highlighter to draw an upward-facing arrow alongside each other.]

### Speaker

I'm going to use orange for right, blue for down, green for left, and pink for up.

[The speaker gestures across the grid paper. They mark a small blue dot near a central point of the grid. They pick up the orange highlighter.]

### Speaker

I'm going to begin by marking a start point on my grid paper. It's best to place your start point towards the middle of your grid paper to allow your spirolateral enough space to grow. Now, I will draw lines that match my numbers, making a turn with each number.

[The speaker draws a line moving 4 squares right from the centre point in orange. From this point, they use the blue highlighter to draw a line moving three squares down. They use the green highlighter to continue drawing two squares left, and use the pink highlighter to continue drawing seven squares upward.]

### Speaker

Four right is orange, one, two, three, four. Three down is blue, one, two, three. Green left is two. And seven up is pink, three, four, five, six, seven.

[The speaker alternates between the highlighters to continue drawing the spirolateral, using orange for right, blue for down, green for left, and pink for up. A pink line ends the spirolateral at the top of the grid paper.]

### Speaker

Continue to spiral through your numbers, changing colours with each turn. Right, four. Down, three. Mathematicians notice and wonder. What do you notice about this spirolateral?

[The speaker uses their finger to draw along the lines of the spirolateral. They point to the starting point in the centre.]

### Speaker

Oh, yeah, I also notice that this part of the picture is being repeated over and over. I think if I kept going, my spirolateral would keep growing up on a diagonal, and it appears it could go on forever. I don't ever think it would meet at the starting point again. But I've seen spirolaterals that do meet back at the starting point. So, let's try some different numbers to see if I can make a spirolateral that meets in the middle again.

[A new blank sheet of grid paper appears. The coloured arrows are drawn in the same colours and directions at the top of the paper. The speaker writes the numbers 3, 6 and 9 at the top of the page, then mark a central point in the grid.]

### Speaker

This time, I'm going to choose the numbers three, six and nine.

[The speaker draws a spirolateral starting from the central point. They use orange to draw 3 squares left, blue to draw 6 squares down, green to draw 9 squares left, then pink to draw 3 squares up. They repeat the pattern, alternating colours and directions, until a vertical pink line intersects with a horizontal green line.]

### Speaker

Now, I'm about to redraw over an existing line, and that's OK.

[The speaker draws an orange line over the top of the green line, moving nine squares to the right.]

### Speaker

Orange is nine.

[They continue drawing the spirolateral in alternating colours. The line meets back at the starting point to create a windmill shape. The speaker points to the starting point.]

### Speaker

I'm noticing that this spirolateral looks different to my first. It looks like a windmill. It also joined back at its starting point. And now I wonder if I can get that to work again, to make a spirolateral that joins back at its starting point again.

[A blank sheet of grid paper appears. The coloured arrows are drawn on the top. The speaker writes the numbers 1, 3, 5, 7, and 2 on the top of the paper, and marks a central starting point in the grid. They use alternating colours to draw the spirolateral.]

### Speaker

This time, I'm going to choose one, three, five, seven, and two.

[The speaker connects the last line of the spirolateral back to the starting point. The spirolateral forms a clover shape.]

### Speaker

This spirolateral has joined back to its starting point. It looks different to my first and second spirolaterals.

[Black title on a white background: Let’s try!

On the left side of the screen, a blue rectangle appears with the words “Odd amount of numbers” above it. Inside the rectangle text appears: 1, 4 and 2 3 numbers (odd); 7, 5 and 3 3 numbers (odd); 1, 5, 2, 4, and 3 5 numbers (odd).]

### Speaker

Let's get ready to make our own spirolaterals. Write down three different sets of numbers that have an odd amount, like one, four and two, seven, five and three, one, five, two, four and three. It doesn't matter if the numbers themselves are odd or even, just how many you have. OK, great.

[On the right of the screen, a blue rectangle appears with the words “Even amount of numbers” above it. Inside the rectangle text appears: 1 and 4 2 numbers (even); 5, 6, 1 and 2 4 numbers (even); 8 and 7 2 numbers (even).]

### Speaker

Now let's write down three different sets of numbers that have an even amount, like one and four, five, six, one and two, eight and seven. Now for the fun part. We get to create the different spirolaterals.

[White text on a blue background: Over to you, mathematicians!

The screen changes to 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]

## Instructions

• Write down 3 different sets of numbers that have an odd amount (for example, one, 4, 2; 7, 5, 3; and one, 5, 2, 4, 3. It doesn’t matter if the numbers themselves are odd or even – just how many numbers you have.
• Write down 3 different sets of numbers that have an even amount (for example, one, 4; 5, 6, one, 2; and 8, 7).
• Draw the different spirolaterals!

## Watch

Watch the Spirolaterals part 2 video (1:37).

What is the same and different about these spirolaterals?

### Transcript of Spirolaterals part 2

[Title on a blue background: Our Spirolateral Gallery

In the lower right-hand corner of the screen is the waratah of the NSW Government logo. Small font text in the upper left-hand corner of the screen reads: NSW Department of Education.

Three images of grid paper depict completed spirolaterals. Numbers and coloured arrows appear at the top of each grid. Above each image is a heading. Heading for left image: 4 numbers. Heading for centre image: 3 numbers. Heading for right image: 5 numbers.]

### Speaker

So, mathematicians. Here is a gallery of our completed spirolaterals from today.

[A white border appears around the image on the left. Numbers at the top of the grid read: 4, 3, 2, 7. The spirolateral in the image is composed of orange, blue, green and pink lines. The lines move from a central starting point in diagonal spiralling pattern outward to the right.]

### Speaker

We can see that our first spirolateral that was composed with four numbers, an even amount, did not return to its starting point, and radiated outwards.

[The white border travels across the screen to surround the image in the centre. Numbers at the top of the grid read: 3, 6, 9. The spirolateral in the image creates a closed windmill shape.]

### Speaker

Our second spirolateral that was composed with three numbers, an odd amount, did return to its starting point and crossed over some previous lines. This one created a windmill-like shape.

[The white border travels across the screen to surround the image on the right. Numbers at the top of the grid read: 1, 3, 5, 7, 2. The spirolateral in the image creates a closed clover-like shape.]

### Speaker

Our final spirolateral was composed with five numbers, another odd amount, and also returned to its starting point.

[Black text over a white background: What’s (some of) the things we’re starting to notice? When we create a spirolateral with an even amount of numbers, it doesn’t return to its starting point.

A small image of the diagonal spirolateral.

Text: When we create a spirolateral with an odd amount of numbers, it does return to its starting point.

Small images of the closed windmill and clover spirolaterals.]

### Speaker

When we asked some kids what was the same and what was different about these spirolaterals, they noticed that when we create a spirolateral with an even amount of numbers, it doesn't return to its starting point. They also discovered that when we create a spirolateral with an odd amount of numbers, it does return to its starting point.

[Text: Is this always true? Can you find an example where an odd amount of numbers forms a spiral that doesn’t return to its starting point?]

### Speaker

So now we're wondering, is this always true? Can you find an example where an odd amount of numbers forms a spiral that doesn't return to its starting point?

[White 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]

## Instructions

Can you find an example of where an odd amount of numbers forms a spiral that doesn’t return to its starting point?

## Watch

Watch the Spirolaterals part 3 video (0:39).

What happens to a spirolateral if you use different paper?

### Transcript of Spirolaterals part 3

[Black text on white background: What happens if we use different paper?

In the lower right-hand corner of the screen is the waratah of the NSW Government logo. Small font text in the upper left-hand corner of the screen reads: NSW Department of Education.

Text: Test out your spirolaterals using the same numbers but on triangular paper. This time you’ll need 3 different coloured markers.

An image of triangular mathematical grid paper. Three coloured highlighters sit on the paper and coloured arrows point in three diagonal directions at the top.

Text: Does our conjecture about odd or even amounts and whether they make a spirolateral return to its starting point hold true for triangular paper as well?]

### Speaker

Now we're wondering what happens when we use different paper like this triangular paper. Test out your spirolaterals using the same numbers, but this time on triangular paper. You will only need three different coloured markers as triangles only have three sides. Does our conjecture about odd or even amounts and whether they make a spirolateral return to its starting point, hold true for triangular paper as well?

[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]

You will need:

## Watch

Watch the Spirolaterals part 4 video (0:53).

What's (some of) the mathematics when making spirolaterals?

### Transcript of Spirolaterals part 4

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

In the lower right-hand corner of the screen is the waratah of the NSW Government logo. Small font text in the upper left-hand corner of the screen reads: NSW Department of Education]

### Speaker

So what's some of the mathematics?

[Black text over a white background: What’s (some of) the maths? As mathematicians, we notice and wonder! When we wonder, we can develop a conjecture that we can test out.]

### Speaker

As mathematicians, we notice and wonder! We wonder when we develop a conjecture. Like a hypothesis in science, you might also call this a hunch, or a theory, that we can test out.

[An image of three sheets of grid paper appears containing coloured spirolaterals. One spirolateral spirals outward diagonally away from the starting point. The other two spirolaterals make closed shapes that return to their starting point. Text: When we create a spirolateral with an even amount of numbers, it doesn’t return to it’s starting point. When we create a spirolateral with an odd amount of numbers, it does return to its starting point.]

### Speaker

So, when we noticed this…

[Text: Does an odd amount of numbers in a spirolateral always make it return to its starting point?]

### Speaker

And then decided to test out this…

[An image of a coloured spirolateral on triangular grid paper appears. The numbers 1, 3, 5, 7, 2 are at the top of the page, along with three directional arrows in coloured markers. Text: We tested out using different paper.]

### Speaker

And then tested it out using different paper, we were thinking and working like 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]

## Instructions

• What happens if we use a different paper?
• Does our conjecture about odd and even amounts, and whether they make a spirolateral return to its starting point, hold true for triangular paper as well?