SWBAT create and solve their own linear programming problem. SWBAT present and explain their work.

Students gain insight into linear programming problems by working in groups to create and solve their own problems.

10 minutes

This lesson is the start of the closing sequence of this Systems of Equations unit. Students work together in groups to create and solve their own linear programming problems and then present their work to the class. This end-of-unit project can be used as an assessment as each student will be completing his/her own write-up of the problem. This lesson helps students to understand the elements of linear programming as they develop a context for applying linear programming techniques.

**To get ideas about how to share student work at your school, check out my Sharing Projects movie.**

In Day 1 of this lesson, students look back at linear programming problems that have already solved in class and identify important characteristics of those problems (constraints, variables, what they are trying to maximize or minimize). By the end of today’s class, students should have started to brainstorm a context for the problem situation they will create.

I begin class by explaining to students that today they will begin work on a project where they are the math problem writers/creators. This work will be a final project of the unit and will count as an assessment. I emphasize that this project is an opportunity for students to “show what they know” and will demonstrate their learning over the course of the unit. Each group will create a linear programming problem and present its solution to the class. They will begin by reflecting on previous problems they have seen and indentifying the variables, constraints, and factor that was maximized or minimized.

45 minutes

Students work independently on Brainstorming Ideas for Linear Programming Problems. They should work on Questions 1 to 2 independently and then get back into their groups to brainstorm about Questions 3 and 4. This activity helps students to see the structure of the other linear programming problems they have worked on in the past. It prepares them to think about writing their own problems by helping them to recognize the pieces that each problem has in common (2 variables, between 2 and 4 constraints, something that is minimized or maximized).

Things I watch for:

- I encourage students to define their variables precisely. For example, rather than writing “Boston students and out-of-district students”, I encourage them to write “the number of Boston students and the number of out-of-district students.”
- Constraints vs. profit. Some students may confuse what the problems are trying to maximize and minimize with one of the constraints. I help them to identify the key words that lead to a constraint and why there is no limit on profit or cost. I try to elicit that this difference also reflects the distinction between an equation and an inequality.

As students begin brainstorming situations for their problems in their groups, I am careful with how much I help. At this stage, students are just brainstorming ideas and once they try to set up the problems, they may recognize that they have to make some changes. A part of Day 2’s lesson plan will focus on linearity, so I think carefully about how much I want to keep students “on track” during this portion of the lesson.

5 minutes

I have students finish class today by reporting out on where they are in the brainstorming process. If some groups are struggling with ideas, I have one group share how they came up with a situation. I want to get a sense of where students are by the end of class today so I can begin Day 2 with everyone together.

**Homework**: When I start Day 2 of the lesson, I want to have students share out their rough ideas for a situation they have thought of where they want to maximize or minimize something. I may have to assign students to complete Questions 3 & 4 on Brainstorming Ideas for Linear Programming Problems so everyone can participate in the discussion on the next class day.