# About how many paper clips

A thinking mathematically targeted teaching opportunity focused around estimation and measuring length.

## 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
- MA1-RWN-01
- MA1-RWN-02
- MA1-GM-02

## Collect resources

You will need:

- pencils or markers
- your mathematics workbook.

## About how many paper clips – part 1

Watch About how many paper clips – part 1 video (4:27).

### Speaker

Hello little mathematicians, welcome back.

I hope you're having a really lovely day. I've got this piece of paper here for us to do some thinking with and I found these paper clips at home and what I was wondering or thinking about is that we could use our paper clips and this piece of paper to warm up our mathematical imaginations and so we know when read stories, it's really important to imagine what's happening in stories in our brains cause it helps us make meaning from them. And the same happens in mathematics, it's really important for us to work on and practice our mathematical imaginations.

[Screen shows an A4 piece of paper, in landscape orientation and a small paperclip holder with different sized and coloured paperclips in it. Presenter picks up the paperclip holder and empties the paperclips onto the piece of paper, sitting the paperclip holder to the right of the paper.]

### Speaker

So, I wonder if you can help me by thinking about, about how many paperclips long is this edge of my piece of paper?

[Presenter picks up one of the longer paper clips and places it horizontally along the edge of the paper, starting in the bottom left-hand corner. Presenter slides their finger along the bottom edge of the page from left to right indicating the length of the paper that needs measuring – from where they have placed the paperclip to the other end of the page. They then move the paper clip back into the pile.]

### Speaker

What are you thinking? Ah okay, well let's test out your theories. So, it's one paper clip and 2 paper clips.

And yes, I was being very careful not to overlap them or leave big spaces when I'm measuring. You've got to be very accurate. Two paperclips, what are you thinking now?

[Presenter moves 2 longer paper clips to the bottom edge of the page, and lines them up along the edge starting in the left-hand corner. The presenter makes sure the shorter ends of the paperclips are touching. They highlight the paperclips are not overlapping by demonstrating what overlapped paperclips would look like, and that there is not a big space between the paperclips, by moving them apart and showing what spaced out paperclips would look like. The presenter then puts them back where they were originally, end-to-end along the bottom of the page, starting in the left-hand corner.]

### Speaker

Okay, I like how you're using evidence to refine your estimates.

[Presenter moves a third long paperclip to the bottom edge of the page and line it up end-to-end with second paperclip. They then move a shorter paperclip and line it up end-to-end. The presenter starts to move a fifth paperclip but then stops.]

### Speaker

3 paperclips. Four paper clips. Fa...

Ah, you're right mathematicians. Good spotting. I can't measure the length of my piece of paper using different sized paper clips.

[Presenter points to paper clips along the bottom edge of the paper, indicating the different sizes. They move the shorter paper clip above the third longer paperclip pointing at the difference in size between the longer and shorter paperclips.]

### Speaker

So, you're right. What we should have done is sorted them out first. So, I'm looking for ones that are the same length.

These ones are not the same length as those big ones.

[Presenter moves the 3 longer paperclips from the bottom edge of the page into a group on the paper and lines them up horizontally one above the other to show that they are same length.]

### Speaker

They are definitely shorter.

[Presenter lines up the shorter paperclips horizontally one above the other to show that they are the same length.]

### Speaker

They're all the same length about. I think this one is like the big ones. Can you see any other big ones? Yeah, thank you. This one.

[Presenter moves the pile of paperclips around and finds more longer paperclips and places them above the other longer paperclips.]

### Speaker

It's good sorting task, isn't it? I'm sorting them based on their, attribute of length.

[Presenter finds one another long paperclip and places it at the top of the longer paperclips column and pushes them all along their edges, so they align.]

### Speaker

So, that's why some of them in here you can see ones that are yellow and blue and pink because we're not worried about the colour, we're worried about how long they are.

[Presenter moves the shorter coloured paperclips to the column with the others and indicates the different colours using their finger.]

### Speaker

That's really important when we're measuring length.

Green one. This one, this one, I think this one’s a long one. Oh, oh my gosh is that the same length? No, it's a bit shorter than the long ones and it's...

[Presenter continues sorting the paperclips and then finds a paper clip which is not in the same shape as the other standard paperclips, it is in the shape of a plane.]

### Speaker

Oh, it's about the same length, isn't it? It's a little bit bigger than these ones here, so I'd love to be able to use my plane paper clip, but I'll put that away for now.

[Presenter compares length of plane paper clip with other paperclips by lining it up next to the longer and shorter columns to see if it is the same length. Presenter then places the plane paperclip back into the paper clip holder as it is not the same size as the others.]

### Speaker

So, you're right, what I should be using is all the same size, so let's go back to using these ones cause that's where we started.

[Presenter pushes the shorter paper clips to the top of the page and moves the larger paper clips down.]

### Speaker

So, we had worked out I think, that we had 3 of them.

[Presenter now moves 3 longer paperclips to the bottom edge of the page end-to-end, starting in the left-hand corner as before.]

### Speaker

And now how many more do you think I need to measure the length of my line?

[Presenter indicates the length of the paper with their fingers.]

### Speaker

Yes, some of you are thinking about three more cause that looks like that has about the space to fit 3 more.

[Presenter uses 2 fingers to roughly measure the length of the 3 paperclips and then moves their fingers to the right to see if that is how many paperclips it will take to measure the paper.]

### Speaker

And some of you think it's four cause it's a little bit bigger than three. You think? OK, let's work it out to check.

[Presenter indicates the length of the paper using their fingers again. Presenter checks the 3 paperclips are neatly lined up along the edge of the page, touching at the ends, then adds one more paperclip horizontally next to the third paper clip. They use 2 fingers to roughly measure the length of the paper to the right of the 4 paperclips, still yet to be measured.]

### Speaker

What are you thinking now? Now you think just two cause now when you see that you think I can't fit three more paper clips that length in there, let's see.

[Presenter adds 2 more paperclips end-to-end along the edge of the page and points to them as they reach the end of the page.]

### Speaker

Nice visualising mathematicians. Six paperclips long.

[Presenter points to each end of the paper indicating it is 6 paperclips long.]

### Speaker

But you know, I wonder now because I have these shorter ones about how many shorter paperclips do you think I would need to go the length of my paper?

[Presenter places 3 shorter paperclips end-to-end horizontally under the row of longer paperclips. They use their fingers to indicate the length of the 3 paperclips and then the length of the paper to the right, still yet to be measured.]

### Speaker

Can you pick up your pencil and paper now and draw a picture and show me about how many small paperclips will it take to go the length or the distance of the edge of my paper.

Over to you mathematicians.

[End of transcript]

### Instructions

Draw your estimation of how many small paper clips it will take to measure the length of the paper.

## About how many paper clips – part 2

Watch About how many paper clips – part 2 video (3:17).

[Screen reads: You will need … a pencil or pen and some paper.]

### Speaker

Welcome back mathematicians.

Can you please show me your drawings?

[Screen shows a blue A4 piece of paper placed horizontally on screen. On the left side of the paper there are 9 multicoloured paper clips and on the right side of the paper there are 2 large, white paperclips. On the top right corner of the paper there is a paper clip holder. On the bottom edge of the page going from left to right there are 6 larger paper clips which span the same length as the paper. Under the first 2 larger paper clips in the left-hand corner are 3 smaller paper clips.]

### Speaker

Thank you, okay, well, let's gather some more evidence now and check and see if you would refine your thinking any further.

And you know I noticed something while you were busy drawing, I noticed something going on just over here in this section.

[Screen shows the presenter using a small orange piece of paper to cover the rest of the paperclips. She leaves the 2 large paperclips and the 3 smaller paperclips on the bottom left edge of the page visible.]

### Speaker

I'll cover this part, so we know to look here and look.

[Screen shows the presenter pointing to the 2 large paper clips and then the 3 smaller paper clips underneath.]

### Speaker

There's 2 big paper clips and they are the same length as 3 smaller paper clips.

[Screen shows the presenter removing the sticky note. She points to the third and fourth larger paper clips and gestures to the space below. The presenter traces along the edge of the 2 large paperclips and signals the 3 smaller paperclips that would be equivalent in length to the 2 larger paper clips.]

### Speaker

Yeah, and so what I started to wonder is that does that mean that I would have another one, 2, 3 paperclips along this distance and another one, 2, 3 paperclips along this distance?

[Presenter points to the middle 2 larger paper clips distance, pointing out one, 2 and 3 spaces where the smaller paper clips could fill in the distance then does the same for the last 2 larger paper clips.]

### Speaker

And then I started using my mathematical imagination to say, well, if there's 3 here, then there'd be another 3 and another 3, and I know 3 and 3 combines to make 6 because it's like patterns on a dice.

[Presenter points to first 3 smaller paper clips under the larger paper clips, then moves along to the right pointing out the distance of the next 2 larger paper clips. Underneath she taps along 3 times to signify the 3 smaller paper clips and and repeats for the final 2 large paperclips.]

### Speaker

And there's 2 3s and that's 6 and then 3 more 6, 7, 8, 9, so I think I could use my mathematical imagination and this idea that for 2 big paperclips there were 3 small ones to help me reason.

Shall we check now together? That would be great. Okay, so another 3.

[Presenter now places 3 smaller paper clips under the next 2 larger paper clips.]

### Speaker

That's right, I'm gonna make sure line them up nicely where they're touching, no big gaps, no overlaps. Oops they're a bit fiddly?

[Presenter then places 3 more smaller paper clips underneath the next 2 larger paper clips].

### Speaker

And 3 more. And look at that.

9 small post, paper clips one, 2, 3, 4, 5, 6, 7, 8, 9, and 6 big paperclips, one, 2, 3, 4, 5, 6.

[Presenter points to the first smaller paper clip and then the last small paper clip. She then uses one-to-one correspondence to count out the 9 small paperclips. She then points to each of the 6 larger paper clips.]

### Speaker

And for each of my 2 big paperclips, there's 3 smaller ones.

[Presenter points to 2 larger papers clips and points to the 3 smaller paper clips underneath.]

### Speaker

What an amazing mathematical discovery and good warming up of your mathematical imaginations.

Let's get ready for our next challenge. And before you leave me what’re some of the mathematics.

[Written on screen, What’s (some of) the mathematics?]

### Speaker

Yeah, we noticed some interesting things. You can measure the length of the edge of the paper using different size paper clips. The paper is either 6 big paper clips long or 9 small paperclips long.

And really important, we notice this relationship that for each group of 2 big paper paperclips, there were 3 smaller paper clips.

[Screen shows 2 big paper clips with 3 smaller paper clips underneath them.]

### Speaker

Really cool noticing today. Nice work mathematicians.

[End of transcript]

### Share/submit

- What did you notice?
- Share your ideas in your workbook.