## Products - Pathways to Calculus

06.25.2024

**Overview****Table of Contents****Sample Material****About the Author**s

**Precalculus: Pathways to Calculus: A Problem Solving Approach**

This text was designed to develop students’ conceptual knowledge, problem solving abilities and skills that are foundational for success in calculus. The text and course materials are organized into eight modules. As students read the text they encounter conceptual explanations and questions, and many engaging examples. The online text also includes short videos that illustrate how to solve novel problems, while providing dynamic illustrations and conceptual explanations. Many practice problems are included in the text. The worked out solutions are initially hidden and can be viewed by selecting a *click to reveal* option after attempting the problem. Linked interactive animations are also included to help students make sense of the words in an applied problem by first visualizing the relevant quantities and how they are related.

Teacher support materials include cognitively scaffolded worksheets (with detailed teacher notes) that are designed to keep students’ minds active in making critical connections for understanding the course’s key ideas. Companion (and optional) PowerPoint slides with linked animations and illustrations are provided to support teachers in providing engaging lectures and discussions about the worksheets’ key ideas. A professional development website (still under development) provides videos of teachers using the materials in their own classrooms. Sample exams are also provided for each module.

A research based and validated assessment tool, the *Precalculus Concept Assessment* (PCA), is included in the suite of materials. The PCA has been validated and may be used to place students in calculus or assess pre-post-course shifts of student learning in precalculus or college algebra.

**Module 1: Evaluating, Simplifying and Solving: A Conceptual and Practice Oriented Review**

Methods and rationale for evaluating and simplifying expressions, and solving equations are reviewed. Practice problems and detailed solutions provide all students opportunity to gain fluency with procedural aspects of algebra.

**Module 2: Quantity, Variable, Proportion and Linearity **

This module begins by investigating what is involved in identifying and relating quantities in the context of learning to solve novel problems. The idea of variable is introduced as a way of representing the values that quantities (e.g., distance, time) can assume. These foundational ideas are leveraged when reviewing ideas of proportionality, constant rate of change and average rate of change. Opportunities to practice simplifying and evaluating expressions and formulas appear throughout. The module ends by introducing the distance formula and using the Pythagorean theorem to compute distances that cannot be directly measured.

**Module 3: Functions: Formalizing Relationships Between Quantities **

This module introduces the idea of function as a more formal way of describing and representing how input quantities are related to output quantities. Connections among function representations (words, tables, formulas, graphs) are reviewed and used in the context of solving novel problems. The idea of function composition is introduced as a chaining together of two function processes for the purpose of relating two quantities that cannot be directly related by a simple formula. The idea of function inverse is introduced as a reversal of the function process. Methods for composing and inverting functions that are represented as formulas, graphs and tables are introduced with emphasis on using function composition and function inverse purposefully to solve novel problems. Notational aspects of representing functions are also addressed.

**Module 4: Exponential and Logarithmic Functions: A Multiplicative Approach **

Exponential functions are introduced by contrasting growth that builds on itself from growth that accumulates by adding constant amounts. The multiplicative growth of an exponential function is explored in the context of population growth, compound interest, radioactive decay and other applications. The idea of logarithm and logarithmic function are introduced. Materials support students in understanding a logarithmic function as the inverse of the exponential function. Students are provided plenty of practice in using these ideas to model two quantities and solve applied problems. Notational issues of representing both exponential and logarithmic functions are addressed from the perspective of the meaning conveyed by the symbols.

**Module 5: Polynomial and Power Functions: Promoting Meaningful Connections Across Function Representations**

The general form of polynomial functions is introduced. The proportional growth patterns of linear polynomial functions are contrasted with those of non-linear polynomial functions; ones in which equal changes in the input quantity do not always result in equal changes in the output quantity. These explorations lay the groundwork for understanding ideas of *changing rate of change*, concavity and inflection points on a graph. Methods for graphing polynomial functions are introduced including approaches for finding a function’s roots, maximums and minimums. The module concludes by examining growth patterns of quadratic functions and exploring special methods (completing the square and quadratic formula) for determining a quadratic function’s zeros and extreme values.

**Module 6: Rational Functions and Introduction to Limit**

Growth patterns of rational functions are explored by continuing to reason about how the values of two quantities change together. The long-run behavior of rational functions is examined by investigating what the function gets close to as the input quantity grows without bound in both the negative and positive directions. Examining a function’s behavior as it gets closer and closer to some value(s) of the input variable that make the function undefined provides a conceptual approach to understanding the behavior of rational functions near these function values. Methods for determine asymptotes and the behavior of rational functions emerge out of these explorations.

**Module 7: Angle Measure and Introduction to Trigonometric Functions in the Context of the Unit Circle**

Foundational ideas of angle and angle-measure are developed by investigating approaches for measuring the openness of two rays. Methods for modeling the behavior of periodic motion are introduced in the context of co-varying an angle measure with a linear measurement that maps out a periodic motion, laying the groundwork for introducing the trigonometric functions of sine and cosine. The module concludes by exploring the meaning of period, amplitude, and translations of both the sine and cosine functions.

**Module 8: Right Triangle Trigonometry **

The relationship between right triangle and unit circle trigonometry is made explicit by initially exploring the right triangle relationships defined by the sine, cosine and tangent functions in a unit circle context. The triangle relationships defined by the sine, cosine and tangent functions are used to determine the values of unknown quantities in various applied problems. We conclude the chapter by deriving various *trigonometric identities* that relate the trigonometric functions to one another.

**Sample Materials**

**Dr. Marilyn Carlson** is a Professor in the School of Mathematical and Statistical Sciences at Arizona State University. Her teaching and research career began as a lecturer of mathematics at the Haskell Indian Nations University in 1978. She completed her master’s degree in computer science at the University of Kansas in 1985. She was Director of First Year Mathematics at the University of Kansas from 1985-1995 and completed her Ph.D. in mathematics education at the University of Kansas in 1995. She also served as Director of First Year Mathematics at Arizona State University from 1995-1999. These experiences began her curriculum development and research into the learning experiences that lead to students’ continued mathematics course taking and learning. Dr. Carlson is a frequent invited speaker and the author of more than 50 published and presented research papers that report results of investigations into what is involved in knowing, learning and understanding key ideas of college algebra, precalculus and beginning calculus. She received a National Science Foundation CAREER award, was a member of the Eisenhower Advisory Board for the State of Arizona, served as coordinator of the Special Interest Group for Research in Undergraduate Mathematics Education, served on a National Research Council panel investigating advanced mathematics and science programs in U.S. high schools, and has participated in policy deliberations at state and national levels. Most recently, the Mathematical Association of America awarded Dr. Carlson the 2008 Selden Award for Research in Undergraduate Mathematics Education. Dr. Carlson has been the principal investigator of research and outreach projects funded by the NSF and the Eisenhower program. She led the development of the Ph.D. concentration in mathematics education in the Department of Mathematics at Arizona State University, and has taught a wide range of undergraduate courses in mathematics and graduate courses in mathematics education. Over the past 8 years Dr. Carlson has served as dissertation advisor for 11 Ph.D. students in mathematics education.

**Dr. Micheal Oehrtman**