## The Activities

**Topics: Multiplication, Division:**Perfectly Perilous Math, Challenge 3 — if you spent 50 cents every second, how many days does it take to spend 1 million dollars (to nearest day).**Topic: Combinations:**We continued the pumpkin activity from last week. Last week, they had made a bunch of pumpkins but hadn’t checked for duplicates. This week, we tried to come up with strategies for finding all the duplicates. Also, I started with the simplest version of the problem (circular eyes, nose, and mouth), and then gradually added elements (non-symmetric parts, multiple choices), computing how many different combinations there were each time.**Topics: Logic, Measurement:**Perfectly Perilous Math, Challenge 4 — if you have one bucket that holds 3 quarts, and one that holds 5, how can you measure*exactly*4 quarts using just those two buckets? We worked as a group to solve this problem using various containers and beads (instead of water).

## How Did It Go?

We had all five kids this week. All the activities this week were group activities, which turned out to be a problem. Most of the time, only two or three of the kids were actually working on solving the problem; the others were drawing, making paper airplanes, or otherwise not paying attention. Kid A grabbed and crumpled another Kid B’s paper, and later in circle (unrelatedly) Kid B threw a container of beads at Kid A, which we all had to help pick up. I think another issue was that the problems were all pretty hard; the kids working on them did a good job, but it discouraged the other kids. One of the kids who drew during the Million Dollars activity said they couldn’t help because they didn’t know how to do multiplication, and then was very attentive for the remaining two problems.

#### Spend a Million Dollars

One of the kids is a lot better at large multiplication than the others, so that kid did most of the computations. Another kid helped figure out what the right numbers to multiply were. The kids figured out how many seconds were in a day, but then they were planning to multiply by 50 (cents) — they didn’t realize that 50 cents was half a dollar. They also needed help figuring out how to divide 1 million by 43,200 — they know the method of repeated addition/multiplication, but I think their number sense suffers a lot past 1,000. They got pretty close to getting the right answer, but I had to a help a lot for the final few steps.

#### Pumpkin Combinations

One of the kids suggested sorting the mouths by orientation, and then by type of mouth. A different kid decided to implement this, but changed it to have 1 column for each of the combinations of 4 mouths. Some of the other kids helped sort (there were about 35 pumpkins total), and eventually the pumpkins were all in columns. There were still 8-9 pumpkins in each column, and they mostly stopped making progress at this point, besides finding a few duplicates ad hoc. I suggested putting the circle eyes above the triangle eyes, but they didn’t take to this idea.

Then I switched gears a bit and started with a problem with circle eyes, nose, and oval mouth. This meant there was only one possible pumpkin, which the kids figured out right away. Then I made the nose triangular. There were answers of both 2 and 4; the 4 answer was reasonable because they showed how you could put the straight edge in any of the 4 directions (although two of them make pretty weird noses). I said we should only allow the two directions, and then added triangular eyes. They got 4 right away. Then I added a smiley/frowny mouth, and it got much harder. I got answers of 4, 6, and 8. 6 is an interesting answer, because you could think of that if you considered varying each dimension one at a time. I had the kids try to write out all the combinations — I really should have had them to all 4 for the previous problem first, because the key insight here is that if you group the faces by smiling vs. frowning, you have the same set of 4 faces from the round mouth case, once for each mouth. In the end, one kid was able to figure out that there should be 32 different answers for the previous week’s problem, but wasn’t able to figure out which faces were still missing.

#### Tricky Buckets

The beads worked okay, although one of our containers was more like 2.6 than 3, so the measurements didn’t quite come out right. Also, we ended up picking up a bunch of beads because of the bead-throwing incident. One of the kids figured out a solution right away using a third bucket; and then they were able to come up with the 2-bucket solution with only a bit of help. I asked a follow-up question about a 5 and 7 bucket (trying to make each of 1, 2, 3); the kid who had done the best on the first part was able to make progress and solved 2 and 3 but not 1.