Cuisenaire Rod Exploration Day 1

This year for my middle school interventions I have decided to focus on units coordination.  At this point I am envisioning that most of our work with be around multiplicative and fractional reasoning.  Although now that I am starting I may have to do some additive reasoning first with a few students.  

I like to start the year focusing on characteristics of mathematicians (taken from Tracy Zager's Becoming the Math Teacher You Wish You'd Had, I wanted some open activities that allowed us to notice, question, and explore.  I chose to use the Cuisenaire Rods and some activities that I learned from Simon Gregg.  

Here was the first day's lesson plan:

Making the counting numbers.

Start with 3

How many ways can we make 3?

Ask Questions: Are 2,1 and 1,2 considered different arrangements?

Prove: Have we found them all?  How do you know?

Move to 4

How many ways can we make 4?

Ask questions: How does that compare to 3?  

Move to 5

Intuition: How many combinations do you think there will be?

How many ways can we make 5?

Work together and alone: As it gets to be more combination will they each make combinations or work to find different ones to make the entire collection?

What do you notice?  What do you wonder?

• Look for patterns

• Make predictions

• Notice relationships (a 3 rod can be replaced with 3 other possibilities).

• Generalization: Is there a way to figure it out without trying to build them all?

Here are the results of some of our explorations:

I let the groups decide if order mattered.  When it did we found lots of different ways to arrange the combinations and looked for patterns so we could try to predict:

 

We discovered that if there are 3 blocks there are 3 ways to arrange them. 

If there are 4 blocks there are 4 ways to arrange them.

We didn't have time but this would be a great Always/Sometimes/Never exploration.  (I'll let you try it and see).

For the groups that decided order doesn’t matter there aren’t as many combinations but it was harder to explain/prove we had found them all (maybe because we didn’t spend as much time thinking about all the possibilities and arranging them to make sure we had them all). 

From 5-6, a student realized they could use all the combinations of 5 and then add one to each.  Then looked for additional combinations.  

We noticed a pattern, but our pattern broke.  

Reflection:

This activity on the surface seems very elementary.  I had a moment of doubt before my first group of students.  "Am I really going to ask 8th graders what numbers add up to 4?"  Thankfully I pushed that doubt aside, knowing the power of the Cuisenaire rods.  I loved every minute of this.  It was challenging to try to find all combinations.  It reminded me a lot of the work we do with probability before students know how to make probability tables or trees.  We needed an organized list, a way to organize our combinations to make sure we had them all.  It was very cool to see the different ways that students chose to arrange their collection and explain how they knew they didn't miss any.  

It was interesting to see the predictions and how they would change them once they started making their collection without any prompting.  

Even after class I continued to explore on my own because I was so fascinated by it all.  Here are some resources Simon shared with me as I explored.

Here is a blog post with a proof that confirms our doubling pattern. 

It also shows how Pascal's Triangle shows up in the pattern.

As for our broken pattern, I still am not sure about that, but it turns out that Rananujan worked on that and his work is celebrated in this beautiful children's book.