This is a little different from other poster problems in that after the class discussion, students amend their posters to say what range they would choose if they got to play the game again, and why.
Facilitate a discussion of the different approaches. Select a sequence of posters that will help students move from their current thinking (Levels 1–3) up to Level 4 or 5. Be sure to include, for example, a wide-range, low-scoring poster where the students were thoughtful but discovered through experience that a wide range results in many “captures” but a low score. If such a poster doesn’t exist, you can take anyone’s data and ask, “Suppose this group used a really wide range such as the 16 range (4 points per capture). How many points would they have gotten?” This models the idea of a “what-if” calculation, and is easy to do.
Level 1:
The group chooses a range by guessing, with little reference to data they collected before. The range may be off-center. (The center should be at 14, or close to it.)
Level 2:
Students choose the range of the box in the box plot, simply because it was the box. This may still be off-center.
Level 3:
Students deliberately center the range even though past data may have been off-center. They choose a range using the box or otherwise give a one-sided reason for the choice (e.g., we want to capture more or we want a higher score per capture), but do not explain the trade-off well.
Level 4:
Students center their range, which is between 4 and 10 wide (10–16 points per catch). Their score will probably be over 400. They clearly articulate the trade-off: you get fewer points per catch for a wide range, but you’ll capture more—so there’s a “sweet spot” in the middle where you get the most points.
Level 5:
Students use 4-dice data distribution to make “what-if” calculations: would they get more points if they had chosen a different range? They actually calculate these potential scores and adjust their range accordingly.
Questions to ask across presentations
- What’s the median value? How much does it vary from poster to poster? (Should be 13–15.)
- How can you use the box plot to help you decide on your range? How much does the box vary? (Look at the different box plots and compare. Note that the min and max will vary more than the quartiles.)
- If you picked the same range as the box in the box plot, how many would you catch? What would your score be? (You would catch a bit more than half—25 to 30 rolls. Why half? Because the box contains the middle half of the data. Score depends on the width of the box.)
- What range would you pick to be pretty sure of catching almost all of the rolls? (Wide, e.g., 6–22 or 4 points per catch)
- Why is that range a bad choice? (You catch them all, but it’s not enough points. 4 times 50 is only 200 points.)
- What is the trade-off in this game? (A wide range gets many catches but few points per catch. A narrow range gets a lot of points per catch, but not that many catches.)
- How did you use the data you collected before? The data could be from
The prediction you made during Step 2, based on rolling fewer dice, or, even better,
The distribution of sums of four dice you rolled before actually playing the game.
Focusing on Variability
Ask students to look at the distributions of rolls on all the posters.
- What’s the same about the distributions? What’s different? (They all have more in the middle and fewer on the ends, but the details vary widely.)
- Suppose we picked a simple range, like 10–15. (20 points per catch.) What are the different scores we would get with each of these sets of rolls? (Each group could calculate and report back. Scores should vary considerably.)
- So: how much of this game is luck, and how much is skill? (Lots of luck, but you have to pick a reasonable range in the middle. Some students may think there is more skill when they’re really only lucky.)
- Suppose we did 500 rolls instead of 50. How would the distribution change? What would the shape of the distribution be? (If there are 10 posters with graphs, consider adding them all up.)
Enhancing the posters
Now challenge the groups:
Seeing all the other data and the other groups’ posters, imagine you’re going to play the game again. What range would you choose, and why?
Write your range—the minimum and the maximum, and how many points you get for each capture—and your prediction for your score.
Then explain, clearly, why you chose the range you did.
(The instructions are on their sheet,
Handout #3.)
Optional Contest
If there’s time (or if some students finish early, or if you want a homework task) you can give students
Handout #6: Fifty More Rolls (or use
Slide #1, which has the same graphic).
The handout has a record of 50 more rolls of four dice. They can calculate the score their new range would have gotten, and compare that to other groups’s scores with the same data.
Another challenge is to figure out what range would give the best score with that data.