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**Overview:**

This lesson helps students learn to find the feasible region in a linear programming problem, to graph a family of profit lines, and to find the optimum point – the point on the feasible region that will produce the greatest profit. The lesson plan is based on the activity Patricia Valdez presents in the video for Part II of this workshop.

**Time Allotment:**

Two 50-minute periods

**Subject Matter:**

Systems of linear inequalities

Feasible regions

Profit lines

Optimum points

**Learning Objectives:**

Students will be able to:

- Graph the feasible region in a linear programming problem.
- Graph a family of profit lines for a given linear programming problem.
- Determine the optimum point or solution to a linear programming problem.

**Standards:**

*Principles and Standards for School Mathematics*, National Council of Teachers of Mathematics (NCTM), 2000:

NCTM Algebra Standard for Grades 6-8

http://standards.nctm.org/document/chapter6/alg.htm

NCTM Algebra Standard for Grades 9-12

http://standards.nctm.org/document/chapter7/alg.htm

**Supplies:**

Teachers will need the following:

- Gridded flip-chart paper
- Markers and tape
- Rulers, scissors, and grid paper

Students will need the following:

- Notebook or journal
- Pens/pencils
- Textbook
- Rulers, scissors, and grid paper
- Calculators

**Steps**

**Introductory Activity:**

**1.**Select a student to read yesterday’s homework assignment from “Picturing Pictures.” You might want to select several students and ask each one to read one part of the assignment. Students must have completed the graph of the feasible region in the homework assignment in order to be able to find the optimum point in today’s activity. Assign each group the task of preparing the answer from one part of yesterday’s assignment and writing their solution on gridded flip-chart paper. For example, one group’s solution will illustrate the time constraint that states Hassan has only enough time to paint 16 pictures; another will show the cost constraint which limits the number of pictures he can paint due to the price of materials. The students should hang their solutions on the wall, and one student from each group should present the group’s solution to the class.

**2.**As students present, ask questions to make sure that they understand the problem, and to help the rest of the class follow their reasoning. Students should be able to articulate their thinking and the process they used to solve the problem.

**3.**Ask a student to present the graph of the feasible region. This graph combines all of the graphs described in Step 2 onto one grid. Make sure that the profit equation is written on this chart paper, as it is the cornerstone of today’s lesson. Leave this graph up on the board throughout class.

**Learning Activities:**

**1.** Begin today’s lesson, “Profitable Pictures,” by asking several students to each read one part of the instructions. After each student has read, ask questions of the class to make sure that they understand the problem.

**2.**Ask students to begin working in groups on problem #2 and to use the profit equation of 40P + 100W = 1000.

**3.**Circulate around the room, helping the groups. When you see a group finish the problem, ask one student from the group to write their solution on the board. The student should also graph the profit equation on the graph that contains the feasible region. (See Step 3 of the Introductory Activity).

**4.**When the student finishes writing the group’s solution on the board, ask him or her to present it to the class. Pose questions to make sure the class understands the meaning of the solution and the reasoning behind it. Also, mention that there are multiple methods for solving the problem, and that it doesn’t have to be done with the particular method presented.

**5.**Instruct the groups to repeat the process using the profit equation

40P + 100W = 500.

**6.**Circulate around the room, offering help when necessary. Find a group that has completed the task, and have one student present the group’s solution to the class and graph the profit equation. (Note: Choose a group that has used a solution method different from the first group so that students can compare the two methods and processes.)

**7.**Repeat the process using the third and final profit equation,

40P + 100W = 600. At the conclusion of this problem, there should be three parallel profit lines on the graph. The three lines should cross through at least a portion of the feasible region.

**8.**Ask the students to focus on the profit lines. They should notice that they are parallel, and that as the profits increase, the lines have a greater

y-intercept and intersect the feasible region higher up in the coordinate plane.

**9.**Discuss why the lines are parallel. The students should note that each equation is for a different profit, and therefore the lines have no points in common. They should also note that all the equations have the same left side (40P + 100W), and the only difference in the three equations is the amount of profit. Some students may observe that since the left side of the equations is the same, the slopes of the three lines must be the same.

**10.**Tell the students that the goal is to find the highest profit possible. Ask the students to cut out a piece of paper in the shape of a right triangle. The triangle is formed by the intersection of the x-axis, the y-axis, and one of the profit lines.

**11.** Ask students to slide the triangle up the y-axis (without turning or rotating it) and to try to locate the last point on the feasible region that the hypotenuse of the triangle will touch before leaving it. This is the optimum point – the point that will produce the maximum profit for Hassan the artist.

**12.**Ask the students to find the coordinates of this corner point (6, 10). They should substitute the numbers from the corner point into the profit equation to find the profit: 40(6) + 100(10) = 1240. Therefore, the largest profit possible is $1240. This profit occurs when Hassan sells six pastels and 10 watercolors.

**Culminating Activity/Assessment:**

Ask students if there are other points on the feasible region that should be checked to determine if their profit is greater than or equal to the point they just located. Identifying the optimum point is a very difficult concept for students and they will need more practice before they gain a good understanding.

The questions below dealing with systems of linear inequalities have been selected from various state and national assessments. Although the lesson above may not fully equip students with the ability to answer all such test questions successfully, students who participate in active lessons like this one will eventually develop the conceptual understanding needed to succeed on these and other state assessment questions.

- Taken from the Texas Assessment of Knowledge and Skills (Spring 2003):

At Northwest Electronics, audiotapes cost $5.00 per package and videotapes cost $10.00 per package. Which inequality best describes the number of packages of audiotapes, a, and the number of packages of videotapes, v that can be purchased for $45.00 or less?

A. 5a + 10v < 45

B. 10a + 5v 45

**C. 5a + 10v 45 (correct answer)**D. 10a + 5v < 45

- Taken from the Virginia Standards of Learning, Algebra II (Spring 2000):

This graph of a linear programming model consists of polygon ABCD and its interior. Under these constraints, at which point does the minimum value of 3x + 2y occur?

**A. A (correct answer)**B. B

C. C

D. D

**Teacher Commentary: Systems of Linear Equations**

This is the paper of an 11th grade student with a fairly low math skills level. After class discussion and examination of his paper, I observed that he was able to demonstrate understanding of the major concepts discussed in class today. He was able to make the connection between the slope of his line of best fit and the speed of his writing. He also explained that “when the time is zero, [that] the letters will be zero,” and was able to see that this should hold true for both the right hand and left hand data.

During class discussion, he was allowed the time to think through and adjust his answer as he was sharing with the class. His group’s conclusion paragraph shows that they were able to understand the three potential scenarios for a linear system: intersecting lines, parallel lines, or two lines sharing the same points. Overall, his work shows an understanding of the lesson objectives. I did notice a fairly common mistake on his paper. For question 3b, he was asked to find an equation for the line of best fit and he forgot to write y = for each equation. This will be something I will need to continue to emphasize.

After reading the students’ responses, I feel that this introductory lesson to linear systems went well. Students have been able to identify different scenarios for linear systems and have been able to make a real-world connection. As we continue to explore the systems unit, the students will have a rationale and purpose for learning how to solve a linear system. I plan to continue to discuss applications of linear systems for the second day of the unit. By having students work in groups and investigate and graph a system, they will be able to make important observations and comparisons of two plans. Subsequent lessons in the unit will lead students to learn the algebraic techniques for solving systems, while continuously stressing the connections to the real world.