# Remainders game

Stage 2 – A thinking mathematically context for practise focused on developing flexible multiplicative strategies and using inverse operations

## 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-MR-01
• MA2-MR-02

## Collect resources

You will need:

• a pencil

• paper

• 24 counters each

• a dice

• 6 squares of paper.

## Watch

Watch Remainders game video (7:23).

Collect remainders when creating groups of objects.

### Transcript of Remainders game

(Duration: 7 minutes and 23 seconds)

[Title over a navy-blue background: Remainders game. In the lower left-hand corner of the screen is the waratah of the NSW Government logo. Small font text in the upper left-hand corner reads: NSW Department of Education.

A number of items are laid out over a large white sheet of paper. In the middle, is a stack of yellow square of paper and a blue, 6-sided dice. On either side, are two sheets of blue paper. A black marker pen rests on the sheet of paper on the left. Above each sheet of paper, is a handful of dry pasta shells.]

Hi, Barbara.

Hi, Michelle.

### Michelle

How are you today?

### Barbara

I'm very well. How are you?

### Michelle

Great. We are gonna play a game called the Remainders Game.

OK.

### Michelle

OK. So we each have 24 counters, and we're using dried pasta, non-edible.

### Barbara

Well, if you cook it, it is. (LAUGHS)

I don't know.

### Barbara

That can be the prize.

### Michelle

And we have six squares of paper that we can use to help us do some calculating if we need it. And I have six because we're using a one to six-sided dice. OK, so we could change, like if we use a spinner to five we would just have five, for example, so we can change it as we like. So, you roll and I'll tell you how to play.

[On the right of screen, Barbara rolls a 3.]

### Barbara

So, I've rolled a three.

### Michelle

So, what we need to think about is dividing your 24 counters into three equal groups.

[Michelle arranges 3 yellow squares of paper on Barbara’s side of the work area.]

### Michelle

So, you can put these out if you like, they are your groups, to help you move the pasta into if you want to do that or to help you visualise. You can also just use what you know if you have some number facts as well.

### Barbara

So, I know that three fives are 15.

### Michelle

OK.

[Barbara takes 5 pasta shells from her pile and puts them on one of the squares of paper. She places another 5 shells on each of the other 2 squares of paper.]

### Barbara

So, I can make those straight away without having to count them out.

Three fives.

### Barbara

OK, I should get that right. OK, that's five and that's five. And now I've got nine left.

[Barbara adds 3 more shells to each of the squares of paper.]

### Michelle

Oh yeah, another three in each group, is that what you are thinking?

### Barbara

Yeah, exactly.

[On the sheet of paper on the left side of the screen, Michelle writes “24 ÷ 3 = 8”.]

### Barbara

So, if I was gonna do 24 divided by three, the answer is eight.

### Michelle

Eight in each group, and for you there's no remainder. So, you don't get to keep any of the pasta just yet. So, put it back into your pile.

[Barbara and Michelle put the pasta shells back into a pile.]

### Barbara

OK, so in this game we want remainders, right?

### Michelle

You want reminders. So, I'm gonna put this here while I roll.

[Michelle moves her sheet of paper. She rolls the dice. It lands on 4.]

### Michelle

And I've got a four, and I actually already know that 24 divided by four will have no remainders.

### Barbara

How do you know that?

### Michelle

Because half of 24 is 12, and half of 12 is six and that's the same as dividing by four. So, no remainders for me either, but you need to record my move.

[On her sheet of paper, Barbara writes “24 ÷ 4 = 6”.]

### Michelle

So, 24 shared into four equal groups, this is why mathematicians invented symbols, is equivalent in value to six in each group.

### Barbara

OK, so no remainders.

### Michelle

No remainders for me either. Your go.

[Barbara moves her sheet of paper off screen, and Michelle brings her own sheet of paper back to the work area. Barbara rolls the dice. It lands on 5.]

### Barbara

Five. Oh, five makes me happy because if I have five groups...

[Barbara lays out 5 yellow squares and arranges them like the dots on the 5 face of a dice.]

### Barbara

I'll put these out but I don't think I need to move the pasta this time. I'll put it like this.

### Michelle

Oh yeah, like a dice, nice.

### Barbara

OK, so what I'm imagining is four in each one and that's gonna give me 20 and then I'm gonna have four left over.

Oh, yes.

### Barbara

Do you agree? Do you want me to make it?

### Michelle

No, I agree with you.

[Michelle writes “24 ÷ 5 = 4”.]

### Michelle

So, you're saying 24 shared equally into five equal groups means you'd have four, because four fives is 20.

### Barbara

Yep, that's right.

[Michelle moves 21 of Barbara’s pasta shells, leaving 4 behind.]

### Michelle

And there'd be four left over, this many left over.

### Barbara

That's right, yeah.

[Michelle adds to the equation. It now reads “24 ÷ 5 = 4 r 4.”]

### Michelle

And that would mean that you can't put an equal number in each group. So, they're the leftovers. So, we call them remainders.

### Barbara

Four in each group and then four remainders.

[Michelle moves the 4 shells over to the right of screen.]

### Michelle

So, what that means now is that you get to keep those four counters.

### Barbara

OK, great.

[Michelle moves the remaining 21 pasta shells on Barbara’s side back to their original position.]

### Michelle

And they still are in play.

### Barbara

Oh, so I've got less counters now.

### Michelle

So, next time you start, you're starting with 20.

[On her sheet of paper, Michelle writes “20”.]

### Barbara

Oh, OK. Well, that was clever writing it down so we don't forget.

[Michelle moves her sheet of paper off screen. She rolls the dice. It lands on 5.]

### Michelle

Yeah. Alright, my turn. Oh, hey, this is nice. So, I can actually just use your reasoning, which is if I had my fives out like this.

[Michelle arranges 5 yellow squares in the same pattern as before.]

### Michelle

If I had four on each one, that would be 20.

### Barbara

Yes.

[Michelle moves 4 pasta shells away from the main group.]

### Michelle

And because this is 24, the difference between 24 and 20 is four and I can't equally share four into five groups without fractioning them. So, that gives me a remainder of four.

### Barbara

Fantastic. So, 24...

[Barbara writes “24 ÷ 5 = 4 r 4”.]

### Michelle

Shared equally into five equal groups means there is four in each group with four left over.

### Barbara

And we call that a remainder. OK, and now the next one.

[Below the equation, Michelle writes “20”.]

### Michelle

You are starting at 20.

### Barbara

OK, so we are exactly the same at the moment.

Yeah.

### Barbara

I'll even put this one because I know...

### Michelle

[Barbara removes her sheet of paper from the screen as Michelle returns her own to the work area. Barbara rolls the dice. It lands on 5.]

### Michelle

Oh, now five is not a good roll.

### Barbara

No, it's not. It was so good before.

It was.

### Barbara

OK, well, we already covered that, isn't it?

[Michelle writes “20 ÷ 5 = 4”.]

### Barbara

If I have five groups, there will be four in each group and I'll have nothing. There'll be no remainders. Nothing left over.

### Michelle

Partitions equally. OK, my go. Now, I don't want a five. Before I liked a five. Now I'd like a six or three.

[Michelle rolls the dice. It lands on 4.]

### Michelle

And I don't want a four either.

### Barbara

No, you don't want a four.

### Michelle

Because four fives is equivalent to 20, so there's no remainders. No leftovers.

### Barbara

So, 20?

[Barbara writes “20 ÷ 4 = 5”.]

### Michelle

Yeah, shared into four equal groups is five in each group. OK, your turn.

### Barbara

OK. So, I don't want a four, I don't want a five. I'd like a six.

### Michelle

Six would be good, or three. (LAUGHTER)

[Barbara rolls the dice. It lands on 5. Michelle writes “20 ÷ 5 = 4”.]

### Michelle

So, 20 shared into five equal groups is equivalent to four in each group.

### Michelle

That's a good known fact for us now.

[Michelle rolls the dice. It lands on 3.]

### Michelle

Oh, and a three. I like this.

[Michelle lays out 3 yellow squares of paper.]

### Michelle

So, I might put my three groups out just so I can get you to visualise with me. So, what I'm gonna think about is, I know I have 20 left and I know that can't be divisible by three.

### Barbara

OK, how do you know that?

### Michelle

Because it would be 21. Because seven threes is 21. So, 20 can't be divisible by three because there's only a difference of one between 21. So, I will have leftovers. So, from 21, knowing 21, if I take one more group away, that would be 18.

### Barbara

OK, that makes sense.

### Michelle

Yeah, so that would mean that from 18 shared into three groups, I could count up if I wanted. So, I could say 3, 6, 9, 12, 15, 18.

### Barbara

Six in each group.

### Michelle

So, I'm gonna make it now.

[Michelle places 6 pasta shells on each of the 3 yellow squares. Barbara writes “20 ÷ 3 = 6 r 2”. Below that, she writes “18”.

Barbara then rolls the dice. It lands on 5. Michelle writes “20 ÷ 5 += 4”. Michelle then rolls a 4. She lays out 4 yellow squares of paper. She places 4 pasta shells on each of the squares of paper. There are 2 remainders. Barbara writes “18 ÷ 4 = 4 r 2”.

Barbara and Michelle continue to play the game in fast motion. The final equation on Michelle’s sheet of paper reads “12 ÷ 3 = 4”. The final equation on Barbara’s sheet of paper reads “3 ÷ 2 = 1 r 1”.]

### Michelle

Alright. So, Barbara, we've come to realise something in the game that it's the person who gets to two is the winner.

### Barbara

Oh, OK. Yeah, because after that you can't go anymore, right?

### Michelle

You can't go, because even if I rolled a two, then it would be equally divisible. There's no possible remainders I can get.

### Barbara

And a one as well.

### Michelle

And a one is also. So, it's the first person to get down to two.

Oh, OK.

### Michelle

And look how many gos it took us, and look, all of a sudden, all your fives over here, they just disappeared and weren't helping you.

### Barbara

I really wanted a five over here. I think I wasted all my fives.

### Michelle

This is a really good game. And over to you, mathematicians, to adapt.

[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]