## Dirt Bike Dilemma

• Lesson
9-12
2

Students discover the algorithm for solving linear programming problems and gain conceptual understanding by solving a real-world problem and using graphing calculator applications.

This lesson develops conceptual understanding of linear programming by walking students through the process of linear programming. Along the way, students are asked to explain what is happening and why, which allows them to internalize the procedural skill necessary to solve linear programming problems.

The basis of this lesson is the Dirt Bike Dilemma Activity Sheet. Before attempting to use this material in class, be sure to look over the activity sheet and solve the problems on your own.

In particular, you should notice that the activity sheet requires the use of TI Graphing Calculators. If you intend to use this lesson with a different type of calculator or with a spreadsheet program, you will need to modify the activity packet before copying and distributing it to students.

To be prepared for this lesson, you will need to copy the DRTBK program into your calculator. Right click on the DRTBK Program and choose "Save Target As…" Then, save the file to your computer desktop.

• Double click on the TI ConnectTM icon.
• Attach the TI‑83 plus or TI‑84 plus graphing calculator to the computer using the TI GRAPHLINKTM cable. (This USB cable comes with the calculator.)
• Click on Data Explorer or TI Group Explorer.
• Drag the DRTBK icon from the desk top into the TI Data file.
• Click on DRTBK.8xp to highlight it.
• Select Actions from the tool bar.
• Select Send to TI Device.

The computer should show the file being transferred to the calculator.

You will also need a program called Transformation on your calculator. It may already be there. You can determine if the Transformation program is installed by pressing the APPS button and scrolling through the alphabetical list of applications. If the Transformation program is not listed, you will need to install the program. The Transformation Graphing Application can be downloaded from the TI Web site. As before, download the program to your computer, and transfer it to your calculator using a TI GRAPHLINKTM cable and TI ConnectTM software.

Each student will need a TI‑83+ or TI‑84+ graphing calculator containing the DRTBK program and the Transformation Graphing application. If these programs are not installed, take some time at the beginning of class to have students download these programs to their calculators. In addition, each student will need a copy of the Dirt Bike Dilemma activity sheet. Each team will also need some colored pencils and a deck of the Dirt Bike Cards.

Divide the class into teams of three students. One member of the team should be given all of the Wheel cards; this team member is responsible for completing Question 1 on the activity sheet. Similarly, another team member should be given all of the Exhaust Pipe cards and complete Question 2, and the last team member should receive all of the Seat cards and complete Question 3.

This lesson is designed to guide students to discover and consolidate the concepts associated with solving linear programming problems. Your role as teacher is to assess their understanding and provide assistance if they encounter difficulties. Move from one team to another, listening to the discussions. Encourage students to work cooperatively; try to refrain from answering individual student questions, especially those that can be answered by the team.

Read the problem out loud to your students. Ask a student to describe the problem in his or her own words.

The first part of the lesson (Questions 1‑3) asks the students to work independently. Basically, Questions 1 3 deal with the same concepts. Each team member is asked to complete a table and graph, relating the number of Rovers that can be assembled given the number of Riders that have been assembled, based on the number of wheels, exhaust pipes, or seats. The purpose of these questions is to help the student visualize the problem and to come up with the constraints for the linear programming problem that they will solve.

Randomly ask different teams to explain how they arrived at their responses, especially to Questions 7, 9, 10, 12, 13, and 14. If you are not satisfied with their response, ask some probing questions, such as the following:

• What happens if I select a point outside the feasible region?
• Can the corner points also tell me the combination that will give the minimum profit?

Continue to question until you feel that they are making a connection. Visit each group at least once.

Bring the class together after most teams have completed Question 10. Go through the steps with the class of how to set up and use the DRTBK program. (The procedure for using this program is found in Question 11 on the activity sheet.) Also, go through the first three steps of using the Transformation Graphing Apps. (These steps are found in Question 12 on the activity sheet.) When completing the table in Question 11, tell your students if the maximum value occurs more than once, they should write down both combinations.

When all teams have completed Questions 1 through 13, have a whole‑class discussion. Use the questions from the Questions For Students section.

On the board or overhead projector, list all of the responses to Question 14, "List five major steps required to solve a linear programming problem?" After all responses have been collected, allow the class to narrow the list down to the five major steps.

Allow the class to complete Question 14 on the activity sheet. This can be done with the time remaining in class or as a homework assignment. If used as a homework assignment, the solution should be discussed the next day.

### References

• College Preparatory Mathematics. "The Toy Factory." Math 3 (Algebra 2), Second Edition. Sacramento, CA: CPM, 2002.
• Texas Instruments, Incorporated. "Linear Programming: A Flash View." Eightysomething! The Newsletter for Users of TI Graphing Calculators. Austin, TX: TI. Spring 2000: 5.

Assessment Options

1. Students can play the Clue Cards Game as an assessment activity. As students work, circulate and assess their ability to solve linear programming problems.

Make one copy of the clue cards for each team. Copying each set (Dog Food, Painting, and Four Wheelers) onto a different color of card stock can help manage the collection of cards when teams are finished. Cut each sheet into four clue cards. Each team receives three sets of four clues cards. Place a set of cards for each team in an envelope. Each set of four cards contains the information for one linear programming problem.

2. Copy the game board to a transparency sheet, and place it on the overhead projector. Assign a colored chip or marker to each team. As each team correctly solves a set of clues, move their colored chip to the next level.

• The class should be divided into teams of four students.
• Each team should have an envelope containing three sets of clue cards, scratch paper, pencils, graph paper, a ruler, and a graphing calculator (optional).
• To begin, open the envelope and find the Dog Food clue cards. Give one card to each member of each team.
• Students may only look at the clue on the card they received. They may not look at anyone else’s. Each students should read their clue to their team members. Cooperatively, they should solve the problem.
• When a team thinks that they have arrived at the solution to a problem, they should raise their hands. They should not say the answer aloud.
• If a team’s solution is correct, direct them to go to the set of clue cards for Paintings. Then, move their playing piece to the next level, and award the team five points. If the team’s solution is not correct, ask them to look over their work and try again.
• Students should continue this process until they have found the solution to all three sets of clues.

3. A correct solution to Dog Food is worth five points. A correct solution to Paintings is worth ten points, and a correct solution to Four Wheelers is worth fifteen points. The first team to find the solutions to all three sets of clues wins an additional twenty points. The second team wins an additional ten points, and the third team wins an additional five points.

Questions for Students

1. What is a feasible region?

[The feasible region is the region formed by the intersection of all of the constraints.]

2. What is an objective function?

[An objective function is function for which you are trying to find the minimum or maximum value.]

3. Why must the corner points of the feasible region produce the maximum or minimum value of the objective function?

[The corner points of the feasible region produces the maximum or minimum value of the objective function because as the y‑intercept of the objective function line increases (or decreases), the last point it encounters as it leaves the feasible region is one of the corner points.]

4. Are there times when no unique point will minimize or maximize an objective function? If so, when? If not, why not?

[There are times when there is no unique point that will minimize or maximize an objective function. This occurs when the objective function lines are parallel to one of the sides of the feasible region. Therefore, as the y‑intercept of the objective function line increases (or decreases), the last object it encounter is a line segment and not a single point. In this case, there will be multiple points that yield the maximum (or minimum) value.]

5. What are the five major steps necessary for solving linear programming problems?

[The five major steps for solving a linear programming problem are:
1. Determine the inequalities that represent the constraints.
2. Graph the feasible region.
3. Determine the corner points of the feasible region.
4. Determine the objective function.
5. Substitute the coordinates of the corner points into the objective function to determine which yields the maximum (or minimum) value.

Note that student lists may appear differently, but they should contain these same basic ideas.]

Teacher Reflection

• Was students’ level of enthusiasm/involvement high or low? Explain why.
• How did your lesson address auditory, tactile and visual learning styles?
• How did the students demonstrate understanding of the materials presented?
• Did you find it necessary to make adjustments while teaching the lesson? If so, what adjustments, and were these adjustments effective?
• Was your lesson developmentally appropriate? If not, what was inappropriate? What would you do to change it?
• What were some of the ways that the students illustrated that they were actively engaged in the learning process?

### Learning Objectives

Students will:

• Understand the steps required to solve a linear programming problem.
• Solve linear programming problems.

### NCTM Standards and Expectations

• Use symbolic algebra to represent and explain mathematical relationships.
• Draw reasonable conclusions about a situation being modeled.