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:

Cube


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.


Tetrahedron



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.

Cylinder



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.




Discussion

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.