Tables, graphs, and equations all represent models. We use terms such as “symbolic” or “analytic” to refer specifically to the equation form of a function model; “descriptive model” refers to a model that seeks to describe or summarize phenomena, such as a graph. In Module 5, Topic B, students expand on their work in Topic A to complete the modeling cycle for a real-world contextual problem presented as a graph, a data set, or a verbal description. For each, they formulate a function model, perform computations related to solving the problem, interpret the problem and the model, and then, through iterations of revising their models as needed, validate, and report their results.
Students choose and define the quantities of the problem (N-Q.A.2) and the appropriate level of precision for the context (N-Q.A.3). They create 1- and 2-variable equations (A-CED.A.1, A-CED.A.2) to model the context when presented as a graph, as data and as a verbal description. They can distinguish between situations that represent a linear (F-LE.A.1b), quadratic, or exponential (F-LE.A.1c) relationship. For data, they look for first differences to be constant for linear, second differences to be constant for quadratic, and a common ratio for exponential. When there are clear patterns in the data, students will recognize when the pattern represents a linear (arithmetic) or exponential (geometric) sequence (F-BF.A.1a, F-LE.A.2). For graphic presentations, they interpret the key features of the graph, and for both data sets and verbal descriptions, they sketch a graph to show the key features (F-IF.B.4). They calculate and interpret the average rate of change over an interval, estimating when using the graph (F-IF.B.6), and relate the domain of the function to its graph and to its context (F-IF.B.5).
In Topic D, students are formally introduced to the modeling cycle through problems that can be solved by creating equations and inequalities in one variable, systems of equations, and graphing (N-Q.1, A-SSE.1, A-CED.1, A-CED.2, A-REI.3). The End-of-Module Assessment follows Topic D.
In Module 1, Topic A, students explore the main functions that they will work with in Grade 9: linear, quadratic, and exponential. The goal is to introduce students to these functions by having them make graphs of situations (usually based upon time) in which the functions naturally arise (A-CED.2). As they graph, they reason abstractly and quantitatively as they choose and interpret units to solve problems related to the graphs they create (N-Q.1, N-Q.2, N-Q.3).