# Mathematics K–12

ES1 – A thinking mathematically targeted teaching opportunity exploring working systematically to find rectangles which have the same area but may look different

## 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-RWN-01
• MAE-CSQ-01
• MAE-CSQ-02
• MAE-2DS-01

## Collect resources

You will need:

• coloured pencils or markers

## Watch

Watch About how many rectangles part 1 video (3:57).

How many smaller rectangles could cover a larger rectangle?

### Speaker

Hello there, young mathematicians, welcome back!

Today to warm up our brains, we're going to look at this big blue rectangle here and I also have this smaller orange rectangle.

[Screen shows an A4 blue piece of paper in landscape orientation, and a smaller, orange rectangle piece of paper.]

And what I'm wondering is, can you use your mathematical imagination and imagine how many of these smaller rectangles would it take to fill this larger rectangle?

[Presenter picks up smaller orange rectangle piece of paper and places it to the right of the blue piece of paper.]

Can you now pick up your pencils and your book and draw a picture to show me how many orange or smaller rectangles you think it would take to fill the big blue rectangle? Over to you mathematicians!

[Speaker picks up smaller orange rectangle piece of paper and places it back down and points to blue piece of paper.]

Welcome back young mathematicians. How did you go with your drawing? Can you show me what yours looks like?

I see there's a few different ways of thinking, isn't there? This is what mine looks like.

[Presenter shows a horizontal pink piece of paper. On it there is a rectangle drawn with a horizontal line across and a vertical line down the middle, creating 4 smaller squares.]

So, this is my big blue rectangle that's representing this rectangle. And I think 4 of the orange ones will fit onto the blue one.

[Presenter points to indicate the square she has drawn on the pink piece of paper. She then points to the smaller orange piece of paper, indicating that the smaller boxes on her pink paper represent the smaller orange piece of paper.]

So, one here, a second one here, a third one here, and a fourth one here.

[Presenter points and traces around the other 3 remaining square boxes on the pink piece of paper.]

Some of you are thinking the same thing.

[Presenter removes pink piece of paper from screen and picks up orange piece of paper.]

Should we check to see? Okay. Say here's our smaller rectangle. Let's put it over the top of our big one.

[Presenter places smaller orange piece of paper on top of the larger blue piece of paper, in the bottom right corner, aligning it with the bottom and side edges of the blue piece of paper.]

Oh yeah, and now as I'm visualising. I think this rectangle here, if I use my fingers, I think that would definitely fit up the top.

[Presenter uses fingers to roughly measure the size of the orange piece of paper, indicating that it can be used to measure the rest of the area of the blue paper.]

Are you seeing that too? Let's check. So, I only have one rectangle, so I can't just lay them out, so what I might do is very carefully, mark this place with my finger.

[Presenter marks top of orange piece of paper with finger and shifts the orange piece of paper up towards the top edge of the paper. She indicates to the empty space of the left, showing that they are the same size.]

I'm pretty confident that fits there and actually, what I imagine is it's going to be the same on this side 'cause this looks like it's about halfway and if I fold, my piece of paper over...

[Presenter folds blue piece of paper in half vertically, holding orange piece of paper at the front. She then unfolds blue piece of paper and lays back down. The orange piece of paper remains on the top right-hand corner of the page.]

I can check. Yes. Look at that. So now what I see...is that this would take up the same space on this side 'cause they're halves, look, if I move this over.

[Presenter points to orange piece of paper then then points to left side of the blue piece of paper. She moves the orange piece of paper over to show that it fits into the other side.]

Oh, and you think I should fold it this way to prove too? That's good thinking. Let's do that. Fold it the other way in half, and now let's look.

[Presenter now folds blue piece of paper in half horizontally and places it back down. She removes the smaller orange rectangle.]

Yeah, look. 1, 2, 3, 4. 4 small rectangles, cover the area of my bigger rectangle.

[Presenter now moves the small orange piece of paper over the folded squares on blue piece of paper showing that the orange piece of paper fits into each 4 squares created by folding the blue paper.]

A bit like what I imagined in my mind and what I drew in my picture.

[Presenter picks up her pink paper from before, and shows the camera to display her thinking.]

Nice warming up of your mathematical imaginations, little mathematicians!

I wonder what would happen now if my rectangle was only this size?

[Present now folds orange piece of paper in half.]

About how many of these rectangles do you think I would need to cover all of my paper?

[Presenter lays the folded orange piece of paper next to blue piece of paper to the right and points to the squares in the blue paper.]

Over to you mathematicians to think about it and draw a picture. Okay!

[Screen reads, your challenge! About how many of the smaller orange rectangles are needed to fill the area of the large dark blue rectangle? Draw a picture to share your thinking.]

[End of transcript]

## Instructions

• About how many of the smaller orange rectangles are needed to fill the area of the large dark blue rectangle?

• Draw a picture in your student workbook to share your thinking.

## Watch

Watch About how many rectangles part 2 video (1:46).

Continue investigating strategies to explore the area of a rectangle.

### Speaker

[Screen shows a blue A4 piece of paper placed in landscape orientation as well as a smaller orange piece of paper. The large, blue piece of paper has been folded in half vertically and horizontally before being unfolded. It has a vertical and horizontal crease.]

Welcome back young mathematicians! How did you go?

Yeah, I think I noticed something too. When, because we had worked out when our rectangle was this size, that we needed 4 small rectangles to cover up the bigger rectangle. Look, 1, 2, 3, 4.

[Presenter moves orange piece of paper into each of the 4 boxes on the blue paper showing how they fit.]

Yeah, and the folding of our paper helped us see that. And then we folded it in half, our small rectangle, and we said, now how many of them do you need to cover the same piece of paper?

[Presenter picks up orange piece of paper, showing that it is folded in half.]

Yeah, and what I knew too, like you some of you, is that I will now need 2 smaller rectangles for this sized rectangle.

[Presenter shows how the 2 smaller rectangles fit into one larger rectangle.]

So, before I needed 1, 2, 3, 4 small rectangles, to cover the area of my big rectangle.

[Presenter unfolds orange piece of paper and shows how when it is unfolded it fits into each of the 4 rectangles.]

Now I think I need 1, 2, 3, 4, 5, 6, 7, 8 rectangles.

[Presenter now folds orange piece of paper and moves paper over each of the blue rectangles. She shows how each rectangle can fit 2 smaller rectangles, and demonstrates this as she moves between all 4 rectangles.]

And I could count it by twos. Look, 2, 4, 6, 8. Did you see that too? Yeah, and that for each section I needed 2 rectangles.

[Presenter slides folded orange paper within the larger blue rectangle sections, counting up by twos as she goes.]

Nice visualising mathematicians! Now let's get ready for our next challenge!

[End of transcript]

## Discussion

• Discuss with someone else what you discovered about how many orange rectangles are needed to cover the larger blue rectangle.

• Did you notice similar things?