Wednesday, 20 September 2017

8 in a row

There is a book called "Open Ended Mathematics Activities" by the godfather of Australian mathematics education, Professor Peter Sullivan. If you do not have a copy, you need to get one. (It is available through AAMT - try here: 

There is an activity in the book that asks you to imagine a line of 8 students, some of them standing up and some of them sitting down. How many different ways could you organise the standing and sitting children?

We started working on this problem. The kids got several different solutions so I stopped them and asked them to predict how many different ways they thought it could be done. Most answers ranged from 4-8. One student predicted 10 solutions, another 20 and a final student guessed 1000. 

"He probably just means a really big number," said one perceptive student.

I decided to change the task a little bit and break it down into a few smaller steps.

So I took it back to the simplest possible question:

If there are 8 children and only 1 of them is sitting down, what are all the possible positions that the children could be organised in?

The first couple of minutes we were pretty random in our strategies. We were just playing with combinations and seeing what happened.

Then the "A-ha!" moment when one group discovered the advantages of being organised and systematic.

We found that there were 8 possible ways for the line of 8 to be organised with one sitting and 7 standing.

So I asked, what if there were 2 sitting - but they have to be sitting together? (I asked for them to be sitting together to try to simplify the problem and to limit the possibilities. It also revealed some interesting patterns - see below.)

Once we had seen how to "get organised" and systematic, it was a pretty quick journey to finding that there were 7 possible arrangements with 2 sitting together and 6 standing.

Here's one for those Richmond fans...

Three sitting and 5 standing.

Then with a bit of intuitive thinking, some of the students saw a pattern emerging.

The number of possible solutions plus the number of sitting students will always equal 9 (one more than the number of students).

For example:

And yes it proved to be true.

We only needed to go a few more steps before we were convinced.

Yep - 4 sitting plus 5 combinations equals 9.

So adding up all the possible combinations, we found 36 ways to arrange 8 students, some standing and some sitting. But these were only the combinations where the sitters are sitting together.

Now, I wonder how many possible combinations there would be if the sitters could be ANYWHERE in the line?

Wednesday, 23 August 2017

Build a staircase pattern

I needed some pictures of staircase patterns for a presentation I was doing at the recent Canberra Mathematical Association conference. So I put out the Cuisenaire rods out and the kids went for it.

I gave them 5 minutes and looking down, I saw a couple of standard staircases in front of me.

I was just about to say, "Can you do it another way?" when I looked around.

Here is what I saw. 

My creative bunch of crazy kids had already pre-empted my request and they had come up with 17 different staircases.

Here they are:

And then there was....

So in a class of 23 kids, we had 19 different staircases. 

There is so much maths to explore in these patterns. 
- What changes between each step of each staircase?
- How much does it increase (or decrease) by?
- What patterns can you see?
- Can you make predictions about what the 20th step would look like? or the 100th? or the nth?

Interestingly, the adults in my conference workshop didn't come up with quite as many different staircase patterns.

Never undersell the creativity of the students in front of you.

Tuesday, 8 August 2017

Border Patterns

We have been exploring the inspirational PASMAP program in our class this year. We have tried lots of great ideas and this has formed the basis of our focus on patterns. 

A huge thank you to Jo Mulligan and Michael Mitchelmore.

This is the book that has been my inspiration this year.
It is available from ACER.
No - I am not on commission. Wish I was.

So one of the tasks that PASMAP uses is called "border patterns". Students are given an outline of a shape with pre-drawn squares on which they can place Unifix cubes to make patterns.

Sounds simple. Genius often does.

We had made patterns like this before but I was interested to revisit the task and change a few of the variables.

When we did this previously, we had made AB patterns or ABB patterns.

This time I wanted to increase the number of colours to 3 and not specify the length of the unit of repeat. What would the kids do?

Task - Use 3 colours to make a repeating border pattern around this shape.

Yep - can do. Easy.

Here is what happened.

So the students leapt into action. Soon we had cubes and blocks happening.

And then came that awful moment of realisation - 

"Hey, I can't do it."

"I wonder why? What can I change?"

"Look - your pattern has 2 oranges next to each other."

"Nope - that won't work either."

So, ABC didn't work. AABBCC didn't work either. This student found a solution using AABBC. 

Nice thinking.

Can you do it a different way?

And this one is AABCB. A nice variation.

So I asked about the "unsuccessful" borders. Why can't you make a ABC pattern or AABBCC pattern around this border?

One student soon realised that this border required 10 cubes so the unit of repeat couldn't be 3, like in ABC.

"Because 10 is not in the 3 times tables. If I count by 3, I will never say 10."

And from that idea, we soon agreed that the unit of repeat would need to be 2 or 5 (the factors of 10). But since we had to use 3 colours, then we would need to make a unit of repeat that was 5 cubes long - such as AABBC or AABCB etc.

Moving on, we looked at the second border on the page.

Same rules - use three colours and make a repeating border

The first thing the students did was count how many squares were in the border frame. They were learning...

Here are some of their patterns:




You guessed it - the unit of repeat was 7 - because they counted that there were 14 squares around the border.

I wonder how many different ways you could do this?

Friday, 4 August 2017

Free Choice Patterns

One of our favourite days is when I say, "Free choice patterns!"

The kids love it.

But they are very funny kids - the patterns they "choose" to build are often closely related to the prescribed patterns of the previous few days.

Take a look at what we did with Free Choice Patterns yesterday:

Yes - inspired by the 5x5 challenge from earlier in the week, this student has decided to see what happens with a 6x6 grid using 6 different colours...

...and then what to do with a 6x4 grid using 6 colours.

A different student's variation on the 5x5 challenge. This time it is using 4 colours and making the diagonals in pairs.

A-ha! This looks a lot like the patterns of squared numbers that we did previously. 1x1, 2x2, 3x3 etc

And look! It's the one where we counted by 2s, 3s and 4s and put the columns side by side.

I was really interested to see how students took a familiar idea or pattern and took it further, trying new things with it and pushing it to see what would happen.

Of course, some were highly decorative as students explored shape and colour:

With all this pattern making, have we neglected the essentials, things like number skills and operations?

Well, we did do a "lesson" on addition of 2-digit numbers with regrouping this week. That's about as hard as it gets in addition for Year 2. The lesson took about 3 minutes.

This is not an empirical research study - but all bar 3 of the students "got it" in the 3 minutes. Had playing with patterns actually helped lay foundations for number work?

I would say - yes.

As one boy said, "It's a bit like a pattern - you just have to know what comes next..."

Thursday, 3 August 2017

Number Patterns

Having done lots of patterns with shape and colour, I thought it was time to move the attention to number patterns.

I had assumed that the students would know what I meant when I said, "Make a number pattern."

Apparently not.

Or maybe I was the problem - maybe my definition was too narrow and my expectations were based on my own previous experience and knowledge.

Anyway, here is what they produced:

So - we have a pattern of numbers being repeated. There were lots that looked like this. We could substitute the numbers for a colour or a shape and we would be back to where we were with our Unifix and Cuisinaire patterns. 

Yes - it is a pattern.

No - it is not what I was expecting.

But wait, there's more...

Ok - so I can see counting by 2, 3 and 10. This is what I had anticipated. Obviously these students will win at the game of "Guess what the teacher is thinking". 

And some more sophisticated variations on these patterns:

Counting by 7s

Counting by 10 but off the decade

As I circulated and chatted, I came across one student who was mucking around with some rulers. Fortunately, I refrained from intervening - I was about to tell him to put them away.

He had made a simple 0-9 grid:

As I was about to move on, he laid down the rulers:

"Look at that!" he said. "There is a pattern that goes 2, 7, 2, 7 down the middle ruler. And the diagonal rulers are all the even numbers."

Very true - a nice observation.

Note to self: Do not interrupt. Give the students space to play and experiment. They will observe things that will surprise you.