Illuminations: Dirt Bike Dilemma

# Dirt Bike Dilemma

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

### Learning Objectives

 Students will: Understand the steps required to solve a linear programming problem. Solve linear programming problems.

### Materials

 TI 83+ or TI 84+ Graphing Calculators Colored Pencils Dirt Bike Dilemma Activity Sheet Dirt Bike Dilemma Cards, one set per group of three students DRTBK Program for TI Calculators

### Questions for Students

 What is a feasible region? [The feasible region is the region formed by the intersection of all of the constraints.] What is an objective function? [An objective function is function for which you are trying to find the minimum or maximum value.] 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.] 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.] What are the five major steps necessary for solving linear programming problems? [The five major steps for solving a linear programming problem are: Determine the inequalities that represent the constraints. Graph the feasible region. Determine the corner points of the feasible region. Determine the objective function. 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.]

### Assessment Options

 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. 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. 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.

### 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?

### NCTM Standards and Expectations

 Algebra 9-12Use symbolic algebra to represent and explain mathematical relationships. Draw reasonable conclusions about a situation being modeled.

### 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.
 This lesson prepared by Joanne Nelson.

2 periods

### NCTM Resources

 More and Better Mathematics for All Students
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