# Binary Fox Hunting

This week I led the big kids circle. All six kids attended.

## The Activities

1. Topic: Money. Book: Tightwad Tod by Skinner.

2. Topic: Money, Counting. Count a huge box of change.

3. Topic: Binary Search. In pairs the kids played the “Fox Hunting Game” which I made up.  There’s a fox out on a road and the farmer is trying to catch it. The fox is running all around the road (which has squares numbered from 1 – 40). Each turn, the farmer gets to build one fence that will block the fox.  The fox can choose which side of the fence to run to.  Once the fence is built, the fox can never get past it. The game is over when the fox is trapped on one square, and we count how many fences it took to catch the fox.

The fox is trapped between the fences at 26 and 37. Now the farmer is building a new fence at 29 and the fox has to decide whether to run to 27-28, or 30-36.

4. Topic: Geometry. Show the kids the compasses again, and let them draw circles.

## How did it go?

#### Counting Money

The kids were all impressed by my box of 10 years worth of change.  I asked them to guess how much money was in the box, and half the kids guessed \$1000, and the others just said “a lot”.  We reviewed the names of the coins, and how much each was worth.  Then I dumped out the box, and said we’d count it all. Every time a kid collected a dollar of change, I’d check it and then add it back to the box.

First the kids all just grabbed 4 quarters at a time.  We counted about \$30 that way.  Then some kids decided to try other ways to get a dollar.  Some did 10 dimes.  One kid made a dollar out of mixed denominations.  Several kids successfully counted 20 nickels.  However, 100 pennies was quite challenging because there were not that many pennies, and it was very hard for the kids to keep track of how many they had collected. They tried to work together but that also didn’t go so well, so eventually I stepped in to help.

Everyone was excited by my ever-increasing tally of dollars, and kids wanted to keep counted how many they had so far.  In the end, we had counted \$146.69.

#### Binary Fox Hunting

I assigned pairs for this activity, and each kid took several turns being the farmer and the fox.  At first there was a lot of confusion because the kids didn’t understand that the fox was roaming over a whole range of the row, not stuck on one square.  Many farmers thought you had to put the fence whereever the fox was, and some foxes thought you could jump past old fences that were supposed to block you.

I eventually sat with each pair and walked through the game and the choices the farmer and fox can make.  At the end, everyone seemed to understand the rules. It also looked like the foxes were pretty good at choosing the side of the fence that had more open squares. However, the farmers were not very consistent about where they would build the fences.  The best strategy is binary search – split the range in half each time to catch the fox quickly.

I picked up all the kids’ materials, and then we all sat down to discuss strategy.  I asked them “When you were foxes, which side of the fence did you pick?” My daughter and a couple other kids suggested they would choose the side with more numbers.

Then I asked, when they were farmers, how did you pick where to put the fence? Most kids said they picked randomly.  Some said they put the fence whereever the fox was.

Next I played the farmer and demonstrated a very bad fence-placing strategy, first putting a fence on #1 (the fox moved to 2-40), then putting a fence on #2 (the fox moved to 3-40).  I asked the kids how many fences I would have to build to catch the fox, and they said “a lot”.  One kid looked at the number line and said it would take 39 fences.

Even after this demonstration, the kids still couldn’t explain a good strategy for the farmer.  We should play it again.

#### Compasses

We had about 5 minutes left, so I handed out the compasses and showed them to the two kids who hadn’t seen them before.  Again everyone loved trying to draw circles, but 2 of the compasses broke during this circle, so we need sturdier compasses.