I open this lesson by asking my students to share any mistakes they found in the final challenge from yesterday's lesson. I listen as they discuss the problem and ask leading questions like "What do the points used to calculate the possible profits represent?" and "Do all of the points fit all of the constraints?". When we've finished that problem I direct their attention to a system of equations graphed and projected on my front board. I ask for volunteers to identify the maximum and minimum points, which they do fairly easily because the graph gives them a visual perspective and because that's what we've been working on. I have intentionally excluded the inequalities (shading) because I want my students to figure out that these are as important to the modeling process as the linear equations. After the points of intersection are all circled graph 1. I ask my students to pair-share what additional information they need to make a decision about the best answer(s) for this system. (MP1, MP4) After a few minutes I randomly select students to share what they discussed. Generally at least a few students recognize that they need to know whether the points circled fit all the limitations of the problem. There are also some who question having negative values for any linear programming problem. I tell them that the two dashed blue lines represent the minimum and maximum temperature for optimal growth of a specific bacteria, that the red line represents proportions of nutrient A and the black line represents proportions of nutrient B. I continue by saying that the nutrients cannot exceed either limiting line without resulting death of the bacterial culture. I now ask for volunteers to shade the graph appropriately with the result shown in graph 2. To close this piece of the lesson, I ask my students to review the two graphs and pair-share about which points were potential answers and why some of them were discarded even though they were mathematically correct.
You will need copies of the Linear Programming Challenge Problems for this section of the lesson. I tell my students that today's challenge will give them an opportunity to put together all the pieces they've been working on plus a little extra. I say they will be writing systems of equations, graphing and solving the systems, determining which answers are actually viable solutions, and finally explaining why each answer works or why not. I let them select a partner to work with and distribute the Linear Programming Challenge Problems then ask if there are any questions. I finish my directions by telling my students that they will be presenting their work to the class and need to be ready to explain why only certain answers were viable. (MP1, MP2, MP4) While they're working I walk around offering encouragement and redirection as needed.
When everyone is finished, I randomly select teams to present and have them roll a die to determine which problem they get to explain. I have them use the document camera and projector because it's faster since they already have the work and graph done. After each team presents I have the rest of the class critique their results and allow the presenters to respond/explain. (MP3)
To close this lesson I give my students a notecard and ask them to describe a situation where an answer would look good but not be a viable solution to the problem. I tell them they can't use the problems we've been working on, but that those problems should give them a good starting point. (MP1) My does it work video explains why I chose this approach to closing the lesson.