• MAO-WM-01 
• MA3-AR-01
• MA3-MR-01

[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]