After handing out The Music Shop Problem, the students should be given about 5 minutes to begin reading, thinking, and working in a quiet, individual setting. This individual time is critical in order to allow all students to engage with the problem and begin working toward a solution strategy. I would expect all students to complete problem 1 during this time and to begin making progress on problem 2.
The teacher's role during this time is to check in with as many students as possible to ensure that they are correctly understanding both the situation and the question. I expect that as students move on to problem 2, many will begin to struggle because this problem asks them to formulate inequalities to represent various constraints. Not only are they faced with the challenge of abstracting the situation, but there is the added challenge (for many) of working with inequalities. Be ready to assist, but be careful not to tell the students what to do!
Now, give the class the opportunity to work collaboratively on this problem. Since it is very early in the year, I like to assign students to small groups (about 3 students per group); later in the year, I will begin allowing them to self-select their partners. Also, since I do not know the students very well yet, the groups will be formed more-or-less randomly.
The first task for each group is to come to a consensus on the equations & inequalities for problems 1 and 2. This is the stage at which the group is formulating the model (with some very explicit guidance) by creating algebraic equations & inequalties to represent the constraints in the situation. A good beginning is half the work, so take this step slowly! (MP 2)
The next task is for each member of the group to construct a graph of the entire system of inequalities in problem 2. For some reason, many of my students think that a good graph is a tiny graph, so encourage them to make it nice and big and to consider the domain and range carefully before beginning to plot any points. Some students may need a head-start, so I like to get the graph started and have a few copies on hand just in case. Finally, colored pencils may be helpful.
The third and final task for each group is to discuss the meaning of the solution set in the context of the situation. They must carefully decontextualize and interpret the meaning of an ordered pair in order to do this, and it will provide them with crucial practice communicating their own thinking and responding to the reasoning of their peers. (MP 3) Many groups will be tempted to look to you for "the answer", but be careful not to give it away just yet - they're perfectly capable of making sense of this problem and only need encouragement. (MP 1)
A class discussion should follow in which student groups are called on to provide the inequalities, identify particular solutions, and explain the meaning of the solution set.
The teacher should focus on assessing the degree to which students understand the connection between the mathematical model and the real-world situation. To do this, you might ask a student to provide an example of an ordered-pair that violates one or another of the constraints. For instance, "Give me an example of an ordered pair that corresponds to purchasing too many instruments altogether? Too few guitars? Guitars and basses in the wrong ratio?" This kind of questioning tests students' understanding of the boundaries between viable and non-viable options. For strategies and tips on leading classroom discussions, please see my Strategy Folder.
By the end of this class period, all students will be able to explain how the mathematical model was formed, interpret a given ordered pair as representing either a viable or non-viable option, and identify the region of the graph in which the viable options are contained. You might choose to make one or more of these objectives a written exit ticket exercise.