# How many rectangles? – Stage 1

A thinking mathematically targeted teaching opportunity encouraging students to explore arrays as rectangles and investigate how they may look different but be equivalent in area

## 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
• MA1-CSQ-01
• MA1-2DS-02
• MA1-FG-01

You will need:

## Watch

Watch How many rectangles? (Stage 1) Part 1 video (2:45).

Investigate similarities and differences of 2 rectangles.

### Transcript of how many rectangles Stage 1 part 1

[White text on a navy-blue background reads ‘How many rectangles? (Stage 1) Part 1’. Small white text at the bottom reads ‘NSW Mathematics Strategy Professional Learning team (NSWMS PL team). In the bottom right corner, the NSW Government ‘waratah’ logo.]

[Black text on a white background reads ‘You will need…’ Below, black text bullet points (as read by speaker). On the right, in still colour images, a sheet of grid paper and a collection of green sticky notes and small yellow square tiles.]

### Male speaker

For this activity, you'll need 24 square tiles, some grid paper and something to write with. If you don't have square tiles, you might like to cut squares out of paper or use square sticky notes.

[On a white desktop, 2 groups of 12 yellow tiles arranged in 3 rows of 4 on the left and 2 rows of 6 on the right.]

### Male speaker

Hello mathematicians! I'm so excited and happy that you could join me today, as I need your help with a problem. But first, I wanted us to share our thinking. Here, I have 2 rectangles that are made using little squares. I want us to take some time to think about three things.

[The speaker sticks different coloured sticky notes below the tiles with the questions he asks written on them.]

### Male speaker

Firstly, what is the same about these rectangles? What is different about these rectangles? And then starting to think, what do you wonder after looking at these rectangles? If you're with someone today, you might like to share your ideas. Pause the video now and enjoy some thinking time.

[White text on a blue background reads ‘Over to you!’]

### Male speaker

After some thinking time and talking with Penny, there are a few things that we noticed were the same and different about these two rectangles. One thing that we noticed was the same was that both of these rectangles are made using 12 squares. On this rectangle, I can see that I have 3 rows, with 4 in each row. I can see that I have 1 x 4, 2 x 4, 3 x 4, which I know is renamed as 12.

On this rectangle, I can see that I have 2 rows, with 6 in each row. And I know that double 6 is 12. However, when I look at these rectangles next to each other, I notice that there is something different. We noticed that even though both rectangles are made of 12 squares, they look quite different. And this made us wonder. We were wondering... can you make another rectangle that uses 12 squares? Over to you mathematicians.

[White text on a blue background reads ‘Over to you!’]

[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]

## Discuss and reflect

• What is the same about these two rectangles?
• What is different about these two rectangles?
• What are you wondering?
• Can you make another rectangle that uses 12 squares?

## Watch

Watch How many rectangles? (Stage 1) Part 2 video (1:42).

Explore how many different rectangles can be made.

### Transcript of how many rectangles Stage 1 part 2

[White text on a navy-blue background reads ‘How many rectangles? (Stage 1) Part 2’. Small white text at the bottom reads ‘NSW Mathematics Strategy Professional Learning team (NSWMS PL team). In the bottom right corner, the NSW Government ‘waratah’ logo.]

[On a white desktop, 3 groups of 12 yellow tiles arranged in 3 rows of 4 on the left and 2 rows of 6 on the right. Below, 12 tiles in a single row.]

### Male speaker

After some careful thinking, I've made this rectangle. And when I look at it in comparison to my first 2 rectangles, I can see that it uses 12 squares, but it again looks very different. And this made me wonder.

[The speaker sticks different coloured sticky notes below the tiles with the questions he asks written on them.]

### Male speaker

How many different rectangles can you make that use 12 squares? Now because we know mathematicians like to record their ideas, I'm going to use this piece of grid paper to help me keep track of all of the different rectangles I've made.

[The speaker places a sheet of grid paper below the yellow tiles and uses a black marker to draw on it.]

### Male speaker

When I look at this rectangle, I notice that there is one row, with 12 in that row and I can use my grid paper to record this. I'm going to go down one row... and across 12. When I look at my drawing now, I can see that I have one row...

[The speaker uses a red marker to write ‘1’ on the left of the rectangle and ‘twelve’ above it.]

### Male speaker

..with 12 in that row. Over to you now mathematicians. How many different rectangles can you make that use 12 squares?

[White text on a blue background reads ‘Over to you!’]

[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]

## Discuss and reflect

• How many different rectangles can you make?

• How will you know when you have found them all?

## Watch

Watch How many rectangles? (Stage 1) Part 3 video (2:48).

Explore strategies to find all possible rectangles.

### Transcript of how many rectangles Stage 1 part 3

[White text on a navy-blue background reads ‘How many rectangles? (Stage 1) Part 3’. Small white text at the bottom reads ‘NSW Mathematics Strategy Professional Learning team (NSWMS PL team). In the bottom right corner, the NSW Government ‘waratah’ logo.]

[On a white desktop, a group of 12 yellow tiles arranged in 3 rows of 4. On the right, a sheet of grid paper has three different shaped rectangles drawn on it in black marker. Above, a blue sticky note reads ‘What do you wonder?’ and a yellow sticky note reads ‘How many different rectangles can you make?’]

### Male speaker

After exploring the problem for a while and recording and making some rectangles, I got really stuck trying to make new rectangles. After talking to Penny, she helped me discover this really interesting strategy to take a rectangle that I already have and turn it into a new rectangle. And I want to share that strategy with you.

What I can see here is the last rectangle that I've made. I can see that it has 3 rows, with 4 in each row. And I've recorded that here on my piece of paper. If I take my rectangle now and rotate it... I can see that I have transformed it to have 4 rows, with 3 in each row.

[The speaker uses a black marker to draw the new rectangle shape on the sheet of grid paper.]

### Male speaker

If I were to record that here, I would record my 4 rows, with 3 in each row. And I found this really interesting. When I look at these 2 rectangles side by side, I can see that 3 4s has 12 squares and 4 3s has 12 squares. After talking to Michelle, she asked me a question and Michelle was wondering then...

[The speaker sticks down a new yellow note with the question written on it.]

### Male speaker

..are there multiple ways to make rectangles that have 24 squares? And what about rectangles that use 18 squares? Time for you to explore now mathematicians.

[White text on a blue background reads ‘Over to you!’]

[White text on a blue background reads ‘What’s (some of) the mathematics?’]

[Black text on a white background reads ‘What’s (some of) the mathematics? Below, black text bullet points (as read by speaker). Below on the left, 2 colour images of the yellow tile rectangles. On the right, 3 colour images of different dominoes that equal all 7.]

### Male speaker

And so what's some of the mathematics? We know that shapes like rectangles can look different but have the same or an equivalent area. And this reminds us of how numbers, like 7 for example, can look like a collection of 7 and it can also look like a collection of 5 and 2 more, or 4 and 2 and one more.

[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]

## Discuss and reflect

Are there multiple ways to make rectangles with 24 squares? What about 18 squares?