# Leftovers

Stage 3 – A thinking mathematically context for practise focussed 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 © 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
• 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 Leftover video (7:56).

Investigate dividing quantities to get remainders.

### Transcript of Leftover

[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.]

Hello, Barbara.

### Barbara

Hello, Michelle. How are you?

### Michelle

I am great. How are you?

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.

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.

OK.

### Michelle

And what we wanna do is think about 100 divided by something. But we wanna create remainders or leftovers.

Oh, OK.

Yes.

### Barbara

So we don't want it to be, we don't wanna do 100 divided by 10, because...

No.

..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.

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.

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.

Right.

### Michelle

So then would you do 30 divided by seven...

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 =]

OK.

### Barbara

OK. So if I had, ooh, do I feel confident? OK. So I'm gonna think about it as multiplication.

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.]

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.

Yeah.

### Michelle

And you get to keep the two points, Barbara.

[Michelle circles the r 2 and writes B next to it.]

OK, great.

### Michelle

And now we have a new starting number, which is 100 minus the leftovers.

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.]

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.

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.

OK.

### Michelle

So, then I need to think about 98 divided by five.

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.

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]

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.

OK.

### Michelle

Oh, I should have written 19, not 15.

### Barbara

So I'm thinking 10 could be good.

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.]

Yeah.

### Michelle

...Is 11 remainder seven, is that right?

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.

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.

Oh, yeah.

OK, so.

### Michelle

What would be eight nines?

72?

Oh, yeah.

### Barbara

Yeah, that's pretty good.

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.]

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...

OK.

### Michelle

I think, or a six. Actually, three won't be any good because three times 20 would be 60.

### 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.

Oh, OK.

### Michelle

But we then have to calculate who had the largest number of leftovers.

OK.

### Michelle

And it'll be you 'cause you had...

### 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.

OK, yeah.

### Michelle

You could change the starting number, so you could start with 50.

OK.

If you want it.

### Barbara

And make it a little bit...

### Michelle

Yeah, or 20 to get started and get comfortable playing the game.

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.