# Dice collection Stage 2

A thinking mathematically targeted teaching opportunity focussed on thinking about and mentally solving problems in different ways, exploring multiplicative situations and representing data

## 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

## Outcome

- MAO-WM-01
- MA2-RN-01
- MA2-AR-01
- MA2-MR-01

## Collect resources

You will need:

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

## Watch

Watch Dice collection Stage 2 part 1 video (1:29).

[Text over a navy-blue background: Dice collection (Stage 2). 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

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

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 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. Are you ready? Great! Here it is.

[The speaker moves the colourful sheet of paper to reveal 25 coloured dice, arranged in a 5 by 5 grid. All of the green dice have the 3-face showing. All of the orange dice have the 6-face showing. All of the red dice have the 5-face showing. All of the purple dice have the 4-face showing. The first row is arranged in the colour pattern, green, orange, red, orange, green, and the number pattern, 3, 6, 5, 6, 3. The second row is arranged in the colour pattern, purple, green, purple, green, purple, and the number pattern, 4, 3, 4, 3, 4. The third row is arranged in the colour pattern, red, purple, red, purple, red, and the number pattern 5, 6, 5, 6, 5. The fourth row is the same as the second row. The fifth row is the same as the first row.]

Write or draw one thing that you notice about this dice collection.

Have you written one thing down? Great! Now keep looking and write something else that you notice.

When you have lots of ideas, talk to someone if you can, about what you noticed about this dice collection.

[Text over 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

- What do you notice about this dice collection?
- Write down all of your ideas.
- If you can, talk to someone, to share what you noticed about this dice collection.

## Watch

Watch Dice collection Stage 2 part 2 video (13:54).

(Duration: 13 minutes and 53 seconds)

[A 5 by 5 grid of 25 coloured dice is arranged on a white surface in the top corner of the screen. In the grid, all of the green dice have the 3-face showing. All of the orange dice have the 6-face showing. All of the red dice have the 5-face showing. All of the purple dice have the 4-face showing. The first row is arranged in the colour pattern, green, orange, red, orange, green, and the number pattern, 3, 6, 5, 6, 3. The second row is arranged in the colour pattern, purple, green, purple, green, purple, and the number pattern, 4, 3, 4, 3, 4. The third row is arranged in the colour pattern, red, purple, red, purple, red, and the number pattern 5, 6, 5, 6, 5. The fourth row is the same as the second row. The fifth row is the same as the first row.]

### Speaker

Let's talk about some of the things you might have noticed about this dice collection. I am going to record our thinking as we go. I can see that our dice collection is organised into an array.

[The speaker points horizontally along the rows of the grid, then vertically along the columns of the grid.]

### Speaker

It has horizontal rows and vertical columns. So I'm going to use this to help me.

[The speaker uses a black marker pen to write “5 rows” “5 columns” below the grid.

### Speaker

I can see that there are five rows, and five columns.

[DESCRITPION: The speaker points to the rows of dice as she counts. She points to the first row of dice. Then the first 2 rows of dice. Then the first 3 rows of dice. Then the first 4 rows of dice. Then to all 5 rows of dice.]

### Speaker

Or another way of looking at it is, this is one five, two fives, three fives, four fives, five fives.

[The speaker writes “5 fives = 25”.]

### Speaker

And I know that five fives is equal to 25.

You might have counted in multiples of five, like this.

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

### Speaker

Five, ten, 15, 20, 25. Or you might have used your knowledge of doubles.

[The speaker uses a green sheet of paper to cover the bottom 3 rows of dice, leaving only the top 2 rows visible.]

### Speaker

Here is ten.

[She moves the sheet of paper so that the first 4 rows of dice are visible.]

### Speaker

If I double that I get 20…

[She moves the sheet of paper so that all of the dice are visible.]

### Speaker

..and five more is 25.

I wonder how many dots are in each row? And I'm curious to see if there is a pattern there. Mathematicians are always looking to make sense of things and looking for patterns is one way that we do this. Let's start with the top row.

[The speaker covers the bottom 4 rows with the sheet of paper, leaving only the top row visible. She points to each of the dice as she counts the number of dots.]

### Speaker

To find the sum or total of all of these dots in the first row, I'm going to be strategic. I can see that there are two sixes. So that makes 12. And I know that three and three makes another six. So that's 18. And five more is 23.

Now, I'm going to record the sum of each row in a table.

[Using a blue marker pen, the speaker writes “Sum of each row” beside the grid. Below that, she writes “1^{st} row 23 dots”.

### Speaker

So in the first row, there are 23 dots.

[She moves the sheet of paper to reveal the second row of dice. She points to each of the dice as she counts their dots.]

### Speaker

Let's take a look at the second row. So in the second row, I can see four threes. And I know that four threes make 12 and double three is six, so 12 and six is 18.

[In the table beside the grid, she writes “2^{nd} row 18 dots”.]

### Speaker

So there are a total of 18 dots in the second row.

[The speaker moves the sheet of paper to reveal the third row of dice.]

### Speaker

Let's take a look at the third row. How would you work out the total?

[The speaker points to the dice in the third row as she counts them.]

### Speaker

I look at the third row and I can see that there are three fives and I know that that's 15, and double four, or eight more, is 23.

[In the table, the speaker writes “3^{rd} row 23 dots. She moves the sheet of paper to reveal the fourth row of dice.]

### Speaker

Let's take a look at the fourth row.

I can see just by looking at it that the fourth row is exactly the same as the second row.

[In the table, the speaker writes “4^{th} row 18 dots”.]

### Speaker

So the sum of the fourth row, just by noticing that, is also 18 dots.

[The speaker moves the sheet of paper to reveal the last row of dice.]

### Speaker

And take a look at the last row. What do you notice about that? That's right. It is the same as the first row.

[In the table, the speaker writes “5^{th} row 23 dots”.]

### Speaker

And it also has 23 dots.

[The speaker draws a box around the table. She draws a horizontal line under the heading and draws a vertical line to separate the rows and the number of dots into two separate columns.]

### Speaker

Have look at the information in the table that we have created. What do you notice?

Yes. It may be the start of a repeating pattern.

We have 23, 18, 23, 18, 23. If there was another row in our collection, and it had 18 dots, then it may have been a repeating A-B pattern. Where A is 23 and B is 18.

[Beneath the table, the speaker writes “23”, and then writes the letter “A” below that. She also writes “18” and the letter “B” below that. She then writes “23” with the letter “A” below it again, and then writes “18” with the letter “B” below it.]

### Speaker

But as we know, for it to be a pattern, it needs to repeat over and over and over again.

[Below the A-B pattern, the text “Repeating pattern” has been written. The green sheet of paper has been removed from screen.]

### Speaker

Now, let's look at the sum of each column. So let's look at the first one together.

[With the green sheet of paper, the speaker covers the second, third, fourth and fifth column of the grid of dice, leaving only the first column visible. It features the colour pattern, green purple, red, purple, green, and the number pattern, 3, 4, 5, 4, 3.]

### Speaker

What strategy would you use to work out the total number of dots there? I am going to bridge to ten first, to help me work it out. I know that double three is six, and four more is ten, which means I have five and four left, and all together that is 19.

[Beside the table which notes the sum of each row, the speaker writes another heading, “Sum of each column”.]

### Speaker

Also going to record my thinking in a table.

[Beneath the heading, the speaker writes “1^{st} 19 dots”.]

### Speaker

First column is a total of 19 dots.

[In the table, the speaker writes “2^{nd}”. She moves the green sheet of paper to reveal the second column of dice. It features the colour pattern, orange, green, purple, green, orange, and the number pattern, 6, 3, 4, 3, 6.]

### Speaker

In the second, let's take a closer look.

I'm going to double the six first, which will give me 12, and I can see another six by adding three and three together. And that's 18, and four more is 22.

[In the table, the speaker writes “22 dots”.]

### Speaker

I'm noticing that when we add the column so far, the sum of each is different to what we got in the rows. Did you notice that too? I wonder if this will also be in A-B pattern. Let's keep going to find out. Let's take a look at the third column together.

[The speaker moves the sheet of paper to reveal the third column. It features the colour pattern, red, purple, red, purple, red, and the colour pattern, 5, 6, 5, 6, 5.]

### Speaker

I can see three fives. And I know that's 15, and two fours, which is eight. So altogether that makes 23.

[In the table, the speaker writes “3^{rd} 23 dots”. The speaker moves the sheet of paper to reveal the fourth column of dice. It matches the second column.]

### Speaker

And looking at the fourth, it is exactly the same as the second. So I know that the fourth column, there are also 22 dots…

[In the table, the speaker writes “4^{th} 22 dots”.]

### Speaker

..and in the fifth...

[The speaker moves the sheet of paper to reveal the fifth column. It matches the first column.]

### Speaker

Can you see which one the fifth column looks like? Yes. It looks exactly like the first one. And we know that in the first column there were 19 dots.

[In the table, the speaker writes “5^{th} 19 dots”. The speaker draws a box around the table, and then draws lines to separate the heading and the two columns of data.]

### Speaker

Let's take a close look at the second table that we have created.

[The speaker points to each of the rows in the table as she reads the data aloud.]

### Speaker

So the sum of each column can see in the first we had 19 dots, 22 dots, 23, 22 and 19. That's interesting. We start with 19 in column one. The sum then increases in columns two and three, and then it decreases back to 19 in column five. I noticed that this collection uses four different dotted dice. I can see some dotted with threes, with fours, with fives and with sixes. Now I wonder how many there are altogether in our dice collection. I could add up all the rows, or all the columns to find out. But I think I will do it multiplicatively, because there might be a pattern or something to notice here too. I'm going to look at how many there are of each type of dotted dice and multiply. So let's start with the threes. How many threes do you see?

That's right. There are eight threes.

[In red marker pen, the speaker writes “8 threes = 24”.]

### Speaker

And all together that makes 24. Now to the fours, I can see eight fours.

[The speaker writes “8 fours = 32”.]

### Speaker

And I know that eight fours is 32. How many fives do you see? Yes, there are five fives, and that equals…

[The speaker writes “5 fives = 25”.]

### Speaker

..25. And there are four sixes…

[The speaker writes “4 sixes = 24”.]

### Speaker

..which equals 24. So can you work out how many dots there are all together if we add these four totals?

[The totals of each dice number appear in a box. The number 32 is circled. It changes to read 30. The number 24, which appears twice, changes to 25.]

### Speaker

To add these numbers together, I'm going to partition 32 into three tens and two ones, and I will add each one to the two 24s to make them both now 25. I know that three 25s is equivalent to 75, and 30 more is 105.

[The speaker writes “105 dots altogether”.]

### Speaker

Now I'm looking at our dice pattern, and I can see that it's symmetrical. So symmetrical means that one side is the same as the other. It's like a reflection, or that our dice collection has been cut in half and has been flipped.

[The speaker places a thin paintbrush along the middle column of the grid. The pairs of columns on either side of the middle column are the mirror image of each other. ]

### Speaker

And if I use this paintbrush to show that that's where I would place the line of symmetry. And if you take a look, this side of the dice collection, if you flip it over, is the same or a reflection of the other side. So that is what we call 'one line of symmetry.' Can you see another line of symmetry?

Yes.

[The speaker places the paintbrush along the middle row of the grid. The two pairs of rows on either side of the middle row are the mirror image of each other.]

### Speaker

If we put the brush right there across the centre, going horizontally, this is another line of symmetry, because this part of our collection, if we flip it over, is exactly the same as the top part.

[In green marker pen, the speaker writes “2 lines of symmetry”.]

### Speaker

So we can say that our dice collection has two lines of symmetry.

[Beneath the text, she draws a box with a vertical line through the middle. Beside that she draws another box, with a horizontal line through the middle.]

### Speaker

One that goes vertically, and the other horizontally. Well done, mathematicians. You noticed lots of interesting things about our dice collection, and I'm sure that you found many more, too.

[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 together to help you get started. Have fun, mathematicians!

[What’s (some of) the mathematics?]

### Speaker

What's some of the mathematics?

[Text:

- Mathematicians see and think about problems in different ways.

For example, here are 3 different ways to work out how many dice there are.

Below are three images of the 5 by 5 grid of 25 dice used in this video. Text around the first image reads “5 rows” and “5 columns”. Text below the first image reads: Identify that this is an array of 5 fives which equals 25. Text below the second image reads: use skip counting 5, 10, 15, 20, 25. Text below the third image reads: Identify 10, double it to make 20 and add 5 more to make 25.]

### Speaker

Mathematicians see and think about problems in different ways. For example, here are three different ways to work out how many dice there are. The first way is I know that this is an array that has five columns and five rows. And I know that five fives equals 25. Another way is to do skip counting.

[The numbers 5, 10, 15, 20, 25 appear at the ends of the rows of dice as the speaker counts aloud.]

### Speaker

Five, ten, 15, 20, 25.

[Rectangular outlines appear around the rows of dice in the third image. A black rectangle appears around the first and second row, a red rectangle appears around the third and fourth row, and a green rectangle appears around the fifth row.

### Speaker

And another way is to look at the dice collection, identify ten, know that if I double it, I get 20, and if I add five more, that makes 25.

[Text:

- Mathematicians use mental strategies to solve mental problems.

For example,

Use knowledge of doubles to add quantities together.

An image below shows a row of dice with the colour pattern, green, orange, red, orange, green, and the number pattern 3, 6, 5, 6, 3. Arrows point to the two dice which have their 3-face showing. Text above the arrows reads: Double 3 is 6. Arrows point from the two dice which have their 6-face showing. The arrows point to text, reading: Double 6 is 12. The dice in the middle of the row, which has its 5-face showing, is circled.

Text: apply known facts and strategies for multiplication.

An image below shows the same row if dice. Arrows point to the two dices which have their 6-face showing. Text above the arrows reads: 2 sixes are 12. Arrows point to the two dice which have their 3-face showing. Text below the arrows reads: 2 three are 6. The dice in the middle of the row is circled.]

### Speaker

Mathematicians use mental strategies to solve problems. For example, I can use my knowledge of doubles to add quantities together. I can also apply known facts and strategies for multiplication.

[Text:

- Mathematicians notice, explain and justify patterns.

For example, in our dice collection we identified the start of a possible repeating pattern, but needed further information to prove it. We justified that another row was needed to align with our thinking that patterns repeat over and over and over again.

Beside the text are two images. The first shows the table drawn in this video, which noted the sum of each row. The second shows the grid of dice that was used in this video.]

### Speaker

Mathematicians notice, explain, and justify patterns. For example, in our dice collection, we identified the start of a possible repeating pattern, but needed further information to prove it. We justified that another row was needed to align with our thinking that patterns repeat over and over and over again.

[Text:

- Mathematicians record their thinking in an organised way.

An image below shows that grid of dice, the two tables, and all of the text that was written in this video.]

### Speaker

Mathematicians record their thinking in an organised way. We use these tables, numbers, words, and diagrams to do this.

[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?