Ping Pong Redux: A Visual Representation

Before reading this post, you might want to read my previous post on Ping Pong Redux since below is a solution of sorts. Another option, though, is to not read the previous post and instead attempt to backwards engineer the problem. I'd be curious what people came up with (and if the fact that you know the name is too constricting). I'll leave out a key for now.

Ping Pong Redux

Heard this problem from a colleague at lunch earlier this week and loved it.



Dan and a childhood friend loved playing ping pong. They loved playing ping pong so much that they devised a new rule to make games last longer. Scoring and play is normal, except that the score is "reduced" whenever possible. In other words, if the score is 7-4 and I win a point, instead of going to 8-4 the score becomes 2-1. Like in normal ping pong, games go to 21. Note: If you are leading 20-7 and score a point, you do not win. The score would go to 3-1.

There are some great mathematical questions to explore here. I'd recommend thinking of your own before reading the below list of mathematical questions (some of which are Dan's).

• What are all the possible final scores?
• What scores are impossible to get?
• In ordinary scoring rules, the winner has to win by 2. Is that rule necessary with these reducing rules? Explain.
• What strategies might a player use to avoid (or cause) a score being reduced?
• If two players were not evenly matched, do you think these rules would favor the weaker player or the stronger one? Explain.
• Describe as generally as possible games in which no reductions occur. How likely do you think such a game would be?
• How long would you expect a game to last if players were evenly matched (each player had a 50% chance of winning the next point)? 

My Gambling Skills

As a wrap-up to our Games of Probability group project, my 6th graders and I were talking about the probability of getting a "Yahtzee" if you were playing with 2 dice instead of five (if there is one mathematical habit of mind that may be burned into the soul of each of my students this year, let it be the idea to try a simpler case). We were discussing the case where you get your Yahtzee on the third roll. We went over the probability of not getting a Yahtzee on the first roll. I rolled 2 dice and didn't get a Yahtzee. We then went over the probability of not getting a Yahtzee on the second roll. I rolled again and didn't get a Yahtzee. We went over the probability of getting a Yahtzee on the third roll. I rolled two dice and, what do you know, rolled a Yahtzee. 

In the words of @cheesemonkeysf, "I love it whenever the Universe tosses a teacher a freebie."

A Truly Group Project

My fellow 6th grade math teacher and I are trying something crazy this week. We were two weeks into our counting/probability unit and wanted a project that would:

• Help students practice the foundational skills and concepts
• Push students to explore more challenging, unfamiliar, open-ended problems
• Reiterate the importance of group work
• Reiterate the importance of writing down solutions, reasoning, and explanations in an organized fashion. 

So our crazy new idea? We've put together four open ended, challenging probability games. Eight groups of 3 or 4 will spend tomorrow working on one of these games (2 groups for each game in each class). On Tuesday, groups will rotate and work on a different game, starting where the previous group left off. 

The hope is that, as an entire group, we make significant progress on each of the four problems. I'm super excited to see how this works (and I'm sure we'll all learn a lot even if it completely bombs). Below are the games we're introducing, and the investigation questions.

Yahtzee:

Investigate the probabilities associated with this game.  Think about the possible ways to make the required combinations, and then think about the points that you win for each part of the game. Are the points that Milton Bradley assigned to each combination in the game “fair?”  If you feel that they are “unfair” how you would you recommend to the makers of this game that they change the scoring to make it “fair?”

Pig (we're calling this "On a Roll"):
When you think that you understand the game, begin to investigate different strategies that you might use or recommend to others to help players decide when to stop rolling and when to take a chance and roll just one more time.  

Deal or No Deal: 
Play a version of the game at http://bit.ly/dealnodealgame.  Keep track of the revealed suitcases and the banker offers and determine whether or not (with an explanation) you should take each offer. What other questions do you have about this game?

 Drunken Walk (we're calling this "Dizzy Walk"):
1 dimensional version:

Where should you put your house? Would your placement of your house change if you initially rolled a die to determine how many times to flip the coin? Explore other questions you have about this game.

 2 dimensional version:

Let’s call the address of where Larry starts (0,0) and the address he would end up at after flipping HHTTT (2,3). How many different ways are there for Larry to get to (3,5)?

Imagine instead that Larry starts at (5,5). This time, instead of flipping a coin, you spin a spinner split into 4 equal areas marked North, East, South, & West. Now where would you put your house? What if you can create the spinner and determine the size of the 4 areas.

   
Here are the nitty gritty details of how to play the games if you're interested (ie our handouts).