Wednesday, 31 May 2017

Numbers under the pattern

Yes - we are still playing with patterns.

It never gets boring because each day a new pattern, idea or interpretation emerges. This is the joy of inquiry in maths.

I had an idea a few nights ago that we tried out in class yesterday. Pictures are below.

I was thinking about the 100's square and how kids see patterns in the numbers. Like skip counting, patterns in the units column, patterns in the tens column etc.

I wondered what would happen if they covered their number patterns with coloured blocks.

Then I wondered what would happen if the grid under the coloured blocks wasn't the standard 100's grid.

What if it was only the even numbers?

Or what if it was the counting by 3's pattern?

Or what if it was counting by 10's?

Or what if we didn't start counting at 0 or 1 but chose some other number as the starting point?

And what if there weren't 10 numbers in a row, but maybe 5 or 7 or ...?

(Yes - I am a wonderer)

So I banged out a set of different grids for the kids to explore.

Here are the grids I made:


And then I asked the kids to look at any particular grid and find a pattern. Then I asked them to show their pattern by placing the coloured cubes on top of the numbers.

Finally I wondered if other people could work out what the numbers were underneath the cubes by looking at the pattern they made. And if they couldn't, would revealing a few of the numbers help give us a clue?

Here's what they did:

I made this pattern. Could you guess what numbers are underneath? What if I showed you a few of them?

Here's my pattern. It's actually a few patterns together. Let me show you one number. Any ideas?

What if I show you a few more numbers? Can you tell me what the patterns are?


This is two patterns put together. If I show you some of the red numbers, can you work out what the blue numbers are?

Does this help? And why have I put a red cube on top of a blue one in the fourth row?

"I am counting by 18. I just started with a different number for each colour cube." 
(from an 8 year old - I was impressed.)

So have a look at this pattern. What do you see? What might the yellow numbers be? And the red ones? What would the next line look like?

Here are a few of the numbers uncovered. Can you work out what the other numbers on the grid would be? 

Now I've taken away a few more. Does that help? Could you reconstruct the pattern and replace the missing cubes correctly?

This was a lot of fun. We had a lot of different things happening.

Next step will be to get the students to create their own grids to go under the cubes.

Or to create 3D grids.

Or to create grids that are not rectangular.

Or to....

Monday, 29 May 2017

A Calendar Puzzle

Here is a puzzle I made of the 2017 calendar.

It was something I had been thinking about for a while. I wasn't sure how it would go. So I put it together and left it for a relief teacher to use while I was away for the day.

It looks like this:

Each shape is a month. They are not in order. You need to cut them out and reassemble them in a long strip as the 2017 calendar. The colours are the seasons. I can probably send you a copy of the document if you are interested or you could make your own. 

My aim was to have a look at the way the months fit together and to show that time doesn't stop at the end of a month. In fact, these divisions and structures we place on time, like weeks and months, are pretty arbitrary. Even the seasons, which are observable in nature, do not necessarily conform to the months that we allocate to them. 

So when I got back to school, I sat down with the kids and asked them about the calendar. They showed me what they had done.

Nice work:

- found the months - tick
- recognised the seasons - tick
- completed the puzzle - tick

So what did they learn? I asked them to tell me what they saw:

Well, the month pieces fit so nicely together because there is no gap between the end of one and the start of the other.

All the months can't start on a Sunday for example because they are all different lengths.

Some months start on the same day of the week or finish on the same day as other months. March and November start on Wednesday.

Next year will start on a Monday.

Every day of the week gets to be the first day of at least one month.

The seasons were different colours. Summer starts in December so that is why it is red and so is January and February but they are not together.

And what questions did they have?

Will all years look like this?

Will there be a pattern of the days every year?

Are some years the same as other years?

How many different ways can you make a calendar?

Who invented the calendar?

Now we have some questions, I wonder who will take action?


Tuesday, 23 May 2017

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.

Tuesday, 16 May 2017

Making a Star Pattern

I asked the kids to make a star pattern this morning.

They knew what I meant but I wanted them to clarify the concept anyway.

They told me:

- it needs to be a star shape

- it needs to be a pattern

- it is not a straight line - by this, they meant it is not linear but it is radial, not words they know (yet) but ones that we will introduce.

Once they had made some patterns I asked them what these patterns had in common and what they had that was different.

They told me:

- all star patterns are joining together in the middle - this was a great observation and will become the focus when we start talking about radial symmetry

- different stars can have different numbers of legs - another good observation. Some of the patterns we made had 4, 5, 6, 8, 10....legs.

Here are pictures of what they produced:

4-pointed star:

This started from the outside and moved inwards. It got a bit tricky when the pieces didn't fit neatly but the idea is there.

5-pointed star:

I liked this because, while 5-pointed stars are quite a common in nature, I wasn't sure that any of the students would come up with this.

6-pointed star:

The pattern blocks are good for this as they make good use of the hexagon as a basic shape.

7-pointed star:

This is one I really wasn't thinking I would see. You can see that it is difficult to make a pattern with seven as your base number but we made a good attempt.

8-pointed star that became 12-pointed:

This one seemed to grow by itself as the pattern radiated outwards.

10-pointed star:

 This is unusual. It has symmetry but I'm not sure that is is perfectly radial. I think it gets back to the idea that the star needs to start at a point in the very centre.

Wednesday, 10 May 2017

Making big shapes out of little shapes

We have been using the PASA material from Jo Mulligan and ACER and came across something really interesting with the kids. 

One question looked at a conventional tangram puzzle and asked:
 - how many of the small triangle will fit into the middle sized triangle?
 - how many of the small triangle will fit into the big triangle?

I thought this would be pretty obvious to the kids.

It wasn't.

They had lots of confusion and lots of struggle.

Seemed to me that we needed to spend some quality time with the pattern blocks.

So we got the blocks out and started to look at how we could use the small shapes to replicate the bigger shapes.

My favourite shape is the yellow hexagon. I was interested to see how the students might construct a similar hexagon using the other shapes.

Here are all the possible solutions they came up with.

I wonder if that is all of them - or can you do it another way?

Then we had to consider other shapes, of course. We have a very large set (super size) of the pattern blocks. Some students got a few of these and tried to remake them using the smaller blocks.

This is just making the same shape using different sized pieces. The combination of the trapezium and the equilateral triangle is interesting though. 

This one is my favourite - it took some patience.

Then we started to explore other material. We found a square can be made from 4 triangles or from 10 of these rectangles.

Someone had fun making this big triangle.

And finally we had some fun using a random selection of shapes to make some squares.

The yellow one is cheating - it is 5 right angles triangles overlapping each other.

Then we had one student attempt to make a square from equilateral triangles but she got stuck.

"This is impossible!" she said.

"Why?" I asked, innocently.

"Well, the corner on the square is too fat. The green corner is always too little."

And there we have it ladies and gentlemen - you can never make a square if all you have is equilateral triangles.

Nice work, Year 2.

Monday, 8 May 2017

Change My Pattern

One of the activities we do with patterns is to change or build onto a pattern that someone else starts.

Firstly, each student makes a pattern using the materials provided.

Then they all stand up and walk around the circle until I say stop. And then they get to work on the pattern in front of them - adding to it, extending it and building on it.

This is very interesting because it means the students need to look at the new pattern, identify what it is doing, identify the unit of repeat and then modify it in a regular and repeating way. 

Here are a few examples of what we did today:

Adding C to an AB pattern:

Adding C and D to an AB pattern:

Adding D to an ABC pattern:

Changing the form of the pattern:

Changing the materials of the pattern:

Dealing with a very long unit of repeat:

Think this was going to trick someone? Think again! This student has identified the really long unit of repeat and has made some very interesting additions to the pattern. 

A great time in 2BF - well done champions!