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, 2021.
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?