Support Materials --

Teaching Math: A Video Library, 9-12

Taxicabs

Content Standards
Probability, Algebra, Discrete Mathematics
Process Standards
Connections, Problem Solving, Reasoning, Communication
School: North Carolina School of Science and Mathematics; 550 students
Location: Durham, North Carolina
Teacher: Dot Doyle
Years Teaching: 18
Students in Classroom: 16
Grade: 11
Taxicabs
Video Overview
Students share their homework on the probability of taxicabs starting downtown and ending downtown after two fares. To find the probability, they place data about the initial distribution of taxicabs into tree diagrams. Ms. Doyle demonstrates how students can organize the information in a transition matrix. Students then work in groups, using a transition matrix to solve the same problem, but with a different initial distribution. At the end of class, students discuss their discoveries and solutions.

The Class Challenge
Students present and discuss tree diagrams they created using the following data about the distribution of taxicabs:

Pick Up Destination

Northside
Downtown
Southside

Northside Downtown Southside

50% 20% 30%

10% 40% 50%

30% 30% 40%

Course and Curriculum
The North Carolina School of Science and Mathematics, which includes only grades 11 – 12, is a state–supported residential magnet school that emphasizes math and science. This 11th–grade course, Contemporary Precalculus through Applications with Topics, is required for all students. Prior to this course, students took Algebra 1, Algebra 2, and Geometry at their former schools.
The course focuses on functions, including polynomial and rational functions, exponential and logarithmic functions, and trigonometric functions. Prior to this lesson, students worked with matrices. The goal of this two–day lesson was to introduce Markov chains in order to use matrices to model a phenomenon. (Markov chains is a process wherein a transition from one state to another is based on probabilities, and the probabilities are constant and independent of previous behavior in the system.)
Ms. Doyle has students sit around a conference table to promote conversation among students and to allow them quick access to the chalkboards on each wall. She occasionally has students sit in different places so they can work with different students in the class. When she regroups them, she has them share one piece of personal information with each other, so they know something about each other outside of class.

A Pre–Viewing Exploration for Teacher Workshops
Using Matrices to Simplify Probability Questions
Objective: To use matrices to solve complicated probability problems.
Materials: One graphing calculator with the capability to multiply matrices for each small group.
Activity
Before you begin the activity, do the Matrices Exercise below.
Solve this problem in small groups.

When the long distance telephone service was demonopolized, three companies became major long-distance carriers: Company A, B, and C. Suppose that after initial selection of services, people decided to switch companies. Of the patrons using Company A, 40% stayed with A, 20% switched to B, and 40% switched to C. Of the patrons using Company B, 30% switched to A, 50% stayed with B, and 20% switched to C. Of the patrons using Company C, 70% switched to A, 20% switched to B, and 10% stayed with C.
Construct a tree diagram to determine the probability of a patron who started with A, either staying with A or returning to A after two switches.

Exploring Content

How could you use matrices to determine the probability after three switches?
What is the transition matrix for the probabilities? (Hint: To raise a matrix to a power, the matrix must be square, i.e. 2 x 2 or 3 x 3. After you have entered the matrix and selected it, consult the calculator manual for steps to select the power.)
Enter the transition matrix on the calculator and determine the probability of staying with Company A after three switches. Multiply the matrix until it reaches a steady state. When does it reach a steady state to the nearest tenth?
Does the distribution of patrons for each company at the start affect the final distribution of patrons for each company? To find the answer, use these three initial distribution matrices: (1 0 0), (.8 .1 .1), and (.5 .3 .2). (Hint: multiply the initial matrix and the steady state matrix.)

Exploration Answers
Activity

Staying with A=(.4)(.4)=.16
A to B to A=(.2)(.3)=.06
A to C to A=(.4)(.7)=.28
.16+.06+.28=.50
The Probability is .50

Exploring Content

Set up a matrix of the probabilities and raise the matrix to the third power.

A B C

A
.4 .2 .4

.3 .5 .2

.7 .2 .1

B
C

The probability is .446 that patrons stay with Company A after three switches.

A B C

A
.446 .278 .276

.437 .305 .258

.473 .278 .249

B
C

The matrix reaches a steady state at about [A]^6.
No, the initial distribution does not matter. Of the patrons who started with Company A, A will have 45% of patrons, Company B will have 28%, and Company C will have 28%.

Topics for Discussion
How does the conference–style environment affect student interaction?

How did Ms. Doyle make transitions between talking to the whole class to having students talk among themselves? When is each situation appropriate? What do you need to think about in planning for those transitions?
What are possible barriers to creating this kind of environment in other classrooms, especially with a larger class? What classroom management issues might arise?
Describe how you would create a conference environment in your classroom.
What are the elements of a worthwhile mathematical task?

What mathematics did the students discover in this lesson that made the task worthwhile?
How can a teacher create an environment in which students use technology tools routinely without becoming dependent on them?
How can technology use deepen students' understanding of mathematics?
How does this lesson make connections within mathematics and to the real world? Brainstorm a list of other real–world situations in which you could use probability and matrices. How could students collect data for these real–world situations?
What would you assign for homework following this lesson?
What evidence did you see of students trying to make sense of the content?

Describe Ms. Doyle's questioning technique for helping students understand the answers provided by multiplying the transition matrix.
What additional questions might you ask to guide students toward choosing to use matrices to continue the problem?
What techniques did Ms. Doyle use to help students work with the probabilities entered in the initial state matrix as well as the long–term results provided by the steady state matrix?
How did Ms. Doyle question students' generalizations to assess the understanding behind them?