Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

 Observing Representation
 Review Matrix Multiplication | Taxicabs | Student Work | Student Work Reflection #1 | Matrix Approach | Interpreting Matrices | Student Work Reflection #2 | Classroom Practice | Observe a Classroom | Your Journal

After the students had worked with a tree diagram approach to the taxicab problem, they went on to explore different situations and different initial distributions in class. It became clear to everyone that tree diagrams became unwieldy and "messy," in the words of one student. The teacher presented a set of questions that opened up the idea of using transition matrices rather than tree diagrams to solve the problem. He also encouraged students to articulate their understanding of the matrix representation and its use.

Recall that the main question was, "What is the probability of starting in Downtown and being three after three trips?" Here is a student's work as she began to represent the problem information in a matrix.

Here is what Louise wrote on the overhead projector, as well as the class's conversation about it:

Louise: I realized that the possibilities from the table could be written as a matrix. At first I wondered how to show the taxicab's starting place in the matrix. In the problem the cab starts by picking up a passenger downtown. Then I saw that those are the numbers in the middle row, the fraction of all the cabs that pick up from Downtown and take someone to the other destinations.

The class took a few moments to point out those same numbers in the first level of the tree diagram and in the original table before Louise continued her explanation.

Louise: First I wanted to figure out the probability of every type of two-fare trip, like NS or SD, to try the matrix method and to know about the positions just before picking up a third passenger.

This is my representation of the problem after two trips:

Mr. Karsky: Let's talk about this matrix for a minute. What is the meaning of the middle column in Matrix R, the product of T•T?

Jenny: It shows the probability of ending up downtown after two trips, given the original starting point of either N, D, or S, each represented by a row. The Downtown to Downtown probability is in the middle, 0.33, same as we got before with the tree diagram.

Mr. Karsky: Why is it appropriate to multiply matrices for this problem?

Susan: You need to find the probability of starting from a downtown location and ending up downtown after two fares. You can end up Downtown in several ways, going to Northside then Downtown, or Downtown then Downtown again, or to Southside then Downtown, like you can see in a tree diagram. You first multiply to find each of those probabilities, because only a fraction of each group at a place in trip 1 ends up in a given place at trip 2. Then you add all three probabilities. The matrix keeps things in order and makes sure the correct terms are multiplied times each other.

Eric: It's like in the tree diagram. A row tells the percentage of cabs that end up in each location if they start in a particular place, like Downtown. Once they go out again from the new location, there are three different probabilities for each of the next ending locations. See, in the diagram you can tell that 50% of cabs starting downtown end up in Southside. But then a cab that's in Southside takes the next rider downtown 3/10 of the time, so that makes 15% go Downtown to Downtown with two fares. But there are two other ways of ending up downtown, through Northside or through Downtown, so you also need to multiply those other cases; for instance 0.1 •0.20 and 0.40 • 0.40.

Mr. Karsky: What about the bottom row of the matrix? What does it tell us?

Anita: I don't think we need it, because it's about starting in Southside and ending up at each of the places.

Jenny: We don't need it now, but it gives probabilities for each of the paths that could go on and end up downtown when we do a third trip.

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