Summer SAVY 2016 (Session 1, Day 2) – Puzzles and Problem Solving
We began class today by delving deeper into the problems studied yesterday. Each day will begin with two questions designed to have the students apply the knowledge learned in the previous day to a new scenario. Today’s questions dealt with the 21-card game and ZipZap, and both questions led to interesting discussions about factors, multiples, and divisibility rules.
We looked at two dice games today, “Petals Around the Rose” and “Polar Bears and Fish.” Both games involve being able to decipher a riddle through understanding various rolls of the dice. These games proved challenging for the students, but we made some headway as we struggled. We learned that the answers were always even. We learned what the roses and the holes in the ice were, and gradually we had built enough knowledge to be able to solve the riddles.
Another problem we investigated today involved determining how many ceiling lights out of 20,000 would be on after one person pulls every chain, another pulls every other chain, the next every 3, and so on until finally the last person just pulls the 20,000th chain. A common theme to several problems we have solved thus far is the creation of a similar but simpler problem to see what patterns develop. Students used playing cards to model these lights and noticed several patterns, some of which led nowhere and others led somewhere but very slowly. Ultimately, we determined that all lights that are perfect square numbers are on. Figuring out how many of those there were out of 20,000 proved to be a challenging exercise for us!
We also interacted with the Monty Hall Problem (http://en.wikipedia.org/wiki/Monty_Hall_problem). Students paired off and worked together to simulate 20 trials of this problem, 10 trials where they stuck with the original door and 10 where they switched to the new door. We compiled all of our data as a class. Sticking won roughly 28% of the time, and switching won 46% of the time. While it appeared that switching was the correct play from our data, we actually had a shockingly low number of wins as switch. Such a result would only happen by chance about once every 7400 trials. Either we made some mistakes playing the game, or we were very, very unlucky!
We wrapped up our day with our first full-fledged ZipZap contest. We still haven’t managed to count past 28 as a class. It is difficult! I was impressed with the way that the groups handled the morning ZipZap question (What are the first five numbers that are ZipZaps?). Most groups had 4 of the 5 numbers correct on their first try and were able to spot their errors.
All in all, it was another successful day in Puzzles and Problem Solving. There was some mathematical frustration, and there was some mathematical celebration. There will be plenty more of each the next three days. Tomorrow we will begin tackling perhaps my favorite all-time math problem. I’m excited!