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...

Creativity in Maths

Is it possible to re-think our approach to maths?

Here is a video put together with help from my son and one of his friends. We did this a while ago but the sentiment still holds true. Hope you like it - it's my first attempt at video…

The quality of the video is a bit dodgy but Blogger has limits on the size of files you can upload. If you want a better version, send me an email and we can work something out.

bruce.ferrington@radford.act.edu.au 


I have also now posted this video on Youtube:

https://www.youtube.com/watch?v=aULT8bU9-_o

Hope you enjoy it!

Number Patterns

I have been giving my kids a challenge each morning when they arrive at school - make a pattern. Sometimes I have told them what sort of pattern I would like them to make. Other times it has been free choice.

I put out some materials - blocks, dice, shapes, counters, beads etc - and watch what happens. I will show you some of these things in the next few days.

But today I put out some number cards. These are cards I made up with the digits 0-9 on them. I asked the kids to make a number pattern.

Whenever we do a task like this, there are a few key individuals who take a lead and other students tend to follow their interpretation of the problem. I am keen to discourage this. I would like to see all students thinking for themselves and not relying on other people to tell them how to interpret life.

Anyway, here are some samples of what we first came up with:

Some numbers with patterns in them...

Not quite was I was expecting. Even this creative version:

We had a conversation about this - stopped working and considered the nature of the task. (This is a common conversation - I often pause the activity after about 10 minutes and refer the students back to the original request - what do you need to do?)

Were these really number patterns - or were they numbers with patterns in them? Is there a difference? How do you define a "number pattern"?

This got a few brains recalibrated and so we got bak to work.

And here is what we got after another 10 minutes:

Ah-ha! Counting up by 100's.

And counting up by 10's.

And then I saw this one:

3 5 7 9

2 4 6 8

1 3 5 7

"Stop!" I said. "Don't go any further!"

I asked the class to look at the pattern.

What would the next number be?

I put it up on the whiteboard and asked students to add their thoughts during the day.

I also intervened/interferred and put the numbers in reverse order to that which their creator had placed them (because when she explained it to me, she started with the bottom row and moved upwards to the top row.)

Here is some of the solutions that the students gave:

So, what IS the next number in the pattern?

How would you explain what is going on? 

Are we adding by 1111? Or are we following an odd/even pattern between each row? Or are we doing consecutive numbers in the digits considered vertically?

Ask your class - they may have another explanation.

Good luck.

World Maths Day - Pattern Hunt

A few days ago I got a tweet from @prashanigamage wondering about ideas for World Maths Day. I suggested a "pattern hunt" - she suggested students write a book celebrating what they know about number.

I've put the two together - we have made a book about patterns we found in the school.

We started with our pattern hunt. Here are a few we found:

Here is an i-movie we made from photos we took in the playground:

We had lots of fun - so much fun that we ended up making a book about the patterns we found.

Here are a few excerpts from the book:

One of our climbing things has white, red and yellow balls on it.

We have some artwork in the playground.

This is a worm with consonant phonemes on it.

Our landscapers were creative with the placement of trees and seats.

Also practical, now that those trees have grown a bit and give us shade.

Our fencing has a regular number of bars on it.

Thanks again to @prashanigamage for the inspiration.

Our focus on patterns continues!

There is no miraculous new way to teach maths.

or.. Some things I learnt in 2016.



During 2016 I was fortunate to work on a project called:

reSolve: Maths by Inquiry

Keep an eye out for it - there is some good work behind this project.

This project is lead by the Australian Association of Mathematics Teachers and the Australian Academy of Science. It is based at the Academy in Canberra.

The main building at the Academy of Science, sometimes known as the Martian Embassy.

I learnt a lot over the year, much of which I will share with you over the course of time. But here are some of the "big learnings" that came through to me:

1. There is no miraculous new way to teach maths 

There is no magic pill, no silver bullet, no "deus ex machina", no program we can import "drag and drop" style into our schools that will save our PISA results or make us Top 5 in TIMMS. 

There is only striving for our best - our best efforts, our best examples, our best pedagogy, our best teaching practice. This is quality education.

2. If I do not give all students full access to the curriculum, then I am perpetuating discrimination and inequity.

There is a temptation to give some students a "watered down" version of the curriculum, to give an over simplified or "dumbed down" education in mathematics. This is not fair. It is not just. It is not equitable. 

Yes - it is difficult to engage all students. Yes - it is difficult to provide access to the full maths curriculum to all students. Yes - it is difficult to pitch lessons at a level that will challenge all students in the class.

But that it our job. It is why we choose to front up to the class each day. It is why we do not work in tele-marketing or data entry. We want to teach and we want to do it well.

3. We need to hear the voice of the child.

This has been simmering away in the back of my mind for a while now. To effectively engage students in learning, we need to give them a voice. We need to listen to their ideas and suggestions, we need to reflect on their observations and we need to examine their strategies to find new and powerful insights.

The great man himself, Peter Sullivan, told me, "99 times out of 100, the teacher needs to shut up. The other 1 time, the teacher needs to shut up."

This is very difficult for many teachers - don't we just love to fill the space with the sound of our own voices? But to stop and listen to the voice of the child - ah! there is deep learning to be had here.


Anyway, that's a few of the things I learnt in 2016.

The challenge is for me to take them into the classroom and see if I can actually do them.

I'll let you know how we get on.

I'M BACK!

Australian readers may appreciate the reference to a cultural icon - Aunty Jack.

Just when you thought it was safe, I have returned from my year off (well, working hard at the Academy of Science, but more of that later) and am back in the classroom.

Look out Year 2 - it is going to be a year full of maths.

So, as every teacher does, I have set a few goals and resolutions for the year:

1. To focus on pattern making and spatial reasoning - get the kids to make, create and extend patterns each day as a "start the day with a bang" kind of activity.

2. Be patient - this is a personal goal. Be patient with the kids. Be patient with colleagues. Be patient the system. Be patient with myself. Lots of opportunity for growth here.

3. Nurture relationships for learning - get alongside the strugglers and hear the story from their perspective. Open doors to growth by listening and finding those "things that make you go WOW!"

4. Get back to the bagpipes.

I'll let you know how we progress with each of these over the year. 

We apologise for the temporary suspension of services

Dear reader,

I am pleased to announce that I have I recently accepted a 12 month appointment as a Resource Writer with the "Mathematics by Inquiry" project being run by the Australian Academy of Science and the Australian Association of Mathematics Teachers.

As a result I will be away from the classroom and will not be able to post my ideas and reflections to this blog.

Thank you for your support over the last three years. Please stay tuned for some exciting developments from the "Mathematics by Inquiry" team. And please continue accessing the ideas I have already posted here on the blog.

All the best,

Bruce Ferrington