You will need to have copies of the lemonade information ready for this part of the lesson or be ready to project it on your board. I begin this lesson by reviewing our discussion yesterday about limits and constraints. I thank my students for their work in helping my friend set up the model she needed to find her most profitable option. I go on to say that we have a new challenge to address today. The question is how much to charge for lemonade to make the best profit. I explain that the lemonade stand sells small and large cups of fresh lemonade. I project the lemonade information on my front board and ask my students to work individually to set up constraints for this problem like they did for others yesterday. (MP1, MP2, MP4) While they're working I walk around offering encouragement and reminders as necessary. When everyone is finished I tell them that today we are not just looking for the constraints, we will actually be solving for the optimal result, starting with this problem. I explain that the next step is to graph all the equations and inequalities we've written as constraints, so I distribute graph paper and rulers and let my students get to work. As always, I walk around offering encouragement and redirection as needed. When everyone has completed this section I select one good graph (I look for a graph that is clearly and correctly drawn and labeled) to project and let everyone check their work against it. I tell my students that we're ready to find how many large and how many small cups of lemonade we should sell to make a maximum profit. I circle the points of intersection on the graph and explain that these represent maximum or minimum points within the limits we've set up. You can see the equations and a sketch of the graph on my educreations video.
You will need copies of the Linear Programming problems handout for this section of the class. Now that we've worked through an entire linear programming problem as a class, I tell my students that they get to work with their back-partner to solve some problems on their own. I remind them to write the constraints, then graph them, and finally use the points of intersection on the graph to find the desired solution. I distribute the Linear Programming problems handout and ask if there are any questions. (MP1, MP2, MP4) While my students are working I walk around offering encouragement and redirection as needed. When everyone is done, I ask them to take a "gallery walk" to check their work against the answers I've posted around the room and ask questions about any they don't understand.
I really want my students to remember how important it is to start with good, meaningful equations so I give them a different kind of challenge to wrap up this lesson. I present them with a Linear Programming Challenge all set up with equations, a graph and a solution - except it's the wrong solution. My challenge to my students is that they figure out why it's wrong, where the mistaken assumption is. (MP1, MP2, MP6) This also sets up the lesson for tomorrow, which focuses on whether the solutions are actually viable answers to the problem.