lsquared on what you must know to take calculus

(lsquared couldn't get this comment to post on Blogger):

Yes, you need to be very good at algebra. Think factoring, equation solving and complex fractions.

You also need to be quite good at trigonometry: knowing right triangle trigonometry is sufficient for almost everything except calculus--for calculus you need to know how to solve equations that have trig functions in them, and you need to know your trig formulas (the double angle formulas are particularly useful). You don't actually do any of the "verify this trig formula" stuff in calculus, but those "verify the trig formula" problems are very useful for teaching the algebraic trigonometry you use in calculus. Oh--and radians. I'm morally convinced that radians were invented to make trig functions work for calculus. Degrees are better for everything else, but in calculus, you have to have radians.

You also need to know logs: solving exponential equations with logs, solving log equations with exponents, and manipulating logarithms.

Back to rational functions. Be able to solve rational equations, simplify rational expressions that are not equations, and graph (by hand, not by calculator) rational functions (using properties of the rational function to graph, not by plotting points) (I may be unusually picky about this).

Sequences and series, including finding sums using the formulas for arithmetic and geometric sequences. Also the binomial theorem.

You don't need matrices, though you do need to be able to solve simultaneous linear equations. If there's a section in the pre-calc book on find partial fraction decompositions, that's directly a calc algorithm.

I think that might be everything you need for pre-calc.

This was an excellent exercise for me. I’m going to have to keep a copy of this to use when counseling students about whether they are ready for calculus or not. Indeed, the homeschooled kid next door wants to get into my calc class in the fall, and I shall be handing this to him I expect. Good luck finding a good class--I think the community college might be a good choice (assuming that they have a pre-calc course).

lsquared on area conversions

lsquared writes:

I'm really wishing for a national curriculum right now--or at least a series of textbooks that goes K-10 instead of K-6 and another for 6-8.

I'm doing math with a friend's daughter. She's homeschooling, so I get to pick the book, and we're using a Singapore text. Our most recent section is converting units of area: cm² to m² and that sort of thing. So, I thought to myself, this is the US, we should also do problems with in² and ft²--I'll go look through the elementary and middle school texts in my library (I have 2-3 full series of each sitting outside my office door at work). Guess what? None of the books teach the topic at all. Aargh! Everyone is assuming someone else is teaching it, and no one is.

The first time I saw anything about area conversions was in the Saxon Math books I used to teach myself math a few years ago.

I learned inch-to-centimeter and inch-to-foot-to-yard conversions in school, but I learned nothing about area or volume conversions, and I continue to find volume conversions slightly confusing.

Speaking of Saxon Math, I've been thinking I need to get back to my books. I had almost finished the second book in the high school series when I was diverted to whatever I was diverted to. At the time, I was finding logs difficult to deal with in the "shuffled" organization of Saxon Math, especially since I wasn't studying every day.

Speaking of logs, I was emailing with Barry G. earlier today and I compiled a list of all the topics I was never, ever exposed to in high school math or college statistics.

I'll post tomorrow when I'm at my other computer.

The list is too long to remember off the top of my head.

lsquared on project based learning

I just had an epiphany. I think I get where these project based peoploe are coming from. When kids complain about learning something hard, and what the kids say is "when am I going to need this in the real world" these people believe that the kids are really articulating what the problem is. Wow.

In my experience (and I know there are teachers who agree with me, as well as others who don't) the complaint "when am I going to need this in the real world"" is secret code for "I don't want to sound stupid, or like this might be my fault, but I don't understand this". I have found that if you attack the not-understanding, the complaining tends to go away. I know from sad experience, that if you attack the "real world" part of the complaint without addressing the not-understanding (and while keeping the same content), the complaints do not go away.

I think the project based people may have come up with a third alternative: if you both address the "real world" part, and you seriously reduce the content, then all the kids who tend to complain cease complaining. Whadaya think?

l squared on "collect and correct"

I teach college, so I feel a little as though I shouldn't have to collect homework...but, of course, I do. If you don't collect it, they don't do it. Collecting the homework may be the single most effective thing I do to improve student learning. I don't check all the problems, just 2-4 representative ones, but it still makes a big difference.

Steve H on collecting and correcting