# Double decker bus

Double decker bus is a thinking mathematically targeted teaching opportunity focused on exploring efficient strategies to combine quantities.

In partnership with ReSolve.

## 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
- MA1-RWN-01
- MA1-RWN-02
- MA1-CSQ-01

## Collect resources

You will need:

- pencils or markers
- your mathematics workbook.

## Double decker bus part 1

Watch Double decker bus part 1 video (7:42).

[In a vector-style graphic, a red double decker bus on a city street. Black text on the bus reads ‘reSolve’. A yellow building has the NSW Government red ‘waratah’ logo on it. At the top, black text reads ‘The reSolve Double Decker Bus’.]

### Speaker 1

Hello there, mathematicians. Welcome back. Today I invited my great mathematical friend, Kristen Tripet from reSolve at the Australian Academy of Science, to send us a task today. And she sent the reSolve Double Decker Bus.

[Black text on a white background reads ‘The reSolve Double Decker Bus’. Grey text below reads ‘How many children?’ At the bottom, the NSW Government logo and a yellow banner that has the ‘reSolve’ logo in white on it.]

### Speaker 1

So, let's have a look at what the problem is. Where would you like to sit on the reSolve double decker bus?

[The red double decker bus on the city street from earlier. Cartoon boys and girls gradually appear in the empty windows of the bus when mentioned.]

### Speaker 2

Luke wants to sit up the top, right at the front. Lara wants to sit at the top, but up the back. Leo would like to drive the bus and James wants to stand at the door. Slowly, the bus fills with all the other children.

[The red double decker bus from earlier. Black text alongside reads ‘7 children on the top’ and ‘8 children on the bottom’. Further text below (read by speaker).]

### Speaker 2

There are 7 children on the top of the double decker bus, and 8 children on the bottom. How many children altogether?

### Speaker 1

So, over to you, mathematicians. How many children altogether? Get out your pencils, your paper, and write down your ideas. And remember, because we're thinking like mathematicians, you might have more than one different way to solve this problem. So, press pause here, and over to you to do some thinking.

OK, mathematicians, welcome back. How did you go?

[The red double decker bus on the city street from earlier.]

### Speaker 1

OK, let's have a look at some of the strategies for solving the double decker bus problem.

[Black text on a white background reads ‘The reSolve Double Decker Bus – Looking at strategies.]

### Speaker 1

Yep, and some of these might be like the ways that you've thought, and you might have had some different strategies, too. So, let's have a look.

[Black text on a white background reads ‘Zoe’s Strategy’. On the left, a cartoon girl has pig-tails and wears a purple shirt. A speech bubble above her head reads ‘I counted all the children on the bus’. On the right, the red double decker bus has children in the windows. White numerals are placed below each child (as explained by speaker.]

### Speaker 1

Here's what Zoe was thinking. She thought, "I could count all of the children on the bus."

[On the left, Zoe has two speech bubbles. The top bubble reads ‘I counted 7 beads on the top and 8 on the bottom’. The bottom bubble reads ‘Then I counted how many beads there were altogether’. The bubbles are positioned between a Rekenrek calculating frame at the top and another at the bottom (as explained by speaker). The Rekenrek frame has two parallel wires that have red beads and white beads on each wire that slide from left to right.]

### Speaker 1

And what Zoe did, if we can use a Rekenrek to model her thinking, was that she counted everything first. So she counted 1, 2, 3, 4, 5, 6, 7, on the top of the bus. See that? And then she counted on the bottom of the bus that there were 8 children, and then she counted them again all together. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15. And so, she used a count-all strategy, and we can have a look at what that looks like on the Rekenrek. Let's see it happening in action. So, here's Zoe. 1, 2, 3, 4, 5, 6, 7. 1, 2, 3, 4, 5, 6, 7, 8. And then she counted them again. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15. 15 to what she counted all to work out.

[Below the Rekenrek frame red text reads ‘Yikes! That is a lot of counting’.]

### Speaker 1

And yes, that is quite a lot of counting, isn't it? Did you come up with a counting strategy? OK. Should we see another one? Excellent idea. Let's have a look at what Ahmed was thinking about.

[Black text on a white background reads ‘Ahmed’s Strategy’. On the left, a cartoon boy has black hair and wears a blue shirt. A speech bubble below him reads ‘I made a group of 10 using all 7 children on the top and 3 children from the bottom. There were 5 more children on the bottom’. On the right, the red double decker bus has children in the windows. Above the bus, blue text reads ‘7 and 3 make 10’ and below the bus, green text reads ‘…and 5 more make 15’.]

### Speaker 1

Aha, did you think about this, too? Ahmed was thinking about this idea of collecting a 10. So, he thought, if I could think about 7 at the top, and if I, you know, collect 3 more from down the bottom, that makes a group of 10. And then there's 5 left, which would make 15.

[Ahmed’s strategy is shown on the Rekenrek counting frame (as explained by speaker).]

### Speaker 1

So, if we model his strategy on a Rekenrek, here's what it would look like in pictures, where he had 7 on the top row of the Rekenrek, and 8 down the bottom. And he thought, if I move this chunk of 3 across, and bring the other chunk of 3, my 7 and my 8 becomes 10 and 5. Let's have a look at that. So, here's the 7 and the 8. And then what he was thinking was, hold on a second, this chunk of 3 down the bottom, I could exchange for a chunk of 3 at the top. And now I have one 10 at the top and 5 more down the bottom. Yes, which makes 15.

So, let's have a look at this in a different representation.

[On the left, a vector-style image of a set of balance scales. There are white counters mark 1-10 on each side. 4 yellow weights are positioned on the right of it. The weights move to a number on the scale when mentioned by speaker. Mathematics on the right (as read by speaker).]

### Speaker 1

Here we have a whole bunch of numbers that could look a little confusing, but also a balance scale. Yes. So, here's the 7, and here's the 8 to represent what was on the top and the bottom rows of the bus. Then what happened was, he kept the 7, but thought about 8 as 3 and 5 more, and then said, well, I know 3 and 7 actually could be exchanged to put a number on a different peg, so I could swap my 3 and my 7, and change that into a 10. And so, now I can see that 7 and 8 is equivalent in value to 5 and 10, and we call one 10 and 5 more 15. And I can tell that because the sides of my balance scale now balance out. Did you use a strategy like this too? Where you thought about grouping 10 and some more? Oh, OK. You had a different way? Let's see if it's like Penny's thinking.

[Black text on a white background reads ‘Penny’s Strategy’. On the left, a cartoon girl has brown hair and wears a red shirt. A speech bubble above her head reads ‘7 children from the top and 7 children from the bottom make 14. There is just one more child on bottom’. On the right, the red double decker bus has children in the windows.]

### Speaker 1

So, what Penny was thinking about, is that there's 7 children at the top and there's actually 7 children at the bottom. Yes, because 8 is composed of 7 and one more. So, then she said, well, I know double 7 is 14 and one more makes 15. That's right. So, she used a near-double strategy.

[Penny’s strategy is shown on the Rekenrek counting frame (as explained by speaker).]

### Speaker 1

And let's have a look at what this would look like using the Rekenrek. So, here's her collecting her 7 because she knows it's made up of 5 and 2. And then she collected the 8 as 5 and 3, and then she partitioned off the one, went double 7 is 14, put the one back to get 15.

So, there are 3 really interesting ways of solving the double decker bus problem from Kristen at reSolve. Yes, and you might have had another different strategy. So, let's talk quickly about what the mathematics was that we saw today.

[White text on a blue background reads ‘What’s some of the mathematics?’ In the bottom right, a white NSW Government ‘waratah’ logo.]

### Speaker 1

We know this is a really important question to ask as a mathematician.

[Black text on a white background reads ‘What’s (some of) the mathematics?’ Below, black text bullet points (read by speaker) along with colour images of ‘Zoe’s strategy’ and ‘Ahmed’s strategy’ from earlier.]

### Speaker 1

So, there's a couple of really important points that we should talk about. There are lots of different strategies that you can use to solve the same problem. We found 3 today that we talked about, but you might have had others, which is really awesome.

Some strategies are more efficient than others. So, with Zoe's strategy, even if she just counted all the people on the bus that would have taken her 15 steps, whereas Ahmed's strategy took 3 steps. And that's how we know something's efficient. It's not based on how fast you do it, it's based on the number of steps you take.

[Black text on a white background reads ‘What’s (some of) the mathematics?’ Below, black text bullet points (read by speaker) along with 2 colour images of the balance scales from earlier (as explained by speaker).]

### Speaker 1

And here's the other really cool thing that we saw today, that as the mathematicians, we get to be in charge of the numbers. Yeah. So, even when I see something like 8 and 7, I can rethink that as 5 and 10 or 3 and 5 and 7. So, I can use numbers in a way that makes sense to my brain and connect to knowledge that I already have, so I can use it.

Alright, mathematicians, great work today. I look forward to chatting to you soon. And until then, happy mathing.

[The NSW Government waratah logo turns briefly in the middle of various circles coloured blue, red, white and black. A copyright symbol and small blue text below it reads ‘State of New South Wales (Department of Education), 2021.’]

[End of transcript]

## Instructions

- How many children altogether?
- Explain how you worked out the total number of children.
- Are there other ways you can solve this problem?

## Double decker bus part 2

Watch Double decker bus part 2 video (2:18).

[Black text on a white background reads ‘The reSolve Double Decker Bus’. Grey text below reads ‘Which strategy would you use?’ At the bottom, the NSW Government red ‘waratah’ logo and a yellow banner that has a ‘reSolve’ logo in white on it.]

### Speaker

Hello there, mathematicians! Welcome back to another delightful day of reasoning, and problem solving, and creative thinking. So my friend Kristen Tripet sent a follow up question for us for today. And before we get into it, let's remind ourselves of some things that we realised yesterday.

[Black text on a white background reads ‘What’s (some of) the mathematics?’ Below, black text bullet points (read by speaker) along with colour images labelled ‘Zoe’s strategy took 15 steps’ and ‘Ahmed’s strategy took 3 steps’. A cartoon image of each child is positioned alongside 2 Rekenrek calculating frames that have red beads and white beads together on 2 parallel wires. Further explained by speaker.]

### Speaker

So we noticed yesterday that there's lots of different strategies that you could use to solve the same problem. And some strategies are more efficient than others. Yes, well-remembered, that efficiency is about the number of steps that you have to take, not about how fast something is. Well not just about that. Mm hmm. And the other thing that was really helpful for us yesterday that you're going to have to use today, is this idea that as the mathematicians, we get to be in charge of the numbers.

[Black text on a white background reads ‘What’s (some of) the mathematics?’ Below, black text bullet points (read by speaker) along with 2 colour images of some green balance scales with white number markers and yellow weights on them (as explained by speaker).]

### Speaker

Yep, so we can see 8 and 7, and rethink of that as 5 and 10. We can use these ideas of equivalence, or we can see 8 and 7 and rethink of that as 7, and 3, and 5. Mm hmm. Because we know 8 can be broken up into chunks of 3 and 5.

OK, here's Kristen's challenge for you today, mathematicians.

[Black text reads ‘Which strategy would you use if there was:’ Bullet points on the left read by speaker. On the right, a vector-style image of a red double decker bus. Black text on the side reads ‘reSolve’.]

### Speaker

Which strategy would you use if you were solving these problems? What if on the bus, there are 7 on the top and 6 on the bottom? What strategy would you use? What if there are 5 on the top and 8 on the bottom? What if there are 12 on the top and 9 on the bottom? And what about 9 on the top and 8 on the bottom? And what about 7 on the top and 9 on the bottom?

So mathematicians, I'd like you to choose 3 of your favourite scenarios or questions there, and have a go of thinking about what are 2, at least 2, different strategies that you could use to solve your 3 favourite problems? OK, over to you mathematicians!

[White text on a blue background reads ‘Over to you, mathematicians!’ In the bottom right, a white NSW Government ‘waratah’ logo.]

[The NSW Government waratah logo turns briefly in the middle of various circles coloured blue, red, white and black. A copyright symbol and small blue text below it reads ‘State of New South Wales (Department of Education), 2021.’]

[End of transcript]

## Instructions

- Choose 3 of your favourite scenarios.
- Draw a picture to show what strategy you would use to solve the 3 problems you choose.
- What are some different ways to sort your collection?
- Some strategies are more efficient than others. Remember how Zoe’s strategy took 15 steps and Ahmed’s strategy 3 steps. You might like to use what we learnt to help you here