# Decimats (understanding fractions)

Stage 2 and 3 – A thinking mathematically targeted teaching opportunity focused on supporting students to make sense of decimal place value

Game from Anne Roche (APMC, 2010)

## 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-RN-01
- MA2-RN-02
- MA2-PF-01

- MAO-WM-01
- MA3-RN-01
- MA3-RN-02
- MA3-RN-03
- MA3-AR-01

## Collect resources

You will need:

- 2 sheets of paper per player
- different coloured pencils or markers
- 1 spinner (PDF 141KB) labelled tenths and hundredths
- 1 six-sided dice or 1-6 spinner (PDF 141KB) (you could also use playing cards)
- 1 paper clip for the spinner.

## Watch

Watch Colour in decimats – part 1 video (8:57) to learn how to make your game board.

[Text on a white background: Play again with thousandths.

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

### Speaker

You might like to play Colour in Decimats again, but this time with thousandths.

[An image appears on screen. It depicts a Decimats game board. A piece of paper has been divided into ten even segments. One of the tenths segments has been partitioned into ten smaller segments. One of those hundredths segments has been further partitioned into ten smaller segments.

Below are four blue cards with text: “ones”, “tenths”, “hundredths”, and “thousandths”.]

### Speaker

If you do, you'll need to create a different gameboard that looks like this. What do you notice?

[A red square appears around the tenth segment that has been partitioned into hundredths. The red square shrinks and then outlines the hundredth segment that has been partitioned into thousandths.]

### Speaker

Yes, I've partitioned one of my hundredths into ten equal parts or thousandths.

[An image of a spinner appears on screen. Three of the segment are labelled, “thousandths”, two of the segments are labelled “hundredths”, and one of the segments is labelled “tenths”.]

### Speaker

You'll also need to adjust your spinner, so that you have three sections labelled thousandths, two sections labelled hundredths, and one section labelled tenths. It's over to you, mathematicians!

[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

- Make your game board.

## Watch

Watch Colour in decimats – part 2 video (14:01) to learn how to play.

(Duration: 14 minutes and 1 second)

[Text over a blue background: Let’s play! In the bottom right-hand corner of the screen, is the waratah of the NSW Government logo. Text in the upper left-hand corner of the screen: NSW Department of Education.

Viewed from above, are the materials to play ‘Colour in Decimat’, arranged over a large, orange sheet of paper on a table. At the top of the board, are two identical sheets of paper. Lines have been drawn on the paper that divide the sheets evenly into tenths. On both sheets, the segment in the upper left-hand corner has further been divided into hundredths. Between these two sheets of paper, is a purple die, with its 5-face facing upward. Beside the sheet of paper on the right, is a small translucent sheet of paper.

Below each sheet of paper, is an identical set of three yellow cards with text. Text on the cards in each set reads: “ones”, “tenths”, and “hundredths”. Below each set of yellow cards, is another, larger white sheet of paper. A table, divided into three columns, has been drawn on each sheet of paper. On each table, the first column is headed: “What I rolled”. The column below has been divided into two sub-columns, headed: “fraction” and “decimal”. The second column of each table is headed: “What I filled in”. Text in the cell below reads: “words”. The final column is headed: “Total”. Text in the cell below reads: “decimal”.

Finally, there is a square sheet of paper between these two tables. On it, is a pie chart, divided into six even segments. Text in two opposite segments reads: “tenths”. Text in the four remaining segments reads: “hundredths”. A paper clip rests on the centre of this square of paper.]

### Speaker

Alright, now that we've got our game boards all set up, we are ready to play Colour in Decimat, you ready mathematicians? OK, well, I didn't have anybody here to play with today and I was feeling a bit lonely. But then along came my friend…

[An animated Iron Man, crouched in an action pose, appears on screen.]

### Speaker

..Iron Man and he agreed to play with me, which I thought was really nice of him because, you know, he's a busy guy.

[Iron Man moves over to the small, translucent sheet of paper in the upper right-hand corner of the screen. A red square briefly outlines the side of the game board. A blue square then briefly outlines the left side of the game board.]

### Speaker

And just so, he's going to go over here and this is going to be his game board and this is going to be my game board. Now, Iron Man, the aim of the game is to be the first player to fill in the entire hole.

[The speaker waves her hand over the sheet of paper in the upper left-hand corner of the screen.]

### Speaker

So the first player to fill in the hole rectangle, alright? Or after about ten goes, the player who's filled in the most of their game board, got it? Alright, would you like to go first or should I? Oh, thank you so much. He said that I can go first, which is really lovely.

[The speaker places a pen through the end of the paper clip, and centres it in the middle of the pie chart. She flicks the paper clip, which spins around the tip of the pen. It lands on a segment with the text, “hundredths”.]

### Speaker

I'm going to spin the spinner, and I have spun hundredths. So now I need to roll my dice to find out how many hundredths I get to fill in.

[The speaker rolls a 2.]

### Speaker

And I've rowed two, so I'm going to choose a colour. It's important that we choose a different colour every roll because that helps us keep track of our thinking and we can go back and check if we need.

[Using a purple marker pen, the speaker writes “2/100” in the first sub-column, beneath the text, “fraction”, on her score sheet.]

### Speaker

So I'm going to record this as a fraction, two hundredths. Great, and then to help me record it as a decimal, I can use what I know about our place value system. And in this two hundredths, I don't have any ones…

[In the second sub-column, under the text “decimal”, the speaker writes a zero.]

### Speaker

..and so I'm going to write a zero there to show that I have zero ones.

[Beside the zero, the speaker draws a decimal point.]

### Speaker

Now, the next part is really important. I'm going to put, yeah, a decimal point, and a decimal point helps to separate our ones, our tens, our hundreds, all of our whole numbers over this side, from our fractional quantities, our tens, our hundreds, our thousands, and so on. So I put the decimal point there to show the difference, and then I asked myself, do have any tenths in two hundredths? No, I don't, do I?

[Beside the decimal point, the speaker writes another zero.]

### Speaker

So I'm going to put a zero in there to show that I have zero tenths. But finally hundredths, do I have hundredths into hundredths?

[Beside the zero, the speaker writes “2”. Expressed as a decimal, the number reads, “0.02”.]

### Speaker

Yes, I have two of them. And so I can rename two hundredths and write it like this. We might read this as 0.02, but it's the same in value as two hundredths. Alright, so now my job is to colour in two hundredths on here.

[The speaker points to the sheet of paper in the upper left-hand corner. She points to the whole sheet of paper, then to some of the tenth segments, and then to some of the hundredth segments.]

### Speaker

So I need to remind myself that this was the whole, and then each of these rectangles were the tenths, right? And so, OK, these are the little ones are hundredths. So I think I'm going to start by colouring in these two.

[The speaker colours in two of the hundredth segments; the two segments in the upper left-hand corner of the sheet of paper.]

### Speaker

Oh, a bit outside the lines. OK, coloured in my two hundredths.

[On the scorecard, in the “What I filled in” column, the speaker writes “two-hundredths”.]

### Speaker

So I'm going to write that in words, two hundredths. And then what's the total that I filled in on my game board? Two hundredths. I'm going to write that as a decimal.

[On the scorecard, in the “Total” column, the speaker writes “0.02”.]

### Speaker

So remember it's zero ones, zero tenths with a decimal point to separate, but two hundredths. OK, so it's over to Iron Man and Iron Man's... well, he's a picture, isn't he, so he can't, he's having a bit of trouble spinning the spinner, and so I'm going to do it for him like this.

[The speaker spins the paper clip. It lands on a segment with the text “hundredths”. She rolls the dice. It lands on 5.]

### Speaker

Great and he has spun hundredths as well. Let's find out how many hundredths he gets to colour. Oh five, so going to colour five hundredths.

[Under the “What I rolled” column on Iron Man’s scorecard, the speaker writes the fraction “5/100” in blue pen. They then write it as a decimal, “0.05”. On the sheet in the top right-hand corner, the speaker colours in five of the hundredths segments. On Iron Man’s scorecard, the speaker writes “five-hundredths” in the “What I filled in” column. In the “Total” column, they write “0.05”.]

### Speaker

Alright, so we're going to record that as a fraction, which is five hundredths.

[The speaker spins the paper clip. It lands on a segment with the text “tenths”.]

### Speaker

Tenths, ah, yes! That's great, because I can see the tenths are so much bigger. They're ten times bigger than my hundredths. So that's a really good spin for me to cover or fill in most of my board.

[Using a black marker pen, the speaker draws a horizontal line below the first row of information on her scorecard.]

### Speaker

Alright, I'm now going to choose a different colour and I'm going to record this. So I've got... oh, I haven't even rolled yet to see how many tenths.

[The speaker rolls a 5.]

### Speaker

What have I - oh five! Five tenths, fantastic.

[Using a green marker pen, the speaker writes “5/10” in the fraction sub-column of her scorecard.]

### Speaker

So five tenths as a fraction like that.

[In the decimal sub-column, the speaker writes “0.5”.]

### Speaker

Now this time we still have zero ones and I use the decimal point. And then in my fractional quantities I've actually got five tenths. So I can write it like that, we can also rename this one as one half. Interesting fact, I'm going to choose... I think I'm going to fill in...

[On the sheet of paper in the upper left-hand corner, the speaker colours in the five tenths segments on the bottom of the page.]

### Speaker

I'm going to fill in this whole bottom row. Five tenths... oh, this feels good. I think Iron Man is getting a bit nervous, are you? Or you think you can still win?

[The speaker writes “five-tenths” in the “What I filled in” column on her scorecard.]

### Speaker

Alright, so I have filled in five tenths. Now in total, how much of my hole have I filled in? Let's have a look. So I haven't filled in the hole, so I still haven't filled in any ones. How many tenths have I shaded in? One, two, three... five, to the five tenths, right.

[The speaker writes “0.52” in the “Total” column.]

### Speaker

So five tenths, and then how many hundreds have I shaded in... I've coloured in two. So another way that I could rename this is to look at this as being fifty-two hundredths because in each of these tenths, there are ten hundredths.

[Beside the decimal in the “Total” column, the speaker writes “=52/100”.]

### Speaker

Might write that here it's equivalent in value to fifty-two hundredths. Alright, Ironman, it's over to you.

[The speaker spins the paper clip. It lands on “hundredths”.]

### Speaker

Oh, hundredths, oh, sorry Ironman, I know you were hoping for tenths. Let's see how many hundredths you get to colouring.

[The speaker rolls a 6.]

### Speaker

Six, alright, well, that's better than nothing, isn't it?

[The speaker draws a line beneath the first row of information on Iron Man’s scorecard.]

### Speaker

So Ironman is going to draw a line to keep track of his thinking. I'm going to choose a different colour now.

[Using a red marker pen, the speaker writes “6/100” in the fraction sub-column on Iron Man’s scorecard. In the decimal sub-column, she writes “0.06”.]

### Speaker

So we have rolled six hundredths which we can rename as zero ones, zero tenths, but six hundredths. Great, and now, Ironman, where do you... oh, you've got a bit of a problem here, don't you. Can you say what Iron Man's problem is?

[The speaker points her pen to each of the five remaining hundredths segments of Iron Man’s gamecard.]

### Speaker

Yes, he's only got one, two, three, four, five hundredths here left to shade in, but he needs to shade in six of those hundredths. So what do you think he could do? Yes, you're right mathematicians, because we know that inside each of these tenths there are actually ten hundredths, aren't there? So he can partition another one of these tents into hundredths.

[Using a black marker pen, the speaker draws a line vertically down the centre of the segment in the bottom right-hand corner of Iron Man’s gamecard. She then draws four horizontal lines, partitioning the segment into ten smaller segments.]

### Speaker

So let's partition this one here and halve it... and then divided into ten equal parts. Great, and now we've got enough hundredths to work with. So we're going to... how do you want to do it, Iron Man? You're going to fill in these five first.

[Using a red marker pen, the speaker colours in the five remaining hundredth segments in the upper left-hand corner.]

### Speaker

And see what's happening here, Iron Man's now filled in an entire tenth, hasn't he, or ten hundredths. And then we've got one more hundredth we need colour in, so let's choose this one.

[The speaker fills in one of the newly partitioned hundredths segments. She writes “five-hundredths and one hundredth” in the “What I filled in” column on Iron Man’s scorecard.]]

### Speaker

Alright, so what Iron Man has filled in here is a little bit different. He's filed in five hundredths... and one hundredth. So he's actually partitioned the six hundredths into five and one because we know that six is made up of five and one. Great, so then what in total has he filled in? So we notice here that he's actually completed a whole tenth, right and we've got one hundredth, but we could also look at that as being ten tenths and one hundredths.

[In the “Total” column of Iron Man’s scorecard, the speaker writes “11/100 = 1/10 and 1/100”.]

### Speaker

So he's filled in total hundredths... eleven hundredths, but that's equivalent in value to one tenth, and one hundredth, isn't it?

[The speaker points to the yellow cards above Iron Man’s scorecard. In the “Total” column, she writes “0.11”.]

### Speaker

So, if we use our place value to help us out, we've still got zero ones, decimal point, and then we've actually got one tenth, one for tenth, and one little hundredth, so we can rename it to 0.11. Who do you think's winning at the moment? Yeah, I think I am, too.

[The footage speeds up. The speaker continues to play the game, filling in each of the cards accordingly. She scores one tenth. She colours in one tenth segment blue. On her scorecard, she writes: “1/10”, “0.1”, “one-tenth” and the total as “0.62”. Iron Man then scores 6 tenths. She colours in 4 segments green, then another 2 segments green as well. On Iron Man’s scorecard, she writes: “6/10”, “0.6”, “four-tenths and two-tenths”, and the total as “0.71”. She then scores 6 hundredths. She partitions a tenth segment into ten hundredths. Using a red marker pen, she colours in three of the original hundredth segments, then three from the newly partitioned segment. On her scorecard she writes: “6/100”, “0.06”, “three-hundredths and three-hundredths”, and the total as “0.68”. Iron Man rolls a one hundredth. Using a purple marker, she colours in one hundredths segment on Iron Man’s gamecard. On Iron Man’s scorecard, she writes: “1/100”, “0.01”, “one-hundredth” and the total as “0.72”.

The speaker rolls 6 hundredths. On her gamecard, she colours in 5 hundredth segments, and another single hundredth segment. On her scorecard she writes: “6/100”, “0.06”, “five-hundredths and one-hundredth” and the total as “0.74”. Iron Man rolls 6 hundredths. She colours in 2 lots of 3 hundredths segments. On Iron Man’s scorecard, she writes: “6/100”, “0.06”, “three-hundredths and three-hundredths” and the total as “0.78”.]

### Speaker

So, Iron Man and I have been playing for a while, and have a look at our game boards. Yes, I think so too mathematicians, I think it's really close. Now I've just spun tenths and then rolled four, so four tenths. But looking at my game board, I've got a bit of a problem. Can you say what it is, mathematicians?

[The speaker points to the empty segments of her gamecard.]

### Speaker

Yeah, I only have two tenths left to fill and then four and two hundredths, so six hundredths. So the space I've got available is two tenths and six hundredths or twenty six hundredths. And I need to be able to fill forty hundredths, so four tenths, and I don't have enough room, so guess what that means? Aha, it does. I missed my turn and it's back over to Iron Man. Oh, and look how close we are, who do you think's winning at the moment, mathematicians? Yes, I agree, Iron Man, I think you are winning because have a look at this.

[The speaker points to the remaining empty segments of Iron Man’s gamecard, then the empty segments of her own gamecard.]

### Speaker

He's got two tenths left to fill and so do I. But then I've got six hundredths and he's only got, yes two. So that's a difference of four hundredths, Iron Man, you are winning by four little hundredths. Alright, we're going to continue playing and battle it out to find out who's going to be the ultimate champion of Colour in Decimat. And it's over to you mathematicians, have fun playing with mathematics.

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

### Speaker

So, what's some of the mathematics?

[Text over a white background: Our place value system is called a base-ten system. It’s a base-ten system because each time we get a collection of ten we regroup and rename it. Sometimes, we might talk about how 10 tenths is renamed as 1 whole and 10 hundredths is renamed as 1 tenth.

· We can see this clearly in our Decimat when 1 whole is divided into ten equal parts (tenths) and 1 tenth is divided into ten equal parts (hundredths).

· This means there are 10 hundredths inside 1 tenth and 10 tenths inside 1 one.]

### Speaker

Our place value system is called a base-ten-system, and it's a base-ten-system because each time we get a collection of ten, we regroup and rename it. Sometimes we might talk about how ten tenths is renamed as one whole, and ten hundredths is renamed as one tenth.

[An image appears on-screen. It depicts a blank sheet of paper and a yellow card with text, “ones”.]

### Speaker

We can see this really clearly in our decimat when one whole is divided into ten equal parts which we rename is tenths.

[Another image appears. It depicts a sheet of paper, divided into ten equal segments above two yellow cards with text, “ones” and “tenths”. Another image appears. It depicts a sheet of paper that has been divided into ten equal segments. One of those segments has been divided into a further ten equal segments. Yellow cards below feature the text: “ones”, “tenths”, and “hundredths”.]

### Speaker

And one tenth is divided into ten equal parts or hundredths. This means there are ten hundredths inside one tenth and ten tenths inside one one.

[Text over a white background: We can represent decimal fractions in different ways and they still have the same value. For example:

5-tenths is equivalent in value to 0.5 and 5/10 and 50/100.

An image shows a partially coloured Decimat gameboard. On the gamecard, 5 tenths segments have been coloured green and 2 hundredths segments have been coloured purple. The first line of the scorecard reads: “2/100”, “0.02”, “two-hundredths”, and the total is “0.02”. The second line of the scorecard is incomplete. It reads: “5/10”, “0.5” and “five-tenths”.]

### Speaker

We also know that we can represent decimal fractions in lots of different ways and they still have the same value.

[In the image, the 5 green segments on the gamecard are outlined by a yellow square. On the scorecard, yellow circles appear around the text: “five-tenths”, “5/10” and “0.5”.]

### Speaker

For example, five tenths is equivalent in value to 0.5 and five tenths written as a fraction.

[Black lines appear over the green segments on the gamecard. They divide each of the segments of tenths into hundredths.]

### Speaker

And we can also see that five tenths is equivalent to fifty hundredths.

[Text over a white background: As mathematician, we can think flexibly about numbers by partitioning and renaming to help us work with them. For example:

· We can partition six0hundredths into five-hundredths and one-hundredth

· We can also rename eleven-hundredths as one-tenth and one-hundredth or 0.11

An image on screen shows Iron Man’s partially completed Decimate gameboard. On the gamecard, 5 hundredth segments have been coloured blue. Corresponding text on the scorecard reads: “5/100”, “0.05”, “five-hundredths” and “0.05”. 5 hundredths segments and an additional one hundredth segment shave all been coloured red. Corresponding text on the scorecard reads: “6/100”, “0.06”, “five-hundredths and one-hundredth” and “11/100 = 1/10 and 1/100 0.11”.]

### Speaker

As mathematicians, we can think flexibly about numbers by partitioning and renaming to help us work with them.

[A yellow circle appears around the fraction “6/100”. Yellow squares appear over the 6 red hundredth segments on the gamecard.]

### Speaker

For example, we can partition six hundredths into five hundredths and one more hundredths.

[Yellow squares appear around all of the coloured hundredth segments on the gamecard. A yellow circle appears on the scorecard, around the fraction “11/100”. The green squares fade away. AA green square appears around a whole, coloured tenth and another green square appears around a single coloured hundredth. A green circle appears over the text on the total column of the scorecard, “11/100 = 1/10 and 1/100 0.11”.]

### Speaker

We can also rename our eleven hundredths as one tenth and one hundredth, or 0.11.

[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]

## How to play

- Take turns to roll the dice and spin the spinner and fill in the game board. For example,
If a 2 is rolled and ‘hundredths’ are spun, we record our move as a fraction (2/100) and as a decimal (0.02).

Colour in on the game board.

Complete the 'What I filled' column in words. For example, 'two-hundredths'.

Calculate the total and record as a decimal.

Use a different coloured marker or pen to fill in the game board for each turn.

The winner is the first player to fill in 1-whole (their entire gameboard) or the player who’s gameboard is closest to 1-whole after 10 spins.

If a player spins a fraction that won’t fit into the available space, they miss their turn.

You can partition a roll in equivalent ways. For example, I rolled 3-tenths but I only have 2-tenths left empty, and some hundredths too.

I can partition my 3-tenths as 2-tenths, 8-hundredths and 2-hundredths more to colour in 3-tenths of my gameboard in total.

- Looking for another way to play? Try this!

## Watch

Watch Colour in decimats – part 3 video (0:46) to learn how to play.

[Text on a white background: Play again with thousandths.

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

### Speaker

You might like to play Colour in Decimats again, but this time with thousandths.

[An image appears on screen. It depicts a Decimats game board. A piece of paper has been divided into ten even segments. One of the tenths segments has been partitioned into ten smaller segments. One of those hundredths segments has been further partitioned into ten smaller segments.

Below are four blue cards with text: “ones”, “tenths”, “hundredths”, and “thousandths”.]

### Speaker

If you do, you'll need to create a different gameboard that looks like this. What do you notice?

[A red square appears around the tenth segment that has been partitioned into hundredths. The red square shrinks and then outlines the hundredth segment that has been partitioned into thousandths.]

### Speaker

Yes, I've partitioned one of my hundredths into ten equal parts or thousandths.

[An image of a spinner appears on screen. Three of the segment are labelled, “thousandths”, two of the segments are labelled “hundredths”, and one of the segments is labelled “tenths”.]

### Speaker

You'll also need to adjust your spinner, so that you have three sections labelled thousandths, two sections labelled hundredths, and one section labelled tenths. It's over to you, mathematicians!

[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]

## Discuss and reflect

What was the difference between the final game board totals for each player? (By how much did the player win?)

How could you prove it?

What are some similarities and differences between your two game boards?