# Bakery – multiplicative strategies (reSolve)

Stage 3 – A thinking mathematically targeted teaching opportunity focused on developing flexible multiplicative strategies, communicating and reasoning

In partnership with reSolve

## 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

## Outcomes

- MAO-WM-01
- MA3-MR-01

## Collect resources

You will need:

a pencil

your student workbook

cupcake array student sheet from the downloadable resources (PDF 14.9 MB)

## Watch

Watch the reSolve bakery video part 1 (3:03).

[Title over a purple background: re(Solve) Maths by Inquiry. Below the title, is a line of 8 illustrated cupcakes, all decorated with different coloured icing and toppings. In the lower right-hand corner of the screen is the logo of the Australian Academy of Science and the small font text: reSolve: mathematics by Inquiry is funded by the Australian Government Department of Education.]

**Speaker**

Hello there, mathematicians! Now this problem was sent to me by my friend Kristen Tripet at reSolve with the Australian Academy of Science. And in the same way that I like to send you puzzles and problems, and I like to send them out to your teachers as well, I like to be sent them also. And so Kristen sent us this one to explore over the next few days.

[Text on a white background: reSolve Bakery. How many cupcakes? In the lower right-hand corner of the screen is the red waratah and blue text of the NSW Government logo.]

**Speaker**

So it's all about the reSolve Bakery, and you know, which is good because I really like cupcakes.

[Text: Charlie is a baker who has his own cupcake shop. It is a small shop, but very popular! Each day he bakes fresh cupcakes to be sold and the cakes are baked in a tin that looks like this. Beside the text is an image of a cupcake baking tray. It has 4 rows, each with 6 cupcake moulds. Text continues: How many cakes can be baked at one time in this tin? How do you know?]

**Speaker**

So here's a context for us to consider. Charlie is a baker who has his own cupcake shop. It is a small shop, but very popular. Each day he bakes fresh cupcakes to be sold and the cakes are baked in a tin that looks like this. How many cakes can be baked at one time in this tin? Ah, and what's a strategy you used to work that out? OK, so you might have known here that when you, that there's four rows, and there's six in each row, and you might have a known fact that four sixes are 24. Uh huh. Mm hmm, you might have looked at it as two sixes, which is 12, and another two sixes, which is another 12, and you join those and it's 24.

Yeah, you could have actually partitioned it the other way and said, well, I know six fours is 24, and then two fours more, four twos more. Mm hmm. OK, let's keep going with the problem.

[Text: Each day, eight different flavours of cupcakes are made, one tray full of cupcakes for each flavour. There is one tray of chocolate, one of vanilla, and one of red velvet. There is one tray of strawberry, white chocolate raspberry and peppermint-choc. There is even a tray of chocolate marshmallow and one of salted caramel. Below the text is the line of 8 cupcakes, each with different icing and toppings.]

**Speaker**

Each day, eight different flavours of cupcakes are made, one full tray of cupcakes for each flavour. There is one tray of chocolate, one of vanilla, and one of red velvet. There is one tray of strawberry, white chocolate, raspberry, and peppermint choc. There is even a tray of chocolate marshmallow, and one of salted caramel. Oh, this task is making me feel hungry. Do you have a favourite? Mmm, they look really beautiful.

[Text: Charlie bakes eight trays of different flavoured cakes each day. How many cupcakes does Charlie bake each day? Below the text, are 8 groups of 24 cupcakes. Each group of cupcakes has different icing and toppings and is arranged in 4 rows of 6.]

**Speaker**

OK, so here's Charlie's dilemma. Charlie bakes eight trays of different flavoured cupcakes each day. How many cupcakes does Charlie bake?

[Additional text reads: Create a poster to show how you solved the problem. You might like to use a copy of the cupcake array to help explain how your strategy works.]

**Speaker**

Yeah, so, over to you mathematicians now, to create a poster to show how you solve the problem. You might like to use a copy of the cupcake array to help explain how your strategy works. Yes, and because we love being George Polya, I'd like you to think about more than one different way that you could solve the problem.

Yep, you don't need to have the pictures of the cupcakes if you don't need them. But over to you and we'll come back to this tomorrow. Have fun exploring!

[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

Charlie bakes eight trays of different flavoured cakes each day. How many individual cupcakes does Charlie bake each day?

Create a poster to show how you solved the problem. You might like to use a copy of the cupcake array to help explain how your strategy works.

## Collect resources

You will need:

your thinking from reSolve bakery part 1

a pencil

your students workbook.

## Watch

Watch the reSolve bakery video part 2 (6:52).

[Large white text on a purple background reads ‘re(Solve)’. Further vertical white text at the end of it reads ‘MATHS BY INQUIRY’. Below, eight colourful cupcakes with different coloured icing and other toppings. In the bottom right, a black logo has a curved dome-like building and black text that reads ‘Australian Academy of Science’. On the left of it, small black text reads ‘reSolve: Mathematics by Inquiry is funded by the Australian Government Department of Education’.]

**Speaker**

Hi there mathematicians. Welcome back to our task sent to us by Kristen Tripet at reSolve.

[8 groups of cupcakes are arranged according to likeness. Each group is 4 rows by 6 columns. Black text above (as read by speaker). At the bottom, the NSW Government red ‘waratah’ logo and the ‘reSolve – Maths by Inquiry’ logo from earlier on a thin purple banner. Further steps explained by speaker. She uses a ‘spotlight’ tool to highlight areas as mentioned.]

**Speaker**

So, this was the problem she posed for us, remember? And it was really about how many cupcakes does Charlie bake each day? So, we know from the image that there's 8 x 12. Look, here's one 12, a second 12... a third 12, a 4 12, 5, 6, 7, and 8 x 12. And we had to work out how many individual cupcakes is that? How many altogether? What's the product?

[TEXT: reSolve Bakery – Looking at strategies]

[The array of cupcakes from earlier is divided up into groups of 5. Above the array the mathematics is written in black pen. Further information and additional methods/arrays explained by speaker.]

**Speaker**

So, let's have a look at some strategies. So, one thing you could have done because you had the image was actually to count, count all of them by ones. Or you could have done something like skip counting by 5s. And this was pretty easy to do because we had the picture. But it's also pretty inefficient in this context of the problem if we have other knowledge and understanding that we could use.

You could have also have counted in a different way by 10s, maybe instead of by 5s. And look, if you had partitioned up the array, you might have done something like this 10, 20, 30, 40, and you could have kept going.

But we still really wondered are there some other more efficient strategies we could have used, you know if we had the skills or understanding of the mathematics to do it? So, let's have a look at how else we could have thought about this problem.

[In the top left, a cartoon image of a red-haired boy who wears a blue shirt. Black text alongside him (read by speaker). Below, the array of cupcakes (further explained by speaker).]

**Speaker**

So, here's Theo, and this is how he solved the problem. Can you make sense of his thinking from this diagram?

Yeah, OK. So, like you too, I can say that I think he's partitioned some of the cupcakes from the tray, anyway, or he's partitioned what looks for us to be like an array. So, let's have a look because then there's all the symbols around the outside which might make your mind boggle a little bit. So, let's have a look at what he did. Originally this was the array and it was 8x24 or 8 24s. And he was like, well, I'm not entirely sure I know that. So, what he decided to do is use what he knows. And he partitioned out and said, well, I know 8 x 24 is equivalent in value to 8 x 20 and 8 x 4. So, what that means is that 8 x 20 over there on the left, that's 160, 8 x 4 is 32. And if I join that together, I get 192.

Ah-ha. Alright, how else could we have thought about this problem? Let me show you another way.

[The same array of cupcakes has a ‘tree’ of mathematics below it (as explained by speaker). On the right, a second cartoon boy wears glasses and has a green shirt.]

**Speaker**

This is Toby. And what do you think he did to solve the problem?

OK, shall we have a look together? OK, let's see. So, Toby sees the 24, each collection of 24 inside the larger array.

[The 8 groups of the cupcake array are shaded in different colours. Then it becomes 4 groups, then 2 groups and finally just one group/colour (as explained by speaker).]

**Speaker**

Then he doubles, he joins those together. So, he doubles 24 to find 48. Then he joined those together, he doubled 48 to get 96. And then he doubles one more time. So, he has doubled three times to work out it's 192.

OK. Let's have a look at a third different strategy we could have used to solve the problem.

[The same array of cupcakes is grouped by a purple box, red box and green box around it. Above the purple box it reads ‘24 x 2 = 48’; the red ’48 x 2 = 96’; and the green ’96 x 2 = 192’ (as explained by speaker). On the right, a cartoon girl has pig-tails and wears a purple shirt. A purple block of colour fills in each group as mentioned by the speaker.]

**Speaker**

This is Mai, and let's see that, see if you can make sense of what she did. You might be noticing that, too. It's very similar to Toby's way of thinking. Yeah. She started with the 24 and then doubled that to find 48, doubled 48 to get 96, and doubled the 96 to find 192. Yes, so, let's have a look at those two strategies for a moment. What do you notice here that is similar about these two strategies?

[Toby’s strategy alongside Mai’s strategy.]

**Speaker**

OK, and what are some things you noticed that are different? OK, I'm gonna get you to write your ideas in your notebook, and I'm going to give you today's challenge. So, here you go. Thanks to Kristen Tripet at Resolve. Here's the next part of the challenge for us.

[The ‘reSolve cupcake’ slide from earlier and the ‘reSolve Bakery’ slide. Above the line of colourful cupcakes there is a black text question (as read by speaker).]

**Speaker**

Charlie's cupcake shop might be, might only be small, but he takes a lot of orders. His cakes are used for school fundraisers, and they're also a favourite at birthday celebrations. Today is a big day as there are a lot of cakes to bake.

[Black text (read by speaker). On the left, a notepad has a list written on it detailing each person’s order.]

**Speaker**

Amy has put us in a special order. She would like Charlie to bake 2 special flavours for a very special birthday celebration. Orange Jaffa and Cookies and Cream. She would like 2 trays of each. Barry has ordered 2 trays of original flavour for his school's fete. Demi has ordered cupcakes to serve after a show in the town hall. She has ordered one tray of each original flavour. She likes the sound of Orange Jaffa and Cookies and Cream, and so, she has ordered 2 trays of these as well. Charlie also needs to make an extra tray of the 8 original flavours to be sold in his shop. That is 4 trays of each flavour, 4 trays of 10 flavours. That is 40 trays of cakes. 40 trays of 24 cakes. Oh, how many cakes does Charlie need to bake?

Oh, is your brain sweating? Great, because that's the goal.

[Black text (read by speaker). On the right, a sheet of grid paper is divided up by black lines (as explained by speaker).]

**Speaker**

So, Kristen gives this tip. She says 40 trays of cakes can be represented as a grid. Each large rectangle represents one tray of cupcakes. Each small grid square is that rectangle, in that rectangle represents an individual cake. How many cakes does Charlie bake? Create a poster to show how you solve the problem. You might like to use a copy of the grid array to help explain how your strategy works. You might also like to think about the strategies we explored today and how you could use those to help you solve this. Over to you mathematicians.

[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 are these two strategies similar and how are they different?

- How many cakes does Charlie need to bake?
Create a poster to show how you solved the problem. You might like to use a copy of the grid paper in your student workbook to help explain how your strategy works.

## Collect resources

You will need:

- your thinking from reSolve bakery parts 1 and 2
a pencil

your student workbook.

## Watch

Watch the reSolve bakery video part 3 (4:22).

Title over a purple background: re(Solve) Maths by Inquiry. Below the title, is a line of 8 illustrated cupcakes, all decorated with different coloured icing and toppings. In the lower right-hand corner of the screen is the logo of the Australian Academy of Science and the small font text: reSolve: mathematics by Inquiry is funded by the Australian Government Department of Education.]

**Speaker**

OK mathematicians, welcome back. Let's explore some thinking around this problem sent to us by our friends, Kristen Tripet and Ruqiyah, at reSolve.

Text on a white background: Amy has put in a special order. She would like Charlie to bake 2 special flavours for a very special birthday celebration, Orange Jaffa and Cookies and Cream. She would like 2 trays of each.

Barry has ordered 2 trays of each original flavour for his school’s fete.

Demi has ordered cupcakes to serve after a show in the town hall. She has ordered 1 tray of each original flavour. She likes the sound of Orange Jaffa and Cookies & Cream and so she has ordered 2 trays of these as well.

Charlie also needs to make an extra tray of the 8 original flavours to be sold in his shop.

That is 4 trays of each flavour! 4 trays of 10 flavours.

That is 40 trays of cakes! 40 trays of 24 cakes.

How many cakes does Charlie need to bake?

Beside the text is an image of a page in a lined notebook. The text reads:

AMY

2 trays of Orange Jaffa

2 trays of Cookies & Cream

BARRY

2 trays each of the 8 original flavours

DEMI

1 tray each of the 8 original flavours

2 trays of Orange Jaffa

2 trays of Cookies & Cream

CHARLIE’S SHOP

1 tray each of the 8 original flavours]

**Speaker**

So this was the problem that we were asked to solve.

Text: 40 trays can be represented as a grid.

Each large rectangle represents one tray of cupcakes Each small grid square in that rectangle represents an individual cake.

How many cakes does Charlie bake?

Create a post to show how you solved the problem. You might like to use a copy of the grid array to help explain how your strategy works.

Beside the text is an image with rectangles marked on a section of grid paper. Overall, there are 4 rows of 10 marked rectangles. Each rectangle contains 24 gridded squares, arranged in 6 rows of 4.

Additional text: 40 twenty fours = 24 forties = ?

24 x 40 = ?]

**Speaker**

And really what this was getting at, was this idea of, you know, what is 40 twenty fours, or 24 forties, which is the same as saying 24x40.

Text: resolve Bakery. Looking at strategies.]

**Speaker**

So let's have a look at some strategies.

Text: This is how Suin solved the problem. Can you make sense of her thinking? Beside this text, is an icon depicting a smiling woman. Below, the marked grid of rectangles has been broken into 4 distinct rows if 10. To the left of the rows, the “240” appears 4 times. To the left of that, then the number “480” appears twice. To the left of that, the number “960” appears just once. To the right of the rectangles are equations:

10 x 24 = 240

4 x 240 = 960

This can also be written as

(10 x 24) x 4 = 960]

**Speaker**

So here's how Suin solved the problem. Can you make sense of her thinking? Let's have a look together.

The text and equations surrounding the rows if rectangles disappear. The individual rows of marked rectangles come together to form one large rectangle. An equation beside the rectangle: 40 x 24 = ?]

**Speaker**

So there's 40 twenty fours.

DESCRITPION: The equation disappears. The rows split up again. An equation above and to the left of the rows: 40 x 24 =

A multiplication appears beside each row: 10 x 24.]

**Speaker**

Yes, and she partitioned the four tens into 110 each to get 10 twenty fours.

The equation now reads 40 x 24 = 10 x 24 + 10 x 24 + 10 x 24 + 10 x 24]

**Speaker**

Yes, and she did that. Three, four times in which she knew that she could rewrite that as, instead of 10x24 + 10x24 + 10x24 + 10x24…

The equation changes to read: 40 x 24 = (10 x 24) x 4]

**Speaker**

..you'd be more efficient as a mathematician to record that as 10x24 and do that four times. So she did. She renamed ten twenty fours as 240. Then she used her knowledge of doubling…

The top 2 rows join, and the bottom 2 rows join, and form 2 shapes with 2 rows each. Beside each shape, the number “480” appears.]

**Speaker**

..to join those together first, because 240 is 480…

The 2 shapes join and form one large rectangle containing 4 rows. The numbers “480” disappear and are replaced by one instance of the number “960”.]

**Speaker**

..and then joined and doubled the remaining portion to get 960.

The rows separate again, and the original text and equations representing Suin’s thinking reappear around them.]

**Speaker**

Yes, and so that's an insight into Suin's brain and how she thought about the problem and what her poster looks like. Shall we have a look at another strategy? OK.

Text: 24 x 40 = ? Beside the equation, is the 4 by 10 array of rectangles.]

**Speaker**

So here's our starting problem. And here's another way you could have thought about it. Using the same idea as of partitioning...

The array splits in half, into 2 pairs of rows. A multiplication appears beside each pair: 12 x 40.]

**Speaker**

..you could change 24 forties into two lots of 12 forties…

The pairs of rows join so that there is just one long pair of 2 rows on screen. A multiplication above it reads: 12 x 80. An equation at the top of the screen reads: 24 x 40 = 12 x 80]

**Speaker**

..and then use that to double one number and have the other. So 24 forties can become 12 eighties and I can rename it. And then what I'm going to use is my knowledge of tens and renaming. So, when I partition now, the 12 eighties into 10 eighties, and two eighties more.

The 2 rows of small, grided squares at the bottom of the array are coloured orange. A multiplication above reads: 10 x 80. A multiplication below reads: 2 x 80. The equation at the top of the screen reads: 24 x 40 = 12 x 80 = 10 x 80 + 2 x 80.

The multiplication above the array changes to read: 10 x 80 = 800. The multiplication below the equation changes to read: 2 x 80 = 160.]

**Speaker**

And 10 eighties is just renamed 800, and two eighties is a number factor? No. Double 80 is 160.

All of the grided squares in the array turn orange. Within the array, the numbers “800 + 160” briefly appear, and then are replaced by “960”. The equation at the top of the screen reads: 24 x 40 = 12 x 80 = 10 x 80 + 2 x 80 = 960.]

**Speaker**

Then I join those two quantities together to get 960. So that's another strategy you could have used. Yes, and sometimes it's really tricky from a poster to work out what's happening inside someone else's brain.

Text over a white background: reSolve Bakery. Cupcake boxes.]

**Speaker**

All right. Let's have a look at the next challenge that we've been sent by Kristen and Ruqiyah.

Text: Charlie has a box that holds 12 cakes and he has a box that holds 10. Inside each box is a flat cardboard tray. The tray fits snugly in the boxes and has circles cut out of it so the cakes have a place to safely sit.

Charlie was folding up boxes for 10 and 12 cakes. He put the tray into the box for 10 cupcakes. He noticed that the packaging said one side of the tray was 25cm and the other side was 25cm. It was a square. He looked at the tray for 12 cupcakes. It measured 22cm on one side and 28cm on the other.

Beside the text, is a square with 12 circles in it, arranged in 3 rows of 4. Measurements on its sides indicate that it is 28cm long and 22cm wide. Another square has 12 circles in it, arranged in 2 rows of 3 and 2 rows of 2. Measurements on its sides indicate that it is 25cm long and 25cm wide.]

**Speaker**

Here it is, cupcake boxes. Charlie has a box that holds 12 cakes, and he has a box that holds ten cakes. Inside each box is a flat cardboard tray. The tray fits snugly in the boxes and has circles cut out of it, so the cakes have places to safely sit. Charlie was folding up boxes for ten and 12 cakes. He put the tray into the box for ten cupcakes. He noticed that the packaging said one side of the tray was 25 centimetres and the other side was also 25 centimetres. It was a square. He looked at the tray for 12 cupcakes. It measured 22 centimetres on one side and 28 centimetres on the other.

Text: Both sets of side lengths added to 50! Charlie was surprised. Does this mean that both trays would be the same size? Surely, he thought, the tray that held 12 cupcakes would have a bigger area than the tray that held 10 cupcakes?!

Do you think the trays have the same area? If not, which tray do you predict has the biggest area? Select an efficient strategy to determine which area is larger.

Below the text, the two squares appear side by side.]

**Speaker**

Both sets of side lengths added to 50. Charlie was surprised. Does this mean that both trays would be the same size? Surely, he thought, the tray that held 12 cupcakes would have a bigger area than the tray that held ten cupcakes. Do you think the trays have the same area? If not, which tray do you predict has the biggest area? Select an efficient strategy to determine which area is larger. Over to you mathematicians.

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

- Do you think the trays have the same area? If not, which tray do you predict has the biggest area?
Select an efficient strategy to determine which area is larger.

Record your thinking in your student workbook.