Part I
“Homework: Inverse Variation”
That’s right, no homework on direct variation—it’s time to develop the second one. They should be able to do it pretty well on their own, in analogy to what happened in class. But you will spend a fair amount of the next day debriefing them on inverse variation, just as you did on direct. The defining property is that when the independent variable doubles, the dependent variable chops in half. Again, it is true to say “when one goes up, the other goes down”—but it is not enough. 1/x2 has that property, and so does 10-x, and neither of those is inverse variation.
Examples are a bit harder to think of, off the top of your head. But there is an easy and systematic way to find them. I always warn the students that I will ask for an example of inverse variation on the test, and then (now that I have their attention) I explain to them how to do it. Inverse variation is y=k/x (graphs as a hyperbola). This equation can be rewritten as xy=k. This is useful for two reasons. First, it gives you the ability to spot inverse variation—if the product is always roughly the same, it’s inverse. Second, it gives you the ability to generate inverse variation problems, by thinking of any time that two things multiply to give a third thing, and then holding that third thing constant.
For instance: the number of test questions I have to grade is the number of students, times the number of questions on the test. That’s obvious, right?
If I want to turn that into a direct variation problem, I hold one of the two multiplying variables constant. What do I mean “hold it constant?” I mean, pick a number. For instance, suppose there are twenty students in my class. Now the dependent variable (number of questions I have to grade) varies directly with the independent variable (number of questions I put on the test).
On the other hand, if I want to turn that exact same scenario into an inverse variation problem, I hold the big variable constant. For instance, suppose I know that I am only capable of grading 200 problems in a night. So I have to decide how many questions to put on the test, based on how many students I have. You see? Double the number of students, and the number of questions on the test drops in half.
Algebraically, if I call t the number of questions on the test, s the number of students, and g the number of questions I have to grade, then g=ts. In the first case, I set s=20 so I had the direct variation equation g=20t. In the second case I set g=200 so I had the inverse variation equation t=200/s.
I explain all that to my class, slowly and carefully. They need to know it because the actual test I give them will have a question where they have to make up an inverse variation problem, and I always tell them so. That question, if nothing else, will come as no surprise at all.
Part II
“Homework: Direct and Inverse Variation”
This is on the long side, and introduces a few new ideas: the idea of being proportional to the square (or square root) of a variable, and the idea of dependence on multiple variables. So you should ideally hand it out in the middle of class, so they have time to work on it before the homework—and be prepared to spend a lot of time going over it the next day.
"This is the "teacher's guide" book in Kenny Felder's "Advanced Algebra II" series. This text was created with a focus on 'doing' and 'understanding' algebra concepts rather than simply hearing […]"