Intent

In this activity, students continue to apply the technique of considering a large number of trials to analyze probabilistic situations involving points or payoffs.

Mathematics

Students build on the notion of considering a large number of trials to analyze situations with weighted probabilities. They continue to use an area model to think about probability and consider why it is possible for an outcome with a lower probability to result in more points over the long run.

Progression

After working individually, students share methods and demonstrate their reasoning. The method of comparing the total points accumulated after some large number of trials is emphasized. Students see that different “large numbers of games” yield the same winner and that a convenient number can be selected for this analysis. These strategies are further developed and drawn upon through the remainder of the unit.

Approximate Time

25 minutes for activity (at home or in class)

15 minutes for discussion

Classroom Organization

Individuals, followed by whole-class discussion

Materials

transparencies of the rug diagrams (optional)

[link to BLM download]

Doing the Activity

Explain that this activity uses the rugs from Rug Games, so students can use the probabilities found for that activity. However, the situation is now more complex, because each region now has points associated with it.

Discussing and Debriefing the Activity

You might assign one or two groups to each rug and have them prepare a presentation of their solution to initiate class discussion. During this time, observe groups sharing methods, and note who has done the homework and how well students seem to have understood the ideas in the assignment.

If groups are having difficulty, you might suggest they pick a convenient number of games to play and see which color would earn the most points if the darts landed according to probability. This “large number of trials” method is used throughout the unit, so be sure it is among the methods presented.

Some students may break up the area into equal-size sections, write the appropriate number of points in each section, add up the total points, and divide by the number of sections (as if each equal section got one dart).

Following up on the “large number of trials” method, ask, Did you all use the same number of games? Since the answer will likely be no, you can bring out that the predicted winning color is the same no matter how many games are played, even though the total number of points earned for each color will vary with the number of games.

As students begin to value this approach, help them to see that there is often a convenient “large number of games” to select. Ask, Do some numbers of games work better than others? As needed, you can ask about the advantages of choosing certain numbers in a particular situation. For example, for rug A the probabilities are 715715 size 12{ { {7} over {"15"} } } {} and 815815 size 12{ { {8} over {"15"} } } {}, so it would be convenient to choose a large number that is a multiple of 15.

As with the discussion of Spinner Give and Take, you may want to bring out that although the results usually won’t follow the probabilities exactly, the fraction of darts landing in each color should be close to the theoretical probabilities in the long run. If the total number of darts is large, slight variations from the theoretical probabilities won’t change which color yields the most points. A numerical example might help students grasp this idea.