# Finding numbers on a number line

A thinking mathematically targeted teaching opportunity focussed on subitising a collection of dots

## 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
- MA2-RN-01
- MA2-RN-02

- MAO-WM-01
- MA3-RN-01
- MA3-RN-02
- MA3-RQF-02

## Collect resources

You will need:

eyeballs

thinking brains

something to write with

strips of paper

## Watch

Watch Finding numbers on a number line video (8:13).

[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 pink number line is across the bottom of a sheet. It has arrows on both ends and has 5 points at different sections. 2 of the points are labelled 50 and 150. 3 of the points are labelled with a question mark and a letter: A, B and C.]

### Speaker

Hello, mathematicians. I love receiving challenging tasks and trying them out and really getting my brain nice and sweaty. And my friend Michelle sent me this task to try out with a group of kids, and they loved it. We had two really different ways of looking at the problem, very different, but both with really robust thinking. So, I'm gonna show you how they thought about this problem. We're gonna separate the two types of thinking into the blue team and the pink team.

[The speaker places a Bluey toy in the upper left-hand of the sheet.]

### Speaker

And the blue team will be represented by Bluey. OK, Bluey.

[She places a Batman toy under the Bluey toy.]

### Speaker

And the thinking of the pink team will be represented by Batman. So, I'm gonna take you through their thinking and see what you think. So, let's look at this problem. I need to find what point A is.

[She points to point B.]

### Speaker

What point B is.

[She points to point B.]

### Speaker

And what point C is. There are some things that I know though.

[She touches point 50.]

### Speaker

I know that that is 50.

[She touches point 100.]

### Speaker

And I also know that that is 150. I'm going to take you through the thinking of the pink team first. Now they thought that number lines often start with zero. So, they decided that point A would be zero.

[She places a pink post-it note labelled 0 over point A.]

### Speaker

So, I'll just place that over here next to Batman.

[She touches point 50 and point A. Then B.]

### Speaker

And then they thought, well, it looks like the 50 is halfway between point A and point B, but they had to prove it. And they used this really clever strategy of paper folding.

[She places a green strip of paper against the number line. She marks on the strip where point A and point B aligns. Then she folds the paper in half.]

### Speaker

So, what they did was, what they wanted to find out was if indeed the 50 was halfway between point A and point B, so they got a bit of paper. They marked point A and they marked point B. And then they folded it in half.

[She joins the two dots on the paper together. She unfolds the paper and places it against the number line. She aligns the dots under point A and point B.]

### Speaker

So, we're gonna join the dots here together to know that they're lined up, and when they opened it up, they saw that, yes, the 50 was halfway between point A and point B, which made them think that since 50 is five tens, double five tens is ten tens. So, this should be 100.

[She places a pink post-it note labelled 100 over point B.]

### Speaker

And they decided that 100 was point B. Now, the blue team strongly disagreed with the conclusion reached by the pink team because they thought...

[She touches point A, then point 50, then point 100.]

### Speaker

Well, if this is zero and this is 50 and this 100.

[She places her thumb on point 50 and her forefinger on point 100.]

### Speaker

Then this section here represents five tens or 50.

[She places her thumb on point 100 and her forefinger on a point halfway between point B and C. Then points to the halfway point.]

### Speaker

Which would put 150 over here. And we actually know that 150 is all the way over there.

[She touches point 150.]

### Speaker

That was part of the original problem. But thinking about halfway points, they thought

[She touches point C.]

### Speaker

Well, it could be that point C is halfway between 50 and 150. And they really liked the way that the pink team set about solving the problem, so they thought they'd try the same thing. So, they got a strip of paper and placed it underneath the number line.

[She places a strip of paper against the number line. She marks on the paper where point 50 and point 150 are.]

### Speaker

And they marked 50 and 150. And then they set about seeing if C is actually the halfway point. So, let's check that now.

[She takes the paper off the number line.]

### Speaker

Fold it in half and make sure that the two marks met.

[She places the paper against the number line, aligning points 50 and 100 on the paper to those points on the number line.

### Speaker

And yep, that's right, they could conclude that C is actually the halfway point.

[She touches point 50 then point 150.]

### Speaker

Now, here we've got 50 or five tens, and 150 or 15 tens. Now, the difference between 15 and five is ten. So, half of ten is five.

[She touches point C.]

### Speaker

Which means that this point here has to be five tens more than 50. So, 100.

[She writes 100 under point C.]

### Speaker

So, they decided that point C is actually 100.

[She places a blue post-it note labelled 100 on top of the sheet, above point C.]

### Speaker

Now, at this point, the pink team thought your reasoning is sound. We are gonna have to change our mind. We agree this must be 100.

[She moves the pink post-it note labelled 100 from above B, to under the blue post-it note labelled 100.]

### Speaker

And we agree this cannot be zero.

[She removes the post-it note labelled 0 from point A.]

### Speaker

But then, what is A and what is B? So, at this point, both groups were quite certain, really, that point C was 100. And the blue team thought, well, let's use this certainty to try to work out what point B is.

[She places a label ‘100’ under point C.]

### Speaker

And the pink team thought, well, why don't we use the number line that we've already got. They actually suggested, we don't need the whole number line, let's just use the part that we need.

[She cuts the number line at 100.]

### Speaker

So, that's what they did. They split the number line at 100 and at 50.

[She aligns the paper marks to points 50 and 100. She points to 50 and 100 on the number line.]

### Speaker

So, they thought, OK, well, we know that this is 50 and this is 100, and they are both landmark numbers, they are multiples of ten. And the pink team thought, well, it might be useful to look at the other landmark numbers that could help us. Now, we know that there are five tens in between 50 and 100.

[She touches point 50 and 100. Then circles with her finger the section between the two.]

### Speaker

So, they're gonna need to divide this section of the number line into five equal parts, into fifths. There's a few ways that you can fifth, that you can divide into five equal groups.

[She removes the paper.]

### Speaker

But they decided they would use two and a half rolls, a technique that they'd learned from Professor Di Siemon.

[She rolls the paper around her right forefinger and middle finger.]

### Speaker

So, they did one roll. We need to have two full rolls and then another half.

[She rolls the paper until the sections were in equal five parts. Then she folds it.]

### Speaker

And because you're going at two and a half, it means that where you end is on the opposite of where you start. So, you want to fiddle around with it for a little while. It actually took me a while to be able to get confident using this technique. And then at the point there, you press it down

[She opens the paper up.]

### Speaker

And that gives you five equal pieces.

[She places the paper against the number line, aligning the numbers with the marks.]

### Speaker

Put them back in here and we can actually mark them.

[She marks on the number line where the creases where. She labels the first mark 6 tens, the second 7 tens, the third 8 tens and the last 9 tens.]

### Speaker

So, we've got six tens, seven tens, eight tens and nine tens.

[She touches the marks 6 tens and 7 tens.]

### Speaker

But really, it's this part here that interests us. Six tens and seven tens.

[She touches point B.]

### Speaker

And actually, this B seems to be halfway between six tens and seven tens. So, 60 and 70. Six tens and five more is 65. So, at this point, both teams agree that point B would be 65.

[She labels point B ‘65’.]

### Speaker

They actually felt that they trusted this with a level of certainty, so they labelled it here. So, because there's certainty about point C being 100, I'm gonna take this Post-It out.

[She removes the post-it note labelled C.]

### Speaker

And now we have certainty about B being 65. So, I'll take this one out.

[She removes the post-it note labelled B.]

### Speaker

And the pink team learned something very important here, it's that number lines don't always start at zero. Which made them think, well, what is point A? It's not zero.

[She touches point A.]

### Speaker

It can't be zero. But what is point A? So, over to you to find out what point A is. And if you'd like to challenge yourself, maybe you can find what zero is.

[Text over a navy-blue background: Over to you! What is point A and where is 0?]

### Speaker

Over to you, mathematicians.

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

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

Bullet points below read:

· Mathematicians share ideas and ask questions.

· Number lines don't always have to start from zero.

· Using landmark numbers can help when solving problems.

· You can find landmark numbers by partitioning, breaking apart using fraction strategies.

On the right-hand side of the bullet points is an image that shows the number line.]

### Speaker

What's some of the mathematics? Mathematicians share ideas and ask questions. Number lines don't always have to start from zero. Using landmark numbers can help when solving problems. And you can find landmark numbers by partitioning, breaking apart using fraction strategies.

[ 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 point A on our number line?
- How can you prove it?

Can you find 0 on our number line?

How can you prove it?