# How many rectangles? – Early Stage 1

A thinking mathematically targeted teaching opportunity encouraging students to work systematically as they explore rectangles

## 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
• MAE-2DS-01
• MAE-GM-01

You will need:

## Watch

Watch the how many rectangles Early Stage 1 part 1 video (2:59).

Investigate similarities and differences of 2 rectangles.

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

[White text on a navy-blue background reads ‘How many rectangles? (Early 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 9 green squares of paper and 1 orange square and a collection of 9 yellow square tiles and one red square tile.]

### Male speaker

You'll need 10 square tiles, grid paper, and something to write with. If you don't have square tiles, you might like to cut out squares or use square sticky notes.

[On a white desktop, 2 sets of 10 tiles, 9 yellow and 1 red in each, arranged in 2 rectangles. The left rectangle has the red square in the bottom right corner and the right rectangle has the red square in the middle of the top row.]

### Male speaker

Hello, mathematicians. I'm so excited and happy that you can join me today as I need your help with a problem. But first, I want us to share our thinking. Here, I have 2 rectangles that are made up of little squares. I want us to take some time to think about three things. Firstly, what is the same about these 2 rectangles?

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

### Male speaker

What is different about the rectangles? And then starting to think about, what are you now wondering, after looking at these two rectangles? If you have someone with you 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 taking some time to talk to Penny and share our thinking, there are a few things that we noticed were the same about these 2 rectangles. Firstly, we notice that both rectangles are made using 10 smaller squares. Now, I could count these squares to make sure there's 10 or you may have noticed what we noticed, and it's that the squares are arranged to look like a 10-frame. I can see 5 in the top row and 5 in the bottom row. And I know this mathematical pattern that 5 of something and 5 of something is renamed as 10.

We also noticed that both rectangles have 9 yellow squares and one red square. However, when we looked closely at the 2 rectangles next to each other, we noticed that something is different. We noticed that the red square is in a different position. On this rectangle, the red square is in the bottom right-hand corner and in this rectangle, the red square is in the middle of the top row. And this got me wondering, using the squares that you have with you today, can you make a different rectangle? Over to you now, mathematicians. Can you make a different rectangle?

[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

• What is the same?
• What is different?
• What are you wondering?
• Can you make a different rectangle?

## Watch

Watch the how many rectangles Early Stage 1 part 2 video (2:29).

Explore how many different rectangles can be made.

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

[White text on a navy-blue background reads ‘How many rectangles? (Early 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 sets of 10 tiles, 9 yellow and 1 red in each, arranged in 3 rectangles. The left rectangle has the red square in the bottom right corner and the right rectangle has the red square in the middle of the top row. The third rectangle, below on the left, has the red square in the bottom left corner.]

### Male speaker

After some careful thinking, I made this rectangle. When I look at it next to our first 2 rectangles, I can see that the red square is in a different position. On this rectangle, the red square is in the bottom left-hand corner. Now you may be wondering what I'm wondering, which is, how many different rectangles can you make?

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

### Male speaker

Because we know mathematicians like to record their ideas, I'm going to use this piece of grid paper to help me keep a track of all the rectangles I've made. When I look at my rectangle, I notice again that it looks like a 10 frame, so that's where I'm going to start.

[The speaker draws a 10 frame on the sheet of grid paper with a black marker.]

### Male speaker

I'm going to trace around the outside. Draw my 4 lines going down, and then my one line going across. When I check that, I can see that they're the same, and I can see that I have 5 in the top row and 5 more in the bottom.

I'm gonna start by drawing my red square. When I look at my rectangle, I notice that my red square is in the bottom left-hand corner so I'm going to start with that. I'm going to go to my drawing, and I'm going to colour in the bottom left-hand corner square red. I now can see, and I notice, that all of the other squares are yellow.

[In fast-motion footage, the speaker colours the remaining squares in yellow.]

### Male speaker

Now, when I look at this, I can see that I've taken what I've made and I have drawn it to help keep a track of how many I can make.

So now, over to you, mathematicians. How many different rectangles can you make, and how will you know if you've found them all?

[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

• How many different rectangles can you make?
• How will you know that you have found them all?
• Record your rectangles using the grid paper.

## Watch

Watch the how many rectangles Early Stage 1 part 3 video (4:13).

Are there even more rectangles you can make?

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

[White text on a navy-blue background reads ‘How many rectangles? (Early 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, 3 sets of 10 tiles, 9 yellow and 1 red in each, arranged in 3 rectangles. The left rectangle has the red square in the bottom right corner and the right rectangle has the red square in the middle of the top row. The third rectangle, below on the left, has the red square in the bottom left corner.]

### Male speaker

After some careful thinking, I made this rectangle. When I look at it next to our first 2 rectangles, I can see that the red square is in a different position. On this rectangle, the red square is in the bottom left-hand corner. Now, you may be wondering what I'm wondering, which is, how many different rectangles can you make?

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

### Male speaker

Because we know mathematicians like to record their ideas, I'm going to use this piece of grid paper to help me keep a track of all the rectangles I've made. When I look at my rectangle, I notice again that it looks like a 10-frame so that's where I'm going to start.

[The speaker draws a 10 frame on the sheet of grid paper with a black marker.]

### Male speaker

I'm going to trace around the outside, draw my 4 lines going down, and then my one line going across. When I check that, I can see that they're the same. And I can see that I have 5 in the top row and 5 more in the bottom.

I'm gonna start by drawing my red square. When I look at my rectangle, I notice that my red square is in the bottom left-hand corner so I'm going to start with that. I'm going to go to my drawing, and I'm going to colour in the bottom left-hand corner square red. I now can see, and I notice, that all of the other squares are yellow.

[In fast-motion footage, the speaker colours the remaining squares in yellow.]

### Male speaker

Now, when I look at this, I can see that I've taken what I've made and I have drawn it to help keep track of how many I can make. So now, over to you, mathematicians. How many different rectangles can you make? And how will you know if you've found them all?

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

### Male speaker

After exploring the problem for a little while, I found myself getting really stuck trying to create new rectangles. After talking to Penny, she helped me discover a really interesting strategy to make new rectangles. and I want to share that with you.

[A single rectangle made up of square tiles has 9 yellow tiles and one red tile in the top left corner. A sheet of grid paper on the right has 5 different yellow and red tile arrangements drawn on it. Above, a blue sticky note has black text that reads ‘What are we now wondering?’ and a yellow sticky note reads ‘How many different rectangles can you make?’]

### Male speaker

What I can see here is the last rectangle that I've made and I've recorded it on my paper. Penny's strategy works like this. If I were to take my red square and move it down one position… I can see that I've made a new rectangle that I haven't recorded yet. If I were to continue to do this over and over and over again, I should see every possible rectangle that I can make.

However, after talking to Sarah, she showed me a different strategy. Sarah's strategy was to move my five squares at the bottom up to the top row. And now what I can see is I've made a rectangle that is long and skinny. I wonder using Penny's and Sarah's strategies, are there even more rectangles that you can make?

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

[A blue text header on a white background reads ‘What’s (some of) the mathematics?’ Further text below reads ‘When we first started to solve the problem, we made new rectangles by randomly placing the red square in a new position’. A colour image of 3 different rectangles. Further text reads ‘Penny helped us to work systematically by starting in one position and moving the red square around the rectangle one box at a time. This helped us work out how many different solutions there were because we could more confidently keep track of our thinking. Working like a mathematician includes working systematically and this is important because it can help us solve problems’ A second colour image has 3 additional rectangles.]

[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

• How many different rectangles can you make?
• Record your rectangles using the grid paper.
• Using Sarah or Penny's strategy, are there even more rectangles you can make?