Use projects, real-world activities, and games to bring precalculus to life for students.
By Donna Iadipaolo
High school precalculus is meant to be an intermediary step to what is largely viewed as the most pressing mathematical challenge - calculus. All high school students should be made aware that calculus is a required course for many careers including engineering, medicine, and business. Consequently, the purpose of calculus is to build upon advanced algebra, analytic geometry, and trigonometry in order to prepare students for studying calculus. Often times, students start a precalculus class by delving into the theory and practice of the behavior of different kinds of functions, in addition to a detailed exploration of trigonometry. Also, concepts such as vectors, matrices, and polar coordinates are introduced in precalculus.
Making Math Relevant
When I took precalculus as a student in high school, it all seemed abstract. The material was never contextualized to apply to real-world purposes. Even though I earned an A, mathematics, and particularly precalculus, didn't seem relevant to me. I remember repeatedly doing math problem after math problem, rather than engaging in projects and games.
Using an investigative approach to trigonometric concepts, students can conduct an experiment in which they change the coefficients and take note of consequential changes in amplitude, period, frequency, and phase shift of the functions. They might also apply certain technology, such as Ti-83 and the Apple program “Grapher” to note detailed transformations in the graph.
In the book “Projects for Precalculus,” author Professor Todd Swanson promotes precalculus projects that utilize real-world data and analyze more experimental situations. For instance, students may examine various functions that demonstrate how the rate of different crickets’ chirps increases as the temperature rises. Students might investigate how the period of a pendulum is dependent upon the length of the string holding the mass; an observation first quantified by the Italian Renaissance mathematician, scientist, and nonconformist Galileo. Students may research what the flaw of the Ancient Greek musical scale developed by Pythagoras is and determine how this inconsistency has been mathematically corrected. Keeping with the musical theme, students might examine how the frequency of a guitar string is proportional to its length. Furthermore, the concept of a limit and the definition of a limit might be explored in the meaning and behavior of exponential functions that model human or animal populations.
Using the National Common Core Standards as a model, teachers could create other real-world investigations and projects to entice students. For instance, one precalculus standard states that students “understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.” Students might love to ponder how the measurements of earthquakes, sound, and chemical solutions arose. Therefore, creating precalculus projects that focus on the calculations of concrete decibels, pH, and measures of a Richter magnitude scale would be motivating.
Math Games in the Classroom
Games would enliven a precalculus class as well. For instance, “Jeopardy Precalculus” might bring excitement to new vocabulary terms such as: continuity, asymptote, discrete, discontinuous, even/odd function. Better yet, students could create their own games to test these precalculus vocabulary terms. Another game might be for students to organize teams and then race to draw what the graphical representation of the function might look like.
More Precalculus Lessons:
How do credit card companies make their money? In this lesson, students can examine real-world connections to exponential functions.
Students explore various representations of the functions of sin and cosine. Students change the coefficients for the amplitude, period, frequency, and phase shift and note the change in the graph.
This lesson has students explore the Fundamental Theorem of Algebra. In this Fundamental Theorem of Algebra lesson, students find the roots of a polynomial on their Ti-Nspire using the Fundamental Theorem of Algebra.
Students identify different locations on a unit circle. Students identify the different trig properties and rewrite the equation of a circle in standard form.