# The Pre-Calculus Split

In the math department at Alzar School, there is always a wide variety of needs from students coming from all over the country.  Particularly at the pre-calculus level, every district seems to handle curriculum a little differently.  Pre-calculus expands on nearly every topic that was presented in geometry and algebra 2 classes with more analytical techniques, applied problems, and precise theorems.

This semester’s student group brought the greatest variation of student strengths and needs.  For the first unit, six students were divided into four groups.  One student took an analytical look at functions to make connections between tables of values, equations, graphs, and how algebraic changes to an equation are manifested as transformations of a graph and changes to the related table of values.  A second student worked on solving complicated systems of equations where as many as four unknown quantities were related through a series of four equations.  Through a very concrete and somewhat tedious method, the value of each variable could be determined.  To alleviate some of the tedium, we explored ways computer algebra systems can help us make the calculations.

Two students continued their deeper exploration of trigonometry.  Trigonometry is typically introduced as the Pythagorean Theorem and the sine, cosine, and tangent ratios as they apply to right triangles.  We recently went on to use trigonometric identities and formulas to see the many ways these ratios are related to each other and solve some complicated algebraic problems.

The final group of two students left behind everything they once knew about graphs and y=f(x) equations.  All prior learning in algebra was put in a box labeled: “rectangular equations” and we took a new spin on the idea of a function.  First up was the idea of parametric equations.  With parametric equations, a two-dimensional graph is described by two equations, not one.  The value of y is no longer based on x, but both x and y are independent functions of a third variable: typically t.  These types of equations are particularly valuable for describing projectile motion of falling objects where force vectors are split and categorized by vertical (y-direction) and horizontal (x-direction) forces.  As a project, we developed a set of equations to describe the path of a nail stuck in a wheel as the wheel rolls along a surface.  It doesn’t simply spin in a circle, because the wheel is also moving horizontally.  For those of you who lived a childhood on the edge, this is also the path you took as you climbed inside an abandoned tractor tire and had your friends roll you down the hill.

The second type of functional relationship we explored were polar equations.  In this coordinate system, vertical and horizontal position are no longer relevant.  We instead graph r (the straight-line distance of a point from the origin) as a function of theta (the angle between that straight line and the horizontal axis).  The resulting graphs are a variety of interesting rose-curves, spirals, and ellipses.  This class of equation is very useful in analyzing orbits, such as planetary motion, atomic particle physics, and magnetic fields.

As we work through the semester, the class arrangements will continue to shift slightly so students can work together in new groups to explore different types of problems instead of being bored by unnecessary remediation.

Dan Thurber, Math Teacher