## Sunday, September 09, 2007

### Piecewise Functions

The following demonstrates why I don't do "formal" lesson plans before my lesson. Well, this and the fact that I'm a last-minute (post-last-minute?) person. I eventually do get the plans down on paper, since I use them the following year to see what works and doesn't work.

I was introducing piecewise functions to my precalculus class. I started with an example of a power company in town and suggested that they may offer incentives to people who use less electricity, so if you use less than or equal to 500 kW per month, they charge you 10 cents per kW, and if you use more than 500 kW per month, they charge you 15 cents per kW. I then gave them the disclaimer that I had no idea how much a kW was and if this was at all reasonable. (future assignment for the class and me: find out how many minutes of light that is). Then to make sure they got it, I asked them to put on paper in calculator-ready form, how much my bill would be if I used 600 kW last month. I wasn't even thinking about the ambiguity, but a great question came up: is it 15 cents for everything since you went over, or is it only 15 cents for the 100 kW over? (we went with the 2nd option).

Then we started drawing the graph, and decided that there would be an initial service fee of \$20 even if you didn't use any kW. I asked them to discuss in their groups the axes labels and what the left part of the graph would look like (before 500 kW). We also did the right side. Now here is where in the past I would have breezed through this and shown how it was piecewise and then went on with another "math-class" piecewise function to work with. But a student pipes up with, "ooh! I know the equation of the left part of the graph". This got us on the path of finding the 2 line equations, and seeing what and why the slopes are what they are, and discussing point-slope form (the forgotten quiet nice guy in the class of line equations) vs. the ever-popular-player-guy "slope-intercept" form of the line equation. We set them up and discussed that those equations go on forever, so how do we show someone where it's restricted to. We then got it into an f(x) piecewise format and worked with finding f(32) and f(701) and describing what they mean.

Yea! For unexpected improvements to my as-yet-unwritten lesson plans.

1. Anonymous3:36 PM

Great idea! I'm stealing this as we're starting piecewise functions soon!

I too just experienced "reintroducing" the point slope form. Not sure why the students don't seem to like it.

2. Anonymous3:51 PM

Yea! I don't exactly remember, but I believe you can get it done in one class period (48 minutes or so). I had block scheduling, and we did some review before hand.

Good luck,

3. i tend to play down the point-slope form
to the extent of: "it's there in the text
and if you know it already
or want to learn it, by all means
feel free".

but that's just because my students--
college students with weak math backgrounds--
struggle hard enough with slope-intercept
that the added burden of a special-purpose
formula doesn't seem worth it.
better to spend more time on getting
*one* thing understood well
than just pile on confusion
in the name of "coverage".

obviously, with more capable students
(or more time in the quarter ...
a semester's worth, say ...)
i'd very likely act differently.

one issue that arises frequently:
"x" and "y" *stay* "x" and "y"
while "x_1" and "y_1" are replaced
with numerical values.
this shouldn't be confusing,
but it is.

i like to capitalize --
yes, believe it or not --
constants (e.g. y = Mx+B)
and wish this were an industry standard.

4. Anonymous6:27 PM

I like your idea of capitalizing the constants or "things" that are replaced with numbers. That's a good visual. I think I'll start adding that to my repetoire. Thanks.

I have to get my kids ready for next year and the AP exam where some of the answers are given in point slope form, so that's why I'm stressing it.

5. Anonymous7:07 AM

Thank you sooo much!! You just saved my grade!!!

6. Great idea! I'm excited to try this in my algebra II class. Thanks for the idea.

7. pmorourke9:31 AM

My middle range students have been struggling with the Algebra basics, like point-slope and such. I think they began to understand when I explained that the point-slope form comes from the slope formula (which most of them know REALLY well). I showed them that in the slope formula, if you multiply each side by the (x2-x1), you get m(x2-x1)=y2-y1. Since there are three parts to the slope formula (one of which is always missing): slope, point1 (x1,y1), and point2 (x2,y2), when we have two points we use m = (y2-y1)/(x2-x1) because we are missing the m. When we have one point and a slope, the second point is missing that's why it's just x & y not x2 & y2.

That seemed to clear up the point-slope for a bunch more of them.

As for the piecewise, THANK YOU! I have been looking for more examples to explain to them.