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 (2022) © NSW Education Standards Authority (NESA) for and on behalf of the Crown in right of the State of New South Wales, 2024.
Outcomes
- 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.
(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?