Rationale
Common Core State Standards. Every teacher across the United States has heard of them. "Forty-three states, DC and four territories" 1 are at various stages of implementing them. As we transition to these standards and their associated assessments, we (teachers) are looking for resources to help us make our students successful. This curriculum unit will serve as a guide for implementing the Standards for Mathematical Practice while providing meaningful instruction in the Mathematical Content Domain of High School Functions. This unit emphasizes domain and range in the study of functions and their inverses. I am writing it primarily for my precalculus students, but much of it will apply to any algebra course.
I teach in a vocational school district consisting of four high schools that draw from middle schools in all six districts in New Castle County, Delaware. We use the Core-Plus: Contemporary Mathematics in Context (CPMP) textbook series, which is an integrated math curriculum. We teach math courses in 90-minute block periods every day for one 18-week semester; students have three to four courses per semester, including their chosen career area for 90 – 180 minutes per day for the entire school year. With some exceptions for the very high and very low achievers based on a district placement test, students take two semesters of Core-Plus (1 & 2) in their freshman year, Core-Plus 3 in their sophomore year, and a "Trigonometry" transition course in their junior year. I had been teaching a traditional precalculus course for seniors, and advanced juniors for several years. For a multitude of reasons beyond my control, I started the 2013 school year with a new precalculus course guide that followed the newly published Core-Plus Mathematics: Preparation for Calculus, Course 4. I had limited access to the new textbook prior to the start of the school year, but enough to be aware of major topics incoming students had never studied. Not surprisingly, we (all district teachers teaching this new precalculus course) found more holes as the semester went along. My goal for this curriculum unit is to patch some of those holes in the content while integrating CCSS Mathematical Practice Standards.
The most glaring hole came when I began teaching how to solve trigonometric equations. First of all, my students only had a crash course in graphs of trigonometric functions. Secondly, while students have used inverse operations to solve equations, they never learned about inverse functions. Fortunately, our course alignment has been adjusted so that both of these topics will be taught in prerequisite courses. However, the most challenging part of solving trigonometric equations came when I tried to explain why there were an infinite number of solutions and how to find and denote them based on the calculator's response to sin -1, cos -1, or tan -1 of some value. Over the years, even my most advanced students have struggled with inverse trigonometric functions because these functions, being periodic, are not one-to-one unless the domain is restricted. As a result, I chose to write this curriculum unit to focus on domains and ranges of functions to prepare students to both operate on functions (add, subtract, multiply, divide, and compose) and find inverse functions. Throughout the unit I will use and/or design tasks that both lead to a deep content understanding of functions and incorporate the eight CCSS-M Practice Standards.
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