It wasn’t anything innovative, and could use a lot of work to refine it, but maybe you’ll find something you can work with? I link to a couple more sheets I created above if you want to see what came after… how I introduced end behavior and horizontal asymptotes, and how I introduced graphing. View this document on Scribd Taking Things Further Out of this came nice discussions of holes and vertical asymptotes. I started with the admonition that no equations should be used and everything needed to be thought of graphically if it was going to be an effective exercise. It took my kids about a class period to do this first packet (they finished the rest up at home). I felt wildly successful with this when it came to the introductory materials for rational functions. My goal - gentle reader, to remind you - is to do very little explaining and have the kids figure as much out on their own as they can. doc form My Awesome IntroductionĪlthough there are definitely ways I can improve this, here is how I started off rational functions. ![]() Of course once this is done, you can throw in the other stuff… Thus the output is getting bigger and bigger and bigger.)Īnd of course kids need to understand how the third and fourth terms are (graphically) creating x-intercepts in the final graph. ![]() (The explanation I’m looking for says that at x-values closer and closer to 1, the denominator is getting smaller and smaller, but the numerator is staying at 1. Not just because we’re dividing by zero so things go crazy and explode, but being able to articulate precisely why the function blows up. Kids need to understand that the second term is creating the function to blow up in a certain way at. I expect them to say that for any value other than 3, the fraction will evaluate to be 1 (thus it will not affect the rest of the multiplication), but when is 3, we clearly get something undefined. Kids need to understand what the first term is doing - not just “as a rule” but conceptually/graphically. So for example, when we see the rational function listed above, we find it easier to view it as: I want them to see how they are constructed term by term by term. Second, I want kids to build up rational functions, instead of breaking them down. ![]() To the point where on the first day, students do not see a single equation, and are asked (entreated!) not to write a single equation down. To counter this, I made two major changes to how I approach/introduce rational functions this year.įirst, initially, I focus heavily on the graphical side of things. The answers I have heard from kids are procedural, and rarely have any deep stuff underneath. ![]() You might recognize students don’t know what a hole truly is and why it appears in a graph… or they might not understand why vertical asymptotes appear… at least not on a deep level. And every time I dug a little deeper to see what they truly understood about these equations, it became clear that a procedure to “solve” these questions was taking the place of understanding what was going on. … and then spend the rest of the time asking kids some questions, like “what’s the x-intercept(s)?” “what’s the y-intercepts?” “what’s the vertical asymptotes?” And so from one big equation, you pull out all this individual stuff…īut from what I’ve seen when most kids approach rational equations, it is all very procedural. My impression is that most people introduce rational functions by showing something like…
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |