*This is Book 2 for Years 1 + 2.*

*Do yourself a favour and get a copy.*

We have spent a lot of time playing with patterns. I wanted to explore the link between the pattern and the numbers behind it. This exploration was going to be related to some of the PASMAP ideas, so I can't take credit for them but I think we explored a few interesting related ideas as well.

This "lesson" or "sequence" went over about 4 days. It started with a counting pattern. It ended up looking at the relationship between square and triangular numbers.

This is going to take a few posts...

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__Counting by 3's__

Anyway, we started off looking at what happens when we count by 3's. This is a good pattern to look at because it is slightly more difficult than counting by 2's or 10's but not so difficult as to be out of reach.

So I asked the students to show a counting by 3 pattern using our favourite Cuisinaire rods. Here is what they produced:

*Using the light green as 3, this is a nice representation of what happens when we count up by 3.*

*Notice the outline of the pattern is starting to look like a triangle...*

*We did start to run out of light green rods so this student did some interesting substitutions:*

*i) using a red/white combination to replace the light green*

*ii) using the dark green to represent 2x light green*

*This student is showing both additive and multiplicative thinking - great fluency!*

*A-ha! The triangle shape is starting to emerge!*

*Definitely a triangle...*

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__Exploring the Counting by 3's Pattern__

The PASMAP program encourages a pedagogical model based on five components:

1. Modelling

2. Representing

3. Visualising

4. Generalising

5. Sustaining

We were heading into the "Representing" phase.

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__The "Zero Row" - an important aside__

So of course, when we started talking about the counting by 3's pattern, this was where we had to start.

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__Representing the Pattern__

Talking through what we had made on the floor, I asked the students to describe each level of their pattern. The pattern itself was not a new one - they had seen it before - but I was interested in seeing what it would reveal.

We came up with this representation:

Firstly, I used some Unifix cubes and some Blutack to show the pattern on the whiteboard.

Column 1: the representation

Column 2: number of groups

Column 3: how many in each group

Column 4: total number of cubes

Yes - we included the zero row.

Then I asked, "Are there any mathematical symbols we could put in between the numbers to make number sentences?"

"Yes - we could put a plus sign!"

Really? Well we tried it and guess what? It didn't work.

So, next guess?

"We could do a times sign, like 2 times 3 equals 6."

So we gave that a go. Note that we were happy with 2 x 3 and above but didn't want to commit to 1 x 3 or 0 x 3.

Yet.

By working backwards, we were able to use the pattern and agree that 1 x 3 = 3 and that 0 x 3 = 0.

"Can we see any more patterns in there?" asked the annoying teacher, not satisfied with the hard work the students had put in.

*Anita Chin's fabulous "Make it, say it, draw it, write it" magnets on the white board*

So we wrote out the counting by 3's pattern up to 51. Then we looked at what we saw:

- there are 4 numbers less than 10

- there are 3 numbers in the 10's, 3 numbers in the 20's

- then back to 4 numbers in the 40's etc

- the units pattern goes 0, 3, 6, 9, 2, 5, 8, 1, 4, 7 and back to 0

- the pattern goes odd, even, odd, even

Using these patterns, I speculated about what numbers we would find after 51, without needing to actually count by 3's.

Here is what we predicted:

- the number after 51 will be even

- there will be 3 numbers in the 50's

- 60 will be the next number to end in a zero

The I asked, "Did you notice that numbers in the pattern can be reversed and the new number they make is also in the pattern? Try 15 - reverse it and you get 51. 24 becomes 42. 21 becomes 12."

I wonder if that is always true?

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