The key to this lesson is helping students develop an understanding of the two different methods they might use to graph inequalities. Depending on how the previous lesson finished, I might have identified students who used either Carlos' or Clarita's method already and have them highlight their approach. I find using an actual student in my class to be more effective than the worksheet, but if no one has developed a specific approach, the worksheet examples work just fine. See:
I start class by either highlighting student approaches for the previous class or reading through today's task together. I let students know that every time we graph an inequality, we will not have to graph all three types of points in different colors (like we did in the last lesson). Or task for today is to understand some methods that can help us graph inequalities more efficiently, while still maintaining our conceptual understanding.
I like to begin this task by having different students volunteer to demonstrate Carlos' and Clarita's methods at the board. This year, some of my students found slope intercept form in our previous lesson, but did not start from a table. This part of Carlos' method may be new to them and also might have with understanding where the rate of change comes from for this problem. I try to elicit from students the key piece that for every dog added, they have room for 4 less cats (It will be helpful here if students have had dogs on the y axis for the previous lessons). It is also important that students start with 0 dogs and see how much space they have for cats. This rate of change is easier to see in the Space constraint and a little harder to see in the Start Up Costs constraint which comes next.
Clarita's method may seem a little more straight forward to students as she is accounting for how much space cats and dogs use up at the same time in a standard form equation. The key work here comes in transforming her equation into Carlos' and then back again to see the equivalency of the two.
After we look together at each method, I let students work in small groups on transforming the equations. They then go on to try each method again on the Start Up Costs constraint.
Once most students are through Question 4, I gather everyone back together for a group discussion. I will likely spend a fair amount of time asking students how they used Carlos' method to get an equation in slope-intercept form. Students may have difficulty seeing that for every 2 dogs added, they can afford 5 less cats. Many students will say that for every additional dog, they can afford 2.5 less cats. This makes for an interesting discussion point around the math context vs. the real world context.
Again, we focus on the equivalence of the two equations, and the math steps we need to take to go from one equation to the other. Students often struggle to transform the slope intercept form into the standard form that looks exactly like Clarita's equation. It may be strange to them to multiply both sides of the equation by 32, even though they already have a standard form equation. I talk about how we divided both sides of the equation by 32 to go in the other direction, so we are simply taking the opposite steps.
Lastly, and perhaps most importantly, we look at how we might approach graphing differently depending on which of these equivalent equations we prefer. Again, I try to elicit student strategies from our previous lesson. Some students will have noticed they can over over and down a fixed amount each time to find the next point that is exactly on the boundary line. Other students will have thought about housing only cats or only dogs and knowing both of those points were on the boundary. For my class, it is important to solidify these two methods as useful approaches to graphing inequalities in two variables. I might summarize at the end of class, saying something like, "ok, next time we want to graph an inequality like this, we can either think about the x-intercepts (maximizing one variable) or we can think about slope-intercept form."
Students generally feel strongly about which method they prefer. I like to do a reflection at the end of class where they share which method they like and why. Depending on time, they could do a written exit ticket, or we might do an activity where they get up and move to a side of the room depending on their method. Different representatives from each group can share out their reasoning.
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