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)
By the end of the 45 minutes period, the class had just begun graphing the system of inequalities! The homework assignment was to complete the graph; tomorrow we’ll work out what the solutions set is and what it means. It’s only the 2nd day, and I’m already behind!
For the graph, I’ve found it is very helpful for many students to make two simplifications. First, replace B and G with y and x, since students are more familiar with these variables. Second, replace the inequality with an equality. This allows students to identify individual points to graph and helps them to see the boundary line of the solution set. After the boundary line has been graphed, reassert the inequality and determine which side of the line represents the solution set. Finally, reinterpret this solutions set in terms of B and G to make sure that it makes sense of the given constraint. This is a longer process, but I found that many of my students were simply stumped by an inequality like B + G < 50.
In the final analysis, the primary struggle my students faced was keeping track of the meaning of the symbols being used (see Student Work). Many students would interpret something like “17 guitars” as “17G” rather than “G = 17”. They were thinking of the symbol G as a unit label, like "cm" or "sec", rather than as a variable standing in place of a number. They also had trouble choosing the operation that correctly captured the scenario. For instance, the total number of instruments might be written as a product rather than a sum. In this case, I say, "Suppose Jake bought 12 guitars and 18 basses, how would you use your equation in this case?" As they began to explain to me what they would do, it usually became clear that multiplying didn't make sense.
One student became very upset when I refused to allow him to leave class to look for his misplaced pencils. Instead, I gave him one of my extra pencils to use for the period. He turned his back on his group and stared at the wall, fuming. After giving him a minute or two, I spoke with him about the relative importance of being present in class versus having his own pencils, reiterated that I was not going to allow him to miss class to look for lost pencils, and then suggested that he go get a drink of water to clear his head. A few minutes later he was back in class, not happy, but working.
He returned to me after school for some help, so I guess he isn’t holding a grudge. However, my suspicion is that he is confused by the modeling problem and doesn’t want to face that confusion. He deliberately avoided asking any questions about the classwork assignment and only wanted help with the Weekly Workout problems.
Follow Up: This student was back in class the next day with a much better attitude. He had made sense of the model (perhaps with help at home), and had set himself the challenge of explicitly identifying all of the feasible solutions in one big table (see the attached student work). Not the best use of his time, perhaps, but he clearly understands how to interpret the model in context, and, more importantly, he's excited to be making sense of the problem.
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.
As I mentioned, my class didn't get to this part of the lesson until the next day.
Students are much more confused by the graphing of the system than I thought they’d be. (One young lady approached me privately before class to admit that she was totally lost. “Remember,” she said, “there’s a reason I’m only taking Geometry and Algebra 2 my senior year.” I guess she meant that I shouldn't expect too much from her, but we'll see about that!) So when class began, we carefully discussed how to turn a verbal expression into an algebraic one, using the sentence, "the number of guitars is at least twice the number of basses", as our example. This was student-directed in the sense that I asked students to explain the strategies they used or the ways they thought about the process. It took about 10 minutes, but I think it was helpful for very many students.
I found that when students came to the final problem (identifying feasible options for Jake) many were not making use of the graph. Instead of identifying points inside the feasible region and then interpreting them in context, they were guessing at solutions and then checking them by hand against each constraint. I called for everyone’s attention to point out the usefulness of the graphical solution. By study hall, I found that one student who was struggling previously seemed to have made very clear sense of this “feasible region” and was able to recognize the point of least cost without prompting (see Student Work C). This is progress!