Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

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RepresentationSession 05 OverviewTab atab btab ctab dtab eReference
Part A

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


Reflect on the problem and think about some questions you would ask this student. Then look at the questions listed below. For each question, think about an answer the student might provide; then select "Show Answer" to reveal a sample response.

Question: How would you move students from the tree diagram, which can be messy but easily grasped, to the matrix, which is elegant but not intuitive?

Show Answer
Sample Answer:
One way might be to ask students to consider the limitations of the tree diagram as a representation for multiple-level problems. Ask, "If you had to consider a large number of varying trips, how might you simplify the task?" Students might suggest matrices (this might be a factor of how recently they studied them) or you might suggest it and begin to show how it would work. You might also pose a question about how to find out what ultimately happens to the distribution of taxicabs in this situation.

Question: How does this problem encourage the development of the understanding of the matrix representation and its uses and of the use of a scientific calculator to perform matrix arithmetic?

Show Answer
Sample Answer:
To use the calculator for these problems, a student needs to know certain conventions for representing the information in the calculator. The student must also have a basic understanding of the meaning of specific rows and columns.

Question: How are the matrix and tree-diagram solutions alike? How are they different?

Show Answer
Sample Answer:
They are similar because both involve multiplying values with specific meanings by probability values that are related to them. The tree diagram shows the initial probabilities and then shows the values after each round of rides. It is fairly easy to trace the branches that relate to any particular condition, such as ending up downtown at least once in two rounds. In that way, it provides a sense of the general situation of three possible destinations and multiple possible rides. The matrix solution is neater, especially after more than two rounds of rides. But it is more abstract, and individual values cannot easily be traced backwards if a calculator is used to find the solution.

On the other hand, the matrix solution method makes the required mathematics more efficient and facilitates working with a variety of situations, because individual rows, columns, or entries can be easily examined. Also, it is relatively easy to expand the problem and observe the results for four, or even 10, 20, or 100 trips, or to experiment with having more than three possible destinations or different initial distributions. With a matrix method, a calculator can quickly provide a result, whereas with the tree diagram, many individual calculations need to be performed.


Question: How does this problem promote reflection on the meaning and ability to critique representations?

Show Answer
Sample Answer:
Students see that the tree-diagram representation, which may be more familiar and comfortable, is actually cumbersome when considering several rounds. They are motivated to work with the matrix representation because of its orderliness, their ability to use previously acquired skill with multiplying matrices, and the extendability of this representation. They are also encouraged to talk about their solutions and the meaning of various aspects of the problem.

Notice that the initial matrix the students drew embodies the problem statement. It does not represent a solution. It is only of use when the operation of multiplication is correctly applied. The solution matrix, S, contains the solution, but correct calculation and interpretation of this representation are also necessary parts of a solution. Teachers should help students understand what both the matrix and the matrix multiplication represent in this problem-solving task.


Next  Observe a class at work on this problem

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