From the Grade 8, High School, Functions Progression Document, pp. 7-8:

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1 Time Frame: Approximately 2-3 weeks Connections to Previous Learning: Students use their knowledge of expressions and equations to evaluate functions and to rewrite when necessary. They also use rigid transformations and coordinate geometry as examples of functions and relations (i.e. Under a particular translation, the vertex of the original triangle is mapped from the point (2, 1) to (5, 7) or, after a particular reflection, from (2,1) to (2, -1)). Focus of the Unit: Students understand that functions have exactly one output for every input and that functions can be expressed and described in multiple ways. Students become fluent with function notation and will be able to evaluate functions (i.e. f( 3)) at various inputs. Students explore a variety of functions and representations and see function examples from algebra, geometry, and real world experiences. Connections to Subsequent Learning: Students need a solid understanding of functions as they use different functions to model new phenomena and investigate particular function types such as linear and exponential functions. In future courses, students rely on their understanding of functions to analyze quadratic, logarithmic, and trigonometric functions. From the Grade 8, High School, Functions Progression Document, pp. 7-8: Interpreting Functions Understand the concept of a function and use function notation Building on semi-formal notions of functions from Grade 8, students in high school begin to use formal notation and language for functions. Now the input/output relationship is a correspondence between two sets: the domain and the range. The domain is the set of input values, and the range is the set of output values. A key advantage of function notation is that the correspondence is built into the notation. For example, f (5) is shorthand for the output value of f when the input value is 5. Students sometimes interpret the parentheses in function notation as indicating multiplication. Because they might have seen numerical expressions like 3 (4), meaning 3 times 4, students can interpret f(x)as f times x. This can lead to false generalizations of the distributive property, such replacing f(x+3) with f(x) +f(3). Work with interpreting function notation in terms of the graph of f can help students avoid this confusion with the symbols (see example to right). 3/23/2014 8:58:39 PM Adapted from UbD framework Page 1

2 Although it is common to say the function f(x), the notation f(x) refers to a single output value when the input value is x. To talk about the function as a whole, write f, or perhaps the function f, where f(x) = 3x + 4. The x is merely a placeholder, so f(t) = 3 t+ 4 describes exactly the same function. Later, students can make interpretations like those in the following table: Notice that a common preoccupation of high school mathematics distinguishing function from relations is not in the Standards. Time normally spent on exercises involving the vertical line test, or searching lists of ordered pairs to find two with the same x-coordinate and different y-coordinate, can be reallocated elsewhere. Indeed, the vertical line test is problematical, since it makes it difficult to discuss questions such as is x a function of y when presented with a graph of y against x (an important question for students thinking about inverse functions). The core question when investigating functions is: Does each element of the domain correspond to exactly one element in the range? The graphic on the next page shows a discussion of the square root function oriented around this question. To promote fluency with function notation, students interpret function notation in contexts. For example, if his a function that relates Kristin s height in inches to her age in years, then the statement h(7) = 49 means, When Kristin was 7 years old, she was 49 inches tall. The value of h(12) is the answer to How tall was Kristin when she was 12 years old. And the solution of h(x) = 60 is the answer to How old was Kristen when she was 60 inches tall? Sometimes, especially in real-world contexts, there is no expression (or closed formula) for a function. In those cases, it is common to use a graph or a table of values to (partially) represent the function. A sequence is a function whose domain is a subset of the integers. In fact; many patterns explored in grades K-8 can be considered sequences. For example, the sequence 4, 7, 10, 13, might be described as a plus 3 pattern because terms are computed by adding 3 to the previous term. To show how the sequence can be considered a function, we need an index that indicates which term of the sequence we are talking about, and which serves as an input value to the function. Deciding that the 4 corresponds to an index value of 1, we make a table showing the correspondence, as in the margin. The sequence can be describe recursively by the rule f(1) = 4, f(n + 1) = f(n) + 3 for 2. Notice that the 3/23/2014 8:58:39 PM Adapted from UbD framework Page 2

3 recursive definition requires both a starting value and a rule for computing subsequent terms. The sequence can also be described with the closed formula f(n) =3n + 1, for integers n 1. Notice that the domain is included as part of the description. A graph of the sequence consists of discrete dots, because the specification does not indicate what happens between the dots. + In advanced courses, students may use subscript notation for sequences. Desired Outcomes Standard(s): Functions understand the concept of a function and use function notation. F.IF.1 Understand that a function from one set (called the domain) to another set (call 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 is the graph of the equation y=f(x). F.IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1), f(n+1) = f(n) + f(n-1) for n 1. Interpret functions that arise in Applications in terms of the context. F.IF.4For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. F.IF.5Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. F.IF.6Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Transfer: Students will be able to develop concepts and procedures of functions to analyze graphs, equations, and charts. Ex. Given a graph you can find the domain and range of the graph. EX. Given an equation you can determine if it is a function. 3/23/2014 8:58:39 PM Adapted from UbD framework Page 3

4 WIDA Standard: (English Language Learners) English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics. English language learners benefit from: explicit vocabulary instruction with regard to key features of graphs and tables. explicit instruction with regard to the relationship between graphs or tables and the equations and contexts they represent. Understandings: Students will understand that Functions have exactly one output for each input. Functions can be defined explicitly or recursively. Function notation is used to evaluate and interpret inputs and outputs of functions. Sequences are functions with a domain as a subset of the integer. A function has key features that can be represented and interpreted from a graph, table or quantitative relationship. Essential Questions: What is function notation and how can it be used and interpreted? What are functions and how can they be defined? What are sequences and how are their domains defined? How can you represent a function and what are the key features of each representation? 3/23/2014 8:58:39 PM Adapted from UbD framework Page 4

5 Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.) 1. Make sense of problems and persevere in solving them. * 2. Reason abstractly and quantitatively. Students determine and interpret specific quantities such as f(2) for any function. They reason about the continuous nature of f(x), its general properties and behavior as well as any particular input, output pair. 3. Construct viable arguments and critique the reasoning of others. * 4. Model with mathematics. Students use functions as models for many real-world situations. For example, they may use C(x) =.75x as the cost of manufacturing yo-yos or they may use L( x) x 2 64 as the length of a ramp given its height x. In all such cases, students explain the meaning of any outcomes in relation to the situation being modeled (i.e yo-yos would cost $1250). 5. Use appropriate tools strategically. 6. Attend to precision. * 7. Look for and make use of structure. Functions are a critical concept in mathematics; students develop a solid understanding of their relationship to other mathematics. Students see how functions can be represented as tables, graphs, equations, and real-world models. Functions can be identified or categorized based on particular properties or characteristics. * 8. Look for and express regularity in repeated reasoning. Prerequisite Skills/Concepts: Student should already be able to: Evaluate expressions such as (-3) 2 +4(-3) + 8 Understand, variables, independent and dependent quantities. Advanced Skills/Concepts: Some students may be ready to: Students will create a function from given inputs and outputs. Students will create a real world problem involving functions. 3/23/2014 8:58:39 PM Adapted from UbD framework Page 5

6 Knowledge: Students will know The basic definition of a function. Domain and range of a function represented in a graph, equation, table or real-world context. Sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Skills: Students will be able to Use function notation and interpret statements that use function notation in terms of a context. Identify functions from a variety of representations. Evaluate f(x) for many functions. Translate between symbolic representations of functions and tables or graphs. Find outputs given inputs and inputs given outputs. Relate the domain of a function to its graph and to the context. Interpret key features of a function represented as a graph or a table. Sketch graphs showing key features given a verbal description of the relationship. Calculate and interpret the average rate of change of a function over a specified interval. Estimate the rate of change from a graph. Academic Vocabulary: Critical Terms: Function inputs outputs domain range independent variable dependent variable Supplemental Terms: Representations Evaluate 3/23/2014 8:58:39 PM Adapted from UbD framework Page 6

7 Pre-s Formative s Summative s Self-s #1 Math 1 Unit 3 Pre #3 Math 1 Unit 3 Lesson Pre Give #1 Math 1 Unit 3 Pre #4 Problem Solving with Tables #5 Meat-a-Morphosis Highlights #6 Reasoning with Functions #7 Function Machine Mission #8 Avalanche Review Activity & Avalanche Review KEY Sample Lesson Sequence #9 Lesson Summative #10 Math 1 Unit 3 Summative #2 Take a Stand 1. Intro to Function Notation Model Lesson (F.IF.1, F.IF.2, F.IF.4, F.IF.5, F.IF.6) 2. Sequences (F.IF.3) Investigate, develop, and teach the Arithmetic and Geometric formulas and notation including making sense of the basic components of the formulas including an, a1, r, n, and d. Give #10 Math 1 Unit 3 Summative 3/23/2014 8:58:39 PM Adapted from UbD framework Page 7