VDOE Mathematics Institute • Grade Band 9-12 • Functions • K-12 Mathematics Institutes • Fall 2010
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Overview of Vertical Progression • Middle School (Function Analysis) • 7.12 … represent relationships with tables, graphs, rules and words • 8.14 … make connections between any two representations (tables, graphs, words, rules)
Overview of Vertical Progression Algebra I (Function Analysis) A.7 The student will investigate and analyze function (linear and quadratic) families and their characteristics both algebraically and graphically, including a) determining whether a relation is a function; b) domain and range; c) zeros of a function; d) x- and y-intercepts; e) finding the values of a function for elements in its domain; and f) making connections between and among multiple representations of functions including concrete, verbal, numeric, graphic, and algebraic.
Overview of Vertical Progression Algebra, Functions and Data Analysis (Function Analysis) AFDA.1 The student will investigate and analyze function (linear, quadratic, exponential, and logarithmic) families and their characteristics. Key concepts include a) continuity; b) local and absolute maxima and minima; c) domain and range; d) zeros; e) intercepts; f) intervals in which the function is increasing/decreasing; g) end behaviors; and h) asymptotes.
Overview of Vertical Progression Algebra, Functions and Data Analysis (Function Analysis) AFDA.4 The student will transfer between and analyze multiple representations of functions, including algebraic formulas, graphs, tables, and words. Students will select and use appropriate representations for analysis, interpretation, and prediction.
Overview of Vertical Progression Algebra 2 (Function Analysis) AII.7 The student will investigate and analyze functions algebraically and graphically. Key concepts include a) domain and range, including limited and discontinuous domains and ranges; b) zeros; c) x- and y-intercepts; d) intervals in which a function is increasing or decreasing; e) asymptotes; f) end behavior; g) inverse of a function; and h) composition of multiple functions. Graphing calculators will be used as a tool to assist in investigation of functions.
Vocabulary • The new 2009 SOL mathematics standards focus on the use of appropriate and accurate mathematics vocabulary.
Wordle – Algebra I, Algebra II, Algebra, Functions & Data Analysis, and Geometry
Reasoning with Functions • Key elements of reasoning and sense • making with functions include: • Using multiple representations of functions • Modeling by using families of functions • Analyzing the effects of different parameters • Adapted from Focus in High School Mathematics: • Reasoning and Sense Making, NCTM, 2009
Using Multiple Representations of Functions • Tables • Graphs or diagrams • Symbolic representations • Verbal descriptions
Algebra Tiles ~ Adding Add the polynomials. (x – 2) + (x + 1) = 2x - 1 17
Algebra Tiles ~ Multiplying (x + 2)(x + 3) x + 2 x + 3 18
Multiply the polynomials using tiles. Create an array of the polynomials (x + 2) (x + 3) x2 + 5x + 6 19
Algebra Tiles ~ Factoring Work backwards from the array. (x – 2) (x – 1) x2 - 3x + 2 20
Polynomial Division • A.2 The student will perform operations on polynomials, including • a) applying the laws of exponents to perform operations on expressions; • b) adding, subtracting, multiplying, and dividing polynomials; and • c) factoring completely first- and second-degree binomials and trinomials in one or two variables. Graphing calculators will be used as a tool for factoring and for confirming algebraic factorizations.
Polynomial Division • Divide (x2 + 5x + 6) by (x + 3) • Common factors only will be used……no long division! Let’s look at division using Algebra Tiles
Represent the polynomials using tiles. x2 + 5x + 6 x + 3
Factor the numerator and denominator. (x + 2) (x + 3) (x + 2)(x + 3) x2 + 5x + 6
Represent the polynomials using tiles. (x + 2)(x + 3) x2 + 5x + 6 (x + 3) Reduce fraction by simplifying like factors to equal 1. x + 2 is the answer
Points of Interest for A.2 from the Curriculum Framework • Operations with polynomials can be represented concretely, pictorially, and symbolically. • VDOE Algeblocks Training Video http://www.vdoe.whro.org/A_Blocks05/index.html
Algeblocks Example (2x + 5) + (x – 4) = 3x + 1
Modeling by Using Families of Functions • Recognize the characteristics of different families of functions • Recognize the common features of each function family • Recognize how different data patterns can be modeled using each family
Analyzing the Effects of Parameters • Different, but equivalent algebraic expressions can be used to define the same function • Writing functions in different forms helps identify features of the function • Graphical transformations can be observed by changes in parameters
Overview of Functions Looking at Patterns • Time vs. Distance Graphs allow students to relate observable patterns in one real world variable (distance) in terms of another real world variable (time).
Slope and Linear Functions • Students can begin to conceptualize slope and look at multiple representations of the same relationship given real world data, tables and graphs.
Exploring Slope using Graphs & Tables +200 +200 +200 +200 +200 +15.87 +15.86 +15.87 +15.86 +16.13 +15.86 +15.86 +15.87 +15.87 The cost is approximately $15.87 for every 200kWh of electricity. Students can then determine that the cost is about $ 0.08 per kWh of electricity.
Exploring Functions • As students progress through high school mathematics, the concept of a function and its characteristics become more complex. Exploring families of functions allow students to compare and contrast the attributes of various functions.
Function Families • Linear: Absolute Value:
Function Families • Quadratic Square Root
Function Families Cube Root Rational:
Function Families • Polynomial: Exponential:
Function Families • Logarithmic:
Function Transformations f(x) = |x| g(x) = |x| + 2 h(x) = |x| - 3 Vertical Transformations
Function Transformations f(x) = |x| g(x) = |x - 2| h(x) = |x + 3| Horizontal Transformations