It's important to give students some time to work quietly and individually on a problem before there can be fruitful collaboration. It's during this time that students make sense of the problem (MP 1) and begin considering strategies for solving it. For more information about what this time looks like in my classroom, please see my Strategy Video on Individual Time.
The specific aims for each student during these five minutes are to formulate the inequality that represents the minimum cost and to graph it. They will begin considering how this new equation changes the solution set and how to optimize the situation in the next section of the lesson.
Use GeoGebra to model the cost equation (have cost preset to $36,000) from the The Music Shop Problem.
As a class, discuss what will happen if the cost is increased or decreased. You can use GeoGebra to display the system on the whiteboard, but do NOT make it dynamic by adjusting the cost slider yet (see the GeoGebra file in the resources). Instead, encourage students to make conjectures about what should happen as the cost is adjusted, to listen to one another, and to critique one another (MP 3). Students should be able to see that the slope will remain constant (m = -750/900), so the "cost lines" will all be parallel. They should also be able to explain that changing the cost will cause the x- and y-intercepts to change. From these two observations, they will be able to predict what will happen to the graph as the cost is adjusted up or down, and they should be able to give a pretty good estimate of the optimum solution.
Now, it's time to make use of the slider to confirm what the students have already discovered. The dynamic nature of the program helps to make the solution to the problem more intuitive. They can literally watch the feasible region shrink until it finally disappears, and they can point to the optimum solution on the graph. All that remains is to identify its coordinates, which some students may already have done. It's worth noting here that the optimum solution is not the vertex of the feasible region since this point does not have integer coordinates. The true solution will be a nearby lattice point.
An important final question is to ask students to identify the point of least cost. This will test their ability to apply the same reasoning in a slightly different way, since the direction of the inequality for the cost line will be reversed.
Since students have been focused on using the cost line to identify the point of greatest cost, it will be natural for some to use the cost line to find the least cost, as well. As they'll quickly see, the least cost occurs at the point (17, 5), which is a vertex of the feasible region. Of course, other students will take a step back and point out that this is just common sense: Jake Garcia said that these were the minimum numbers of each instrument he would buy. Of course buying the minimum number of instruments results in the least cost! I guess sometimes you really need a mathematical model, but sometimes you don't!