Up next, students will learn to evaluate composite functions. First, they will do a worksheet with function notation practice problems. I set the problems up in groups of 4 as follows: f(2), f(-4), f(a), f(2x - 5). I think that sneaking up this way on the idea of plugging in an expression for x will help students better understand how to evaluate f(g(x)) as an expression. I remember having a lot of difficulty when I first learned this concept, and this method helps make it clearer for me anyway...
Then, we'll use this dual lens model. I hope it will help them visualize what "the output of f is the input of g" means. After the model, we'll go through the concept of composing functions, and do some example problems together.
In an upcoming class, I will give students a chance to do function composition when given graphs or tables instead of equations.
Thinking Thursday: Explain a Math Trick
2 days ago
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5 comments:
Not only do kids have trouble with function composition, the young adults in introductory abstract algebra classes seem to have the worst time with it. Luckily you are working on maps from R to R, so the lens approach will probably serve you well.
One thing that's worth doing is asking them to figure out whether or not map composition is associative, and then commutative.
Thanks for your response. For commutativity, they can switch the order of the lenses and see if the final projection is the same or not. This should help them understand why they get different results when switching the order of the functions.
But I'm not clear on what you mean by associativity. In the mapping model, you have to start at the initial input and follow the order of the lenses. If you try to start somewhere else, there will be no data to use.
Could you give an example of what you mean, because it sounds interesting and I don't think I'm getting it..
Not the same anonymous, but I think what s/he means is to take three lenses, call them f,g,h, and see that the mapping
x -> f(x) -> g(f(x)) -> h(g(f(x)))
describes both (h*(g*f))(x) and ((h*g)*f)(x), where * represents composition.
Have you tried without the lens? As a student, I found the lenses distracting, and have shied away from using them.
I've done ok (but admittedly with a very different population).
I use the lens just for the first day as a way to introduce the concept. We also look at composition in other ways. I'm not sure yet if it is helpful or not, but I think different students will remember different methaphors for the same concept. On the first quiz, I ask them to use the lens model, but I don't assess them on it by the time we get to the unit test or final exam. At that point, I just care if they have mastered the skill of composition.
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