THE IMPORTANCE OF FUNCTIONS IN COMMON CORE MATHEMATICS

1. THE IMPORTANCE OF FUNCTIONS IN COMMON CORE MATHEMATICS John F. Mahoney Benjamin Banneker Academic HS Washington, DC johnf.mahoney@gmail.com

2. What is a function? Write your definition!

3. Why are functions important? • The function concept • Covariation and rate of change • Families of functions • Combining and transforming functions • Multiple representations of functions Source: Thomas Cooney, et al, “Developing Essential Understanding of Functions Grades 9 – 12”, NCTM, 2010

4. 1. The function concept • Functions are single-valued mappings from one set – the domain of the function – to another – its range. • Functions apply to a wide range of situations. They don’t have to follow a pattern or be continuous. • The domain and range of functions don’t have to be numbers.

5. 2. Covariation and rate of change • Patterns in how two variables change together indicate membership in a particular family of functions and determine the type of formula a function has. • A rate of change describes the covariation between two variables. • A function’s rate of change helps determine what kinds of real world phenomena the function can model

6. 3. Families of functionsSlide 1 • Members of a family of functions share the same type of rate of change • Linear functions are characterized by a constant rate of change • Quadratic functions are characterized by a linear rate of change. • Exponential functions are characterized by a rate of change proportional to the value of the function.

7. 3. Families of functionsSlide 2 • Trig functions are fundamental examples of periodic functions • Arithmetic sequences are linear functions whose domains are positive integers. • Geometric sequences are exponential functions whose domains are positive integers.

8. 4. Combining and transforming functions • Functions that have the same domain can be added, subtracted, multiplied, or divided • Under appropriate conditions, functions can be composed • Composing a functions with “shifting” or “scaling” functions changes the formula and graph of a function in predictable ways. • Under appropriate conditions, functions have inverses

9. 5. Multiple representations of functions • Functions can be represented algebraically, graphically, verbally, and tabularly. • Changing the way a function is represented does not change the function, although different representations highlight different characteristics – and some may only show part of a function • Some representations of a function may be more useful than others • Links between algebraic and graphical representations of functions are important in studying relationships and change

10. CCSSM & Functions • Functions are a major component of 8th grade math, Algebra 1, and Algebra 2 • Before (and during) 8th grade, students need to become comfortable with: • Expressions • Variables • Equality • Solving equations informally, symbolically, graphically, and numerically • Proportional reasoning

11. How will students be tested? “Evidence-Centered Design supports the development of assessment tasks that address the Standards for Mathematical Practice. Thus, the kind of mathematics instruction called for in the CCSSM—engaging students in work that helps them develop mathematical habits of mind—can be reflected in powerful ways in the assessment system.” Source: PARCC 

12. The y-coordinate of the y-intercept of f(x) The y-coordinate of the y-intercept of g(x) http://www.ccsstoolbox.com/parcc/PARCCPrototype_main.html

13. f(3) g(3)

14. Maximum value of f(x) on the interval -5 ≤ x ≤ 5 Maximum value of g(x) on the interval -5 ≤ x ≤ 5

16. Which Standards of Mathematical Practice were emphasized? • Making sense of problems and persevere in solving them • Reason abstractly and quantitatively • Construct viable arguments and critique the reasoning of others • Model with mathematics • Use appropriate tool strategically • Attend to precision • Look for and make use of structure • Look for and express regularity in repeated reasoning

17. Which Standards of Mathematical Practice were emphasized? • Making sense of problems and persevere in solving them • Reason abstractly and quantitatively • Construct viable arguments and critique the reasoning of others • Model with mathematics • Use appropriate tool strategically • Attend to precision • Look for and make use of structure • Look for and express regularity in repeated reasoning

18. What is a function? • A rule that assigns to each number x (the input) a single value y (the output) • A set of ordered pairs that assigns to each x-value exactly one y-value • A set of operations that are performed on each value that is put into it • It is a relation for which to each domain value there corresponds exactly one range value

19. What is a function? • A relation that matches each element of a first set to an element of a second set in such a way that no element in the first set is assigned to two different elements in the second set • A relation in which for each value of x there is a unique value of y. We say that y is a function of x. The independent variable is x, the dependent variable is y

20. What is a function? • A rule that takes certain numbers as inputs and assigns to each a definite output number • A function is a special type of relation in which no two ordered pairs have the same first coordinate and different second coordinate • A relation in which each first component in the ordered pairs corresponds to exactly one second component

21. What is a function? • A rule or correspondence that assigns to each element of X one and only one element of Y • A relationship between input and output. In a function, the output depends on the input. There is exactly one output for each input. • A relation in which each element of the domain is paired with exactly one element of the range.

22. What is a function? • A set of ordered pairs that satisfies this condition: There are no two ordered pairs with the same input and different outputs • A rule that assigns to each element of a set A a unique element of a set B. • For any sets A and B, f: A→B, is a subset f of the Cartesian product A  B such that for every a A appears once and only once as the first element in an ordered pair (a, b) in f.

23. What is a function? • A mapping or correspondence between one set called the domain and a second set called the range such that for every member of the domain there corresponds exactly one member in the range. • One quantity, H, is a function of another, t, if each value of t has a unique value of H associated with it.

24. Two Groups Input Output x-value y-value Ordered Pairs Independent Dependent Domain Range The core ideas that define a function Source: Gregorio Ponce, “Critical Juncture Ahead! Proceed with Caution to Introduce the Concept of Function”, Mathematics Teacher, Sept. 2007

25. The core ideas that define a function • A Pattern: • That Assigns • That Matches • That Corresponds • That Operates • That Relates • A Set of Ordered Pairs • A Correspondence • A Rule • A Set of Operations

26. A Special Requirement For each x, only one y Vertical Line Test For each input, a single output No two ordered pairs With same x and different y For each domain value, exactly one range value The core ideas that define a function

28. Function Notation • f(x) was first used by Leonhard Euler in 1734 • Other common notations • Function notation is not required in 8th grade CCSSM

29. “A Visual Approach to Functions”Frances Van Dyke; Key Curriculum, 2002

30. Motion Detector

31. Motion Detector

32. Common Core - Functions • Interpreting Functions: 4 Eighth grade, 9 HSstandards • Building Functions: 1 Eighth grade, 4 - 5* HSstandards • Linear, Quadratic, and Exponential Models: 5 HS standards • Trigonometric Functions: 4 - 9* HSstandards * Six of the function standards are for the fourth year of HS math

33. INTERPRETING FUNCTIONS Part 1

34. INTERPRETING FUNCTIONS Part 2

35. BUILDING FUNCTIONS

36. LINEAR, QUADRATIC, AND EXPONENTIAL MODELS

37. TRIGONOMETRIC FUNCTIONS

44. Which Standards of Mathematical Practice were emphasized? • Making sense of problems and persevere in solving them • Reason abstractly and quantitatively • Construct viable arguments and critique the reasoning of others • Model with mathematics • Use appropriate tool strategically • Attend to precision • Look for and make use of structure • Look for and express regularity in repeated reasoning

45. Which Standards of Mathematical Practice were emphasized? • Making sense of problems and persevere in solving them • Reason abstractly and quantitatively • Construct viable arguments and critique the reasoning of others • Model with mathematics • Use appropriate tool strategically • Attend to precision • Look for and make use of structure • Look for and express regularity in repeated reasoning

47. A solution to part a