THE IMPORTANCE OF FUNCTIONS IN COMMON CORE MATHEMATICS

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THE IMPORTANCE OF FUNCTIONS IN COMMON CORE
MATHEMATICS

• John F. Mahoney
• Benjamin Banneker Academic HS
• Washington, DC
• johnf.mahoney_at_gmail.com

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What is a function?

• Write your definition!

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Why are functions important?

1. The function concept
2. Covariation and rate of change
3. Families of functions
4. Combining and transforming functions
5. Multiple representations of functions

Source Thomas Cooney, et al, Developing
Essential Understanding of Functions Grades 9
12, NCTM, 2010
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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 dont have to follow a pattern or be
  continuous.
• The domain and range of functions dont have to
  be numbers.

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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 functions rate of change helps determine what
  kinds of real world phenomena the function can
  model

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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.

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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.

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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

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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

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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

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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
  CCSSMengaging students in work that helps them
  develop mathematical habits of mindcan be
  reflected in powerful ways in the assessment
  system. Source PARCC 

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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_ma
in.html
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f(3) g(3)
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Maximum value of f(x) on the interval -5 x 5
Maximum value of g(x) on the interval -5 x 5
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Which Standards of Mathematical Practice were
emphasized?

1. Making sense of problems and persevere in solving
   them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the
   reasoning of others
4. Model with mathematics
5. Use appropriate tool strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated
   reasoning

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Which Standards of Mathematical Practice were
emphasized?

1. Making sense of problems and persevere in solving
   them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the
   reasoning of others
4. Model with mathematics
5. Use appropriate tool strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated
   reasoning

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What is a function?

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

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What is a function?

1. 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
2. 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

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What is a function?

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

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What is a function?

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

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What is a function?

1. A set of ordered pairs that satisfies this
   condition There are no two ordered pairs with
   the same input and different outputs
2. A rule that assigns to each element of a set A a
   unique element of a set B.
3. 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.

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What is a function?

1. 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.
2. One quantity, H, is a function of another, t, if
   each value of t has a unique value of H
   associated with it.

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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
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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

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The core ideas that define a function
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Function Notation

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

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A Visual Approach to FunctionsFrances Van
Dyke Key Curriculum, 2002
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Motion Detector
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Motion Detector
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Common Core - Functions

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

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INTERPRETING FUNCTIONS Part 1
Eighth Grade Algebra 1 Algebra 2
Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. 8.F.1   Describe quantitatively the functional relationship between two quantities by analyzing a graph. Sketch a graph that exhibits the qualitative features of a function that has been described verbally. 8.F.5 Understand the concept of a function and use function notation. Learn as general principle focus on linear and exponential and on arithmetic and geometric sequences F.IF.1, 2, 3   Interpret functions that arise in applications in terms of a context. Linear, exponential, and quadratic F.IF.4, 5, 6     Interpret functions that arise in applications in terms of a context. Emphasize selection of appropriate models F.IF.4, 5, 6                      
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INTERPRETING FUNCTIONS Part 2
Eighth Grade Algebra 1 Algebra 2
Interpret the equation y mx b as defining a linear function, whose graph is a straight line give examples of functions that are not linear. 8.F.3   Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). 8.F.2         Analyze functions using different representations. Linear, exponential, quadratic, absolute value, step, piecewise defined F.IF.7a, 7b, 7e, 8a, 8b, 9         Analyze functions using different representations. Focus on using key features to guide selection of appropriate type of model function F.IF.7b, 7c, 7e, 8, 9
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BUILDING FUNCTIONS
Eighth Grade Algebra 1 Algebra 2
Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x,y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. 8.F.4   Build a function that models a relationship between two quantities. For F.BF.1, 2, linear, exponential, and quadratic F.BF.1a, 1b, 2   Build new functions from existing functions. Linear, exponential, quadratic, and absolute value for F.BF.4a, linear only F.BF.3, 4a Build a function that models a relationship between two quantities. Include all types of functions studied F.BF.1b   Build new functions from existing functions. Include simple radical, rational, and exponential functions emphasize common effect of each transformation across function types F.BF.3, 4a
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LINEAR, QUADRATIC, AND EXPONENTIAL MODELS
Eighth Grade Algebra 1 Algebra 2
  Construct and compare linear, quadratic, and exponential models and solve problems. F.LE.1a, 1b, 1c, 2, 3   Interpret expressions for functions in terms of the situation they model. Linear and exponential of form f(x)bxk F.LE.5 Construct and compare linear, quadratic, and exponential models and solve problems. Logarithms as solutions for exponentials F.LE.4
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TRIGONOMETRIC FUNCTIONS
  ALGEBRA 2 FOURTH YEAR
  Extend the domain of trigonometric functions using the unit circle. F.TF.1, 2   Model periodic phenomena with trigonometric functions. F.TF.5   Prove and apply trigonometric identities. F.TF.8 Extend the domain of trigonometric functions using the unit circle. () F.TF.3, 4   Model periodic phenomena with trigonometric functions. () F.TF. 6, 7   Prove and apply trigonometric identities. () F.TF. 9
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Which Standards of Mathematical Practice were
emphasized?

1. Making sense of problems and persevere in solving
   them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the
   reasoning of others
4. Model with mathematics
5. Use appropriate tool strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated
   reasoning

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Which Standards of Mathematical Practice were
emphasized?

1. Making sense of problems and persevere in solving
   them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the
   reasoning of others
4. Model with mathematics
5. Use appropriate tool strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated
   reasoning

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A solution to part a