Playing with the Number Line

A lot of the work we have been doing this year in Year 2 has been inspired by a couple of people:

- Tom Lowrie and SERC (STEM Education Research Centre) at University of Canberra

- Jo Mulligan and PASMAP (Pattern and Structure Mathematics Awareness Program) at Macquarie University. 

The following activity is based on a lesson from PASMAP. The kids loved it. Have a look.

Day 1

We started with a blank number line on a piece of paper about 3 or 4 metres long. I included arrows at each and then marked on zero.

I told the kids we wanted to make a number line that went up to 50. Instead of marking in each unit, I thought it would be quicker if we just counted up by 5's. 

But where should we put the tick marks?

And where should we put 50?

We knew we had to make the marks evenly spaced so after trying a few unsatisfactory alternatives (a hand, a pen, a shoe, a small diary) we finally settled on a library book that we could use to mark out our number line.

We knew how to count by 5's so marking it out was pretty straight forward. We did this with a red marker.

"Oh no!" I said. "I didn't want us to count by 5's. I meant count by 3's! Where are we going to mark in the counting by 3's pattern?"

After some negotiation we were able to agree where to put the 3's numbers. We did this with a blue marker. 

Once we had marked in the counting by 3's pattern, it wasn't too hard to convince them to find where the 4's pattern would go. we did that one in green.

Panorama shot of the number line.

This is how the number line started out...

...but as we progressed down the line...

...we soon noticed something...

"Hey! Some of these numbers have two colours!"

"Really?" said I. "I wonder why that is?"

Quick as a flash, one bright young lad said, "Well obviously 12 is going to be green AND blue because 3 x 4 is 12."

Great observation for an 8 year old. And so ended Day 1. 


Day 2

I wanted to reflect a bit on what we had done with our number line and then push on a bit further to see where we might end up.

I had a think about some questions that had been prompted by the initial inquiry into the number line and wrote these onto the paper. 

We have rotating activities for maths groups on a Monday morning, so I made the number line one of our tasks. As the groups rotated, the students got to add their bit of thinking onto the number line.

Here is how it unfolded...

1. What numbers go here?

There were a couple of places on the number line I wanted the students to explore.

Negative numbers are below zero

- and we will get to that in Year 6 in the Australian Curriculum

So what numbers are meant to go in these gaps?

Even though we don't have a tick mark for them (yet), there are still some numbers to account for in these spaces.

And what are the numbers that come after 51?

One student started listing them but quickly realised this was going to be a big job.

Instead of naming all of them, we designed a group called "Numbers above 51".

2. Is our number line structurally sound? Does it work?

There were concerns about some of the spacing. 

We knew they were meant to be evenly spaced.

"This is a problem because some numbers are too close together."

3. The Counting by 7's Pattern

I wanted to throw in the 7's pattern to see how they would go.

Worked it out pretty well.

I am setting them up to find prime numbers and we needed to eliminate multiples of 7.

4. Factors and Common Multiples

The rest of this inquiry looked at places on the number line where a number is part of more than one pattern. 

12 is blue and green, not because the colours are "light", but because 12 has both 3 and 4 as factors. 

So are we going to ever find a number that is blue (3's pattern), green (4's pattern) AND red (5's pattern)?

Took us a while but we got there in the end.

YES - 60 will be blue, green and red.

So I wonder if there will be a number that is blue, green, red AND black??!!

Yes, replied the kids, but it is going to be a long way down the number line. Probably 2 million.

Year 2 Addition with Numberlines

This morning I had the opportunity for some quality time with Year 2. I wanted to see what they could do with numberlines. And how they could use them to explain addition and subtraction.

So we started off with a strip of paper that we wrapped around our heads - seemed like the best place to start.

Using the paper strips, we measured the circumference of our heads and then compared to see who had the biggest. Contrary to expectations that the teacher would have the biggest head because (i) he was older and (ii) he had more brains, we found that while my head has a circumference of 57cm, one of the students' heads had a circumference of 61cm.

We compared strips of paper and found that, yes, one was indeed longer than the other. I asked them if they knew a word that would describe that bit of overlap - but couldn't draw out the word "difference" so I had to tell them.

"Oh yeah, we know that," they said.

Next, we had a general chat about how to draw a numberline and then had a go at representing our head data:

57cm + ? = 61cm

This they did (sorry, no photos of this bit) as it was quite an easy calculation to make. The answer (4cm) was never going to be a surprise but I wanted to see them using a known fact before we explored any further.

Then I asked them to use a numberline to show me:

57 + 14 = ?

And so the fun began!

Here is what we came up with...

So, in one jump we go straight from 57 to 71. This is mathematically correct but shows me nothing about using a numberline or the process behind the operation. I asked for more information - and got a blank stare.

Another student started with this idea - what if I jump by 10 straight to 67? But, seeing that space was running out, she wiped out the line and started again...

...to produce this one. Add one jump of 10 and then another jump of 4. Good work!

 But then we had this solution. According to this representation, 57 + 14 = 70. I put it to the group that this one was the correct answer and that 71 was incorrect, which threw them a bit.

"Look!" I said. "There are 14 jumps under the number line. So 57 + 14 must equal 70."

They knew it couldn't be but couldn't see why.

Here's where it gets messy. As the numbers get closer together, there student runs out of room and ends up skipping one of the jumps that he is numbering, so the jump between 69 and 70 is not labelled. Yes - we do need to be careful when construct a numberline. 

So what?

This was just an introductory activity to get an idea of what the Year 2 students understood about numberlines. 

As a result:

1. we talked about some of the important features, such as labels, direction, accuracy and infinity

2. we talked about how to make "provable" jumps, not just blind leaps to apparently random answers

3. we modelled addition on a numberline

4. we laid the groundwork for future explorations using numberlines for other applications

5. we had some fun and I got to meet a few students who I didn't know before. Looking forward to getting back in to see them on a regular basis.