SWBAT write inequalities using constraints, graph inequalities and identify a feasible region, write a profit function, and find maximum profit.

This lesson gives students the opportunity to synthesize what they have learned before they begin to create their own linear programming problems.

5 minutes

I remind students that they are working on the Summer School Linear Programming Problem again today. The goal for this lesson is for students to finish their work on this problem and be ready to present it to the class.

Students began working on this problem in the previous lesson. I like to start the class by having them reflect on what worked well in their groups yesterday. This can help set the tone for a second productive day. I find that sometimes two days of group work in the same group can lead to some off track behavior or a loosening of classroom norms. So, the real purpose of this Lesson Beginning activity is to set the tone for the day’s work. Students could report out from their groups or I might have each student free write individually: **what made their group successful on the previous day?**

25 minutes

At this point in the lesson, students will go back to their homogenous groups to continue their work. Students should have graphed the three inequalities and found the feasible region by the end of yesterday’s lesson (perhaps with some homework) and are ready to graph the profit line (or in this case, cost line).

Things I watch for:

- Students often struggle when they get to the profit line (or in this case, cost line) piece of the work. I remind them that they can choose any number of total costs that the school will spend. They may initially choose a number that is too small; I help them find a better line by encouraging them to choose a larger amount of cost. A number that works nicely for the cost line is $144,000.

As groups finish, I have them prepare presentations on this assignment.

25 minutes

Students will close the class today by presenting their work on this problem. This is a good opportunity for students to explain the process, how they understand the problem, and use mathematical language. I like to focus on **SMP3: Construct viable arguments and critique the reasoning of others** here. For the presenters, I focus on the progression they describe to explain their problem solving process. For the observers, I focus on asking clarifying questions that help deepen understanding.

I prompt students to:

- Explain how they wrote their inequalities. What inequalities were confusing to write? We spend some time on D > B.
- Explain how they sketched their graphs. How did they decide on an appropriate scale? How did they know which side of the line to shade? How did they graph the inequality D > B.
- Explain how they knew where the feasible region was.
- Explain how they came up with their profit/cost line. How did they decide on a total number of profit?
- Explain how they knew where the minimum cost would be. Explain the difference between finding a minimum and a maximum.

At the end of class I let students know that they will be starting to write their own linear programming problems in the next class. This work was in preparation for that process.