It’s all over but the grading for me and my mind’s elsewhere. I’m doing a 103 and a 153 next quarter and have started work on those. Most of the blogging for the 148 classes now ending was in VME.
I’d like to thank my life-support system, Madonn’. Thanks for letting me hang on. Also all my students and anyone who commented here or even ever seriously considered it. Without you, no blog.
The world of mathblogs is expanding vigorously at all levels. It’s a great time to be alive.
JD’s Carnival of π.
Let P_1 = (x_1, y_1) and P_2 = (x_2, y_2). The distance between P_1 and P_2 is then
Humans as opposed to computers would do well to understand that this is the square root of the sum of squares because of the Pythagorean Theorem; my habit in in-class computations is to write out the radical with the parens and superscripts first and fill in the differences after reminding everybody of this connection. Every now and then somebody gets it.
There’s a “formula” given in the same section for the midpoint of the line segment connecting P_1 to P_2: but I consider it foolishness to think of this as a formula. Average the x co-ordinates for the “new” x; average the y’s for the y.
One immediate consequence of the distance formula (or of the Pythagorean Theorem directly…) is that x^2 + y^2 = 1 is the equation of the unit circle; shifts-and-scalings lead to the circle equation
(for the circle with center (H,K) and radius R). One is prepared either to put an appropriate polynomial in x and y into this form (by completing the square) and “read off” the center and radius, or (what is much easier) to write out the appropriate equation for a given center and radius.
The financial section of the exp-and-log chapter is fraught with formulii. And typing ‘em out is no walk in the park, as I might have mentioned from time to time. But let’s see. There’s the old 
Better still, we’re on the net. Google “precalculus formula sheet” and look around. Finding the right ones from the best few you find sounds like a darn good exercise. I should have started here.
I posted some remarks during the review for Exam II (about rational functions and inverse functions) in the old blog. Then there’s this post about the “simplest” Rational Function, some discussion of the Factor Theorem, the review problems of 1/30, and a summary of Transformations.
“Quadratic Formula Lore” was one of my most popular posts. I don’t know whether my decision about Transformation notation counts as “review” (since these notations are “unofficial” for the course and won’t Be On The Test). I made up some hints about using the net and the library. Near the beginning, before I hit my stride, I posted this short rant about defining “relation”.
Let the record show that this was a very productive quarter for me (and that I’m proud of it). What the hell, I’m feeling generous… let the record even show that I had the time of my life doing it.
Posting about the Final will cease by Monday at 8:00 AM (so everybody gets the same info from here). Go do some math problems.
Barry Garelick’s (PDF) “Discovery learning in math: Exercises versus problems” at the Third Education Group’s Review (I can’t quite read this site… there’s a sidebar covering up some text). Spotted in KTM.
Old fave Gilbert Highet noticed at KTM.
Dollars and sense at Social Maths.
Problem solved (probably), at Gowers’s.
Sez here I’m doing a 103 next quarter (8:00—9:50 AM MW) and a 153 (1:00—3:15). I’ve blogged about 153 before (badly) and probably will again real soon (it’s not for me to say how well).
The final’s soon. Somewhat to my surprise, the co-ordinators haven’t prepared a Practice Test (they recently took over the whole final exam; in the old regime one had to write half the Final for one’s own sections). Which, believe me, makes perfect sense: lots of good teachers consider this kind of thing a bad idea (the whole point of a final exam on their model being to cover the entire course).
So it’s on me and I don’t care if it’s a bad idea just now: my students have grown to expect such things and I hate to disappoint them. At the same time, I’ll preserve a little of the roll-your-own flavor: part of the point is to get students to read the book, not just to consider certain problem types. Anywhow, from top to bottom, we did, and will soon do again, problems like these.
Reflections. Given a graph, sketch its various reflections (origin, x-axis, y-axis); keep track of one or more points (given as ordered pairs); show algebraically that a function is “even” or “odd”.
Information From A Graph.Given the graph of a function, write down its Domain, its Range, its local Maxima or Minima (collectively Extrema; know the singular forms of the three plural nouns just mentioned… and in general, for heck sake, learn these words I keep slinging around… the point isn’t just to be able to do the calculations, but to talk about them…), the sets (subsets of the Reals, natch—specifically, of the domain… so we’re talking about x co-ordinates… behave appropriately here I beg of you) on which the given function is increasing or decreasing (or, I suppose, constant, though this strikes me as somewhat affected and I typically distain to give its definition), the sets on which it’s positive or negative, its various intercepts, and whatever I’ve forgotten to mention. This is stuff you mostly all knew before you even took 148 so just get the bloody notations right: what’s a point, what’s a set of x co-ordinates, what have you. Because if you can’t write this stuff down just by looking at the graph and be essentially correct about almost all of it, you aren’t ready to “talk about” graphs of functions…
Piecewise-defined Functions. Graph it if we give you the algebra; write down the code if we give you the graph.
Transformations. We did a lot of this and if I had my way we’d've done even more. Be able to name and calculate “shifts” (I actually prefer to call these “translations”), “shrinks” and “stretches” (“scalings”) as applied to algebraic “formulas” and to graphs (and their individual points).
Quadratic Functions. We looked in some detail at some Optimize-the-Area problems and some Optimize-the-Revenue problems, so prepare for these: in either case, the point is to set up the right quadratic and look at its vertex. We did more review in here than the syllabus called for… complete the square, QF (the Quadratic Formula), what have you. You can thank me later.
Zeros of a Polynomial. With real coefficients. In the easiest cases, one can read the factored form of a polynomial directly from its graph; in the more interesting cases we’ll have to “divide away” the factors—(x-R), say—associated to certain roots (R such that f(R)=0, also known as zeros and, loosely, as x intercepts [strictly speaking, the root is a number and the intercept is a point (i.e., an ordered pair of numbers); some students appear to feel that clinging to the belief that details like this are never important is some higher moral duty... as if thinking clearly is some sort of sacrilege...]); typically one finally cleans up by applying QF to get the last two roots. These can be real or complex. One should also be prepared to construct polynomials given certain roots (and their multiplicities): remember here that complex roots of real polynomials occur in conjugate pairs.
Rational Functions. A single problem of the form “graph y = R(x)“, where R is a rational function, involves lots of sub-problems. Find zeros and vertical asymptotes (factor numerator and denominator first if necessary; we reviewed this very little, alas); put appropriate features in the graph. Determine the “end behavior” (horizontal, oblique, or no asymptote); sketch. Find a “new” point & make the appropriate connections. There’s an inequality to solve under this heading too: I recommend solving these graphically even though the exam itself says “solve algebraically”. If they give you ruled paper, write the other way.
Composite Functions. Compute ‘em. But also, evidently because it’s slightly tricky, give their domains. The point here is to consider, not only the computed “formula”, but the domain of the right-hand “factor” in the composite (g in 
Inverse Functions. Be able to compute the formula for the inverse of a function (given by a formula). First of all, in textbook fashion: to invert f, put x = f(y) and solve for y; the result is (a “formula” for) 
And exponential stuff. But that’s recent. Ongoing, even. And the coffeeshop’s ready and waiting. Gotta go.
Check out “why 21st century skills” at KTM.
Evidently it was World Math Day yesterday.
Here’s the second Math Teachers at Play carny.
A virtual Spirograph at (new to me) 10-Minute Math.
The function ![[t \mapsto A_0 2^{t/K}]](http://s1.wordpress.com/latex.php?latex=%5Bt+%5Cmapsto+A_0+2%5E%7Bt%2FK%7D%5D&bg=ffffff&fg=000000&s=0 "[t \mapsto A_0 2^{t/K}]")
Well, I suppose one should plot the co-ordinate pairs having t equal to (say) 0, 32, and 64 and sketch the curve; look at the calculator; that kind of thing. But we’ll assume everybody gets it. You probably oughta be looking at a source with graphs in it if you don’t already understand this theory pretty well; I’m pretty helpless with online graphics.
Now, one could define the same exponential function in terms of its “tripling time” instead of doubling time; indeed putting f(x) = 3*10^6 in our example yields 3 = 2^(t/32); taking logs (base two) on both sides gives us t/32 = log_2 (3) and hence t = 32 log_2 (3)…about 50.72 years. So g(x) = 10^6 * 3^(t/50.72) is essentially the same function as f (to make g exactly equal to f, replace “50.72″ with the exact value (the “formula”; for some reason students tend to believe that decimals are more “exact” than the algebra they approximate [and have a hard time letting go of this idea]).
What I have in mind is some alternative universe (or future class) where I would hold back on introducing the “exponential growth rate” for at least one section and have students work on exercises about doubling times (and half-lives; I intended to have mentioned by now that when K is negative one has an exponential decay rather than an exponential growth and that -K is then called the half-life of the quantity being measured [usually in this context, "number of molecules of a certain radioactive isotope"]). Things are already hard enough to assimilate without some weird number unknown to elementary mathematics being thrown into the mix.
Passing back and forth between the bases of 2 and 3 could, at least for some students, make it easier to see what’s going on in transforming to the base e… and this “interchangiblity” of the bases seems to be something that students typically seem not to quite “get”. Anyhow, the graphs I’m thinking of here—and’ll have on the board in a few hours—would be w(x)= 2^x and h(x)= 3^x (on the same co-ordinate axes). If we think of “beginning at” 2^x and “going to” 3^x, we’re applying a transformation. This one looks like a horizontal “shrinking” (note in particular that the fixed point is on the y-axis). And indeed it is. Assuming this for a moment, we can conclude that the functions w and h are related by h(x) = w(Lx) for some “scaling factor” L.
I’ll interupt myself to remark that we know this from work in the first week of the course… adds and subtracts are shifts, multiplies and divides are scalings, and sign changes are reflections; things that happen to y go up and down and things that happen to x go side-to-side; finally (and most trickily) such side-to-side moves occur “backward” in the sense that, for example, adding a positive number to the “argument” (replacing f(x) with f(x+3), say) calls for subtracting from the x co-ordinates in computing points on the “new” graph (shifting left by three units in our example).
Returning to our exponential functions, we’ve used geometry to see that 3^x can be written as 2^(Lx) for some L (the scaling factor). Some “simple” algebra (perfectly natural and straightforward once one has done the exercises) tells us that L = log_2 (3) here; indeed 
