I begin today's class by letting students know that today we will take a look at all of our Pet Sitters constraints in one picture. I find students are usually pretty enthusiastic about this task and are motivated to get to a final set of points that will satisfy all of the inequalities simultaneously. I start class by reading through the first four questions of All For One, One For All.
Though I like the outcome of the set of tasks this activity guides students through, I don't necessarily follow it to the letter. Rather than having students find points that satisfy and don't satisfy for the first two graphs, I focus on having students color a set of points that satisfies the first inequality and then on the same set of axes, using a different color, color the set of points that satisfies the second inequality. I find this change in instruction to be helpful to students and a more intuitive way to approach the task since our task has been on graphing now, rather than finding specific points.
Next students get to work in small groups or pairs. Ultimately, I want them to focus on graphing all four inequality boundary lines on the same set of axes and think about where the shading needs to be. They can either follow the activity and find the points first and then complete the graph or make the graph first and then find the points that answer the questions. The main point, of course, is for students to understand clearly that a point can satisfy more than one constraint simultaneously or satisfy some of the constraints but not all. Most students will intuitively get the idea of the intersection of all the shading to be the points that satisfy all four constraints.
I find Questions #8 and #9 to be confusing to students given the progression of the work. Question 8 wants students to create a feasible region on a coordinate graph that shows all four quadrants, while Question 9 wants to bring in the idea of only having a positive number of cats and dogs and restricting the feasible region to the first quadrant. I find having students work to find the feasible region in the context of the situation to make the most sense and then address the term "half plane" again the discussion section of the lesson.
Once each student has a complete graph of the feasible region, we discuss the finished graph together at the board. Depending on time, I will ask one pair or group of students to present their graph. I ask them to explain how they found the overlapping shading areas, going through each of the inequalities. As we discuss, I try to elicit key ideas like: What does an individual line represent? Why do we shade on one side of the line and not the other? What does the final graph tell you?
I follow the PowerPoint to address other key issues. We address the first quadrant issue and talk about how if we were just talking about inequalities without a context, each shaded area would continue and would be a half plane. Now is also a good time to talk about points that are not whole numbers and if they make sense in the context of the problem versus with the inequalities. I introduce the term "feasible region" and we talk about what the word feasible means.
In closing, I ask students to reflect on today's work with an exit ticket. The exit ticket prompt is: How does seeing this combined graph compare to reading the verbal description of the Pet Sitters situation?
I really want to drive home the importance of being able to get a visual picture of all the points that satisfy the four constraints at one time. I might remind students of the work we did in the first lesson if this unit where they played around with numbers and tried to find combinations that would work for all the constraints. I try to show them the beauty of the graph doing all of that work for us, and showing us all the points at once!
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