Monday, 31 July 2017

Colour Patterns in Multiplication

Here is something cool we did in our class. It started off quite simple and then ended up with some questions that we are still thinking about.

So I was playing around with the Unifix cubes and I thought to myself, "What if we used the cubes to represent counting pattens and then we lined them up side by side? What would that look like?"

So I made a number line and we started laying out the patterns. 

The top pattern is the counting by 2s pattern - the white are the odd numbers and the reds are the multiples of 2.

The second row is counting by 3s, the one below is the counting by 4s - I think you get the idea.

The kids thought this was starting to look good so we went a bit further.

We decided that starting at zero was important. And we decided that zero should be red - because it fitted the pattern (because zero is a multiple of all numbers).

And so I asked, "What patterns can you see?"

And here is what we found:

We found some nice looking diagonals and also some verticals. 

Could we use these patterns to predict multiples of numbers that we hadn't modelled yet? 

And why don't some numbers have any red at all? Could there be something special about them?

And what would happened if we extended our lines further? 

Some many questions...

Friday, 28 July 2017

Squares and triangles - Part 2

The day after we played with the "counting by 3's" pattern, I was keen to push on further.

Sometimes "lessons" seem to go longer than 45 minutes. I could see this one was going to take a few days.

Make a square pattern

When the kids come into the room each morning, they are greeted with a pattern challenge.

"Make a square pattern," said the sign on the white board.

So here is what they did.

This is not quite what I had in mind but we never throw away a student's attempt - there is always fertile ground to explore. I hadn't said anything about "area" or "perimeter" or what I expected the square to look like. So this was an interesting conversation piece. Even though we were not going to pursue this type of square. I wanted to direct the thinking towards arrays.

Interesting. A nice square array (6x6) showing how (6x6) can be made up of 9 lots of (2x2).

Lots of squares in evidence here. 

A nice sequence from (1x1) up to (6x6).

This is nice. Can you see (1x1)? (3x3)? (5x5)? (7x7)? And why is it the odd-numbered squares? 

Then I asked the students to draw their squares and write about them. This example is pretty exception for a Year 2 student. 

So we started to look at the pattern of square numbers. An interesting conversation but one I left hanging - I wonder what the next square number is? And the one after that? And the one after that?

The Next Day

Before I had a chance to say anything the next morning, two different students handed me their workings for all the square numbers up to 10x10 (Student A) and 12x12 (Student B). They had both decided to see where the pattern went.

Squares and Triangles

As students came into the room on Day 3, they found a challenge question: "Can a square be made up of two triangles?"

And now we came to the exciting bit where I stepped back and watched what they would do. Would they use any of their previous experience from the previous two days to tackle this question.

I already knew that a square number is the sum of two triangular numbers. But I wasn't about to tell them that. I wanted to observe the thinking.

Here is what unfolded.

Everybody rushed into making triangles. Lots of good looking triangles. Lots of pairs of identical triangles. But we just couldn't get them to fit these triangles together to make a square!

(Hence, probably demonstrating that the two triangular numbers you add together to make a square number need to be different)

It was decided - this myth was busted! No, you cannot make a square out of two triangles.

I finally intervened.

"Hang on - check out the question again. Instead of starting with the triangles, what if you started with the square first?" I asked.

But once again the answer was definitive: you cannot make two triangles out of a square unless you want to cut some counters in half.

Then there came that beautiful moment in teaching, when one lone, brave wavering hand is raised by the shy girl towards the back.

"I..I..think you can..."

It was so simple. Here is her diagram:

And her explanation:

Yes - you can divide the square into two triangles - all you have to do is "miss" a bit to one side of the diagonal and draw your cutting line just in between the counters in the array.

That teaching moment was the highlight of my week. 

Wednesday, 26 July 2017

Squares and Triangles - Part 1

Throughout the year, I have been using Jo Mulligan's "Patterns and Structure Mathematics Awareness Program" (PASMAP) as a resource to stimulate some direction in my maths inquiry with Year 2. It is a great initiative that brings a clear focus onto the significance of spatial reasoning in all areas of mathematics education. The kids love it.

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...

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...

Exploring the Counting by 3's Pattern

As mentioned, I wanted to take the conversation deeper and to get the students to start thinking about representing the pattern using numbers. 

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.

The "Zero Row" - an important aside

During the recent holidays, I attended the AAMT national conference in Canberra (Actually, we were hosting it). Several times during the week, I heard speakers refer to the "zero row" - the very first step in a pattern. 

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

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. 


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?