The purpose of today's lesson is for students to synthesize what they have learned about systems of inequalities and systems of equations in order to solve the original unit problem (how many cats and dogs they should board at their pet sitting operation in order to maximize profit). Students have one piece left to learn: how to create profit lines to determine where the maximum profit will be in the feasible region.
I like to begin class by letting students know they have done a lot of work to reach this point. Today they will be finding the answer to the question they first looked at on the first day of the unit in Introduction to Inequalities: Working with Constraints. We start by reading through Pet Sitters Revisited together. Most of the information in this problem should be familiar to students as we have been working with the same content for most of the unit. Students will need to have their feasible region graphs from the Finding the Feasible Region lesson. (This lesson could also be taken out of the context of the unit, students would have to write and graph inequalities and shade a feasible region from the given information).
The new information to students in the problem is about how much they charge each day for cats and dogs and how much they need to spend each day for food and supplies. I like to address this issue right at the start of class because it can sometimes be confusing for students and this calculation about profit is not the point of the lesson. We talk about how profit is how much you make minus how much you spend so we determine right away that each dog they board will make them $16 and each cat boarded will make them $6.
Using this introduction and information, I like to have students take a look at their graphs and make a prediction about what combination of cats and dogs they think will yield the highest profit. I think letting them play around with the numbers helps them understand the context of the work before they learn how to generate profit lines. I also think it can help students notice that the edges of the feasible region will be the most profitable and given them a sense of why this is so. I might give them five minutes or so to guess-and-check and then share out their predictions.
Next, I let students get to work on Pet Sitters Revisited in small groups or pairs to find the maximum profit. I find it is almost always necessary to guide this part of the work but asking students to first come up with three combinations that would make them $120. I ask them to plot these three points in one color. Then I ask students to find three more points that would make them $180. I again ask them to plot these three points in a different color. Then I ask students what they notice about the points. Students generally notice right away that the points are collinear. It may take them longer to notice that the two lines are also parallel.
Depending on where students are with this activity, I might work with small groups or bring everyone back together to discuss what's happening with the profit. I want to elicit from students what the profit equation would be, and have them give us an idea about why the profit lines are parallel. From there, we turn our attention to the highest profit. If we know that profit lines are parallel, how can we find the maximum profit in the feasible region? I am looking for students to see that if we slide the line over to the edge of the feasible region, the final place the line touches the region should be the maximum profit.
In this case, this point exists where the Space and Start Up Costs intersect. Here is where I want students to remember their work with Systems of Equations. I'll ask students what that point is. Students will be able to guess at the point but will see that we cannot tell what its exact coordinates are from the graph. I then ask students how we can find out exactly what that point is. Hopefully, someone will volunteer the idea of using system of equations for the boundary lines to find the x and y coordinates. Students will then solve on their own to find the exact point that is on both of those lines.
I like the way the numbers end up in the system of equations because the system can be solved using substitution or elimination. I look for students to present how they have solved the system both ways and this can be a good review for students leading up to the end of the unit. Because the answer comes out to a fractional amount of cats and dogs we talk about how this is the maximum profit in terms of the feasible region but won't work out for Carlos and Clarita. I ask students to figure out what whole number of cats and dogs will make the most money knowing what they know now.
Depending on the pace of class, this task has a nice opportunity built into it to get students writing about math. If I have time, or want to assign a graded homework assignment, I tell students to imagine they have been hired to work for Carlos and Clarita as business consultants. Carlos and Clarita want to know what combination of cats and dogs they should board and the students now know all the math they need to advise them in their business venture. I ask students to write a business report to Carlos and Clarita explaining their answers. This gives students a nice opportunity to summarize what they have learned and to justify their answer with mathematical proof.
I remind students that they are trying “convince” the business owners that their answer gives them the most profit. This is a great opportunity for students to practice Mathematical Practice 3: Construct viable arguments and critique the reasoning of others. Students can include in their report a “logical progression of statements” that explores the process they went through to develop their answer. They can also “justify their own conclusions and communicate them to others.”
I find students enjoy writing in a “business tone.” I encourage students to have fun with the assignment and really get into the role of business consultant. There are opportunities here to work across the curriculum with an English teacher about writing for different audiences.
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