Leftovers
Stage 3 – a thinking mathematically context for practise focused on developing flexible multiplicative strategies and using inverse operations.
Adapted from Burns, M. (2015). About Teaching Mathematics, 4th ed., Math Solutions.
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
- MA3-AR-01
- MA3-MR-01
Collect resources
You will need:
- measuring tools (for example, a ruler, tape measure, mug, handspan or a teaspoon)
- an object to indicate your height (a stick, spoon or rope)
- writing materials (paper and pencil) .
Watch
Watch the Leftovers video (7:56).
[A title over a navy-blue background: Leftovers. Below the title is text in slightly smaller font: From Marilyn Burns. Small font text in the upper left-hand corner reads: NSW Department of Education. In the lower left-hand corner is the white waratah of the NSW Government logo.
A large white sheet of paper on table.]
Michelle
Hello, Barbara.
Barbara
Hello, Michelle. How are you?
Michelle
I am great. How are you?
Barbara
I'm great.
Michelle
And hello to the other mathematicians out there today.
Barbara
Hello mathematicians out there.
Michelle
We think, we hope you're great also. We are gonna play a game today adapted from Marilyn Burns. It's called leftovers.
Barbara
Oh, OK.
Michelle
It's pretty cool. So we're gonna play a version from 100 and we just need these numbers at the top.
[Michelle writes 1 – 10 at the top of the sheet.]
Michelle
One, two, three, four, five, six, seven, eight, nine and 10.
Barbara
OK.
[Below 1 – 10, Michelle writes 100.]
Michelle
And we're starting at 100.
Barbara
OK.
Michelle
And what we wanna do is think about 100 divided by something. But we wanna create remainders or leftovers.
Barbara
Oh, OK.
Michelle
Yes.
Barbara
So we don't want it to be, we don't wanna do 100 divided by 10, because...
Michelle
No.
Barbara
..then you get.
Michelle
Yes, you'd have no remainders or leftovers.
Barbara
OK. So you want the leftovers?
Michelle
Yeah. So, let's just play and then we'll figure out how to get better as we play.
Barbara
OK, sounds good.
Michelle
So, do you wanna go first?
Barbara
Yeah. OK, so I'm not gonna do 10 or five. I might do 100 divided by, maybe seven.
[Under the 100, Michelle writes 100 ÷ 7 = ]
Michelle
OK. So 100 divided by seven. And how would you work that out?
[Michelle places a blue paper below the text.]
Michelle
I'm gonna use this as a working out piece of paper.
[On the blue paper, Michelle writes 70 ÷ 7 = 10.]
Barbara
OK. So, I know that 70 divided by seven is 10.
Michelle
Yeah.
Barbara
So I guess I have to just try, maybe I can build up until I get as close to 100 as I can.
Michelle
Yeah. 'Cause you can partition the number, right?
Barbara
Yeah.
[Above 70, Michelle writes 100. Below 100, she writes 30. She draws lines from 100 to 30 and 70.]
Michelle
So what you're saying is that 100 is made up of 70 and 30 more.
Barbara
Right.
Michelle
So then would you do 30 divided by seven...
Barbara
Yeah.
Michelle
..or would you repartition the 30?
Barbara
No, I think I would do 30 divided by seven because then I can just work it out and see how many are leftover. I feel quite confident to do that.
[Below the equation, Michelle writes, 30 ÷ 7 =]
Michelle
OK.
Barbara
OK. So if I had, ooh, do I feel confident? OK. So I'm gonna think about it as multiplication.
Michelle
Yeah.
Barbara
So seven threes are 21, so I can do more than that. So, I don't want to do that yet.
Michelle
Yeah. If you added 21 and seven more, that would be 28.
Barbara
Oh, that's what I wanna do, yeah.
Michelle
So what you're thinking actually, is it that can be partitioned into 28 and two more?
[Under 30, Michelle writes 28 and 2.]
Barbara
Yeah.
Michelle
And then 28 divided by seven is four.
[Michelle writes 28 ÷ 7 = 4.]
Michelle
Yeah? 'Cause seven times four is 28.
Barbara
So I've got two leftover.
[Michelle crosses out the equation above.]
Michelle
And then there's the remainder of two.
[Michelle writes 2, and circles it.]
Barbara
OK.
[Michelle moves the paper to the left side.]
Michelle
Yeah. So that means 100 divided by seven is four, remainder two.
[Next to 100 ÷ 7, she writes: = 4 r 2.
Barbara
No, no, it's 14.
[Michelle adds a 1 in front of 4.]
Michelle
14, remainder two.
Barbara
Yeah.
Michelle
And you get to keep the two points, Barbara.
[Michelle circles the r 2 and writes B next to it.]
Barbara
OK, great.
Michelle
And now we have a new starting number, which is 100 minus the leftovers.
Barbara
OK.
Michelle
So it's 98 now…
[Under the equation, Michelle writes 98.]
Michelle
…and we can't use seven again.
[She crosses out 7.]
Barbara
OK.
Michelle
So, I now need to think of 98 divided by what would leave a lot of remainders. So what I know is that it definitely won't be equally partitioned by five.
Barbara
No, it won't.
Michelle
Because if I divided 98, if the number ends in a zero or five, I know it can be partitioned equally by five, divided, so I might use five.
Barbara
OK.
Michelle
So, then I need to think about 98 divided by five.
Barbara
OK, yeah.
Michelle
And what I would use is my knowledge to know that if I had, if I was thinking about 50, I would need 10 fives to make 50. So it's gotta be less than 20.
Barbara
OK, yeah.
Michelle
So I think it would be 19.
Barbara
Yeah, because it won't...
Michelle
And maybe three leftover.
Barbara
OK, yep, that makes sense.
Michelle
So, I'll record it 'cause I'm over here, but technically you would record it. So 98 divided by five is 15 with three leftover.
[Next to 98, Michelle writes ÷ 5 = 15 r 3. She circles the r 3]
Barbara
It sounds good.
Michelle
And I get to keep the three…
[Next to the equation, Michelle writes M. Under the equation, Michelle writes 95.]
Michelle
…and now we start at 95.
Barbara
OK.
Michelle
Oh, I should have written 19, not 15.
Barbara
So I'm thinking 10 could be good.
Michelle
Oh, yeah.
Barbara
Then I would have five leftover. But I'm thinking, could I do better than that? So 10 and nine would be the same effect, I would have five leftover. What about eight? Could I do better with eight? So, I could do...
Michelle
Well, eight 10s would be 80.
Barbara
Yeah. And then I could...
Michelle
11 10, 11 eights would be 88. And that would leave a difference of seven.
Barbara
Yeah, which is better. Yeah.
Michelle
OK. So 95 divided by eight...
[Next to 98, Michelle writes ÷ 8 = 11 r 7. She circles the r 7, and writes B next to it.]
Barbara
Yeah.
Michelle
...Is 11 remainder seven, is that right?
Barbara
Yeah.
Michelle
Is that what we said? So, oh, you've now got nine points.
Barbara
Yeah. That's pretty good.
Michelle
Can you see that? So we can't use the eight anymore…
[She crosses out 7.]
Michelle
…and we need to do 95 minus seven.
Barbara
OK. So that would be 88?
[Under the equation, Michelle writes 88 ÷.]
Michelle
88. Well and then I need to think of what I can subtract. So I might use your 10 because I would have leftovers eight.
Barbara
Yeah. That's a good, that's a good strategy.
[Next to 88, Michelle writes ÷ 10 = 8 r 8. She circles the r 8, and writes M next to it.]
Michelle
Yeah. So that equals eight remainder eight, leftovers eight, and now my points. And so now we're at 80.
[Under the equation, Michelle writes 80. She crosses out 10.]
Michelle
And we've used 10.
Barbara
Oh, OK. I think I've got a good idea.
Michelle
OK.
Barbara
Because if I had 81, it would be divisible. Oh no, that would be nine times nine. I'm thinking that nine could be a good one.
Michelle
Oh, yeah.
Barbara
OK, so.
Michelle
What would be eight nines?
Barbara
72?
Michelle
Oh, yeah.
Barbara
Yeah, that's pretty good.
Michelle
Yeah.
Barbara
OK, so, I'm gonna go with that because that's...
Michelle
So divided by nine?
Barbara
Divided by nine.
[Next to 80 ÷, Michelle writes 9 = ]
Michelle
So 80 divided by nine is...
Barbara
We said we wanted 72, so it's eight.
[Next to =, Michelle writes 8 r. She circles the r 8, and writes B next to it.]
Barbara
And then we have a remainder of eight.
Michelle
That's another good remainder.
[She crosses out 9.]
Michelle
So that's nine no longer, and 80 minus eight is 72.
[Under the equation, Michelle writes 72.]
Barbara
OK.
Michelle
So that's where I need to start. I won't use four, because I know eight 10s, eight nines are 72. So I know that 16 fours would be 72. So there'd be no remainders.
Barbara
And you don't wanna use a two either.
Michelle
And I don't wanna use a two. So I'll use a three...
Barbara
OK.
Michelle
I think, or a six. Actually, three won't be any good because three times 20 would be 60.
Barbara
And then 12...
Michelle
And then 12 more, so that doesn't work. And actually, six is no good either. So I'm gonna end up with zero remainders in whatever I do here. Yes. So this is the end of our game, actually, 'cause there's no other moves we can make.
Barbara
Oh, OK.
Michelle
But we then have to calculate who had the largest number of leftovers.
Barbara
OK.
Michelle
And it'll be you 'cause you had...
Barbara
I've had more turns, yeah.
Michelle
So my total score was 11…
[Michelle writes 11 M.]
…because eight and three. And your total score was...
Barbara
17, is it? So 15?
Michelle
Oh, yeah. I was thinking of the two eights together, 17.
[Michelle writes 17 B.]
Michelle
So congratulations Barbara. You won the first round of leftovers.
Barbara
I like this game. My brain feels really sweaty.
Michelle
Yeah. And so you can play if you like, all the way from one to 20.
Barbara
OK, yeah.
Michelle
You could change the starting number, so you could start with 50.
Barbara
OK.
Michelle
If you want it.
Barbara
And make it a little bit...
Michelle
Yeah, or 20 to get started and get comfortable playing the game.
Barbara
Or 200.
Michelle
Or 200. So have fun mathematicians playing leftovers inspired by Marilyn Burns. Oh, and I better fix that error in our recording.
[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
- Write the numbers 1-10 (or 1-20) along the top of your paper.
Record your starting number (we used 100 but you can change the starting number to any number you like).
Player 1 chooses a divisor that will result in leftovers (remainders).
Player 1 works out the solution to their problem (in this case, Barbara worked out 100/7).
Player 1 collects the leftovers (remainders) as points.
The chosen number (in this case, 7) is crossed off the list of options.
A new starting number is determined by subtracting the leftovers from the previous starting number (e.g. 100 - 2 = 98).
Play continues until there are no more moves that can be made.
The winner is the person with the most leftovers.