# How many rectangles? – Stage 2 and 3

A thinking mathematically targeted teaching opportunity encouraging students to explore arrays as rectangles and investigate area and perimeter

## 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-AR-01
• MA2-MR-01
• MA2-GM-02
• MA2-2DS-03

• MAO-WM-01
• MA3-AR-01
• MA3-MR-01
• MA3-GM-02
• MA3-2DS-02

You will need:

## Watch

Watch the how many rectangles Stages 2 and 3 part 1 video (4:05).

Investigate similarities and differences of 2 rectangles.

### Transcript of how many rectangles Stages 2 and 3 part 1 video

[White text on a navy-blue background reads ‘How many rectangles? (Stage 2 and Stage 3) 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, a sheet of 1 cm grid paper.]

### Male speaker

For this task, you'll need 1cm grid paper and something to write with.

[On a white desktop, a green sheet of grid paper and a red sheet of grid paper cut into two different rectangles.]

### Male speaker

Hello, mathematicians. I'm so excited that you could join me today as I need your help with a problem, but first, I wanted us to have a chance to share our thinking. Here, I have a drawing of two different rectangles on 1cm grid paper. Before we start though, I want us to take some time to think about three things.

[The speaker sticks different coloured sticky notes above 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 starting to think a little bit about, what are you now wondering after looking at these rectangles? If you have someone with you today, you might like to share your ideas. So 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 to Penny, we noticed that there are some things that were the same and some things that were different about these rectangles. One thing that we noticed was the same is that they both have the same area. Now, I can prove to you that both of these rectangles have the same area by taking my green rectangle and partitioning it by cutting it up the middle.

[The speaker cuts the green rectangle in half with some blue-handled scissors.]

### Male speaker

I can then overlay it on top of my red rectangle to prove to you that they take up the same amount of space. Both Penny and I found this really, very interesting. And we noticed that even though these 2 rectangles have the same or an equivalent area, that they both look very different. This green rectangle is made up of 6 rows, with 8 in each row, and I can rename 6 x 8 as 48. This red rectangle, however, is made up of 4 rows, with 12 in each row, and I can also rename 4 x 12 as 48. And this got us thinking. How many different rectangles can you make that have an area of 48 square centimetres?

Because we know mathematicians like to record their thinking, today we were going to record our thinking on this piece of 1cm grid paper. What you can see here is I've already recorded my 6 x 8 here, by writing 6 x 8 is equivalent to 48. So I'm going to record my red rectangle now. When I look at my red rectangle, I notice again that it is 4 x 12 so that's how I will record it. 4 rows, with 12 in each row. I'm also going to write 4 x 12 is equivalent to 48.

This is a strategy you might like to use to keep all of your rectangles that you've made. Happy exploring, 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

• What do you notice is the same about these rectangles?
• What do you notice is different about these rectangles?
• What are you wondering?
• How many different rectangles can you make that have an area of 48cm2?
• Record your thinking and ideas on the 1cm grid paper.

## Watch

Watch the how many rectangles Stages 2 and 3 part 2 video (2:52).

Explore features of rectangles with the same area.

### Transcript of how many rectangles Stages 2 and 3 part 2 video

[White text on a navy-blue background reads ‘How many rectangles? (Stage 2 and Stage 3) 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.]

[A collection of coloured sticky notes have various questions written on them. On the right, a sheet of 1cm grid paper has a green rectangle and a red rectangle drawn on it.]

### Male speaker

After exploring for a little while and making different rectangles, Sharon helped me discover something really interesting about these 2 rectangles. When I have a look at this rectangle, I notice that it has 2 sides that have a length of 8 centimetres... and 2 sides that have a length of 6 centimetres.

[The speaker writes the measurements on the grid paper with a light blue marker pen.]

### Male speaker

If I combine all of these lengths, that gives me the distance around the outside of the shape which we call the perimeter. When I combine these lengths, I get a perimeter of 28 centimetres. When I look at my red rectangle, however, I notice that I have 2 sides that have a length of 12 centimetres and 2 sides that have a length of 4 centimetres. When I combine all of these lengths, I find that I have a perimeter of 32 centimetres. And this made us very, very intrigued.

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

### Male speaker

We noticed then that even though the rectangles have the same or an equivalent area, they have very different perimeters. And that made us wonder. What is the largest possible perimeter and what is the smallest possible perimeter of a rectangle that has an area of 48 square centimetres?

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

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

[A blue text header on a white background reads ‘What’s (some of) the mathematics?’ Further black text bullet points below (as read by speaker). Colour images of 4 different dominoes that all equal 7 and 2 colour images of the green and red rectangles drawn on the 1cm grid paper.]

### Male speaker

So, what's some of the mathematics? We know that shapes like rectangles can look different but have the same or 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. We also discovered that even though a shape is the same on an equivalent area, its perimeter can actually be very different.

[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 largest possible perimeter of a rectangle that has an area of 48cm2?
• What is the smallest possible perimeter of a rectangle that has an area of 48cm2?