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 (2022) © NSW Education Standards Authority (NESA) for and on behalf of the Crown in right of the State of New South Wales, 2024.
Outcomes
- 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).
(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.]
Michelle
Hi, Barbara.
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.
Barbara
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)
Michelle
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.
Michelle
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.
Michelle
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.
Michelle
Yeah.
Barbara
I'll even put this one because I know...
Michelle
OK, your turn.
[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.
Michelle
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.
Barbara
OK, your turn.
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.
Barbara
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]
Instructions
- Start with a collection of 24 things each.
- Players take it in turns to roll the dice to determine how many groups their collection needs to be shared into.
- The player works out the solution to their division problem and explain their thinking to their partner who records their move.
- If the product cannot be evenly divided, players keep the remainders, and the collection of counters they were working with is reduced.
- The player works out the solution to their division problem and explain their thinking to their partner who records their move.
- The player who reduces their collection to only 2 counters is declared the winner.