Intent

Students are introduced to rug diagrams as a way to think about randomness and probabilities.

Mathematics

Students are introduced to an area model (using a rug metaphor) for determining theoretical probabilities. Area models employ the visual imagery of geometry to help students understand some abstract concepts, and, in particular, are useful for working with equally likely outcomes. The term area model will be introduced formally in the activity The Theory of One-and-One

Progression

Until now, students have worked with probability empirically or in the abstract. This activity introduces an area model as a concrete tool for thinking about theoretical probability. Initially, this model is presented using the metaphor of a rug.

Approximate Time

35 minutes

Classroom Organization

Groups, followed by whole-class discussion

Materials

Handouts and transparencies of the rug diagrams [link to BLM pdf, p. 1-2]

Doing the Activity

One way to introduce the activity is to tell this tale before students read the instructions.

I have a rug at my house, and there is a trap door in the ceiling directly over the rug. The trap door is the same shape and size as the rug. From time to time, the trap door opens and a dart drops directly down onto the rug. The process is quite random, which means that every point of the rug has as good a chance of getting hit as any other.

Now, of course, my guests never sit directly on the rug (it’s dangerous!), but they like to sit nearby and guess which part of the rug the next random dart will hit. To keep things interesting, I have a variety of rugs of the same size that I can put out on different occasions.

Have groups look at the first rug in the activity. Ask, Which color would you predict the dart is most likely to hit on this rug?

As groups explain their predictions, you may want to have them show any additional lines they drew on the rug diagrams. For example, they might draw a diagram like this.

Figure 1

If students haven’t yet used the language of probability to express their ideas, ask, What is the probability of gray being hit? Of white being hit?

Ask students to write the probabilities using the standard notation introduced in What Are the Chances? Thus they would write P(white)=512P(white)=512 size 12{P \( "white" \) = { {5} over {"12"} } } {}. and P(gray)=712P(gray)=712 size 12{P \( "gray" \) = { {7} over {"12"} } } {}.

As groups work through the rest of the activity, ask them to defend their predictions using area calculations and to use the standard notation for writing probabilities. Encourage the use of arguments based on the notion of equally likely. In the diagram above, a dart is equally likely to land in each of the 12 rectangles.

You might have the first groups to finish the activity create new rugs for each other.

Discussing and Debriefing the Activity

When students share their solutions to the rug problems, focus the conversation on understanding the presenter’s reasoning rather than on whether the conclusion matches one’s own.

Some students will probably break the rugs into equal-size pieces. You can relate this subdivision to the idea of equally likely outcomes. For instance, rug A can be divided as shown below into 15 same-size pieces, each with an equal chance of being struck by a dart. Because 8 of the 15 pieces are gray, P(gray)=815P(gray)=815 size 12{P \( "gray" \) = { {8} over {"15"} } } {}. Similarly, P(white)=715P(white)=715 size 12{P \( "white" \) = { {7} over {"15"} } } {}.

Figure 2

You can informally introduce the language of area into the discussion. For example, you can say that the gray region of rug A has a larger area than the white region.

You might remind students of the unit problem by mentioning that rug diagrams will eventually help them to evaluate strategies for the game of Pig.