Lesson Plan 2: Hassan's Pictures - Linear Programming and Profit Lines
Teachers will need the following:
Students will need the following:
- Gridded flip-chart paper
- Markers and tape
- Rulers, scissors, and grid paper
- Notebook or journal
- Rulers, scissors, and grid paper
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.
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.
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.