June 20, 2014 at 12:48 am #2129SharmaKeymaster
(For full lesson with tables and handouts, see link at bottom or attached document)
Materials: (for groups of 2)
6-sided dice (approx. 8)
4-sided dice (tetrahedral dice) (approx. 16)
8-sided dice (approx. 8)
12-sided dice (approx. 4)
20-sided dice (approx. 4)
By analyzing sums of different sets of dice that all have a maximum sum of 12, students will understand and find sample spaces to represent all possible outcomes. They will use the sample space to determine which set of dice had the greater theoretical probability of winning as well as to calculate probability. Finally, students will described how increased trials leads to the experimental data being closer to the theoretical probability.
Part 1: Predict and Experiment
Pass out the activity sheet. Show the students the top row and model what each set of dice looks like and works. First, show an 8-sided die and then show them a 4-sided die as well as how to read this die (the number is read on the bottom of any edge). Explain that for this set, they will roll each die and find the sum. Do one sample roll. Next, show them what three 4-sided dice would be by rolling 3 and finding the sum. Next, show a 12-sided die and explain that, in this case, they will just get what they roll. Next, show two standard dice (6-sided), roll them and demonstrate how to find the sum. Finally, show them a 20 sided die. Roll that and THEN the 8-sided die and SUBTRACT the number on the 8-sided die FROM the number rolled on the 20-sided die (yes, this means it could be negative, but don’t tell them that- let them figure it out!).
Now, give each pair 2 minutes to discuss and then predict the order the dice will finish when each set is rolled 100 time and the sums are added together. Have groups raised their hands to vote for which set will be first, second, etc. Then, let each group pick a set to do the experiment with (or assign them). Ideally, you want 3-4 groups investigating each set of dice and it is nice if they can investigate the one they thought would win.
Once each pair gets their dice, have them roll 20 times and record the sums in the table provided. Then, have each group that had the same set come together to share just the sum of their 20 rolls. Have a volunteer from each group come up to the document camera and record their data on your table (for the sum, mean and mode). Note: there will be a different mode for each small group, but just 1 combined sum and mean.
Ask the class which number (sum, mean or mode) tells you which set won. (It should be the mean, or the sum if all sets had an equal number of trials). Give the class 5-8 minutes to work on the 4 analysis questions and then discuss them as a class. Your goal here is to het the students to agree that they need to list all possible outcomes to verify if it was “fair” (recall equally likely outcomes conversation from 2 coins, 3 people). Your secondary goal is to help students see that more trials means the data will be closer to the theoretical probability.
Part 2: Showing Possible Outcomes
In order to a) verify if the experimental winner should have been the winner and to be able to calculate probability, we will need to show all possible outcomes for each set of dice (this is called the sample space). Have each group first try to do this for the set of dice they rolled and once they are successful, assign them a second set to do this for. Here are some notes to help you guide them:
v All of them can be analyzed with a Tree Diagram
v All of them (expect 1 to 12) can be analyzed with a table (which is often easier)
v All of them can be analyzed with a list, but it gets messy and easy to forget.
v The tree is the most likely choice for the three 4-sided dice.
v The three 4-sided dice CAN be analyzed with a table IF you do the first table as comparing one 4-sided with another 4-sided (you’ll end up with 16 outcomes) and then make a second table to compare those 16 with the third 4-sided dice.
Have each pair also answer the two questions in the text box so that you can compare the theoretical probability with the experimental results.
Have a student come up to show their sample space for each set of dice and have any group who used a different method also come up to explain the method.
Part 3: Calculating Probabilities
NOTE: BEFORE doing this (day 2), make a copy of the nicest/cleanest sample space from part 2 for EACH set of dice for EACH pair of students to have for part 3.
Pass out the tables or tree diagrams for each set of dice to each pair. Model 1-2 problems as a class, and then have them complete part 3.
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