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 din'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!

Sunday, 7 May 2017

What is the place of facts in mathematics?


Friday Night Reflections

Last Friday evening I caught up with a few friends who I hadn't had a chance to spend any quality time with recently.

After we had solved the world's problems and analysed the youth of today, one of my friends came out with a very provocative statement:

"I think 'facts' are given too much importance in science," he said, or words to that effect. 

Now, let me say that the gathered assembly that totalled 4 people comprised science or maths educators (x4), well educated science graduates (x3), current or former president's of state professional associations (x2), and of course the mandatory partridge in a pear tree. You can use a Venn diagram to solve that one kids.

So this provocative statement about 'facts' soon generated some passionate discussion.

And three days later I am still thinking about it. 

And I think I agree - 'facts' ARE given too much importance:

  • not because I am pushing some sort of anti-intellectual glorification of ignorance agenda

  • not because of any Trump-esque view that 'facts' are subjective (although context and interpretation are always critical in understanding facts)

  • not because I am advocating a free-form content-less maths curriculum

BUT (and here is my big BUT)

  • because I believe that 'facts' - and this includes 'maths facts' - are just a starting point. 

What are these 'facts' of which you speak?

When I am talking about 'maths facts', these are a few of the things to which I am referring:
  • addition and subtraction facts
  • times tables facts
  • formulae
  • processes and operations
  • arithmetic
  • symbols and representations
  • notation
These things are often presented as "being the mathematics". When you Google for an image of "mathematics" you get chalk boards (do people still use them??) covered in squiggly notation and complex formulae. These are the things that are considered by many to "be" mathematics.

I don't think they are.

Just as an umlaut isn't the essence of German language, so too a square root sign isn't mathematics.

I think we have lost sight of the purpose of mathematics - to make life better/simpler/more beautiful. This purpose has been obscured by......'facts'.

So what then IS the place of facts in mathematics?

Let me back pedal a little to explain my position:
  • yes - it is useful to know the 'facts' - it is more efficient to know them rather than having to invent them each time I want to use them. 
  • yes - I still teach them to the kids in my class (where 'teach' means explore, play, investigate, pull apart and reconstruct - as well as memorise)
  • yes - a component of assessment needs to consider the mastery of 'facts' and 'skills'.

BUT (and here is my big BUT again)

1. These maths 'facts' are just the starting point. 

They are not the end in themselves. If my assessment is purely based on a student's ability to crunch numbers, recall formulae and perform calculations, then it is a pretty thin type of assessment.

2. 'Facts' need to be useful to make sense of a context. 

And the context is useful to make sense of the facts. 
Here is a fact - water boils at 100 degrees celsius.
Ah - except when...

3. There are bigger things than 'facts'

There are some 'big ideas' that are at the heart of education, things that we need to explore and consider on a daily basis. Things such as:
  • connection
  • equality
  • consistency
  • representation
  • form
  • function
  • cause and effect
  • (insert your favourite concepts here)
These are the things that I am hoping to get my 8 year olds to explore and dissect in the short time that they are with me. It is these things that I want them to know about. These are things that I want them to start analysing - using 'facts', but developing understanding.

Thanks to my PLN and the provocation last Friday night. Hopefully some of this makes sense - or at least provokes some thinking.

Thursday, 4 May 2017

Symmetrical Patterns

One of the ideas the kids have been playing with in our pattern making sessions is "symmetry".

I want to show you a few examples of what they have come up with. This is pretty visual - I think the images speak volumes:

It might start quite simply...

...but it doesn't take long before we start to get more complex ideas.

And then we can head into the 3rd dimension.

And someone always has to go over the top...

But just making pretty patterns - do the kids really understand anything about symmetry? Have they learnt any real maths? Or are they just having fun?

So then, I asked them, what is symmetry?

And the kids told me:

- It is about balance.

- I look for the line of equal.

- I know if I put a bit here, I have to put the same bit on the other end.

Insightful 7 year olds. Love it. Pretty sure they have learnt something about maths today.

Wednesday, 3 May 2017

Patterns! Patterns! Patterns!

This year we have started each morning with patterns.

The kids come in and we have about 15 minutes of making patterns - sometimes I nominate a feature to explore, other times the kids get to make their own. Sometimes they work in groups, sometimes they work alone. 

It always starts off looking like this...

Yesterday after 15 minutes it looked like this...

Busy weren't they? 

Let's look a bit closer at what the kids produced...

A nice mix of materials. 
I do like the red and blue "zipper" pattern made from diamonds and trapezia.

More nice things including an interesting spiral arrangement. 

Love what can happen with a hexagon.

This looks like a screen shot from "Space Invaders" or Galaga"

So much happening here:
1,1 patterns
2,1 patterns
1, 1, 1 patterns
3, 1, 1 patterns
3, 1 patterns

This 15 minutes each day has become significant for my class for several reasons:

1. It is fun - it encourages the kids to get to school on time. They know if they turn up 10 minutes late they will miss something that they enjoy.

2. They are learning from the minute they walk in the door - at the start of the year, there were students who couldn't make a simple pattern AB repeat. Now they are making much more complicated patterns for themselves or building on a pattern started by someone else.

3. It is connected - when we make patterns, we are using language, we are learning social skills, we are developing mathematical understanding, we are building concepts that will apply across all subjects, such as shape, growth, prediction, expectation, observation, generalisation, perspective, form, and so much more. 

Have a go - see what your kids can do.

Tuesday, 2 May 2017

Polygons and Non-polygons

We are moving in to looking at 2D shapes. Our conversation started with a bit of a provocation.

My colleague and I drew up a chart on the white board and we had a stack of pictures of different shapes. Without saying anything, we started placing our shapes into either category - polygons or non-polygons.

We ended up with this:

The idea was to get the kids to think about what we had used as criteria for making decisions about placing items into categories. 

...and the ideas started coming. 

Here is our conversation and the ideas from the kids:

Question: So what is a "polygon"?

Piper – It’s probably a shape.
Sam – They’re shapes that don’t have any round bits on them. All of the lines are straight.
Sarah – They have straight sides.
Josiah – The oval is not a polygon because it has no straight sides.
Andrew – Polygons have straight and non polygons have curved sides.
Rupert – Some of them have four or five sides.
Liam – They have 3 or more sides.
Carys – They can’t have 2 sides; it would have to be curved.There are only two points on the straight line to get joined up so you need a curve.
Alexander – If a side is not joined up then it is not a polygon.
Andrew – The sides of a polygon don’t have to be the same size.
Molly – The sides can’t cross over.

So we distilled it down to one statement:

A polygon is a 2D closed shape with 3 or more straight sides that do not cross over each other.

Happy with that?

Well, then we started throwing out a few more examples and asked the kids to place them in the appropriate group.

One of our very perceptive students was given this picture and sat for a significant time pondering where to place it:

I was confused. Why was she hesitating?

I soon found out.

"Well," she said, "It has most of the things. It is a 2D shape, the lines all join up so it is a closed shape. They don't cross over. It has more than 2 sides. The only thing it doesn't have is straight sides. So it has 4 out 5 things for a polygon."

Nice thinking Abbey!

This raised the important point: being a polygon means you have to have all 5 criteria:
1. 2D shape
2. Closed - no gaps
3. Straight sides
4. 3 or more sides
5. No crossing over

If you fail one of these things, you are not a polygon.

So I went looking for examples that were "almost" polygons: 4 out of 5 of these things but only missing one of them.

Like this one:

straight sides - check
3 or more sides - check
closed shape - check
no crossing over - check
2D shape - uh-oh!

straight sides - check
no crossing over - check
closed shape - check
2D shape - nope
3 or more sides - nope

I wonder what other shapes we could find that are "almost" polygons? I think this question helps use to examine shapes using our definition and make appropriate decisions when classifying shapes. 

And the kids went off to search for new shapes to look at...