Thursday, May 18, 2017

5/16 Expected Values

My planning process this week went something like this: after last week's talk I either wanted to do some group white-boarding or find a new game to explore. I also was thinking more about combinatorics. I've never done anything on combinations (n choose m) and I mulled choosing that as a theme. Then in the middle of the week the Math Counts finals occurred. Watching the live stream was fun for me and I thought the kids would like that too. So initially, I thought about showing pieces of the video and then pausing and have everyone do the problems on the whiteboard. But after some more thought, I worried that it would emphasize the speed of the competitors too much and I also wanted to dig into the Chicken problem more deeply.

Finally this was the structure I ended up with:

We warmed up with some individual skyscraper puzzles from I really like doing these and they went over well engaging everyone. I cut this short after everyone had at least finished the first one of the set I provided in the interest of time.

After watching the videos I transcribed a few of the problems I liked and thought would be good to try on the board:

Whiteboard Problems from the video:

  • Caroline is going to flip 10 fair coins. If she flips  heads, she will be paid $. What is the expected value of her payout?

  • Sammy is lost and starts to wander aimlessly. Each minute, he walks one meter forward with probability 1/2 , stays where he is with probability 1/3 , and walks one meter backward with probability 1/6. After one hour, what is the expected value for the forward distance (in meters) that Sammy has traveled?

  • A finite geometric sequence of real numbers with more than 5 terms has 1 as both its first and last terms. If the common multiplier is not 1 what is the value of the 4th term?

  • The length of a 45 degree arc on circle p, has the same length as a 60 degree arc on circle q. What is the ratio of the areas of circle p to circle q?

  • The novel Cat Lawyer is 300 pages long and averages 240 wd/pg. The sequel Probably Clause is 60 pages longer and 30 more words per page. Probable Clause has what % more words?

  • Ian is going up a flight of stairs. Each time he takes 1,2 or 3 steps. What is the probability that he steps foot on step 4?

  • How many 6 digit integers are divisible by 1000 but not 400?

But I decided to focus on the final question:

"In a barn, 100 chicks sit peacefully in a circle. Suddenly, each chick randomly pecks the chick immediately to its left or right. What is the expected number of un-pecked chicks?"

To build up to it, I started with discussing expected value and used dice questions as starters in a group.

1 What's the expected value of a single roll of a six sided die?
2. What's the expected value of 2 rolls?

After we went over the concepts as a group, out came the blue markers and  I had everyone work on the followup problems up on the various whiteboards.

3. What's the expected value of the product of 2 rolls?
4. What the expected value of the product plus the sum of the rolls?
5. What do you notice about this? 

The kids all worked on these followup questions in groups over about 20-30 minutes and then we gathered together to discuss what we found.  I had to bring out the linearity of expectation relationship and did not go into the proof at this point since its a bit too complex. [This is always a struggle to resolve whether to dig into every observed pattern and find the reason or accept its probably true in order to reach a target for the day.]

Then I posed the chicken problem expecting everyone would work again for a while. I was surprised but several kids almost immediately shouted out the answer. It turns out the scaffolding may have been a bit too much after all. If repeating I might lead off with the final question, work on it a few minutes and then go into the build up.

At any rate for the last 10 minutes we did watch the video communally.  Before showing it I prefaced it with a short talk about speed and emphasizing both the competitors had really trained to build this up and that it wasn't really important outside contest while the problems on the other hand were fairly interesting.  So hopefully, I didn't damage any of the kids self-conceptions. As expected, the room was fairly rapt watching the competitors even many of the parents on pickup stopped and watched it until the end.

Continuing the MathCounts theme I chose this weeks problem from their site: although I'm not completely keen on the questions.

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