Show me a quarter

I asked the kids to show me a quarter.

Here is what they did.

I'm not going to say anything.

Just I love those kids.

And yes, the yellow line has 3 squares, 

which is a quarter of the total 12 squares used...

Quarter of a Cross

There is a puzzle out there called "Quarter of a Cross". Amie Albrecht (@nomad_penguin) at the University of South Australia suggested that my Year 2 kids might like to give it a go. It is a puzzle that has appeal for learners from a range of ages. Amie had seen what we were doing with fractions last week and thought it would be interesting for my kids.

The puzzle looks like this:

from https://amiealbrecht.com/2016/05/21/quarter-the-cross-and-fraction-talks/ 

So we gave it a go. 

And here is what happened.

The first 10 minutes were spent coming to terms with the problem - what would a quarter look like? 

Well we found lots of things that it wasn't. We soon realised that the cross could be made of five squares, so colouring in one or two of these was not going to make a quarter.

Back to the drawing board.

This was one of my favourite solutions because it showed me the struggle that was going on. It was like a window into the mind of the student.

"So, what if I do this? Hmm, no I don't think so. What about... yes that might work!"

It was the process of deciding on a "proof" - how to show I had found a quarter. Along the way, I am going to need to eliminate a few ideas but this is all part of the learning process.

While doing this task, I pulled the class back in for a "chat" about 3 times. This helped to clarify the question and also to discuss any important information we had discovered. This was shared with the class to help progress the solutions. We also had a break of a few hours and revisited the task in the afternoon. 

And here are a few of the ideas that emerged:

1. You can use one of the "legs" of the cross and add it to 1/4 piece of the inside square. 

What the students had seen was that 1/5 + 1/20 = 1/4 although they were not able to articulate this.

2. Using a line that passes through the centre of the cross

This was a good idea. If you extrapolate from this you will find that this method produces an infinite series of solutions as you rotate around the centre point. We only represented 3 solutions.

3. There is a left handed and a right handed house

Interesting - same idea but reflected.

4. Colour makes everything look good

So Amie - hope you like our pics. I think they look great. The kids had a really good time and it was difficult but we got some good learning out of it.

Now, I wonder what other shapes we could divide up...

Lego Fractions

We are continuing our exploration of fractions. I thought it might be interesting to see how we could represent unit fractions (1/2, 1/3, 1/4 etc) using one of our favourite learning tools - Lego!

So we started off gently. Let's have a look at 1/2...

Then 1/3...

Yep - that was pretty much what I was expecting.

So I asked, "Can you do it another way?"

Never fails...

Here is what 1/2 could look like:

Or this...

Or this...

And 1/3:

And...

And from that, a student worked out 1/1, 1/2, 1/3, 1/4, 1/5...

Yes, I know, we went beyond the Australian Curriculum requirements for Year 2 - but how do you stop the kids when they are on a roll and obviously know what they are doing?

Free Choice Friday

One of our favourite days is Friday - that is the day we get to choose out own patterns that we want to make.

So I thought I would share a few pictures of the patterns the kids made last Friday.

Hope you enjoy them...

A very nice 3D creation.

A staircase using some interesting combinations.

 We make nice star patterns in 2BF.

I love it when they go 3D!

A beautiful pyramid.

A staircase - without any yellow or orange Cuisenaire rods.

The other kids had got to them first.

This was something we did earlier in the year - we loved the diagonal lines.

Yep - it's Pikachu!  He's symmetrical - the border isn't but we fixed it later.

And a symmetrical house!

A Collaborative Pattern

We took it up a level last week. 

We like making patterns. I wanted to make a really long one. We had a whole wall along one side of the classroom.

And I also wanted to worked on collaboration with the students.

So I got a long roll of paper and cut it to size. Then I divided the strip of paper into equal sized squares. There were 18 of them. I pencilled in the vertical, horizontal and diagonal lines of symmetry for each square as a guide for the next step.

I lined the students up and told them they were going to make a symmetrical pattern. 
18 times. 
By adding one piece at a time. 
And by building onto what the previous people had already done. 
And we would be using tangram shapes.

Lots of instructions (poor teaching technique) but we did some modelling and broke it down and scaffolded the first few through the process.

Pretty soon we were cooking with gas.

Getting started was a bit tricky but we recognised the need to be systematic.

Slowly the pattern began to emerge.

The pencilled-in lines of symmetry were very useful.

Our final pattern. Each was slightly different, even though we tried really hard to be consistent.

And the final result was very pleasing.

So - interesting to see all the little pieces come together to make a bigger pattern. 

Just like all the little students come together to make something bigger too...

8 in a row

There is a book called "Open Ended Mathematics Activities" by the godfather of Australian mathematics education, Professor Peter Sullivan. If you do not have a copy, you need to get one. (It is available through AAMT - try here: 
http://www.aamt.edu.au/Webshop/Entire-catalogue/Open-ended-Maths-Activities 

There is an activity in the book that asks you to imagine a line of 8 students, some of them standing up and some of them sitting down. How many different ways could you organise the standing and sitting children?

We started working on this problem. The kids got several different solutions so I stopped them and asked them to predict how many different ways they thought it could be done. Most answers ranged from 4-8. One student predicted 10 solutions, another 20 and a final student guessed 1000. 

"He probably just means a really big number," said one perceptive student.

I decided to change the task a little bit and break it down into a few smaller steps.

So I took it back to the simplest possible question:

If there are 8 children and only 1 of them is sitting down, what are all the possible positions that the children could be organised in?

The first couple of minutes we were pretty random in our strategies. We were just playing with combinations and seeing what happened.

Then the "A-ha!" moment when one group discovered the advantages of being organised and systematic.

We found that there were 8 possible ways for the line of 8 to be organised with one sitting and 7 standing.

So I asked, what if there were 2 sitting - but they have to be sitting together? (I asked for them to be sitting together to try to simplify the problem and to limit the possibilities. It also revealed some interesting patterns - see below.)

Once we had seen how to "get organised" and systematic, it was a pretty quick journey to finding that there were 7 possible arrangements with 2 sitting together and 6 standing.

Here's one for those Richmond fans...

Three sitting and 5 standing.

Then with a bit of intuitive thinking, some of the students saw a pattern emerging.

The number of possible solutions plus the number of sitting students will always equal 9 (one more than the number of students).

For example:

And yes it proved to be true.

We only needed to go a few more steps before we were convinced.

Yep - 4 sitting plus 5 combinations equals 9.

So adding up all the possible combinations, we found 36 ways to arrange 8 students, some standing and some sitting. But these were only the combinations where the sitters are sitting together.

Now, I wonder how many possible combinations there would be if the sitters could be ANYWHERE in the line?