The explorations and tasks developed by the Enhancing Explorations in Functions for Preservice Secondary Mathematics Teachers Project can be accessed by following the links below. These 11 lessons comprise the unit on Explorations on Functions and Equations.
- Lesson 1: Solving Problems Class Handout & Instructor Notes
In this course, you will be learning mathematics by engaging in problem solving. In particular, the activities presented throughout this course will entice you to think deeply about some of the mathematics you’ve encountered previously, about new ideas presented, and about the connections between the two. Furthermore, you will be expected to work collaboratively, to discuss your mathematical ideas, and to justify your reasoning. Let’s get started! - Lesson 2: Conic Sections Prep Work, Class Handout, & Instructor Notes
In Exploration 1.3, you generated the standard equation for a circle. A circle is an example of a conic section. In Lesson 2 we explore conic sections. A conic section can be defined as the curve of intersection of a plane and a right circular (double-napped) cone. The equation you generated for a circle is an analytic (or algebraic) definition. In Lesson 2, we will make connections between locus definitions of the conic sections to their analytic definitions. - Lesson 3: A Qualitative Look at Graphical Representations Class Handout & Instructor Notes
The process of constructing graphs of functions or relations based upon a given process or graphical representations helps us focus on how quantities vary together. This qualitative exploration emphasizes covariation of quantities and using a visual representation to determine patterns and structure. - Lesson 4: Examples of Real World Relationships Between Quantities Class Handout & Instructor Notes
To develop a robust concept of functions and foreshadow the mathematical modeling that lies ahead, we consider some real world relationships between quantities. This exploration emphasizes relations (and functions) with a focus on multiple representations, input/output correspondence, and the power of functions to predict. Some of the relations explored here will be revisited later in the course, in explorations focusing on concepts of linear regression, parametric equations, and functions defined by sequential patterns. - Lesson 5: What is a function? Class Handout & Instructor Notes
The concept of function is foundational to high school algebra as well as higher-level mathematics. However, functions are not straightforward to teach or to learn. The purpose of this lesson is to look deeper into the definition of function. - Lesson 6: Functions and Equations Class Handout & Instructor Notes
In school mathematics teaching, using vocabulary in mathematically precise ways is important because there are ambiguities in or common uses of certain terms that can cause confusion when students are confronted with the implied or informal meanings in a mathematical situation. This lesson focuses on the meaning of the term equation and the different constructed meanings associated with the use of the equal sign. Carpenter, Franke, & Levi (2003) assert that a “limited conception of what the equal sign means is one of the major stumbling blocks in learning algebra” (p. 22). - Lesson 7: Complex Roots Visualization Prep Work, Class Handout, & Instructor Notes
…But what happens if you use a different domain for a quadratic function? What if you take your domain to be all complex numbers of the form a+bi, where a and b are real numbers? How does that change ones’ ability to conceptualize the zeros of the quadratic function? How can one see what the graph of a quadratic function “looks like” near complex zeros? - Lesson 8: Spring Mass Lab Prep Work, Class Handout, & Instructor Notes
In this lesson, you will use a sinusoidal function as a mathematical model to represent the data gathered from a vertical spring-mass motion system (ignoring damping). You will also explore the concept of rate of change of a function as it
relates to this real-world data. - Lesson 9: Sequences and Triangular Differences Prep Work, Class Handout, Instructor Notes
Our goal in this lesson is to apply techniques on patterns in data to create function models for the data. We will see that certain patterns in the data lend themselves to particular types of models. That data used in Lesson 10 will be data than can be modeled by common functions taught in secondary school such as linear, quadratic, power, exponential, and logarithmic functions. - Lesson 10: Functions Arising from Patterns Prep Work, Class Handout, Instructor Notes
Building upon the work in Exploration 9.2, we now explore data sets consisting of ordered pairs. The purpose of the next exploration is to identify functions that model the given data sets from derived patterns in the domain of each given data set that result in patterns in the corresponding range of the data set. The patterns we seek in the given data are arithmetic, geometric, or triangular difference patterns. - Lesson 11: Indistinguishable Function Transformations and Function Patterns Handout & Instructor Notes
In Lessons 9 and 10, we see that linear, quadratic, power, exponential, and logarithmic functions have predictable behaviors when comparing inputs and outputs according to certain patterns of consecutive inputs. How does this relate to function transformations or certain phenomena we observe when using graphing technology (e.g. Desmos, GeoGebra, graphing calculators, etc.) to explore transformations of functions in a dynamic manner? In this lesson, we will explore how certain function patterns may seemingly produce a dynamic process that defies the algebraic rules learned previously about transformations of functions.
