# Making halves

ES1 – A thinking mathematically targeted teaching opportunity focussed on making different representations of halves using paper folding

From NRICH

## 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-FG-02
• MAE-GM-03
• MAE-2DS-02

## Collect resources

You will need:

• a few sheets of A4 paper

• scissors

• coloured markers/pencils

• an adult to help with the cutting.

## Watch

Watch Finding halves video (12:44) to learn how to play.

Investigate halves by cutting paper into different shapes.

### Transcript of Making halves video

(Duration: 5 minutes and 39 seconds)

[A title over a navy-blue background: Making halves. Small font text in the lower left-hand corner reads: NSW Mathematics Strategy Professional Learning team (NSWMS PL team). In the lower left-hand corner is the red waratah of the NSW Government logo.

A large pink sheet covers a table. On the left side of the sheet are 2 sheets of yellow and red paper. On the right side is a pair of scissors.]

### Speaker

Welcome back, mathematicians. Are you ready to investigate halves a little bit more? Yeah? Alright, let's get started. You're going to need two pieces of paper.

[The speaker pushes the scissors to the top right corner of the sheet, while spreading the papers across the sheet. She places the yellow paper long-side up and the red paper short-side up.]

### Speaker

But don't worry if yours aren't coloured like mine.

[She places the red paper over the yellow one.]

### Speaker

Mine are coloured just to help me show you. So start by placing one piece of paper over the other so that the corners line up.

[She lines the top right corners of the two papers.]

### Speaker

And then we're going to fold the bottom piece of paper over the edge of this one.

[She folds the yellow paper over the red one. She creates a crease at the fold.]

### Speaker

Alright, and then we're going to cut…

[She takes the scissors, holds the yellow paper and begins to cut at the fold.]

### Speaker

…our piece of paper into... ..a rectangle….

[She pushes the shorter rectangular piece cut towards the top of the sheet. She places the square piece towards the bottom.]

### Speaker

…yeah, and a square. Beautiful. Alright, let's make some more shapes with our piece of paper.

[She spins the square paper around and folds in half.]

### Speaker

I think, let's fold this square in half. And I can see that it's half because my edge and my corners are lining up. We've got two equal parts.

[She takes the scissors.]

### Speaker

And then, we're going to cut…

[She cuts the square at the fold.]

### Speaker

…our square into two smaller rectangles. Whoa, look at this.

[She places the rectangles below the other rectangle.]

### Speaker

We got three rectangles now.

[She takes the rectangle on the right, and folds it in half.]

### Speaker

Let's fold this rectangle in half, whoa, and cut…

[She cuts the rectangle at the fold.]

### Speaker

…our rectangle into two smaller, yes, squares.

[She pushes the squares to the right side of the sheet.]

### Speaker

Fantastic.

[She takes the rectangle below the other. She folds the long-side to the middle.]

Alright, let's fold this rectangle just a little bit of it over like this.

[She cuts the rectangle at the fold. She lets the piece fall.]

### Speaker

Oh, look at that, we're gonna have one long, skinny, smaller rectangle.

[She pushes the skinny rectangle below the rectangle on the sheet, holding onto the larger one.

She folds the larger rectangle into half.]

### Speaker

And now, let's fold this rectangle into two smaller rectangles.

[She cuts the rectangle at the fold.]

### Speaker

Oh, look at all of these different rectangles.

[She moves the two rectangles under the skinny rectangle.]

### Speaker

Yeah, we've got two small rectangles, one long skinny rectangle, a larger rectangle, and two special types of rectangles…

[She moves the squares next the rows of rectangles.]

### Speaker

…called squares. Alright. Now, I want to investigate how I can halve my shapes by using paper folding. So I might start with this…

[She takes the large rectangle at the top and pushes the rest of the shapes up.]

### Speaker

…large rectangle, and I know that my halves need to be equal in size.

[She places the large rectangle down and folds it in half.]

### Speaker

So I'm gonna fold the edges of my paper up so that they join, and that the corners meet up as well, and then fold it down the middle.

[She opens up the paper.]

### Speaker

And when I open up, yes, I should have two equal-sized parts.

[She takes a ruler and marker. She places the ruler down on the fold.]

### Speaker

And I'm gonna draw a line to make it easier to see.

[She draws a line down the fold.]

### Speaker

Fantastic. That's one way that I could halve one of my shapes.

[She moves the rectangle to the right side of the sheet.]

### Speaker

I wonder how we could halve this..

[She takes the skinny rectangle.]

### Speaker

…shape. Alright…

[She puts the paper down long-side up.]

### Speaker

…I might fold it this way, this time.

[She takes one short-side towards the other side, and folds the paper.]

### Speaker

I'm gonna fold it over and press down and open up.

[She opens the paper.]

### Speaker

Oh, hang on, mathematicians. Did you see what just tricked me then?

[She folds the shorter side of the paper over the longer side.]

### Speaker

Let's just double-check if I've got halves or not. No, I don't, do I? Because can you see how these corners…

[She points to the edges of the rectangle’s end.]

### Speaker

…didn't line up with those corners and that edge? Yeah, and I know that my two halves need to be the same size.

[She opens the paper.]

### Speaker

So I need to...

[She aligns the two ends together.]

### Speaker

…ah, that's right. Make sure that they're lined up perfectly.

[She presses on the bend.]

### Speaker

And then, that means that…

[She opens the paper.]

### Speaker

…I've halved my smaller rectangle.

[She takes the ruler and marker. She places the ruler down on the fold and draws a line down the fold.]

### Speaker

Fantastic. So I found two ways of halving my shapes.

[She moves the paper to the right side of the sheet. She moves the squares towards her.]

### Speaker

I wonder what other ways I could find to halve my paper shapes. That sounds like good investigation, mathematicians.

[Text over a blue background: Over to you!]

### Speaker

It's over to you. How many different ways can you halve your paper shapes?

[Text over a blue background: What’s (some of) the mathematics?]

### Speaker

But before you go, mathematicians, let's have a look at some of what we've learnt today.

[A title on a white background reads: What's some of the mathematics? Text below reads: Halves can look different, but as long as they are two equal parts of a whole, they are still halves. Under the text is an image of a various things showing halves from the previous video.]

### Speaker

Remember, yesterday, we learnt that halves can look really different, but as long as they are two equal parts of a whole, they're still halves.

[Next to the image, two images of the different folded rectangles appear.]

### Speaker

And today, we found two other ways that we can show halves.

[Over a grey background, the red waratah of the NSW Government logo appears amongst red, white and blue circles. Text: Copyright State of New South Wales (Department of Education), 2021.]

[End of transcript]

## Instructions

• Watch the video to see how to cut out rectangles of different shapes and sizes.
• Cut some shapes out for your piece of paper.

• How many different ways can you halve the paper shapes you have made?

## Discuss

• Is there more than one way to halve each of your shapes?
• What shape has the most ways of halving?
• How could you show all the ways?