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.

10 minutes

If students have worked on a previous linear programming problem and written about their work, I like to share some student work at the beginning of class that highlights particularly good writing about math thinking. Students can comment on what they notice in the student example and highlight which sentences help the reader understand how the writer found his/her solution.

I have found that students really like to see other students’ work. I leave the name off the student work and allow the author to remain silent about his/her work or allow the author to take credit. Seeing other students’ work allows students to get clearer about what is expected of them in terms of mathematical writing. I have students come up to the Smartboard and highlight sentences that they think show strong mathematical writing.

Next, I introduce today’s task. Students will be working in groups to solve another linear programming problem. This time, they are looking to minimize cost, rather than maximize profit. I have students read the problem aloud in the whole group and ask for clarifying questions.

This lesson is a good opportunity for students to practice their skills in service of reviewing for an assessment or before working on a final project like creating their own linear programming problems.

40 minutes

At this point in the lesson, students will break down into pre-determined homogenous groups. This will allow students to grapple with the problem where they are in the learning process. I think about where I position the groups in the classroom so I can be near groups I know will need more help and I let more advanced groups try to figure things out for themselves. Students will have a chunk of time to work on the problem and I circulate to answer clarifying questions and/or provide guiding hints. I usually position myself to sit with the group(s) I know will need the most guidance. I try to let the other groups wrestle with the problem and use each other as resources.

This problem provides students the opportunity to practice **SMP 4: Model with Mathematics. The complexity of this problem requires students to map relationships between different variables graphically and then analyze those relationships to draw conclusions**.

Things I watch for:

- Some students will have trouble figuring out the scale for their graphs. I encourage them to try to find the highest point that will occur on each axis based on their inequalities. This should help them to realize how high they have to go on each axis.
- Students may have trouble writing the inequality that reflects they have to admit as many out of district students and Boston students. I have them try out numbers to see which pairs would fit to help them decide about the inequality.
- 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.

At the end of this section, students should have graphed the three inequalities and be ready to shade the feasible region. If some groups are further along or behind, I may adjust the homework assignment accordingly.

10 minutes

For today's end of lesson reflection, I ask students to verbalize how solving the Summer School problem compared to the previous Linear Programming problems they worked on. I ask them specifically:

- What was easier?
- What was more difficult this time around?
- What are the steps that you still need to work on?