Rational applications are worth spending more time on than absolute value or square root ones, I find, because they can harness so much algebra AND reality in one single problem. I spent two unhappy years working as a cost accountant, in which this type of problem came up a lot, and it was the only fun part of the job. So I'm giving my kids an assignment that's only on rational application problems, and in order to prepare them, I showed them 3 type of situations that they could expect:
1. The Constant Product situation:
This is the simplest one, in which the two variables involved always multiply to give the same result. They may notice this from inspecting a table of values:
Or by reading between the lines:
In either case, the relationship between the variables can be expressed as xy = a, or y = a/x, in which a is the constant product.
2. The cost-per-person situation:
In which there are two types of costs involved - a fixed and a variable (which I happen to know is how they're referred to in the cost-accounting world):
The fixed is the $3000, and the variable is the $750. The calculation that naturally occurs to students to make is the cost per person, and coming up with a rule for cost per person is fairly easy for most.
The fact that no person can ever pay exactly $750, but always slightly more, makes the concept of asymptote very real. This is a really nice intersection between their intuition and the very abstract concept of asymptotes.
Next level up would be a situation that involves something per something other than cost and people, but which has both a fixed and a variable quantity.
3. The Class Homographic
The name for which only occurred to me moments before I went into class - in which two linear quantities are being divided, so that the resulting function is rational, but its asymptotes would need to be discovered via long division:
Now this is where some really beautiful intersection between algebra and reality happens. They do the long division, which reveals the asymptotes, one of which is at 1000. So is that truly the maximum? Time to discuss. Not to mention that this is a great argument for doing long division in the first place, because it reveals so much about the nature of the function.
Next level up is a situation in which the homographic rule is not given, but two linear relations are, and they need to be divided, for example, the concentration of a solution in which the amount of solute and the amount of solution are both changing linearly.
How will this help them?
So often I hear - Mrs. - I don't even know where to start. Well, we spent the rest of the time looking at other problems and identifying which of the three types each was - which can help point the way.. Which I hope will help them with their assignment - we'll see on Friday when it's due.
At any rate, I just love how Classic Homographic sounds!