## Wednesday, February 20, 2013

### Look How Our Number Wall Grows!

As a little update to my recent Seeing Numbers post, I wanted to show you what an effect these re-imagined factor trees are having on my child.

Here is the first day, focusing on the factors of 20.  As detailed in the Seeing Numbers post, my seven year old used the tree metaphor wonderfully to illustrate different ways to add and multiply numbers to make 20:

The next day she chose 40.  As before, this tree utilized addition and multiplication as she built its branches.  We also noticed, when we made the preliminary wall of factors with our Cuisenaire rods, that we used the same color rods as when we factored 20, just twice as many.

I had other math things planned for today but this morning she asked, "Can we make another number tree?"  Yes, of course!

We settled on 30 and this is when her understanding of what we were doing really took a jump.  After we created wall of C. rods and after I sketched out all the different factor trees for 30 (like I did both times before) she, unbidden, picked which version she wanted to make.  Huge.  Both times prior to today I walked her through the structure of my own tree (more in line with factoring) but did not define what exactly a number tree should be.  As long as she was composing or decomposing numbers somehow on her tree I was set to be happy which is why I was surprised that she moved over to the factoring camp so quickly.

In her 30 tree her artistic vision is grand but her numbers are tiny, lol!  As we looked at her tree closely she showed me the first level of 5 and 6 branches, and then the additional 2 and 3 branches off the 6.  And, as with the other two trees, she added a crowning flower showing the day's number.

These pictures are all on the wall next to our dining table.  She has been admiring them and examining them closely every time she sits down to a meal at which time we also admire each others handiwork. I have invested in some new markers and paper because I never want this to end!

I will be forever grateful to Simon Gregg and his 5th grade students at Toulouse International School for providing us with such a perfect vehicle for exploring numbers in a way that makes sense to us.  I had been looking for a way to look at and explore numbers that involved an expressive visual element and am excited to see where this will take us. We've grown so much already through the process of observing and using the commutative property with the C. rods, applying understanding of groupings to our design, and conceptualizing division in a completely different, very exciting way.

1. Have you had kids trying to find a number that gets them a certain effect? Also curious if they've made/noticed different trees for the same number.

You can play a game like this, too. Take turns adding forks to the tree. First player that can't add a fork loses. (1s not allowed.)

2. My class are 9 - 10 year olds, and I didn't take it much beyond a hundred.

3. 96 was a good number to see the different trees for the same number - lots of ways, but not too big to be intimidating - see my blog post that Malke links to above.

I like the idea of doing it as a game. Would need to be bigger numbers to be going for a while...

4. John, I was going to say what Simon said above. The design possibilities are limited because we're still working with smaller numbers. However, as with other projects we've done, it's very clear to me that one does not need to do these kinds of activities many times before you really start to 'get' the concept (and, in this case, the process of factoring). At that point, the child (and the adult!) has made a personal connection to the process and it becomes much easier to then move toward the more traditional math context.