Support Materials --

Teaching Math: A Video Library, 9-12

Maximizing Profits

Process Standards
Connections, Problem Solving, Reasoning
School: Cardozo Senior High School; 1,000 students
Location: Washington, D.C.
Teacher: Deborah Pearman
Years Teaching: 22
Students in Classroom: 17
Grade: 10 – 12
Maximizing Profit
Video Overview
Ms. Pearman describes the necklaces and bracelets that students have purchased from other students in the school's chapter of the Distributive Education Clubs of America (DECA). The class is asked to find the combination of necklaces and bracelets they must craft and sell to make a maximum profit, given time and materials constraints. The class is divided into small groups to discuss problem–solving strategies, such as finding a verbal model, assigning labels, and writing an algebraic model. After the groups share their strategies with the class, they write, graph, and solve the inequalities and interpret their graphs to find the optimal number. At the end of class, each group explains how it solved the problem.

The Class Challenge
In the video, students use the following information to find the combination of necklaces and bracelets that DECA members should craft and sell to make a maximum profit:

Twenty hours are spent crafting necklaces and bracelets.
There are 10,000 grams of metal beads.
The necklace requires 50 grams of beads and 30 minutes.
The bracelet requires 200 grams of beads and 20 minutes.
Profit for the necklace is $3.50.
Profit for the bracelet is $2.50.

Course and Curriculum
This elective Advanced Algebra course includes trigonometry, polynomials, linear equations and systems, functions, complex numbers, logarithms and exponentials, and sequences. Prior to this course, students took Algebra I. The goals of this lesson were for students to determine strategies for solving the problem and to solve the problem.
In earlier lessons, students found the maximum or minimum value of a function, used a system of linear inequalities to model a real–life situation, studied the graph of an equation, studied slope and rate of change, wrote equations of lines, and graphed linear inequalities in two variables. In this lesson, Ms. Pearman asked students to graph the equations in slope–intercept form, a quick way to graph without a graphing calculator. To find their results, students looked at the inequalities that gave them a bounded figure on their graphs. Then they used the points where the inequalities intersected to get the numbers they used to write the profit function.
Following this lesson, students investigated a similar problem. They used the problem constraints to determine a solution for maximum profit. If a solution did not exist, students determined if the problem was infeasible, had alternative optimal solutions, or was unbounded. In the following unit, students studied matrix algebra (operations, determinants, properties, and identities and inverses) and applications to solving linear systems.

A Pre–Viewing Exploration for Teacher Workshops
Finding the Maximum and Minimum
Objective: To use linear programming to solve a problem.
Materials: Graphing calculator or graph paper for each teacher.
Pre–Activity
Solve this problem individually.
Linear programming can be used to determine the minimum and maximum values of a linear function over a region (often referred to as the critical region). The region is determined by contextual constraints expressed as linear inequalities. If the critical region is a bounded convex polygon, as in the example below, the maximum and minimum values of the linear function occur at vertices of that region. Suppose the critical region was bounded by x + y >= 2, 4y <= x + 8, and 2y >= 3x – 6. The linear function is f(x,y) = 3y + x.

How could you determine the maximum and minimum values of f(x,y) in the critical region?
Is there more than one way to begin the problem? If so, how?
If the vertices of the region are (0,2), (4,3), and (2,0), how could you determine the maximum and minimum values of the function?
What are the maximum and minimum values?

Activity
Solve this problem individually.
A farmer wants to plant cotton and corn on 10 acres of land. Use the information and questions below to determine how many acres he should plant of each crop in order to maximize revenue.

Each acre of cotton requires up to 6 hours of labor per week.
Each acre of corn requires up to 4 hours of labor per week.
He has 48 hours of labor available per week.
Each acre of cotton yields 20 bushels.
Each acre of corn yields 10 bushels.
He would like to plant at least 3 acres of corn this year.
He can sell corn for $4 per bushel.
He can sell cotton for $6 per bushel.

What are the constraints on corn and cotton?
What are the constraints on time?
What are the constraints on acres?
How would you determine the profit function?
Graph the inequalities and find the vertices of the region.
What is the maximum number of acres of corn and cotton the farmer can plant to maximize revenue?

Exploring Content

What is the significance of the points inside the critical region?

Exploration Answers
Pre–Activity

Answers will vary. One possible first step could be to transfer the inequalities into slope–intercept form for easy graphing. Then you could graph the three inequalities, find the vertices, and test the coordinates of the vertices in the function.
Yes. You could graph the lines in different ways. You could use the tabular form to determine points on the lines, use the slope and intercept, or graph on graphing calculator.
You could substitute the coordinates of the vertices in the function. The maximum vertex would give the largest value for the function and the minimum vertex would give the smallest.
The maximum is 13 at (4,3); the minimum is 2 at (2,0).

Activity

Corn is between 3 and 10 acres; 3 <= x <= 10. Cotton is between 0 and 7 acres; 0 <= y < 7.
There are 48 hours available. Four hours are needed for corn and 6 for cotton. 4x + 6y <= 48.
The most the farmer can plant is 10 acres; x + y <= 10.
Use the number of hours of labor, the bushels needed, and the acres. Profit function: P = 4(10)x + 6(20)y; x = acres of corn, y = acres of cotton.
Vertices of the regions (3,6) (6,4) (10,0) (3,0).
Three acres of corn and 6 acres of cotton; (3,6).

Exploring Content

The points inside the critical region represent all possible combinations of acres of corn and cotton, given the constraints, though only one of these combinations gives the maximum profit.

Topics for Discussion
How did students respond to this problem with multiple constraints?

How did Ms. Pearman help students to begin solving the problem? How do you help students to begin solving similar problems?
How might you help students who don't understand how to work within given constraints?
Cite other problems in real–life situations that could be solved using systems of inequalities.
How would you help students improve graphing skills within this lesson?

Why do you think Ms. Pearman asked students to put the equation in slope–intercept form before graphing? What other methods might you use?
Ms. Pearman provided steps for solving word problems. How did students' use of these steps affect their work? How would you use these or similar steps in your own classroom?
Some students had difficulty consistently labeling the scale of the axes in relation to the inequalities they were constructing. How do you respond to similar difficulties?
A group of students thought that because the coordinates of the intersection of the lines were not whole numbers, the lines did not intersect. Discuss how you would respond to this misunderstanding.
How might technology have impacted this lesson?

How might graphing calculators influence the students' understanding of the mathematical concepts in the activity?
How would you present this lesson if your students were using graphing calculators or computers?
What other tools could students use in this lesson?