After introductions and a brief discussion of how we’ve each used dice in our classes in the past, Mark gave out a Roller Derby board – basically a sheet divided into twelve columns, numbered 1-12 – and twelve colored chips to each person and invited everyone to distribute their chips across their board any way they wanted.
Then he broke up the group into pairs, gave out a red and white die to each pair and gave out the rules:
- Each player takes 12 chips and place them into the columns on their board in any way they choose. (Players can have zero, one, or multiple chips in any column.)
- Players take turns rolling two dice and removing a chip from the column with the same number as the total shown on the dice. Every roll applies to both players. If the column is empty, the player does not get to remove a chip.
- The first player to remove all their chips wins
- Keep track of all rolls in a chart with three columns – Red, White, and Sums.
Here are everyone’s first arrangements:
- Who do you think won – Eric v Stephen? Ruben v Tyler? Eric v Tyler?
After each pair rolled about 30 times, Mark asked the group to consider those rolls a trial and to feel free to rearrange their chips and start again (continuing to record the rolls). Mark said he had everyone arrange their chips before he shared the rules because he wanted everyone to place their chips without too much thinking. The goal of this activity is to hone your game strategy and he wanted some random (and perhaps less helpful) choices to be part of the analysis.
Because we had another activity set for our second hour, each pair only got to roll 2-3 rounds before we shared out where we were with our strategies. We ended with more questions and the understanding that we’d need more time to play and experiment with our strategies and revise (or not) accordingly.
Tyler versus Ruben
Tyler and Ruben spent some of their time mapping out all the possible sums. They created a frequency table that shows all 36 combinations after answering their own question about whether rolling a 2 and then a 3 is the same or different from rolling a 3 and then a 2 – they decided it is different, using the fact that they had two different colored dice to illustrate their reasoning. Their frequency table might suggest how we would arrange 36 chips, but we only have 12. How can we adjust our chips using the expected frequency of each possible sum?
Eric versus Steven
Steven and Eric were also trying to use their understanding of probability. Steven reasoned that because there are six ways to roll a sum of 7 (6/36) – making it the most likely outcome – he would place all his chips on 7. Eric favored 7 as well, but he reasoned that there is a 6/36 chance of rolling a 7, but there is a 30/36 chance of rolling something else.
Steven came around to Eric’s way of thinking a bit – not only because Eric won but because of what happened. Eric had six 7s, so he matched Steven for the first six 7s that came up. Steven had to wait for another sum of seven, but Eric had other opportunities to remove chips (both 6s and 8s and there is a 10/36 chance of getting a 6 or an 8).
Then they started coming up with as strategy together, asking, “What is that something else that has a 30/36 chance of happening?”
Steven and Eric ended up starting to talk about proportions to answer the same question that Tyler and Ruben came to – How can we adjust our chips using the expected frequency of each possible sum? Or put another way, you want it to model Tyler’s frequency, but when you only have 12 chips, what constitutes the closest arrangement to the frequency table?
Questions for further exploration
- What is the best strategy for placing your 12 chips?
- Should you always use the same arrangement of chips?
- Does it matter how the other person arranges their chips?
- How often will you win if you place all 12 chips on seven?
- How does the number of chips affect your strategy?
As a final extension, Mark gave out the following:
from the website Would You Rather?
For the second half of the meeting, Steven shared some questions he’s been asking about the TASC.
There were more questions asked than answered as we tried to figure out what information we needed and how to do the math.
- Below is the expected performance table for the math section of the 2015 TASC Official Readiness Test (ORT). If a student guesses for all 17 multiple choice questions on the math ORT (not including the 2 grid or constructed response questions), what is the likelihood of passing the 2015 math ORT if the test is taken once? Twice? Three times?
- Is the table above saying that a student who gets 5 out of 20 questions correct has a better than 1 in 5 chance of passing the math section of the TASC?
- Below is the Expected Performance Table for the 2015 form of the TASC Official Readiness Test. If a student knows 4 answers and guesses at the other 16 questions, what is the probability that she will pass the science test?
- What do the percentage likelihoods of passing charts look like for the other two HSE assessments – the HiSET and G.E.D.?
In attendance: Ruben, Tyler, Eric, Steven, Mark
Programs represented: Hostos Community College, Fifth Avenue Committee, CUNY Adult Literacy PD Team, Touro College