Tuesday, October 03, 2006

Next Lesson: Representational Fluency

The composition lesson went fairly well, though I could have used about 5 more minutes to finish the lecture. Students seemed comfortable with finding things like f(g(2)), but I definitely lost many of them when I tried to finish with finding f(g(x)) as an expression. We'll definitely need to review this a couple of times.

In the next lesson, after reviewing the homework, the Do Now will focus on practicing these concepts (as well as reviewing absolute value inequalities).

Then, I'm going to squeeze in a mini-lesson on using the TI-83+ to graph inequalities and absolute value inequalities. In this case, I mean graphing things like y = (x < 5), where it returns 1 if true and 0 if false. This is a pretty cool way to generate a graph that looks like the number lines we shade by hand. It can solve absolute value problems the same way: y = (abs(2x+1) > 3). After learning this technique, students will check their answers from the Do Now by graphing. I'm hoping that this doesn't take too much time...

Finally, I will give students a handout on Representational Fluency. I've learned from my experiences teaching AP Calculus in previous years how important it is for students to be able to move comfortably between equations, graphs, tables, verbal descriptions, arrow mappings, etc - especially when it comes to the concept of functions. This sheet focuses on graph and table representations of functions - students have to figure out things like f(2) and f(g(-2)) from these representations. I'll post this on ILoveMath .

I'm trying to decide if I should do some work on function decomposition. This is clearly a skill that students will need for Calculus (i.e. working with the chain rule). I also think that decomposing functions might help them understand better what composing functions really means. But I'm also worried about overloading them, and I wonder if they need more time to digest function notation and composition first. It's not in the standards, as far as I can tell, so I wonder if students are expected to understand this idea before getting to Calculus. Any ideas?


Darren said...

Graphing calculators are the scourge of foundational mathematics. I've posted twice about them--both posts are linked in this week's and last week's Carnival of Educations.

Dan Wekselgreene said...

I want my students to be able to do graphical analysis without the calculator. But I also want them to be able to use it as a tool when it's appropriate.

For example, they should be able to solve polynomial inequalities by factoring, making a sign chart, etc. But, if given a higher degree polynomial to factor, I'd want them to graph it first to look for x-intercepts that could be tested with synthetic division (instead of using the rational roots theorem to test a dozen different values).

It's definitely hard to figure out how to make the calculator an aid to learning instead of a crutch that prevents deeper understanding. I don't let my students use their calculators on many of the tests, to ensure that they can do the thinking on their own.

I could be wrong - this is only my second year teaching Algebra 2... I'll just have to see how it goes.

Anonymous said...

For the calculus question....

I'm teaching honors calculus this year and one of the things that we did cover in our review section was function decomposition. They were pretty good at it....but I'm not sure my Algebra 2 students would be. It might require a bit more mathematical maturity than they have. I would assume that it would be covered in a subsequent class--such as pre-calc. (I don't know what your school's sequence is, but that's my two cents!)

Dan Wekselgreene said...

We're still a young school (relatively) - we're in our 7th year of operation right now. We started with a class of 9th graders only, and grew from there. Our curriculum is still being developed - by now, Algebra 1 and Numeracy 1 are relatively well set up and stable, but the other classes have been revamped each year as we've learned more about who our students are and what they need. Also, we switched the order of Geometry and Algebra 2 last year, which has required significant reworking of those curricula.

Pre-calc is being rethought this year, as we now have the first juniors who went through the Algebra 2 honors track in the class. As soon as we find some time, the teacher of that class and I will sit down and really look at the pre-calc objectives - with an eye to what they should already know from Algebra 2, what they need to know for the state standardized test this year, and what will best prepare them for Calculus next year.

I guess my point is that we don't know exactly what is in pre-calc for sure yet... The more I can front-load in Algebra 2, the better it will be for them next year. As long as I don't overload them, of course.

Darren said...

I find the Algebra 2 standards to be fairly "aggressive", like drinking water from a fire hose. How you find time to add in topics that are not in the standards, I don't know but would be most interested in learning.