# Imagining fractions

Imagining fraction Stage 3 is a thinking mathematically targeted teaching opportunity investigating ways to visualise and combine fractions.

## 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
• MA3-AR-01
• MA3-RQF-01
• MA3-RQF-02

You will need:

• a pencil.

## Watch

Watch Imagining fractions video (6:04).

Investigate and visualise the total lemon slices.

### Speaker

Ok mathematicians, it's time to warm up your mathematical imaginations.

So, I'd like you to have a look at the image on the screen now.

[Screen shows 16 clear plastic cups with different sized slices of a lemon in the bottom of each cup. Cups form a square array of 4 by 4, with 4 cups vertical and 4 cups horizontal.

From left to right, the first cup in the first row has one-quarter of a lemon slice. The second cup has one-half of a lemon slice. The third cup has one-half of a lemon slice and the final cup has one-quarter of a lemon slice.

From left to right, the first cup in the second row has one-half of a lemon slice. The second cup has three-quarters of a lemon slice. The third cup has three-quarters of a lemon slice and the final cup has one-half of a lemon slice.

From left to right, the first cup in the third row has one-half of a lemon slice. The second cup has three-quarters of a lemon slice. The third cup has three-quarters of a lemon slice and the final cup has one-half of a lemon slice.

From left to right, the first cup in the final row has one-quarter of the lemon slice. The second cup has one-half of a lemon slice. The third cup has one-half of a lemon slice and the final cup has one-quarter of a lemon slice.]

Uh-huh and I have a question for you, of course. So how many slices or lemon slices are there?

And how might we work it out?

Ok, so to help us really think through this idea a really important question to ask yourself is, what are some things that you notice?

So, what are some things that you notice here in this image?

Uh-huh. Ok, let's share some.

So, some of you have noticed this that there are 16 cups in total. And like some of you, I knew that too because it's a square array of cups 4 by 4 or 4 fours look.

[Screen shows cups now change to having a red square on each of them, indicating how they resemble a square. The squares disappear to show the cups once more.]

Yeah, so because I saw that square array, that's how I knew it was 16.

Yes, and some of you have also noticed that there's no whole slices of lemon in any of the cups. They all have a fractional part.

And that some of the parts are one-half. Can you point to some other one-halfs, one-halves? Ha-ha.

[Screen shows third cup down in the far-left row turns pink to highlight that it contains one-half of a slice of lemon]

Um some are a quarter. Can you see other quarters?

[The first cup in the fourth row is highlighted to show that it contains one-quarter of a slice of lemon.]

Yeah, and some three-quarters. Can you see some other three-quarters?

[The second cup in the third row is now highlighted to show that it contains three-quarters of a slice.]

Great, so noticing things will help us make some decisions about how we could go about working out how many lemon slices there are in total.

So now we might think about, well how could we work it out?

So, let's look at one way of thinking and one way of thinking would be to this, which is we could add everything together like this so reading from the top row top right-hand corner across that row and down one-quarter plus one-half plus one-half plus one-quarter plus one-half plus three-quarters plus three-quarters plus one-half plus one-half plus three-quarters plus three-quarters plus one-half plus one-quarter plus one-half plus one-half plus one-quarter.

It's quite like a tongue twister, but also it makes my brain goggle and I think I could be more efficient than just adding 16 quantities together.

So, let's have a look at another way and in this case, I'm going to ask you to use your mathematical imaginations because I think another way that we could think about this is to collect fractions.

[Screen shows 16 clear plastic cups that form a square array of 4 by 4. There is a piece of a fractional slice of lemon in the bottom of each cup.

In the top row, the first cup has one-quarter of a piece. The second cup has one-half. The third cup has a half. The last cup in the row has a quarter.

The second row of cups from left to right has one-half, then three-quarters, next another three-quarters and the last cup in that row has one-half.

The third row has a half, three-quarters, another three-quarters and then a half.

The fourth row has one-quarter, then one-half, then another half and then one-quarter.]

So, what I mean by that is there's 4 quarters look, 1, 2, 3, 4. And just to make it easier to see, here's what they look like.

[Screen shows each cup in the corners highlighted pink. The fractional lemon slice within them are coloured blue so that they are easier to see.]

And so, the first thing that I could do is imagine in my mind collecting the 4 quarters, and I know that if I did that, it would make 1 and this is what happens in my brain.

[The blue quarters move from each cup to create one whole circle on the left side of the screen. The circles are labelled: 4 quarters… which make 1.]

So, I have 1 and then I might think about some other like quantities which are the halves.

And if I have a look, there are 8 of them.

[The screen shows the cups with the halved slices of lemon highlighted in pink. Each half is covered by a blue semi-circle so that it is easier to visualise.]

And to help you work out what I was doing in my head, this is what the halves look like, and I know something about halves which means I know something about 8 halves and that is that it's 4 in total because for each of the 2-halves, I get a whole.

So, in my brain this is then what happened. I collected the halves and that made 4 wholes.

[The 8 blue semi-circles come together to create 4 whole circles. The circles are labelled: 8 halves… which is 4.]

And then I had to think about what's left, which is the three-quarters, the 4 three-quarters. There's 4 of them.

There they are.

[The final 4 cups in the middle of the screen are highlighted. 4 three-quarter circles cover them.]

And so, then I had to collect the 4 3-quarters, and this is where it got a bit tricky for my brain because now what I had to do, the others were easy to imagine coming together and reforming wholes.

But here I have to partition to reform.

[The blue shapes move over to the right of the screen. Above it reads: 4 three-quarters equals 4 times 3 quarters.]

So, one of my three-quarters I'm now going to re-imagine as one-quarter and one-half and then I can imagine that one-quarter joining across here to make a whole.

[Screen shows one of the three-quarter circles being partitioned into a quarter and a half. The quarter joins one of the three-quarter circles to form a whole circle.]

And then I do the same to another of my three-quarters. I partition it further into one-quarter and one-half.

And slide one piece across to each which leaves 3 so 3, 4 lots of three-quarters is 3 in total.

[The screen shows the third thee-quarter circle being partitioned into a quarter and a half. The one-quarter joins the thee-quarter to create a whole circle, and the 2 halves join together to create one circle.]

Aha and then I just have to collect my wholes together to know that altogether, that means that when I join all those sections, there's 8 whole.

[All of the sections come together to show that the fractional pieces all make 8 whole circles.]

So, here's my challenge to you mathematicians.

Here are 2 different ways that we thought of, one which would just be adding a string of numbers together.

[The screen shows the following image at the top of the screen 3 times: Screen shows 16 clear plastic cups with different sized slices of a lemon in the bottom of each cup. Cups form a square array of 4 by 4, with 4 cups vertical and 4 cups horizontal.

From left to right, the first cup in the first row has one-quarter of a lemon slice. The second cup has one-half of a lemon slice. The third cup has one-half of a lemon slice and the final cup has one-quarter of a lemon slice.

From left to right, the first cup in the second row has one-half of a lemon slice. The second cup has three-quarters of a lemon slice. The third cup has three-quarters of a lemon slice and the final cup has one-half of a lemon slice.

From left to right, the first cup in the third row has one-half of a lemon slice. The second cup has three-quarters of a lemon slice. The third cup has three-quarters of a lemon slice and the final cup has one-half of a lemon slice.

From left to right, the first cup in the final row has one-quarter of the lemon slice. The second cup has one-half of a lemon slice. The third cup has one-half of a lemon slice and the final cup has one-quarter of a lemon slice.

Underneath the first picture, screen reads: One idea: We could add everything together like this: one-quarter plus one-half plus one-half plus one-quarter plus one-half plus three-quarters plus three-quarters plus one-half plus one-half plus three-quarters plus three-quarters plus one-half plus one-quarter plus one-half plus one-half plus one-quarter.

But that makes my brain goggle… and I think I could be more efficient than adding 16 quantities together.

Underneath the second picture the screen reads: Another idea: We could collect the line fractions… So, there are 1 and 4 and 3 is equivalent in value to… 8!

There is also an image of the working out from before, with the fractional pieces all adding together to create 8 whole circles.]

One way where we started to re-imagine the qualities and move things around to help us work with whole my brain 'cause I really like working with landmark numbers and whole numbers and nicer for my brain to work with, they're landmark for me then then fractions.

But I wonder if you could re-imagine, use this imagining strategy on the third image there to think about how you might imagine it differently to what I did because what happened for me was when I got to the three-quarters, it started to get quite tricky for my head and I had to re-partition a fraction which is hard to hold onto all of those ideas.

So, mathematicians over to you now.

What is a different way that you could have imagined the fractional slices of lemon moving?

And draw a picture or pictures to capture your thinking over to you?

[End of transcript]

## Instructions

• What’s a different way you could have imagined the fractional slices of lemon moving?