# Fruit shop – working with arrays (reSolve)

A thinking mathematically targeted teaching opportunity focussed on developing multiplicative strategies by exploring and comparing arrays

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

• MAO-WM-01
•  MA2-MR-01

## Collect resources

You will need:

• 3 colour pencils or markers

## Watch

Watch the Fruit shop video part 1 (11:45).

What do you notice about the arrays of fruit?

### Transcript of Fruit shop video part 1

(Duration: 11 minutes 45 seconds)

[Large white text on a blue background reads ‘re(Solve)’. Further vertical white text at the end of it reads ‘MATHS BY INQUIRY’. Additional white text below reads ‘reSolve Fruit Shop’.]

### Female speaker

Hi there, mathematicians. I have a task for you direct from Canberra and the team at the Australian Academy of Science with Kristen and Ruqiyah.

[Black text on a white background reads ‘reSolve Fruit Shop’. Grey text below reads ‘Working with Arrays’. At the bottom, the NSW Government red ‘waratah’ logo and a blue banner that has a white and black ‘reSolve’ logo on it.]

### Female speaker

And they have sent us their reSolve Fruit Shop task, working with arrays. Let's have a look.

[In a cartoon image, a woman stands alongside some wooden stands of various different varieties of fruit. She has brown hair and wears a green apron. A black sign behind her reads ‘reSolve Fruit Shop’ in white text. A boy and a girl stand in front of her and hold a bag of lemons and a watermelon, respectively. Signage for the various fruit is black text on white. Further explained by speaker.]

### Female speaker

Oh, so here you can see the reSolve Fruit Shop. What are some of the fruit you can see in the background? Mm-hm, I see apricots too 'cause they're one of my favourites actually. They're the little teeny tiny ones. Well, not as teeny tiny as the blueberries. Yes, but the next smallest, I think. Yeah, and the big watermelons. It looks like it's very heavy to hold, doesn't it? OK, do you notice something about all of the boxes and the way that the fruit has been arranged? Yeah, it's pretty neat, isn't it? Yeah, look, let's zoom in.

[A section of the wooden fruit stand has mangoes and red apples grouped neatly. Black text above (as read by speaker).]

### Female speaker

Here are the mangoes and here are the apples. What can you see there that's similar between those 2 collections, and what can you see that's different? Can you record a few of your ideas? It might be a good place to press pause here so you can record your ideas and then we'll share some together, OK.

OK, welcome back, mathematicians. Let's have a look at some ideas together.

[On a white desktop, 2 A4 sheets of white paper have been divided in half using a black marker pen. The speaker picks up a handful of red and yellow counters. Further steps explained.]

### Female speaker

Mathematicians, we thought we'd talk together to look at some different things that you might have seen that's similar or different with the mangoes and the apples at the reSolve Fruit Shop. So, I have my boxes and I just now need to put my fruit in there, and I'm gonna represent the mangoes using the yellow counters. You could use anything that you liked. And I'll use the red for the apples. And there's an array. I feel like I just gave you a thing to think about that they are all arranged as arrays, aren't they? Where there's an equal number of rows and columns.

[The speaker arranges yellow counters on the left sheet of paper and red counters on the right.]

### Female speaker

Look, this one has 2s and 2s, or 3 and 3, and this one there's 4 or 3. And I might have also just told you something else that's similar. Let's have a look. Almost made them. Are you noticing things as I'm putting them down? Excellent. OK, so, what are some things that you notice that's similar between the mangoes and the apples? I might write things that are similar in green.

[The speaker writes ‘Similar’ in the top left of a blank sheet of white paper in green marker pen. On the right, she also writes ‘Different’ in red marker pen. She draws a line down between them in black marker pen to form 2 columns. Further steps explained by speaker as written down.]

### Female speaker

So, similar. And actually, if I was being a mathematician, I would do this as a table, so I will do it like that. Similar and different. Yes, and that helps us organise our ideas, doesn't it? OK. So, some things that are similar. Yes, is it all organised like arrays? They all have an equal number of rows in each box and an equal number of columns. Hmm.

OK. Yes, I know. And if you look down, actually, each column in each box has 3. Yeah, there's 3 here, 3 here, 3 here, 3 here. So, in all boxes, each column has 3. Yes, and that has made you think about something different, isn't it? Because, yes, the rows are different. So, the rows of apples are the same. But the rows of the mangoes versus the apples is different. In the mangoes, there's 2 in each row, and in the apples, there's 4 in each row. The mangoes have 2 in each row, the apples have 4 in each row. Yeah, and I've wrote it here because I thought these ideas sort of could be grouped together. What else is different?

Yeah, the quantity, isn't it? So, in each box of mangoes, there's only 6. And in each box of the apples, there's 12. So, there are 12 apples in each box. A-ha, nice spotting for those mathematicians who realised that if you double the mangoes, double 6 is 12, and there's 6 mangoes, I like your thinking, in each box. What else is something that's similar or different?

[The earlier slide of the mangoes and apples from the cartoon reSolve fruit shop.]

### Female speaker

I'm gonna leave it to you now to note down some other ideas in your notebook, and then we'll come back together. OK, back to you.

[White text on a black background reads ‘Here comes the next question…’]

### Female speaker

Here's the next question from Kristen and Ruqiyah.

[A section of the wooden fruit stand has two boxes of neatly grouped oranges. Black text above (as read by speaker). Further steps explained by speaker.]

### Female speaker

And they said to us, OK, how are the 2 boxes of oranges similar, and how are they different? What are you thinking, mathematicians, what can you see? Yes, they are all orange. We're gonna look for mathematical things that are similar or different. So, quantity or weight or size. Oh, I see this too. I think it's the same quantity. Look, if I take the boxes from being stacked on top of each other and turn this one sideways and then lay them over the top. Yeah, I think it's the same number in each box. It's like the communitive property, look.

[An array of blue circles is arranged in 5 rows of 7 circles each. Black text below (as read by speaker).]

### Female speaker

Here, I have an array. It has 5 7s, which is 35.

[A second array has red circles arranged in 7 rows of 5 circles each.]

### Female speaker

And over here, my array, I would call that 7 5s, which also has a product of 35. And if I turn it around and lay it on the top, I can prove they both have 35. Yeah. So, I think we can use that same imagining to say that both of our orange, both of our boxes of oranges, have 15 in them. Mm-hm.

[The earlier slide with the boxes of oranges.]

### Female speaker

Oh, I saw this 2. They both have arranged the oranges in equal rows and in equal columns. Ah, OK. Yes, we'll come to that difference in a moment. I can also see this that some of you also saw that inside of the 15, you can see 6 and 9.

[The speaker highlights the sections in yellow and blue when mentioned.]

### Female speaker

Look, one 6 and one 9, and it's also in the bottom box. So, that's something else that they have in common. Yes, and to your earlier point that in the top box, we can see 5 rows, one, 2, 3, 4, 5. But in the bottom box, there's only 3 rows. Look, one, 2, 3. So, that's one of the things that's different. Yes, and the same with the columns, isn't it? In the top box, there's 3 columns, and in the bottom box, there's 5 columns. OK, let's have a look at the next challenge.

[A section of the wooden fruit stand has the mangoes and red apples from earlier along with another section that holds punnets of blueberries. Black text above (as read by speaker).]

### Female speaker

So, can you use the number of mangoes in one box to help you work out the number of blueberry punnets? Oh, what are you thinking, mathematicians? So, yes, the punnet is the box that the blueberries sit in. And how can you use what you see with the mangoes to help you think about the punnets? Shall we have a look at this together? Great idea.

[The white desktop and the yellow and red counters from earlier. On the right, small squares of purple paper. Further steps explained by speaker.]

### Female speaker

Mathematicians, I thought we'd talk about this question together. I'll just move this over a little bit. OK, and I'm gonna use these pieces of paper to represent my punnets of blueberries. I'll just make sure that you can see them. Yes, I didn't actually have any... I think you can see that they're there.

OK. So, Kristen has asked us this question of how can we use the information that we can see here about the number of mangoes to help us think about the punnets of blueberries? Hmm. So, what I do in my brain is do some imagining. And what I can see here is that there's a structure of 6 on a dice pattern. And if I look really carefully, I can see that here also, look, because it's 2 rows of 3. So, I know that chunk is 6. And then there would be 3 more. And I know 6 and 3 more is 9. So, that's one way I could have thought about it.

The other thing I could have thought about doing too, I'm gonna use this, I'm gonna make some miniature punnets.

[The speaker uses some scissors to cut a purple square into smaller squares.]

### Female speaker

I could have thought about my mangoes and said, well, if there was one more column of 3s, it would look like this. And so, that would make 9, because 3 3s are 9. And so, that's how I could use the representation of my mangoes to help me work out the punnet, the number of punnets of blueberries that I have. So, 2 strategies that happen in my brain. Would you like to see them again?

OK, so, one was I imagined the 6 that I see here in the mangoes here and the punnets of blueberries. And then I know 6 and 3 more is 9. And the other thing I imagined, and your brain is likely to have been entirely different, which is one of the beautiful things about maths is you get to see inside people's brains, is I imagined what the quantity would be if there was another column of 3. And so, that would give me 3 3s, which is 9.

OK, over to you, mathematicians, to think about how else you could use what you know about the mangoes to think about the blueberry punnets.

[A section of the wooden fruit stand has the mangoes and red apples from earlier and another group of peaches added. Black text above (as read by speaker).]

### Female speaker

OK, and now back to you, mathematicians. I have 2 questions for you to go off and explore now. One is this one. Can you use the number of apples to help you work out the number of peaches? And can you draw some diagrams for us so that we can get inside your brain and see your amazing thinking? And our second question to leave you with today is this one.

[The earlier cartoon of the ‘reSolve Fruit Shop’. Black text reads ‘Can you make any other connections?’]

### Female speaker

Can you make any other connections as you're doing this task, using any of the fruit that you like? I think I'm gonna play with the apricots because they're my favourites, and I think I can see a relationship to the strawberries. OK, mathematicians, over to you.

[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

• How are the mangoes and apples similar and different?
• What else is can you add about something similar or different?

• How are the two boxes of oranges similar and different?

• How else could you use what you know about the mangoes to help you work out the number of blueberry punnets?

• Can you use the number of apples to help you work out the number of peaches? Can you draw some diagrams for us, to share your amazing thinking?

• Can you make any other connections using any of the fruit?

## Watch

Watch the Fruit shop video part 2 (5:35).

Explore connections between arrays of fruit.

### Transcript of Fruit shop video part 2

(Duration: 5 minutes 35 seconds)

[In a cartoon image, a woman stands alongside some wooden stands of various different varieties of fruit. She has brown hair and wears a green apron. A black sign behind her reads ‘reSolve Fruit Shop’ in white text. A boy and a girl stand in front of her and hold a bag of lemons and a watermelon, respectively. Signage for the various fruit is black text on white. On the left, black text reads ‘Can you make any other connections?’ At the bottom, the NSW Government red ‘waratah’ logo and a blue banner that has a white and black ‘reSolve’ logo on it.]

### Female speaker

Hi there mathematicians. When we left you, we left you with this challenge of can you make any other connections? I'll share with you something that I saw.

[On a large white sheet of paper, 3 purple cards have ‘Apricots’, ‘Strawberries’ and ‘Watermelons’ written on them respectively. 16 yellow counters are positioned beneath the Apricots card, 8 red counters beneath the Strawberries card and a ‘C3PO’ Star Wars action figure and some green Lego blocks are placed around the Watermelon card. Further steps explained by speaker.]

### Female speaker

Hi there mathematicians, welcome back. I thought I'd share with you some of the connections that I made. Yes, and I don't have a printer so I've recreated my… the ReSolve Fruit Shop for you. So, over here, I have counters that are representing my apricots, and I have blocks for watermelons because I figure that'll take up a bit more space in the counter and the watermelons so big. And I think the girl in this story, I'm gonna call her Katelyn because I know a Katelyn who loves watermelons. And so, Katelyn is represented by a C-P3O. I'm not sure how she'll feel about that, but we will go with it anyway.

So, here's some things and I recreated these ones because these were the first connections I made. J just to share with you some things I thought so that that might inspire you to think of a few other things. So, one of the first things I noticed is firstly, I really like apricots. So, when I was looking at the apricots, I noticed it's a square number. Yes, because there's 4 rows and 4 in each row. And so therefore it's a square and I really like square numbers. So, I know that 16.

But when I look to the strawberries, I can see half a square number. Yes, because if I had another 2 rows of 4… I think you can still just see that. Let me check, not quite, I will move it up for you. So, here's my row of strawberries, my array of strawberries with 2 4s, and I can tell that they're 4s and if I made another 2 rows of 4, yeah, I the same square array.

[The speaker places a blank square of purple card on top of the yellow ‘Apricot’ counters and then the red ‘Strawberries’.]

### Female speaker

Look, this might just be the right size. Let's see. Oh, yes, my old blueberry punnet sort of shows this and the same… Yeah and it's got the same little bits hanging over. Can you see that? Yes, and so, if I fold this in half, that covers my red portion, which shows to me that there's half as many strawberries as there are apricots. And also, if I doubled my number of strawberries, I could make a square number, which I really like doing.

And then because I really like square numbers, just because I think they're quite fun, I notice something about my watermelons, and that is that there was one hanging in the basket. And if I joined that to here, and if Kate hadn't have bought or be buying a watermelon, there would be another one here. And so I have almost a square number in my watermelons. And so if I had one more, I would have a square number.

[The speaker arranges the green Lego blocks into rows. She uses a black marker to write on a sheet of white paper.]

### Female speaker

So, in my collection of watermelons, what I have are 3 3s minus one, which is 8, and technically, there's one in the basket. But I think that still belongs to the shop. And I think Kate's about to buy one. And so now what there are is 3 3s minus 2 which will leave seven in the shop.

[The speaker moves the green ‘Watermelons’ on top of the yellow ‘Apricot’ counters.]

### Female speaker

But it also made me think about this mathematicians and look, inside of 16, sorry Kate, to take your watermelon. But if I put them up here... Yes, that inside of my square array of 16 is an array of 9. If I had one more watermelon. I know and that made me start thinking about arrays inside arrays, which is sort of what the fruit shop did.

[The earlier cartoon fruit shop image.]

### Female speaker

OK, so I hope this has inspired you to think about a few other things and then we'll look at our next challenge.

[Large white text on a blue background reads ‘re(Solve)’. Further vertical white text at the end of it reads ‘MATHS BY INQUIRY’. Additional white text below reads ‘reSolve Fruit Shop’.]

### Female speaker

OK, and now we're on to our challenge for today.

[Black text on a white background reads ‘reSolve Fruit Shop’. Grey text below reads ‘Lemon Arrays’. At the bottom, the NSW Government red ‘waratah’ logo and a blue banner that has a white and black ‘reSolve’ logo on it.]

### Female speaker

Back into the fruit shop, we go.

[The earlier cartoon fruit shop image.]

### Female speaker

And did you notice something in the fruit shop? Everything is arranged really beautifully. The mangoes, the apples, the peaches, the strawberries – except for the lemons. Look, they're really messy, and they're not in a nice array like all the other fruit.

[The stand of lemons is highlighted from the cartoon fruit shop image. A sign reads ‘LEMONS \$5 per bag 6 in a bag’. The boy holds a bag of lemons in his hands. Black text on the right (read by speaker).]

### Female speaker

So, here comes our challenge for you. There are 4 bags of lemons, the owner of the fruit shop wants to take the lemons out of the bags and arrange them in a box like the oranges, apples, peaches, apricots and mangoes. She wants more than one lemon in each row and column. How could the owner arrange all of the lemons in just one array? Can you find more than one way?

Alright mathematicians, it's back over to you. Your job is to think about what are the different arrays that you could make, arrangements that you could make, using equal rows and equal columns for the lemons. And draw pictures up so the owner of the fruit shop can make some decisions about which one she likes. Yes, and it could be based on how much space she has to put them. OK, enjoy!

[White text on a blue background reads ‘Back 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

• There are four bags of lemons. The owner of the fruit shop wants to take the lemons out of the bags and arrange them in a box like the oranges, apples, peaches, apricots and mangos. She wants more than one lemon in each row and column.

• How could the owner arrange all the lemons in an array? Can you find more than one way?

• Draw pictures of the arrays so the owner can make some decisions about which one she likes.