Imagining fractions

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.

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. 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 1- half. Can you point to some other 1-half, 1-halves? Haha. Um some are a quarter. Can you see other quarters? Yeah, and some 3-quarters, can you see some other 3-quarters?

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 1/4 + 1/2 + 1/2 + 1/4 + 1/2 + 3/4 + 3/4 + 1/2 + 1/2 + 3/4 + 3/4 + 1/2 + 1/4 + 1/2 + 1/2 + 1/4.

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. 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.

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. 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 eight of them.

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.

And then I had to think about what's left, which is the 3-quarters, the 4 3-quarters. There's four of them. There they are. 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.

So one of my 3-quarters I'm now going to re-imagine as 1-quarter and 1-half and then I can imagine that 1-quarter joining across here to make a whole. And then I do the same to another of my 3-quarters. I partition it further into 1-quarter and 1-half. And slide one piece across to each which leaves 3 so 3, 4 lots of 3-quarters is 3 in total.

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.

So here's my challenge to you mathematicians. Here are two different ways that we thought of, one which would just be adding a string of numbers together. 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 3-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?