New Jersey Student Learning Standards Mathematics Algebra 1 Instructional Support Tool for Teachers: Functions This instructional support tool is designed to assist educators in interpreting and implementing the New Jersey Student Learning Standards for Mathematics. It contains explanations or examples of each Algebra 1 course standard to answer questions about the standard’s meaning and how it applies to student knowledge and performance. To ensure that descriptions are helpful and meaningful to teachers, this document also identifies grades 6 to 8 prerequisite standards upon which each Algebra 1 standard builds. It includes the following: sample items aligned to each identified prerequisite; identification of course level connections; instructional considerations and common misconceptions; sample academic vocabulary and associated standards for mathematical practice. Examples are samples only and should not be considered an exhaustive list. This instructional support tool is considered a living document. The New Jersey Department of Education believe that teachers and other educators will find ways to improve the document as they use it. Please send feedback to [email protected] so that the Department may use your input when updating this guide. Please consult the New Jersey Student Learning Standards for Mathematics for more information. 1 Functions Standards Overview Functions describe situations where one quantity determines another. Because we continually make theories about dependencies between quantities in nature and society, functions are important tools in the construction of mathematical models. In school mathematics, functions usually have numerical inputs and outputs and are often defined by an algebraic expression. A function can be described in various ways, such as by a graph (e.g., the trace of a seismograph); by a verbal rule, as in, “I’ll give you a state, you give me the capital city”; by an algebraic expression like f(x) = a + bx; or by a recursive rule. The graph of a function is often a useful way of visualizing the relationship of the function models, and manipulating a mathematical expression for a function can shed light on the function’s properties. Interpreting Functions (F-IF) Understand the concept of a function and use function notations: F-IF.A.1 F-IF.A.2 F-IF.A.3 Interpret functions that arise in applications in terms of the context: F-IF.B.4 F-IF.B.5 F-IF.B.6 Interpret linear models: F-IF.C.7a-b F-IF.C.8a F-IF.C.9 Linear, Quadratic, and Exponential Models (F-LE) Construct and compare linear, quadratic, and exponential models and solve problems: F-LE.A.1 F-LE.A.2 F-LE.A.3 Interpret expressions for functions in terms of the situation they model: F-LE.B.5 Building Functions (F-BF) Build a function that models a relationship between two quantities: F-BF.A.1 Build new functions from existing functions: F-BF.B.3 Mathematical Practices 1. Make sense of problems and persevere in solving them 5. Use appropriate tools strategically 2. Reason abstractly and quantitatively 6. Attend to precision 3. Construct viable arguments and critique the reasoning of others 7. Look for and make use of structure 4. Model with mathematics 8. Look for and express regularity in repeated reasoning 2 Algebra 1 Instructional Support Tool - Functions Standard F-IF.A.1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). (Content Emphases: Major) Explanations and/or Examples Building Blocks (grades 6 to 8 prerequisites): Given the function ( ), students explain that input values are guaranteed to Grade 8 produce unique output values and use the function rule to generate a table or 8.F.A.1 graph. They identify as an element of the domain, the input, and ( ) as an Understand that a function is a rule that assigns to each input exactly element in the range, the output. The domain of a function given by an one output. The graph of a function is the set of ordered pairs consisting algebraic expression, unless otherwise specified, is the largest possible of an input and the corresponding output. domain. Students recognize that the graph of the function, , is the graph of the equation = ( ) and that ( , ( )) is a point on the graph of . 8.F.A.2 Example 1: A pack of pencils cost $0.75. If n number of packs are Compare properties (e.g. rate of change, intercepts, domain and range) of two functions each represented in a different way (algebraically, purchased, then the total purchase price is represented by the function t(n) = graphically, numerically in tables, or by verbal descriptions). For example, 0.75n. given a linear function represented by a table of values and a linear a) Explain why t is a function. function represented by an algebraic expression, determine which function b) What is a reasonable domain and range for the function? has the greater rate of change. Ex. 1 Solution: 8.F.A.3 a) t is a function because any input (element of the domain) yields Interpret the equation y = mx + b as defining a linear function, whose exactly one unique output (element of the range). graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function b) A reasonable domain is any whole number. A reasonable range is of its side length is not linear because its graph contains the points (1,1), any multiple of .75 that is greater than or equal to zero. (2,4) and (3,9), which are not on a straight line. Prerequisite Examples: Grade 8 8.F.A.1: Determine if the graph represents a function. Justify your reasoning. Determine which of the tables represents a function. Explain why. Table A Table B x f(x) x f(x) 0 1 0 0 1 2 1 2 2 2 1 3 3 4 4 5 3 Algebra 1 Instructional Support Tool - Functions Standard F-IF.A.1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). (Content Emphases: Major) Prerequisite Examples: Grade 8 8.F.A.3: 8.F.A.2: Determine which of the functions listed below are linear and which are not Compare the following functions to determine which has the greater rate linear. Explain your reasoning for each. of change: • y = -2x2 + 3 Function 1: f(x) = 2x + 4 • y = 2x Function 2 • A = πr2 x f(x) • y = 0.25 + 0.5(x – 2) -1 6 0 -3 2 3 Grade/Course Level Connections: n/a Instructional Considerations: Common Misconceptions: • Reinforce that “domain” refers to the set of all possible input values. • Students may believe that all relationships having an input and an • Carefully develop the appropriate language associated with output are functions, and therefore, misuse the function terminology. functions. For example, f(2) means the output or value of the • Students may believe that the notation f(x) means to multiply some function f when the input is 2. value f by another value x. The notation alone can be confusing and • Build upon the idea of a function as an input-output machine. needs careful development. For example, f(2) means the output value Provide applied contexts in which to explore functions. For of the function f when the input value is 2. example, examine the amount of money earned when given the number of hours worked on a job. Contrast this with a situation in which a single fee is paid by the “carload” of people, regardless of whether 1, 2 or more people are in the car. • Examine graphs of functions and non-functions. Students should explain why the vertical line test works given their understanding of each element of the domain (x) being assigned to exactly one element of the range. Academic Vocabulary domain, range, function, independent variable, dependent variable, input, output 4 Algebra 1 Instructional Support Tool - FUNCTIONS Standard F-IF.A.1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). (Content Emphases: Major) Standards for Mathematical Practice: (2) Reason abstractly and quantitatively (6) Attend to precision 5 Algebra 1 Instructional Support Tool - Functions Standard F-IF.A.2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. (Content Emphases: Major) Explanations and/or Examples Building Blocks (Grades 6 to 8 prerequisites): Example 1: Evaluate f(2) for the function f(m) = 5(m − 3) + 17. Grade 6 6.EE.A.2c Ex. 1 Solution: f(2) = 12 Evaluate expressions at specific values of their variables. Include m expressions that arise from formulas used in real-world problems.