# Finding numbers on a number line

A thinking mathematically targeted teaching opportunity focused on the distance between numbers on a number line

## 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

## Outcomes

- MAO-WM-01
- MA1-RWN-01
- MA1-RWN-02
- MA1-GM-03

## Collect resources

You will need:

- something to write on
- coloured pencils or markers.

## Watch

Watch Finding numbers on a number line video (5:30).

[Text over a navy-blue background: Finding numbers on a number line. Small font text in the lower left-hand corner reads: NSW Mathematics Strategy Professional Learning Team (NSWMS PL team). In the lower right-hand corner is the white waratah of the NSW Government logo.

A title on a white background reads: You will need…

Bullet points below read:

· eyeballs

· thinking brains

· something to write with

· strips of paper if you would like to practise finding halfway points with strips of paper.

Text over a navy-blue background: Let’s investigate!

A red number line is in the middle of the screen. It has arrows on both ends and has 6 points at different sections. The second point from the left is 2. The last point on the right is 10. At the bottom of the screen are 4 blue square sticky notes: there are 3 smaller ones on the left of a bigger square that has a 0 on it.]

### Speaker

Hello, mathematicians. I've got a number line problem for you here. It's quite a challenging one, actually.

[The speaker traces the number line from left to right.]

### Speaker

I've got my number line going across.

[The speaker touches each arrow.]

### Speaker

These arrows show us that the number line continues forever in both directions.

[She touches the first point on the left.]

### Speaker

But I've got some points that I have to find. There is some information here for me, though.

[She points to the 2 and the 10.]

### Speaker

I know where two is on the number line and I know where ten is on the number line.

[She touches the points without a number.]

### Speaker

My job is to find what these points are. Now, I do have a little clue. I know that one of these points is zero. So, have a look at the number line and see if you have any ideas of where zero could go. You might like to pause so you can have a close look.

[The speaker touches the 2 missing points after 2.]

### Speaker

OK. So, my thinking here is that all of these points seem to be above two, which means that zero must be at this point right here.

[She touches the first point from the left.]

### Speaker

Would you agree? OK, so I might put zero there and I'll just make sure that the little lines match up.

[She places the sticky note with the 0 under the first point.]

### Speaker

OK, so now I've got three more points.

[She touches the points after 2, then 2 and 10.]

### Speaker

They're all between two and ten. So, how could we find what those points are? And I've been thinking about halves and how finding the halfway point can be really useful. And just by looking at this, the halfway point seems to be maybe around here…

[She touches the second point from the left.]

### Speaker

But I'm not sure.

[She places a strip of white paper across the number line.]

### Speaker

So, I might get a strip of paper.

[She gets a marker.]

### Speaker

And what I'll do is, I'll mark where zero is..

[She writes a zero on the section of the strip that is over the zero on the number line.]

### Speaker

And I'll mark where ten is.

[She writes a 10 on the section of the strip that is on the 10 on the number line.]

### Speaker

And I'm gonna fold them so that the two points match up.

[She folds the strip so that 0 and 10 aligns. She creates a crease in the middle of the fold.]

### Speaker

This means that this fold line here is the halfway point.

[She places a point on the fold line.]

### Speaker

Yeah, so, after folding in half, I can see that this is the halfway point between zero and ten, and I know that half of ten is five.

[She writes 5 at the fold line.]

### Speaker

So, this point must be five. So, what do I do is I might mark this point as five.

[She places a sticky note under the second missing point. She writes 5 in it.]

### Speaker

OK.

[She places a sticky note under the first missing point.]

### Speaker

Now, I might use some sticky notes here to help me.

[She places a sticky note under the third missing point.]

### Speaker

I quite like the idea of paper folding.

[She touches the first missing point, then 5 and 2.]

### Speaker

I can see that this point here is less than five and more than two. And by looking at it, it looks like…

[She circles the area between zero and 2, and then between 2 and the first missing point.]

### Speaker

This section is the same size as this section, so I might try that out.

[She places the strip against the number line.]

### Speaker

I'll match up my zero again.

[She marks on the strip where 2 aligns.]

### Speaker

I'll mark my two.

[She marks where the first missing point aligns on the strip.]

And then I'll mark where that point is. And what I might do at this point is I might actually get rid of this part of the number line.

[She cuts the strip where it is marked 5. She takes the section with zero and 2 and aligns it with the number line.]

### Speaker

So, what I'm trying to see is if two is actually halfway between this point and zero. So, let's see.

[She folds the strip in half. A crease is formed at 2.]

### Speaker

Oh, the fold was on the 2.

[ She aligns the strip against the number line. She touches zero, then 2, then the first missing point.]

### Speaker

This tells me that two is the halfway point between zero and this number. So, how can I find what that number is? Well, I've got an idea.

[She places her thumb on the 2 and her forefinger on the zero.]

### Speaker

This section of number line here is two,

[She moves her finger over to 2 and the first missing point.]

### Speaker

And this section of number line here must be two. I can also check it.

[She aligns the strip to zero and 2, and to 2 and the missing point.]

### Speaker

That's two and that's two.

[She removes the strip.]

### Speaker

So, what I could do is I could count by twos.

[She draws her finger from zero to 2, and then from 2 to the missing point.]

### Speaker

That's two, another two makes this four.

[She writes a 4 in the sticky note under the missing point.]

### Speaker

OK, now over to you.

[She points to the last missing point.]

### Speaker

Can you find what this point is? You probably have a hunch or an idea.

[She touches 5 and then 10.]

### Speaker

You know that it's bigger than five. You know that it's less than ten. But how can you prove what number it is? How can you test out an idea and prove it to be true? Over to you, mathematicians.

[Text over a navy-blue background: Over to you. What is the other unknown point on the number line, and how can you prove it?]

### Speaker

Over to you. What is the other unknown point on the number line, and how can you prove it?

[Text over a navy-blue background: What's (some of) the mathematics?]

### Speaker

What's (some of) the mathematics?

[A title on a white background reads: What's (some of) the mathematics?

Bullet points below read:

· Mathematicians look for clues and test out ideas.

· We can use proportional reasoning to find missing numbers on a number line. In this problem, we use doubling as well as the information that we had to find the points that we did not know.]

Next to the bullet points is an image of the strip being placed against the number line.]

### Speaker

Mathematicians look for clues and test out ideas. With this problem, I had the idea that finding the halfway point could prove useful, but I couldn't just assume that it was the halfway point. I had to prove it. To prove it today, I used a strip of paper. We can use proportional reasoning to find missing numbers on a number line. In this problem, we use doubling as well as the information that we had to find the points that we did not know.

[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

What is the other unknown point on the number line?

How can you prove it?