Mathematical Investigations I is a semester-course designed for students who enjoy mathematics but do not wish to take a rigorous pre calculus or calculus course. Because the course material is non-sequential, it is also suitable for students who initially enroll in a pre-calculus or calculus class and then drop out early in the semester. The prerequisites for this course include geometry and a second year of algebra.
Specifically, I wrote this curriculum to involve four areas of investigations.
1) The first area of investigations is historical in nature and offers an introduction to the mathematics of the Babylonians, Egyptians, Greeks, and the Europeans following the Renaissance. Not only are the students exposed to varied and powerful mathematics, but they are introduced to the mathematicians whose lives were a mix of genius, energy, and emotions that contributed to their successes and failures as well.
2) The second area includes materials that are not normally found in traditional mathematics courses such as number theory, infinity, and topics that display mathematics in a simple but eloquent way.
3) To prove that mathematics is alive and not just relegated to discoveries from the past, chaos theory is presented as the third area of exploration. Sensitive dependence on initial conditions, the Chaos Game, and creation of the Mandelbrot Set are investigated by application of calculator technology.
4) The final area of problem solving requires students to apply intuition, mastery of mathematical skills, and various strategies to solve selected problems, which are not usually found in standard mathematical textbooks.
At the conclusion of the first semester that I taught this course during the 1992-93 school year, I created a survey for my students to express their opinions about the topics that were covered. Although there was a mix of attitudes about the focus of the course, more than half of the 20 students who enrolled in this class wrote that this was the best mathematics course that they had ever taken.
After some revisions over the next several years, I felt confident enough to ask a fellow veteran teacher if she would be willing to teach this semester course. She accepted the challenge and has taught it for many semesters including the 2015-16 school year. Throughout this period of time, she has repeatedly told me that “Mathematical Investigations” is her favorite class that she has ever taught.
Because of the success that I experienced with teaching this class and the positive responses that I have received from several other colleagues who have also taught this class, I wrote a second semester of Mathematical Investigations II. After several revisions that I completed in 2012, I asked another teacher, who had taught for five years, if he would be willing to teach this course. He also accepted and he continues to teach this class, which he states is his favorite class to teach.
Of course, these two semester courses are living documents in which the instructor is able to alter the material if so desired to better suit the needs of the students in the class. Consequently, I am placing the Mathematical Investigations I as a free resource through TpT for interested high school teachers. I did not include an answer key nor a couple of hand-outs since I do not expect anyone do use this entire curriculum.