Over the last week I’ve spent some time look at functions and patterns with my Grade 8 class. They’ve been struggling with the idea of creating a rule for a set of values. I’ve also been reading Jo Boaler‘s book Mathematical Mindsets, which made me think about trying an exploratory activity to introduce the concepts around functions.
I’ve looked at the NCTM’s Illuminations site a few times, and struggled to find anything which I could relate to the topic that I was currently teaching, but this time I came across an activity where students made different length trains using Cuisenaire Rods. Unfortunately in my school we didn’t have any rods, so I ended up using snap cubes which I created different length rods with.
Using the activity sheet provided on the website, students began to explore how many ways they can make each length train from length 1 to length 5. As they started to work a lot of them began by looking at the length 5 train which was a hard place to start. I encouraged them to look at the smaller lengths first.
As they began to explore, they discovered that there was a pattern to their work and some of them started to try to figure out the rule that went with the pattern, as I asked them to find how many ways there were to make 6, 10 and 32 length trains (they had quickly realised that working it out with the rods was impossible).
Almost all of the students realised that the values were doubling each time, and quickly wrote out every combination all the way up to 32. Once they had done this, they started thinking about the link between the length of the train and the number of ways to create a train of that length. Someone mentioned that 2 should be the base, and then another said that the pattern was 2 to the exponent of the length of the train.
They tested this theory but discovered that it was flawed, and this led to the realisation that if you divided the answer by 2 then you arrived at the correct answer. I hadn’t been expecting this to be the solution that they came up with. At this point I prompted them to think about which value they were actually finding and they came up with the fact that the exponent should be x-1. We wrote all of this onto a giant post-it sticky note showing the rule and all of the steps. I also wrote on their solution to ensure that I’d treated them as sense makers. (if I’m honest they were more excited about the size of my post-it than they were about the fact they’d found the solution to the problem).
Next class I introduced them to Pascal’s triangle, using a sheet which I had created (you can use any template). I then asked them to write a table of how many ways there were of making a train of each length using 1 block, 2 blocks, 3 blocks, 4 blocks and 5 blocks. As they began to do this and fill out the tables by drawing their patterns, none of them actually realised what they had found, it was only as they began to count and fill in a numerical table that they slowly one by one realised that they had written out the first few rows of Pascal’s triangle. They then made the link to the fact that they now knew how many ways there were of making a train of each length using a certain number of blocks.
We had gone from not understanding rules to a situation where students could conceptually explain what was happening, and identify the patterns in what they were seeing and doing. The final work is now stuck on the classroom wall for them to look back at, and a number of other classes have looked at it and talked about it too.