# How To Make a Square Number

## The Activities

1. Topic: Fractions: Book: Fraction Action by L. Leedy.
2. Topic: Scale: This was a repeat from last week, where we had some line drawings on graph paper and the kids had to scale them up by a factor of 2. This week, the drawings were a bit harder, and for the kids who were better at this, I had them scale up by a factor of 3 instead.
3. Topics: Square Numbers, Sequences, Proofs:  I introduced and explored the fact that the sum of the first n odd numbers is n^2.
4. Topic: Origami: I handed out origami instructions for an origami horse to each kid, with the goal that they would do it from the instructions without me showing them how to do it.

## How Did It Go?

We had 4 kids this week.  This circle went pretty well, although our daughter did get frustrated and had to leave during the origami activity.

#### Fraction Action

The kids enjoyed this book quite a bit.  It wasn’t too complicated, but there was a slightly harder question at the end of each section that tested them a bit.

#### Upscaling Drawings

We did this in the previous circle with only two kids.  They liked it so much we decided to do it again.  The two kids who hadn’t done it before picked it up pretty easily.  Most of the figures weren’t too hard for them, but all the kids had an off-by-one error at least once.  The bow-and-arrow and the crown (not pictured) turned out to be the hardest.  I think the reason the bow-and-arrow was hard is that I didn’t draw a line for the “string”, which meant that in order to draw a nice curve, it was best to count out the top and bottom of the bow before you started drawing, which wasn’t what they normally did.  The crown was the only picture that had a slope that wasn’t 45 degrees, and this was pretty hard for them.  Again, the best solution is to find the two endpoints and connect them, but this wasn’t the way they had been doing the easier ones.

#### Sum of Odd Numbers

I started by having the kids make squares out of Base Ten Blocks for all side lengths from 1 to 10.  I kept track of the area (number of blocks) for each side length in a two-column table.  One of the kids has been practicing multiplication a lot, and already knew all the answers, but they were all still willing to arrange the blocks into squares.  All the kids liked the table and most of them made their own copy.  Next, I asked what they did when they were looking at the Fibonacci sequence, but no one remembered.  So I started writing down the successive differences, and they saw the pattern after the first few and completed it.  Next I made a table of 1, 1 + 3 + 5, 1 + 3 + 5 + 7, …, with the sum in the other column, to show that the squares were the sums of the first k odd numbers.  I was hoping they would be a bit excited by this “coincidence”, but not so much.  Next I made a 3 x 3 square of blue blocks, and asked how many I needed to add to make a 4 x 4 square.  I showed how you could add 3 on each of 2 sides, plus an additional 1, to complete the square.  Most of the kids were distracted by this point, but one of the kids learned the pattern and could answer questions like “How many do you need to add to an 11×11 square to make a 12×12 square?” without help.  Finally, I was hoping to teach them the trick that the sum of the odd numbers up to n was (n + 1) / 2 squared — but everyone just wanted to add up the numbers manually, they weren’t excited about a trick (and didn’t really understand it).  So in the end, there was some progress, but we didn’t succeed in proving anything.

#### Origami

The goal this time was to have them follow the instructions on their own, but they turned out mostly not to be ready.  One of the kids has been practicing quite a bit, and made a lot of progress on their own, but the rest needed lots of help.  Also, I accidentally picked one that required cutting, and the words on the instructions were very small — which was particularly a problem on the step that said “Repeat steps 4-7”.  Another challenge was knowing when to fold only one layer vs. several layers.  Yet another issue was that several steps showed multiple overlapping folds, and you needed to know that you were supposed to do the folds one at a time, unfolding after each.  We should try this again with a slightly easier model.