# Dice collection – Stage 1

A thinking mathematically targeted teaching opportunity focussed on exploring patterns, using additive strategies and quantifying collections using multiplicative thinking

## 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
• MA1-RWN-01
• MA1-RWN-02
• MA1-CSQ-01
• MA1-FG-01

## Collect resources

You will need:

• paper
• pen or marker
• someone to talk to.

## Watch

Watch Dice collection Stage 1 part 1 video (1:51).

What do you notice and wonder about the dice collection?

### Transcript of Dice collection Stage 1 part 1 video

(Duration: 1 minute and 50 seconds)

[Text over a navy-blue background: Dice collection (Stage 1). Small font text in the lower left-hand corner reads: NSW Mathematics Strategy Professional Learning Team (NSWMS PL team). In the lower right-hand corner is the waratah of the NSW Government logo.]

### Speaker

Hello mathematicians! Today, we are going to be looking at and talking about a dice collection.

[Text over a white background: You will need…

· something to write on

· something to write with

· someone to talk to.

Three images beside the text show: A child holding a blank sheet of white paper, a pencil holder with pens and pencils, and an illustration with two people facing each other.]

### Speaker

For this task, you will need something to write on, something to write with, and someone to talk to that you can share your mathematical thinking with.

[A colourful sheet of paper with white, zig-zagging lines.]

### Speaker

Hello mathematicians!

Today, I have something that I would like to show you under this colourful piece of paper. And I want you to think about what you see and what you notice. I would like you to use what you have chosen to write on to record your thinking. You can use pictures, numbers or words to record what you notice.

Great, here it is.

[The speaker moves the colourful sheet of paper. 16 blue and green dice are arranged in a 4 by 4 square. There are 8 green dice and 8 blue dice. All of the green dice have their 2-side facing upward and all of the blue dice have their 4-side facing upward. There are 4 blue dice in the middle of the square. The dice at each of the corners are also blue. The remaining dice are all green. All of the green dice are in pairs, along the outer side of the square. They are all arranged so that, within each pair, the dice form a mirror image of each other, based on the placement of the two dots.]

### Speaker

Have you written down one thing? Great.

Now keep looking and write something else that you notice.

I'm hearing some counting going on, wonderful. Write that down too.

[Text on a blue background: Over to you mathematicians…]

### Speaker

Over to you 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

• Write down all of your ideas.

## Watch

Watch Dice collection Stage 1 part 2 video (10:54).

Investigate patterns and features of the dice collection.

### Transcript of Dice collection Stage 1 part 2 video

(Duration: 10 minutes and 54 seconds)

[The 4 by 4 grid of 16 dice are arranged as they were in the previous video, on a white surface.]

### Speaker

Let's talk about some of things you might have noticed about this dice collection. I'm looking at it and I think I can see a pattern. Can you?

[The speaker points to the dice around the outside of the square. She points to each of the dice as she states its colour.]

### Speaker

I think I see a colour pattern that goes all the way around the outside of our dice collection. Look at the colours around the outside. Blue, green, green, blue, green, green, blue, green, green, blue, green, green.

[The speaker moves all the dice from around the outside of the grid, into one long, horizontal line, maintaining the same colour pattern.]

### Speaker

To see better, I'm going to move all of the dice that are around the outside, all together.

This will help me to see and to find what we call the core of the pattern or the part of it that repeats over and over and over again.

[The speaker points to the first three dice in the line. She states the colour as she points to each dice. She moves the first three dice away from the line, into their own group.]

### Speaker

And I'm thinking that the core of this pattern is blue, green, green.

[She points to the next three dice, which follow the same colour patter as the first three dice. She states the colour of each dice as she points to them. She moves this second lot of three dice into their own group, away from the others.]

### Speaker

Because what we have after it is another blue, green, green…

[She points to the next three dice in the line, which follow the same colour patter as the previous groups of three dice. She states the colour of each dice as she points to them. She moves this third lot of three dice into their own group, away from the others.]

### Speaker

..and then another blue, green, green…

[The speaker points to the final three dice, which follow the same colour pattern as the three previous groups of dice. She states the colour as she points to each dice.]

### Speaker

..and another blue, green, green.

[The speaker arranges the 4 groups of dice into a stack. She points to all of the blue dice at the start of each group, then points to the remaining green dice in each group.]

### Speaker

So I can also prove that this is the pattern core because I can see that if I put them like this, that they are all exactly the same. First dice in each part is blue and the next two dice that follow are green.

Now, mathematicians describe patterns using letters of the alphabet. So actually what we have here is what we call an A-B-B pattern. A is the blue, green is the B, and the second green is another B.

[The speaker points out the colour pattern in each of the groups of dice.]

### Speaker

Look, A-B-B. A-B-B. A-B-B.

[On the white surface beside the dice, the speaker writes “ABB pattern” in black marker pen.]

### Speaker

So I'm going to record that we have a found an A-B-B pattern.

And we know it's a pattern because the core, blue, green, green, or A-B-B, repeats itself over and over and over again. Can you see a pattern with the numbers too?

[All of the blue dice have their 4-face facing upward, while all of the green dice have their 2-face facing upward.]

### Speaker

Yes, that's right. There is a number pattern.

[The speaker states the number showing on each dice as she points to them.]

### Speaker

4-2-2, 4-2-2, 4-2-2, 4-2-2. So this dice collection actually has two ABB patterns, one with the colours blue and green that repeat over and over and over again. And the numbers that are on these dice being 4-2-2 that also repeat over and over and over again.

[With the black marker pen, the speaker writes additional text, so that it now reads “2 ABB patterns

- colour

- number.

The speaker puts the dice back into the 4 by 4 grid.]

### Speaker

What else did you notice? Did you count how many dice you can see altogether in our collection? Let's check.

[With a green sheet of paper, the speaker covers the bottom 3 rows of dice, leaving only the top row showing.]

### Speaker

How many dice are here? Four, that's right.

[The speaker moves the paper to reveal the second row of dice as well.]

### Speaker

And if we double it, how many do you see now? That's right, there are eight dice.

[The speaker covers the top two rows of dice with the sheet of paper, revealing the bottom two rows.]

### Speaker

Now I can see that I have eight here as well. So how many dice are there altogether if we double out? That's right. There are 16 dice altogether.

[The speaker moves the sheet of paper off screen.]

### Speaker

You may have even worked it out by skip counting like this.

[The speaker points along each row as she counts.]

### Speaker

4, 8, 12, 16. So let's record that.

[The speaker writes “16 dice altogether”.]

### Speaker

16 dice altogether.

Now I can see that dice collection is made up of rows.

[The speaker points to each of the rows of dice.]

### Speaker

This is a row. This is a row. Another row, and another row. It is made up of four rows.

[The speaker writes “4 rows”. The speaker covers the dice with the green sheet of paper, and then moves it to reveal one row at a time.]

### Speaker

Or another way that we can say that is, here is one four, here is another four, two fours, three fours, and four fours.

[The speaker writes “4 fours”.]

### Speaker

Hmm, I've seen something like this before. Have you ever heard of an array? Yes, an array has rows that go across and columns that go up and down.

How many columns do you see?

[The speaker writes “This is an array”.]

### Speaker

Yes, that's right. It has four columns.

[The speaker points to each column as she counts.]

### Speaker

One, two, three, four.

[The speaker writes “4 columns”.]

### Speaker

What else do you notice? Yes. There are dots on our dice. Let's look closely at those now. I wonder how many dots there are in each row?

[The speaker moves the bottom 3 rows of dice away from the top row.]

### Speaker

Hmm, how did you work it out?

I did it like this.

[The speaker separates the two dice on the ends of the row, which both have their 4-faces facing up, away from the other two dice.]

### Speaker

I know that four and four make eight…

[The speaker moves one of the dice from the middle of the row, which has its 2-face facing up, to join the other two dice.]

### Speaker

..and two more makes ten…

[The speaker moves the final dice, which also has its 2-face facing up, to join the other dice.]

### Speaker

..and I know that two more again make 12 altogether.

My brain likes to make ten because this helps me to work it out. I wonder how many are in the second row?

[The speaker separates the second row of dice from the bottom two rows of the grid. This row is arranged: green, blue, blue, green, or, 2-face, 4-face, 4-face, 2-face.]

### Speaker

Can you find the ten and then add what is left? Great.

[The speaker moves the two dice with their 4-face showing up.]

### Speaker

Four and four make eight.

[She moves one of the dice with its 2-face showing up.]

### Speaker

Two more make ten…

[The speaker moves the last dice up to join the others.]

### Speaker

..and two more again makes 12.

[The speaker moves the third row of dice, away from the fourth row.]

### Speaker

And have a look at the third row. That's interesting. It's exactly the same as the second row. So if this is 12, this must be 12 as well. And take a look at the last row.

[The last row matches the first row. She moves the two dice with their 4-faces showing, then moves each of the dice which have a 2-face showing.]

### Speaker

Yes, we've got the two fours. And one two, which make the ten. And then two more that make 12. Well, that's interesting. There are 12 dots in each row.

[The speaker writes “12 dots in each row”. The speaker moves all the rows of dice back together.]

### Speaker

Oh, that's interesting. Did I hear someone say it's symmetrical? What a great word to use to describe our dice collection. Because you're right, it is symmetrical. Symmetrical means that one side is the same as the other. And if I separate my dice here like this…

[The speaker separates the grid of dice into two groups of two columns each. Using the black marker pen, she draws a vertical line where she has separated them.]

### Speaker

..and draw a line, which is my line of symmetry, I can check to see if one side is exactly the same as another.

[Using two fingers, the speaker points to the dice on both sides of the line as she states their colours.]

### Speaker

So I have two greens, two blues. Two blues, two greens.

They are exactly the same. It's like I have this side here and I have flipped it over. And it's exactly the same on the other side of my line of symmetry. Can you see another line of symmetry?

Yes.

[The speaker removes the vertical line which separates the two groups of dice. She pushes the dice back together. She divides them horizontally into two groups of two rows each. She draws a horizontal line where she has separated the dice.]

### Speaker

If I rub that line off there and put our collection together, there is another line of symmetry that will go here across our dice collection.

Can you check to see if one side is exactly the same as another?

[using two fingers, the speaker points to the dice on both sides of the horizontal line as she states their colours.]

### Speaker

Can you see? Two green, two blue. Two blue, two green?

Yes, we have found another line of symmetry.

[The speaker writes “2 lines of symmetry”.]

### Speaker

So this dice collection has two lines of symmetry.

Well done, mathematicians! You noticed lots of interesting things about our dice collection. Can you find any more?

[Text over a blue background: Over to you mathematicians…]

### Speaker

Over to you, mathematicians!

[Text on a white background: What do you notice? Record your thinking using numbers, pictures and words. Below is an image of a grid made from coloured dice. The grid is 6 dice across and 4 dice high. The number shown on each face is the same for each colour of dice. All of the green dice have the 1-face showing. All of the orange dice have the 2-face showing. All of the purple dice have the 3-face showing. All of the red dice have the 4-face showing. All of the yellow dice have the 5-face showing. All of the black dice have the 6-face showing. The first row of dice is arranged in the colour pattern: green, orange, purple, red, yellow, black, and the number pattern: 1, 2, 3, 4, 5, 6. The second row of dice is arranged in the colour pattern: black, yellow, red, purple, orange, green, and the number pattern: 6, 5, 4, 3, 2, 1. The third row matches the first row. The fourth row matches the second row.]

### Speaker

Look at this dice collection. What do you notice?

Think about the types of things we noticed in the first collection to help you. Have fun!

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

### Speaker

What's some of the mathematics?

[Text on a white background: What’s (some of) the mathematics?

· A pattern has a core that repeats over and over and over again.

For example, we found a colour pattern core – blue green green that repeats over and over and over again.

An image below shows 4 groups of dice, all arranged into the pattern core, blue green green. The word “blue” is written above each blue dice and the word “green” is written above each green dice. The letter “A” is written below each blue dice and the letter “B” is written below each green.

Text below the image:

· This is what we call an ABB pattern.]

### Speaker

Our pattern has a core that repeats over and over and over again. For example, we found a colour pattern core. Blue, green, green, that repeats over and over and over again in our dice collection. This is what we call an A-B-B pattern, with the blue dice being the A, and the two green dice both being Bs.

[Text:

· Different strategies can be used for counting the total in a collection.

For example, to work out the total number of dice you may have:

Below, are two images of the 4 by 4 grid of blue and green dice used in this video. Text below the first image reads: Identified that there is 4 in the top row. Double 4 is 8. Double 8 is 16. Text below the second image reads: Used skip counting 4, 8, 12, 16.]

### Speaker

Different strategies can be used for counting the total inner collection. For example, to work out the total number of dice, you may have used a doubling strategy.

[A rectangular outlines appear around the rows of dice in the first image as the speaker counts them. A black line appears around the first row. A red line appears around the first two rows. A yellow line appears around the third and fourth row.]

### Speaker

You may have identified that there is four in the top row. That double four is eight and that double eight is 16. Or, you may have used skip counting.

[The numbers 4, 8, 12, 16, appear at the end of the rows of dice in the second image as the speaker counts them.]

### Speaker

4, 8, 12, 16.

[Text:

· Mental strategies, such as bridging to ten, can be used to work out how many altogether.

Below is an image of a row of 4 dice. Its colour pattern is, blue, green, green, blue.]

### Speaker

Mental strategies such as bridging to ten can be used to work out how many altogether.

[Additional text: 4 and 4 and 2 is 10. Arrows point to both of the 4-face dice, and one of the 2-face dice.]

### Speaker

Four and four and two more is ten.

[Additional text: 10 and 2 more is 12. Another arrow appears. It points to the final 2-face dice.]

### Speaker

Ten and two more is 12 altogether.

[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

• Think about the ideas that were shared about the first dice collection. Use those ideas as a starting point to help you to look at this dice collection.

• What do you notice?

• What else do you notice?

• What is different about the two dice collections?

• What is the same about the two dice collections?