The – count ‘em – eleven authors of MATHPOWER 12, the text currently in use in high schools all across British Columbia, are in a pickle. On the one hand, they know that the young people these days are more interested in the rock and roll music and the reality teevee than the book-learning, and that they’d rather play the video games than do the mathematics.
Unfortunately, however, there’s not a whole lot about graphing conics that’s fun and exciting. So our intrepid authors, fingers firmly on the pulses of the jaded young members of their audience, settle for making half-assed connections between analytic geometry and Real LifeTM in the hopes that their teenaged readers will realize just how kewl math can be. Check out the hook they use in Chapter 3.3 – The Circle, to make sure that their readers are intrigued enough stick around until the end of the unit:
The compact disc player is everywhere these days. Developed initially by Philips and Sony, it first came on the market in 1983. By 1986, over one million CD players were being sold each year. Because of the low-cost laser components, the CD player has become one of the most successful electronic devices to date.
If you trace around the outside of a CD, the result is a circle.
A circle is the set or locus of all points in a plane which are equidistant from a fixed point. This fixed point is called the centre. The distance from this centre to any point on the circle is called the radius.
CD players! Cuz, like, music is cool, and think of how much cooler it will be once you know that the outline of your CD is all x^2+y^2=25 and shit.
Chapter 3.5, on the hyperbola, kicks it up a notch by getting all sonic on us:
The Concorde, a supersonic aircraft developed by Britain and France, first began passenger service in 1976. The Concorde travels at twice the speed of sound at an altitude of 17 000 meters. As a passenger on the Concorde on a flight from London to New York City, you would cross five time zones in 3.5 h. Your arrival in New York would be 1.5h prior to your departure London time.
As you saw in the chapter opener, at speeds greater than the speed of sound, air pressure disturbances accumulate in front of the aircraft and a conical shock wave forms. When the Concorde is travelling parallel to the ground, this conical shock wave intersects the ground in the shape of one branch of the hyperbola.
Right below is a graph of a hyperbola, and a description of its focal property.
Actually, this is quite interesting. I know I saw some of this back when I was an undergraduate, but I forgot it, and now I’m curious about the physics of sound. If I were a high school student reacting in what I assume is the authors’ intended way – “hey, cool, sonic booms give hyperbolas, I wonder how that works” – I might be inclined to flip through a few pages in order to learn more.
Unfortunately, there’s nothing more on sonic booms. Nothing. We don’t get to learn about the significance of the foci in the picture, nor are we even told the most basic fact about the sonic boom hyperbola – namely, that the sonic boom is heard simultaneously at all points on the hyperbola.
We do not even, in fact, see how the focal property of the hyperbola allows us to derive the equation for the hyperbola, period – that formula is just handed to us as-is, and we’re left to trust that it gives us the same shape as the one whose focal distances have constant difference.
Actually, we can forget about the focal property entirely: it never comes up in the exercises or in the tests. (We can even, to a certain extent, forget the equation for the hyperbola, and rely on our graphing calculator to remind us when the need arises.) But, check out the funky photo of the Concorde! (Aside: the textbook authors don’t even provide a picture of a sonic boom.)
I have tutored high school math on and off for more than a decade, and during that time, I have noticed a trend toward stripping high school mathematics texts of logic and proof – and, for that matter, of anything that requires sustained and focused attention – and filling in the gaps with pretty pictures, long-winded examples, and graphing calculator applications. No wonder my students regard mathematics as a disjoint collection of facts: their textbooks give no indication otherwise.
Enough about the conics. Let’s skip ahead to a section on combinatorics (7.4 – Pathways and Pascal’s Triangle), because there are enough elementary applications of that material that there’s no need to provide any disjointed, contrived ones. But reading the introduction to this section, I find that an apology is due – to the kid I’m currently tutoring, and to all of the students who perplexed me with their dogged refusal to read the questions before answering them.
Because, see, when I (and my readers) ranted about students who expect to be able to solve word problems without reading them, we were assuming that the authors of those word problems were not insufferable windbags who are being paid by the word. And by “insufferable windbags who are being paid by the word” I mean insufferable windbags who are being paid by the word, because I can’t think of any other explanation for this atrocious introduction to path counting, which I swear to God I’m not embellishing:
The destiny of Ottawa changed forever when Queen Victoria chose it as the capital of the province of Canada in 1857. Construction of the Parliament buildings began in 1860 and was completed approximately six years later, just in time to be reaffirmed as the nation’s capital in 1867.
Prior to European settlement, various aboriginal groups occupied the region, including the Ottawa nation from which the city took its name. With the arrival of the French in the 1600’s, and later the British, the fur trade became the mainstay of the economy in the Ottawa River valley. Despite this fact, Europeans did not settle the Ottawa area until 1800.
Not long after the timber trade began, and with the completion of the Rideau Canal in 1832 by Lieutenant-Colonel John By of the Royal Engineers, a community called Bytown was firmly established as a centre for timber trade. This thriving town was incorporated as the city of Ottawa in 1855.
The simplified street map shows a portion of downtown Ottawa, close to the Parliament buildings. If you start at the corner of Bank and Laurier, and travel only eastward or northward, there are two routes you can take to walk to Corner B. How many routes are possible to walk from the corner of Bank and Laurier to the corner of Rideau and Elgin?
I’m no longer going to tell students to read the entire question before answering it. A clueless student who reads only the last sentence of this question has a fighting chance of answering it correctly. One who reads the whole tome, on the other hand, has a non-negligible chance of processing the quantitative data in a way familiar to everyone who’s graded word problems: Number of pathways = 1857+1860-1867-1600+1800-1832+1855 = 2073.