2022 PA-A-1-1

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PA.A.1.1

PA.A.1.1 Recognize that a function is a relationship between an independent variable and a dependent variable in which the value of the independent variable determines the value of the dependent variable.

In a Nutshell

Students will examine data from a variety of representations to determine if the relationship between dependent and independent variables represents a function. A student should understand that a function is a relationship in which each input value has a unique output value and that the output value is dependent on the independent input value.

Student Actions	Teacher Actions
Develop A Deep and Flexible Conceptual Understanding when identifying and describing independent and dependent relationships in real-life scenarios. Develop a Deep and Flexible Conceptual Understanding as students explore the meaning of the vertical line test as a way to graphically analyze relationships among dependent values and to verify if a graphed line meets expectations for properties of linear functions. Develop Mathematical Reasoning by discussing how the input (independent variable; x) affects the output (dependent variable; y) and make predictions based on patterns of the relationship between the input and output values.	Pose purposeful questions that enable students to create lists of independent relationships in real life and dependent relationships in real life to understand the meaning that those terms have in functions. Implement meaningful tasks that allow students to analyze various representations of relations (tables, graphs, data sets) and derive the idea that functions have only one possible output for every input. Pose purposeful questions that ask for students to describe the input and output helping them to make connections between properties of dependent and independent variables. Elicit and Use Evidence of the students’ understanding of functions to real-world situations. Provide activities that allow students to identify both the independent and dependent variables in various situations.
Key Understandings	Misconceptions
Changing the input affects the output in a specific way Identify independent and dependent variables in real-life situations Understand that in order for a relation to be a function, for every input (independent variable) there is only one possible output (dependent variable). Not all relations represent a function. In order for a graph to show a functional relationship, each independent value must have only one dependent value. A vertical line test could be used to determine if the relation is a function.	Some students will try to classify all relations as functions. Students may interchange the ideas of dependent and independent variables. Students may refer to the independent (x) values to determine a functional relationship instead of the dependent (y) values.
Knowledge Connections
Prior Knowledge	Leads to
Represent relationships between two varying positive quantities involving no more than two operations (6.A.1.2). Identify a relationship between two varying quantities, x and y, as proportional if it can be expressed in the form y/x or y=kx (7.A.1.1).	Distinguish between relations and functions (A1.F.1.1). Identify the dependent variable and independent variable given a function, equation, or graph. (A1.F.1.2).