College Algebra and Precalculus

Quantitative Reasoning about Rates of Change (CoRD)

John Paul Cook, Oklahoma State University
Allison Dorko, Oklahoma State University
Michael Oehrtman, Oklahoma State University
Michael Tallman, Oklahoma State University

The four activities in this module are intended to be used throughout a College Algebra or Precalculus course to develop and reinforce students’ quantitative reasoning about rates of change. They first reinforce thinking about changes in quantities then encourage comparing those changes for two covarying quantities. The activities then focus on working with constant rates as a constant ratio between changes in two covarying quantities.

The Quadratic River Application with Area and Perimeter (ARC)

Narges Dehdashti, Rose State College
Josh Grieshop, Rose State College
Ashley Tomson, Rose State College

In both parts of the College Algebra Quadratic River Application activity, students will work with applying a quadratic function to a scenario. In part one, students will initially access prior knowledge of area and perimeter of rectangles to engage with the material with tables and GeoGebra. Through the structure of the activity, students will then transition to making algebraic representations of area and perimeter given particular constraints of the scenario. Lastly, students will find the length and width that will maximize an area. In part two, students will complete a similar task to reinforce the idea that the vertex of the  quadratic provides maximum area.

Discovering Absolute Value (ARC)

Dee Cooper, Northern Oklahoma College
Christi Hook, Northern Oklahoma College
Cecil Phibbs, Northern Oklahoma College

In An Absolute Value Discovery, Algebra for STEM instructors are provided with problems that introduce students to the concept of absolute value as a distance. The discovery will lead students to further understand absolute value as a distance measured from zero, and that both positive and negative values can have the same absolute value. Students can model these concept on number lines leading to formal equations which they can solve realizing the possibility of more than one solution or no solutions. This Activity can be used as a whole class discovery, in smaller groups or individually by students.

The Number e (ARC)

Michael Fulkerson, University of Central Oklahoma
Kristi Karber, University of Central Oklahoma
Ashley Tomson, Rose State College

This lesson serves as an introduction to the number e and the application of continuously compounded interest. Students begin by exploring the expression  for large values of n, and they discover that the expression approaches a number (2.71828…) as n approaches infinity. For college algebra students, this also serves as an introduction to the idea of a “limit.” Students then explore the relationship between compound interest and continuously compounded interest.

Tray Folding Activity: Polynomials and Optimization (ARC)

Deborah Moore-Russo, University of Oklahoma
Brad Paynter, University of Central Oklahoma

This collaborative activity involving scaffolded contextual optimization is appropriate for algebra or applied algebra students. It is also a good activity for calculus students to introduce optimization with differentiation, but no differentiation is required to complete the activity. The lesson helps facilitate students’ conceptual development reinforcing their quantitative and covariational reasoning as they identify the relevant quantities, namely the dimensions and volume of the tray, with the appropriate units and algebraic expressions for each. This is done in a geometric context so that students can see how a change in one quantity (the height of a tray) affects its volume.

The Rule of 70 (ARC)

Michael Fulkerson, University of Central Oklahoma
Kristi Karber, University of Central Oklahoma
Scott Williams, University of Central Oklahoma

This lesson explores the Rule of 70, a financial mathematics concept that estimates the time it takes for an investment to double at a given annual interest rate, compounded continuously. Students will use both algebraic methods and an interactive Desmos activity to understand and apply the rule, enhancing their grasp of logarithms, exponential growth, and financial literacy.

Fundamental Features of Functions: Activities that Promote Student Problem-Solving for Various Types of Functions (CoRD)

Lee Ann Brown, Oklahoma State University
Katherine Hiebert Brumley, The Jimmy Naifeh Center
Thomas McNamara, Southwestern Oklahoma State University
Melissa Mills, Oklahoma State University
Quan Tran, University of Science and Arts of Oklahoma

This collection of activities investigates fundamental features of mathematical functions from a College Algebra or Pre-Calculus course. Influenced by the work of Dan Meyer, we created interesting problems with varying amounts of information given to the students. Students need to think about what pieces of information they need and use their resources to find reasonable values for the quantities that they deem important. We have written a teacher’s guide for each activity that outlines how we think that the lesson should go and questions that the teacher can pose to emphasize certain aspects of the problem.

Completing the Square (ARC)

Michael Fulkerson, University of Central Oklahoma
Deborah Moore-Russo, University of Oklahoma
Cecil Phibbs, Northern Oklahoma College

In this activity, students learn the algebraic technique of completing the square through hands-on group work with algebra tiles, linking algebraic concepts to geometric shapes. They apply this method to solve a quadratic equation, graph a parabola, find the center and radius of a circle, and ultimately derive the Quadratic Formula. This practical approach helps students understand and use completing the square in a range of mathematical contexts, enhancing their algebra skills.

Multiple Problem-Solving Strategies and Representational Equivalence for Precalculus (CoRD)

Candace Andrews, University of Oklahoma
Carye Chapman, University of Oklahoma
Casey Violette Haskins, University of Oklahoma
Cory Wilson, Oklahoma City Community College

We present eight activities intended to function either as introductory or expository material for specific concepts in a precalculus sequence. Specifically, we focus on functions and their inverses, average rate of change, graphical transformations, solving composed equations, the ambiguous case of the law of sines, and inverse trigonometric functions and their graphs. We also present activities for modeling with contextual applications and working with multiple representations of functions.

Exponents and their Properties (ARC)

Dee Cooper, Northern Oklahoma College
Christi Hook, Northern Oklahoma College
Cecil Phibbs, Northern Oklahoma College

In this activity for exponents and their properties, students will work individually or in groups through problems using exponents. Each group of problems seeks to help students develop a specific exponential rule. These rules known as Properties of Exponents allow students to properly manipulate problems involving multiplication and division of like bases with exponents as well as raising an exponent to another exponent. Each section deals with a different one of these properties and allows students to discover the property through expanding and then simplifying an exponential expression. Part four gives students a chance to apply the properties by identifying common exponential mistakes and fixing them using the correct exponential property.

Four Homework Problems Over Big Ideas (CoRD)

Narges Dehdashti, Rose State College
Mike Fulkerson, University of Central Oklahoma
Cecil Phibbs, Northern Oklahoma College
Ashley Tomson, Rose State College

In this CoRD for a College Algebra course, instructors are provided with four activities in which students explore, synthesize, and make connections of key ideas in College Algebra. In Activity 1, students analyze two different applications and apply key terminology about functions. In Activity 2, students explore function transformations through a Desmos activity. In Activity 3, students connect the important characteristics of polynomials in standard and factored form with graphs and words. In Activity 4, students consider logarithmic and exponential functions to support the ideas of these functions as well as their importance as inverses of each other.

Understanding the Connections between Equations and Graphs (CoRD)

Monty Harper, Oklahoma State University
Deborah Moore-Russo, University of Oklahoma

Exploring Geometric Series (ARC)

Brenden Balch, University of Central Oklahoma
Michael Fulkerson, University of Central Oklahoma
Scott Williams, University of Central Oklahoma

This activity explores the topic of geometric series in a College Algebra or Quantitative Reasoning class. It introduces students to geometric series through geometry, algebra, hands- on exercises, and real-world applications. It aims to provide students with an introduction to the idea of convergence of an infinite series, which is particularly useful for students progressing to Calculus. The activity also includes a bonus homework extension which explores the relationship between centroids of triangles and a geometric series.

The “Why” of Trigonometry (CoRD)

Michael Fulkerson, University of Central Oklahoma
Kristi Karber, University of Central Oklahoma
Scott Williams, University of Central Oklahoma

Trigonometry is of primary importance in any precalculus course. But often students learn the basics of trigonometric functions and identities without really understanding “why” everything works and how everything “fits together.” A frequently heard complaint is that there is too much memorization: “How can I remember all these identities?” This CoRD seeks to address these issues, while developing the targeted topics listed above. It consists of the following four activities: (1) “Why is the Pythagorean Theorem true?” (2) “Special Triangles: 45-45-90, 30-60-90, and 15-75-90” (3) “The Angle Sum Identities for Cosine and Sine” (4) “Avoid Memorization: Deriving Identities”

Modeling Covariation: Cube Painting (ARC)

Carye Chapman, University of Oklahoma
Deb Moore-Russo, University of Oklahoma

This hands-on collaborative activity is designed to help students conceptualize covariation by working with an easy-to-understand situation that allows them to communicate with others while visualizing a geometric context. While working on the activity, students will work with functions in multiple representations, including using functions written as equations to generalize geometric patterns.

Function Formulas and Graphs: Find the Connections! (CoRD)

Monty Harper, Oklahoma State University
Deborah Moore-Russo, University of Oklahoma

This CoRD steps students through a process of thinking about how to connect a function’s formula to its graph. Through five connected lessons, students will discover how algebraic features of a function’s formula correlate with features of its graph. In particular, students will develop a formulation and accompanying set of rules which may be used to arrive at an accurate graph of a given function by applying linear transformations to the graph of its parent function.