Functions and Modeling
The Functions and Modeling pathway is designed to serve degree programs such as Business, Agriculture, Social Sciences, and Health Sciences, focusing on applications and mathematical tools common in these fields. As a gateway, a Functions and Modeling course is not to require any college-level prerequisite yet it should prepare students to move directly into business calculus, if needed. Several institutions offer a Modeling course at a scale equivalent to or exceeding their College Algebra (STEM Prep) offering.
List of institutions offering a Functions and Modeling course.
Functions and Modeling Initiation Workshop
MIP faculty collaborations on the Functions and Modeling pathway began with an Initiation Workshop in June, 2019. At this workshop, faculty identified the central concepts and skills critical for success in the Oklahoma Functions and Modeling Math Pathway (OSRHE Common Code MA205, https://www.okhighered.org/transfer-students/2018-19/mathematics.pdf). Participants of the workshop identified four foundations to guide development, testing, and refinement of inquiry-oriented course materials to support the development of the central concepts and skills in the Functions and Modeling pathway.
Function is the foundational topic in Functions and Modeling. The function concept enables us to identify, analyze, and gain insight into relationships between real-world quantities that vary in tandem, and is a key prerequisite to learning subsequent ideas in this course. Accordingly, students in Functions and Modeling should develop productive understandings of function (both single- and multi-variable) that can be used flexibly amongst various real-world contexts and representations. This involves awareness and use of appropriate conventions like function notation as well as aspects of quantitative reasoning and covariational reasoning. [Explore the Details]
Modeling and Quantitative Reasoning
Modeling is the process of using mathematics to describe, analyze, and gain insight into real life phenomena. It entails identifying and representing quantities and determining relationships among relevant quantities. Modeling requires careful recognition of, and attention to, the relevant quantities involved in the situation and use of either (1) patterns of covariation and/or rate of change to determine a class of functions (e.g. linear, exponential) that best model a relationship, and/or (2) prior knowledge of the relationship (e.g. physical, geometric) between these quantities to devise a model (e.g. recognizing that the volume of a box is a function of its height). Careful attention to the quantities involved is the heart of what is called quantitative reasoning (e.g. Thompson, 2011), and it is an indispensable component of modeling. Three key questions lie at the heart of quantitative reasoning and are instrumental in guiding the integration of quantitative reasoning into the design of instructional tasks: (1) what object is being measured?, (2) what attribute of that object are we measuring?, and (3) what is the unit of measurement? [Explore the Details]
A rate of change is a measure of how much one quantity changes with respect to another. Rates of change are an integral piece of understanding the nature of a function relationship between two quantities. Understanding rate of change provides students with tools with which they can analyze and make inferences about function behavior and hence gain insight into the real-life phenomena modeled by those functions. Rates of change that explicitly appear in Functions and Modeling include constant rate of change, average rate of change, and percentage change, but rates of change provide a means for developing quantitative understandings of function concepts as well – such as limiting value, maximum/minimum, concavity, and characterizations of function classes – suggesting that emphasizing rate of change is a primary goal throughout the entire Functions and Modeling course. [Explore the Details]
Knowing key characteristics of the various function classes (e.g. linear, exponential, rational, polynomial) provides opportunities for students to expand their understandings of the ways in which two quantities can change together (and the associated patterns of change that might emerge). Understanding function classes in terms of these key characteristics can also expand students’ understanding of functions, facilitating a shift from thinking of a function as a procedure (in which each input is ‘plugged in’ to a particular formula to produce an ‘output’) to a broader representation of an entire relationship between two quantities. This has a number of advantages, one of which is that it supports thinking about constructions involving multiple functions (e.g. function composition and combinations of functions). Another benefit of viewing a function as a unified process is that it supports students’ abilities to compare and contrast the behavior of two functions against one another (in a way that is usually not possible if a student has only a computational input-output view of function). Such comparisons are integral to the Functions and Modeling course because the classification of a particular function as similar to particular class of functions provides a tool to analyze, model, and describe the behavior of a variety of real-world situations. [Explore the Details]
Functions and Modeling CoRDs and ARCs
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Development of Linear and Exponential Concepts (CoRD) Ashley Berger, University of Oklahoma
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Our CoRD materials address modeling and quantitative reasoning within the context of a Functions & Modeling course. There are eight activities, which can be divided into two units of four activities. The first unit focuses on developing understandings of various aspects of linear functions, while the second unit focuses on exponential functions. The activities are presented in a specific order, if an instructor wished to utilize them all. However, the activities can be used individually as instructors see fit. Within the instructor materials, there are guided questions and commentary to help instructors implement the activity with the intended targeted understandings in mind, as well as optional extensions for the activities.
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Exploring Average Rate of Change Given by Tables (ARC) Christi Hook, Northern Oklahoma College
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In Exploring AROC Given by Tables, Functions & Modeling instructors are provided with problems involving analyzing rates of change given in tables. A large group activity is provided where the entire classroom discusses, calculates, interprets, and applies rates of change with population values. Students then use rates of change to estimate and predict future population values. An alternative small group activity includes a shorter full-class activity with four small group tasks. With this option, students will consider increasing, decreasing, and constant rates of change with similar explorations. Each option allows students to determine if a limiting value exists for the population data provided in the table.
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Systems of Linear Equations Exploration (ARC) Cecil Phibbs, Northern Oklahoma College
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In this systems of linear equations exploration for a Functions & Modeling course, students work with tables, graphs, and algebra to investigate the definition of a solution to a linear equation and apply that definition to a solution of a system of linear equations. In part one, students explore the definition through making tables of values in a scenario involving money. In part two, students transition to equations and graphs to see the visual representation of a solution and how it applies to a patio scenario. In part three, students find a solution using algebra and check the answer with graphing in a brunch item scenario.
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Water in the Tub: A Review of Functions and their Behavior (ARC) Michael Hardy, Southeastern Oklahoma State University
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In this activity students are presented with an applied scenario of water level in a tub and asked to complete a graph of the situation considering appropriate units of measure for a piecewise function. While engaging in this low floor-high ceiling activity, students are applying the ideas of quantification, selecting appropriate units of measure (for height and time), and using covariational reasoning to represent the context graphically.
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Constant Rate of Change with Linear Models (CoRD) Rebecca Burkala, Rose State College
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In this linear models CoRD for a Functions & Modeling course, students work through four activities which emphasize linear modeling through the key concept of constant rate of change. In the first activity, students connect the idea of constant rate of change to linear functions. In the second activity, students analyze data to determine if it is linear. Through the third activity, students distinguish between exact linear data and approximately linear data. In the fourth activity, students transition to identifying linear versus non-linear data.
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Transformations of Functions (ARC) Ashley Berger, University of Oklahoma
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In the Functions and Modeling class, there is a high emphasis on establishing the relationship between the algebraic and the graphical representations of functions. One way to underline this is through the study of function transformations. The goal of these activities is to provide a more hands-on, experimentation-oriented component that will allow students to more easily generalize the effects of a set of function transformations on a graph. In doing so, students should have concrete, visual examples to help solidify their understanding of the algebraic and graphical relationship between functions.
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Mathematical Representations (ARC) Gary Barksdale, University of Oklahoma
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Students will develop a deeper understanding of representing real-world scenarios through various mathematical representations, focusing on applied situations that involve covariation. Students will consider the temperature change of a spoon over time, seating arrangements as smaller tables are pushed together, and lawn mowing and the time it takes. Besides being introduced to the idea of mathematical models, the students will practice transitioning between verbal, numeric, and graphical representations while recognizing the differences between continuous and discrete data. Students will also explore dynagraphs to help develop a more nuanced understanding of data representation that emphasizes covariation.
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Function Transformations (ARC) Dustin Gaskins, University of Oklahoma
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In the Functions and Modeling class, there an emphasis on understanding functions, specifically as they are models of realistic circumstances. Working purely with mathematical structures, we can understand function composition easily through an understanding of domain, range, and appropriate symbol manipulation; however, models of circumstances with units and objects being quantified requires more detailed analysis. This activity walks through several tasks which provide students with a scaffolded approach to function composition that will allow them to seriously analyze a context’s impact on function composition as well as useful instances of function composition.
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