## The Activities

**Topics: History of Math, Mathematicians:**Mathematicians Are People, Too: Stories from the Lives of Great Mathematicians (Volume One) by L. Reimer and W. Reimer, Chapter 6 (Galileo).**Topics: Order of Operations, Parentheses**: First, I introduced how parentheses work in simple arithmetic expressions involving +, -, and x. Then, I had them practice evaluating expressions with parentheses with 3 or 4 numbers. Finally, I gave them an expression without parentheses and asked them to figure out all the different possible results by adding parentheses in different places.-
**Topics: Gravity, Experiments:**Inspired by the story of Galileo, we dropped various items off our 2nd-story balcony and saw which landed first.

## How Did It Go?

We had all five kids this week.

#### Galileo

The kids were pretty interested, as usual. There were three interesting possible activities in this chapter: dropping things, pendulums, and telescopes. We’ll probably do pendulums and telescopes in future circles. I mentioned early in the chapter that we would be dropping things later in circle, and they got distracted because they found the stuff we were going to drop.

#### Parentheses

The kids were at very different places initially. Two knew about parentheses, and one of those was really good at evaluating expressions with parentheses (unfortunately, they also kept saying how easy it was). They also varied a lot in how long it took the others to understand how parentheses worked. One of the kids didn’t want to try, but then when I helped them specifically they got it. Another kept saying they didn’t understand; I helped them run through an evaluation a couple times, and I think they got it; but then they had trouble with the next part. Part of the reason I wanted to do this activity was because the kids have had trouble with braces in programming; and indeed, parentheses are a rather challenging concept. The idea of evaluating “inside-out” doesn’t make sense without more explanation, and the idea that you should find an expression that is bounded by parentheses and doesn’t contain any other parentheses and replace it by the result is rather tricky.

When we got to the part where the kids needed to add parentheses, most of the progress was made by the kid who was the best at evaluating. However, for the 4-number case we did, one of the other kids found the final possibility. I realized a useful way to think about it is picking two adjacent numbers and replacing them with the result of applying the operation between them. This suggests a different notation, which is drawing non-overlapping circles instead of parentheses (really, that’s what parentheses represent). I think this would help the kids understand what’s going on better.

#### Dropping Things

We did this activity in combination with the Age 6 circle.

We took turns having kids drop things (same kid held both items and dropped at the same time). The others stood below. As it turns out, non-breakable household items really don’t all fall at the same rate — anything with significant surface area will be noticeably slower. So, there are two possible strategies: 1) arrange to have all the “well-behaved” objects first, so you can establish that weight and size of, say, spherical objects doesn’t affect the speed, before moving on to the tricker items, or 2) do things in a more or less random order and let the kids try to figure it out. Both have advantages and disadvantages — in some ways, #2 is more scientific, but #1 might give them better intuition. I did #2 — and the result was that at least some of the kids came away saying that heavier things fall faster. However, at least one of the kids really got the idea that surface area matters, and could even explain why a book falls faster held on edge than flat: because it “torpedoes” through the air if held on edge.

One amusing thing is that one of the items was a bag of (somewhat stale) hot dog buns left over from earlier in the week, and while we were managing the experiment, a couple of the kids started eating the buns.