Thursday, 27 August 2015

Patterns That Grow

As Marilyn Burns says, pattern is the password of mathematics.

I like playing with patterns. I use them to get the kids interested in finding relationships.

So I gave them this provocation:

Here's a pattern - 1, 5, 9, 13….

Based on this, can you tell me if 21 is going to be in this pattern? And then will 45 be in it as well?

So we got the blocks out and started making some patterns. Here is what the kids came up with:

"The pattern makes a cross shape. And 21 makes the same shape so it must be in the pattern."

"And 45 can make a really big cross too."

From talking as a group we were able to work out that yes, indeed, 45 is going to be in our pattern.

In fact one student pointed out that it was like the 4x pattern (4, 8, 12, 16…) but just one more.

It was a good starting point. The conversation was never going to end there.

What other patterns can you make? Can you explain your pattern using numbers? Can you tell me what the next number will be without making the model of it?

Here's a nice pattern - just like the cross but missing a leg.
CAn you see a link to the 3x pattern?

This one adds on a leg and goes 3D.
And now we were thinking about the 5x pattern.
Is Grade 2 too young to start talking about y = 5x + 1?

We had a good talk about this one. Is there a step missing somewhere? Should there be something between 1 and 8? Is 1 part of the pattern?

Another pattern based on squares - ah ha! Square numbers!

And then taking the square into the third dimension!

This is not the end of work with patterns for the year. We will revisit the concept many times but I feel that the kids have a good grounding now in identifying, making and explaining patterns.

And we had a lot of fun.

Wednesday, 26 August 2015

Messing With Their Heads

I know - it's not something they teach you in your training. I know it is abuse of trust. I know I shouldn't be advocating this.


I wanted to see what would happen when I showed my Grade 2 kids this on the white board:

I had just spent Saturday at the Canberra Maths Association conference where I presented a workshop on multiplication. I am interested in kids understanding what multiplication is all about - not just in memorising parrot-fashion their times tables (although memorisation is important, it sure has helped me in life, but it needs to be backed up with understanding.)


I put the "2x tables" as above in front of the kids.

And they were happy to start reading them out, chanting as they went.


About half way down, their confidence started to waiver and the mumblings grew.

By the time we got to the end, with me loudly proclaiming that 12 x 2 = 122, I had very few followers.

And some insolent 8 year olds began to suggest that I had made a few mistakes!


Anyway, I got them to help me correct my errors, which they did:

And order was restored to the universe.

Why did I do this, when I knew that many of my students were uncertain and lacked confidence in multiplication?

Why did I deliberately give them wrong information and incorrect answers that would confuse them?

Well, I was hoping that they would be able to use the knowledge they had to find the mistakes, correct them and re-establish the pattern that they knew underlies the 2x tables.

And they did.

And hopefully no-one went home and told their parents that Mr Ferrington is hopeless at maths and doesn't even know that 3 x 2 = 6.

Wednesday, 17 June 2015

Nets of 3D Objects

I don't think my kids have done much with nets before. It's important to be able to represent 3D objects in this way.

I was keen to see what they could do so we got out some equipment to have a play.

Here are a few of the 3D objects we looked at and some of the drawings the kids did:


Suggestion #1 - like a rectangle with a square off the side.

Suggestion #2 - Like a cross

Suggestion #3 - lots of square bits

Suggestion #4 - A cross made up of 6 smaller squares 

So we had the general idea that the net of a cube is made up of lots of squares - just not too sure about the details.


Suggestion #1 - it's a triangle with some stripes

Suggestion #2 - it's got more than 1 triangle

Suggestion #3 - think I've seen one before with a square and triangles coming off it

So we don't really have a very good idea about this one. Lots of opportunity to learn here.


Suggestion #1 - it's got 2 circle shapes on top of each other

Suggestion #2 - It's definitely got 2 circles somewhere

Suggestion #3 - It's got circles AND rectangles

Suggestion #4 - yep, I've seen this one before

As teachers, we learn so much from the "fails" of our students. The "correct" responses are fine but they don't in themselves give us much insight into the mathematical thinking of the students.

BUT the errors, the mistakes, the ones that aren't quite right - they are the ones that tell us so much about what our kids are thinking and how they "see" mathematics.

Monday, 15 June 2015

How We Organise Ourselves

We have started a new inquiry in the transdisciplinary theme of "How We Organise Ourselves". I find this theme really interesting in terms of mathematical thinking so I got a bit of a provocation going for the class.

I got a bucket load of what we call "paddle pop sticks" - probably have a different name in your town - do Queenslanders really call them by-jingo sticks?

The task was to find out how many sticks there were. The task itself was pretty meaningless - my prime interest was in how the kids would approach the challenge. What evidence would they show of mathematical thinking? How would they organise themselves?

Well, first off they fragmented into friendship groups and started grabbing for as many sticks as they could get - they are 7 years old after all. 

Anyway, they split up and started to count the sticks. They showed several different methods of getting organised. Here is what it looked like:

One group started counting the sticks and making a pile of the ones they had counted.
They were a bit stunned when I asked how they were going to check for accuracy.

It's a bit easier to see if they are spread out a bit.
But you have to do a lot of counting.
And turn a corner when you hit the wall.

Another group thought of lining up the sticks - in groups of 6...

…or groups of 14 - beautifully colour-coordinated.

Other groups had the idea of making bundles of 10.

 Some appeared more organised than others.

I had to drag them back to the original question…so, how many sticks are there?
Each group wrote up their individual totals
 - but it took insight to decide that they needed to add them all together.

An answer! We got 2637 sticks!

And then I threw them a wobbly - can you check that?

What? they exclaimed.

But, I continued, did you get any ideas from the other groups about good ways to organise yourselves?

Groups of 10 was the consensus - but I was still a bit worried about the way they were laid out on the floor. When we hit the wall we had a problem... 

...some chose to go around the corner and others decided to start new rows.

Finally we got the idea that maybe we can organise these sticks into:
a) bundles of 10
b) rows of 100
c) blocks of 1000

And that made it pretty easy to see 2 groups of 1000, 3 rows of 100, 2 bundles of 10 and 2 left over = 2322 sticks.

And you can see it just by looking.

What a good idea.

Wednesday, 6 May 2015

The Typical Student in Year 2

We are currently inquiring into "Who We Are". It is a opportunity to learn a little bit more about life and our central idea is:

Birth, growth and death are part of 
the natural cycle of living things.

To find out about ourselves, we decided to collect some data. We wanted to know what things we had in common and what our differences were. Some of our questions were about physical features, some about our personal preferences.

Here are our 5 questions:

1. Are you a boy or a girl?
2. How old are you?
3. What sport House are you in?
4. What is your favourite colour?
5. What is your favourite school subject?

All students in Year 2 were surveyed. Here are our results:

Now that we had some data, it was time to start playing with it. 

Our "Typical" Year 2 Student

We found that we had one student who was in the highest scoring category for each question.

She was a girl, currently aged 7, in Acacia, liked blue and her favourite subject was PE!

We had found a typical student - she was very excited!

Our "Atypical" Student

We also found we had a student who was in none of the highest scoring categories.

He was a boy, aged 8, in Kurrajong, liked red and loved doing maths!

An atypical student - he was equally excited.

We showed our data about ourselves…

…compared to the typical student…

…and then recorded this in a table and made a statement based on our data.


Much discussion followed once we started playing with the data. Lots of questions started to come from the kids.

Boys could never be the "typical" student in our data set because they were eliminated by the first question. Similarly, girls could never be the "atypical" student, since they would always have their sex in common with our typical student even if they disagreed on everything else.

Interestingly, our "atypical" student has a twin brother but they could be differentiated by their favourite colour - the other twin liked the colour blue - but all their other answers were the same.

We ended up producing a large graph showing how many of the responses each student had in common with our "typical" student.

An interesting distribution and one that brought on more questions. Prior to representing the data in this way, we asked the students to predict which group they thought would be the largest. Most opted for 2 or 3 things in common, agreeing that it might be expected for people have a few things similar but that there was plenty of option for differences.

The kids were engaged, focused and ready to take it further.

I wonder what they will come back with tomorrow once they go home and reflect on what they have done?

Tuesday, 5 May 2015

Friends of 10 - Is there another way?

One of the key things we need to nail early in Year 2 is "Friends of 10". 

0 + 10 = 10
1 + 9 = 10
2 + 8 = 10
3 + 7 = 10
      4 + 6 = 10 etc

Hopefully most of the kids have this concept by the time they get to us - but there will always be a group of recalcitrants. 

For those who have mastered this idea, they will get frustrated if they feel they are spinning their wheels while they wait for the rest to go through the process each day of trying to recall these number combinations.

And this is indeed what happened.

As we were going through the Friends of 10 number pairs, a couple of my students had them written down and were back to me in less than a minute. Several other students were still rolling around on the floor looking for pencils.

So, using one of my favourite questions, I asked them:

"Is there another way?"

This puzzled them.

"What do you mean?"

"Well, I can see you've done the obvious ones: 1+9, 2+8 etc. Is there another way?"

They hadn't actually expected to have to think about Friends of 10. They thought they knew them all. This was unfamiliar territory.

Then a moment of inspiration!

"Oh! Do you mean with negative numbers!!??"

"You show me," I replied, annoyingly.

And here's what they came back with:

There's a nice pattern kids!

"That's great!" I said. "Can you show me another way?"

"Another way?" they asked.

More thinking required.

"Oh! Do you mean with Roman numerals??!!"

"You show me," I replied, annoyingly again.

And here's what they came back with:

Nice work! Just check the spelling...

It was getting to the point where we needed to move on. I suggested that it would be good to see how many other ways they could show Friends of 10, perhaps as a Home Learning activity.

The next day, I was presented with these ideas:

1. Building on the Roman numeral idea, here are a few Friends of 10 using number systems from other cultures.

2. Looking at fractions and decimals

 3. Using more than pairs, thinking outside the box.

 4. Graphic representation - the Friends of 10 Rainbow.

 5. Going further - some students decided to explore Friends of 20 to see what happens.

Some examples of Friends of 20, nicely typed but a bit random 
which makes it hard to see any patterns.

Friends of 20 done as "buddy pairs": 12 + 8 and 8 + 12

Friends of 20 - an incomplete list but could see where it was going. Do I really need to write them all out if I can see that it will repeat itself once I get past 10???

 Friends of 20 - the complete list

So - I was a bit surprised. I thought, as did several of the students, that this Friends of 10 business might be pretty simple. Who would have thought there could be so many options to explore?

Hand it over to the kids - they will think of things that will never occur to you.

And have some fun.