### Things my precalculus students said in class yesterday

- “So we’re just supposed to KNOW that 7x^2 is like 7 TIMES x^2?”
- “There’s no solution [to the equation x^2=3.52534231], because you can’t take the square root of a
*decimal*.” - “Okay, I see how you got 2x=1.46, but how did you get x=0.73 from that?”
- “Last week’s test was totally unfair - you never gave us a homework problem where we had to graph a rational function where the denominator was like x^2-4x - it was always like x^2-4x plus some NUMBER.”
- “Can we just plug those numbers into our calculators instead? Because, like, I’m not very good with fractions.”
- “Miss, I think you made a mistake grading my test. I had (x(x-2)+3x(x+4))/((x-2)(x+4)) , and then I cancelled out the x-2 and the x+4 to get x+3x=4x, but you marked that wrong.”

These are my students who desperately need grades of at least C+ in my class in order to be permitted to take calculus next year. What’s sad is that many of them seem to be under the impression that I am the one thing standing between them and success in university math classes - if only I’d just make my tests easier, they could advance, and surely their calculus prof next year won’t expect them to be able to solve any math problems that they haven’t seen word for word before.

Ideally I’d have taught them well enough that they’d have earned their C+’s and above - but I’m just not a good enough teacher that I can take kids who can’t do grade school math and mold capable university pupils out of them in one semester. As far as I know, there’s only one teacher who was ever able to do that, and they made a movie about him. So, having given up on educating my weakest students, I’ve committed myself to doing the next best thing: ensuring that they don’t get the grades to take calculus next term.

And this is what I’m keeping in mind in designing the final exam. I’m testing them on the material - graphing functions, solving various flavours of equations, factoring polynomials - but I’m specifically designing the questions so as to trip up everyone who hasn’t learned grade 9 math. I *could* ask them to complete the square of a quadratic with integer coefficients, but I *like* fractions. I *could* ask them to solve a problem that reduces to finding the positive and negative square roots of 7 - but I prefer the number 7.5. I could give them a problem in which they’re given the distance in kilometers and the time in hours - but I prefer minutes. I *could* ask them to sketch the graph of f(x)=2^(x+3), but I rather like the graph of g(x)=-2^x - which many calculators interpret as g(x)=(-2)^x.

The students who belong in university math classes will do fine; the exam is easier than my tests. Those who memorized the problems I did in class without bothering to learn any grade school math will do miserably.

I’m impressed that you stick to realistic grading, even when you face having most of your students fail (which certainly wouldn’t be fun for a teacher I’d imagine). I hope you can handle the abuse that might end up coming your way.

I expect most of my students to pass; I anticipate a failure rate of around 1/3, and another 1/3 I expect to pass, but get less than the C+’s they need in order to take calculus. And a fair number of my students have dropped the course. There might, alas, also be some scaling of the final grades at the end of the term. I’d rather not, but recall that I am competing with the Nice Teacher whose class average is 70% despite the fact that his kids can’t add fractions, either.

And I have a method for handling the abuse that comes my way: forward them to the elephantine bureaucracy that is filing a complaint with the department. Have them fill out forms appealing their grades, and be sent to the department head, who has already told me that he supports me. Mwah.

I object to the example of g(x). I read -2^x as the integer -2 raised to the power of x, which is exactly what you don’t want them to read it as. Asking them to know that you actually meant -1 times the integer 2 raised to the power of x is like asking them to realize that by 6^x you meant 2 times 3 to the power of x.

Here there be trolls.

Like l337n00b, many bright ESL students have problems with early grade-school math because of the language barrier. If only more educators would take the time to translate some of the key lessons into l337, more students like l337noob wouldn’t be left behind.

For example, BEDMAS, the dumb but apparently useful acronym used as a memory aid for the order of operations, might look like ths in l337:

8 3 1) |\/| 4 5

Maybe if more effort was spent finding multilingual teaching assistants for classrooms, we could stem the sliding performance of the truly l337.

Out of curiosity, what does the graph of f(x)=(-2)^x look like? I’m a little embarrased to admit that I’m not able to solve this without digging out math textbooks and spending a half hour on it. I mean, it seems like it has to oscillate up and down, because f(x) will vary between positive and negative as x changes (right?), but what is throwing me is: without a calculator, how do you find what f(x) will be for non-integer values of x, e.g., (-2) raised to the 3.14 power, or 4.28 power, etc.

y=(-2)^x appears to give you a nice increasing spiral in complex-y space.

I have a program that gives me nice graphs of real part, imaginary part, magnitude, and angle-from-positive-real-axis, but I can’t figure out how to get it to actually generate a single graph of the complex-valued function.

Pretty pictures are at http://www.eskimo.com/~dj3vande/unpublished/neg2-powers/ .

And to answer wes’s actual question, rational powers of a number (x^(p/q)) are defined as the qth root of x^p, and irrational powers are defined as a limit of rational powers.

So for (-2)^3.14, 3.14 is 157/50, so it’s (50\/-2)^157.

(Note that this means that noninteger powers with even q of negative numbers have nonzero imaginary parts. I’m actually surprised that the graphs I made worked out as nicely as they did, given that I’d've expected (without having trained my intuition appropriately) a less clean distribution of even-q and odd-q powers.)