# We're exploring functions and laying the foundation for future study.

The gap between algebra and calculus can be difficult to size up. Consider the common name for the bridge between the two subjects: “pre-calculus” sounds less like a subject in its own right and more like a basic training to prepare students for the rigors that lie ahead. Yet, precisely because pre-calculus is not a proper subject, it is also often a time to explore bits and pieces of topics that might otherwise go by the wayside. This vagueness of purpose is a challenge to teachers as it is to students; one sees pre-calculus classes that lurch from trigonometry to matrices to probability.

Thanks to the block schedule at Proof School, we’re able to take a more granular approach. Exponentials, Logs, and Trig might be the most straightforwardly descriptive title of any course I’ve taught: this was a class to get students comfortable with the major functions that form the background not only for calculus, but for statistics, physics, and many other endeavors to come.

We began by exploring functions in general. Viewed from the wrong angle, a function can be a bewilderingly complex object, with a formula like f(t) = 2.6*sin(2π(t–0.22))+12.2. (That’s an actual function we would use later in an astronomical application. Can you guess what it models? Ask a student!) Yet most of the functions we encounter repeat a few basic themes; the theme of this function is “wave”. We learned to decode the language of such formulas in terms of graphs and their geometric transformations, allowing students to recognize the basic forms and then parse out the influence of each element in the formula. In this part of the course, students heard echoes of graphing conics with Dr. V in Algebra II.

Next, we took up exponents and logs. Students sometimes experience logarithms as a formality, whereas, to me, they are a means of passage between two mental number lines: one based on addition and differences, the other on multiplication and ratios. Students have both mental number lines already—the second is implicit in geometric sequences, including the familiar progression of tens, hundreds, thousands (and eventually millions, billions, trillions). Exponents let us jump from the additive to the multiplicative number line; logarithms take us on the return trip. To make sure this point would not be lost on the class, I introduced the number lines before the functions themselves, challenging students to slot in the numbers 1 through 10 on a multiplicative scale (hint: the intervals between them decrease, and the distance from 1 to 2 is the same as the distance from 5 to 10). One student observed that if we just knew the positions of all the primes, we could locate all the rational numbers; others exclaimed over the remarkable coincidence that 2^10 is about the same as 10^3. Through one principle—“equal distances on the multiplicative number line represent equal ratios”—students were able to gradually discover and justify all the major properties of logarithms.

A gift to the school of eight multi-scale slide rules proved to be a perfect fit to this approach. I provided no direct instruction on the use of the slide rule, but merely indicated what could be done and let students find the algorithms. As their enthusiasm for this project showed, it is one thing to have a mental model of the two number lines, and another to be able to manipulate that model with one’s hands.

To celebrate the acquisition of these vital ideas, we closed our unit on exponents and logs with a Festival of Applications. Students got to choose among three mini-investigations on population growth, hyperinflation, and the musical scale. Some did all three! They presented to their peers what they had learned, then we embarked on a second round with a “detective work” theme: one group used carbon dating to decide if a painting had been forged; another studied the origins of a surprising pattern in the first digits of data, which has been used to expose financial fraud; and a third group gleefully played the role of coroner, using Newton’s Law of Cooling to pinpoint the time at which one Trollface B. Hedgehog shuffled off his mortal coil.

The remainder of the course was devoted to trigonometry. Continuing the theme of conversion between scales, we started by exploring the relationship between angles and slopes. For instance, if we have two wedges that each have slope 1/2 on a flat surface, what slope do we get by stacking them? The answer is not 1 as one might expect. Evidently slopes do not add as angles do. By working out a series of progressively more general examples, students discovered the double-angle identity for the tangent function—all before the word “tangent” (let alone “sine” or “cosine”) had been spoken in the classroom. Members of the class recognized some very clever uses of this identity, too; by computing that three slopes of 1/2 “add up” to a slope of 11/2, while four slopes of 1/2 “add up” to a negative slope, one student was able to deduce that these wedges must have an angle of between 22.5 and 30 degrees. She subsequently refined the lower bound to 25.7 degrees, using an invented method that could in principle determine the angle to any desired precision. It is thrilling to realize that the arctangent function is not something that only a calculator “knows” how to calculate!

Having whetted their appetites, the class dug into the full set of trig functions (well, not the full set: there are innumerable arcana like the versine and the sagitta that I managed to avoid mentioning, but my restraint didn’t stop one student from trying to work out the properties of the hyperbolic sine and cosine on his own). Though we began with right triangles, our discussion of “slope addition” made it inevitable that questions about angles bigger than 90 degrees would soon arise, leading us to the unit circle and periodic functions. Not wanting to force too many trigonometric rules on students during their first exposure to the subject, but knowing they would uncover some in the course of solving problems, I gave them a blank sheet on which to record these “trig relationships”. Many of the class filled this sheet with gleanings from their own work, from other students’ presentations, and from my own less discreet comments; a couple even requested a second page.

Scientific calculators, provided as a practical tool, turned out to be a tool of discovery as well. I had a nice conversation with one student who had noticed that sin(cos(tan(sin(cos(tan(sin(
sin(sin(cos(cos(cos(tan(tan(tan(x))))))))))))))) always seemed to be the same mysterious number, about 0.017. This sequence of key-presses was likely inspired by a moment of boredom, but it turned out to be as good a first brush with radians as any. (Reader, can you identify the reason for the mysterious 0.017?) Graphing technology was even more useful for gathering inspiration. When I asked the class if they could find a formula to replicate a graph of a “big wave made of little waves”, I wondered if I was asking too much; I never imagined that several students would independently figure out an exact match, down to the numerical parameters I had chosen, and one kid would throw in a summation for good measure to create a genuine fractal sinewave. Of course, the path to a solution did not run straight for most, and we ended up having an impromptu show and tell to discuss some of the weird graphs the class had inadvertently created.

Although most of the class have a few more stepping stones to cross before they are ready for calculus, we did manage to touch on what calculus is all about, through problems investigating the limit of sin(θ)/θ at the origin and the average value of sin(θ) over a quarter-period. Given a hint or two, a few students were able to intuit the area under an arch of the sinewave. We may not have the tools to prove such things yet, but a child’s reach should exceed their grasp!

With so many new ideas in this class, and very little lecture between bouts of exploration and problem-solving, it would have been easy for somebody to fall behind. Indeed, not everybody progressed at the same pace on a given day. Thus, after each segment of the class was “over”, we continued to reinforce the topics from that unit through sporadic Ten-Minute Workouts. Students were able to self-select either a regular or advanced problem on the topic of the day. (These being Proof School kids, many attempted to do both.) I added this component to the course in an ad-hoc way and didn’t have time for it every day, but after seeing the results and hearing positive reactions from students at both ends of the quiz score scale, I’m encouraged to continue using it in the future... and confident that the class’s newfound knowledge of functions will last.

--Austin Shapiro