2011 WORKSHOP on FUNCTIONS

Numbers in parentheses indicate session (1-8) at which topic was covered.

By the end of grade eight, students will understand various numerical representations, including square roots, exponents and scientific notation; use and apply geometric properties of plane figures, including congruence and the Pythagorean theorem; use symbolic algebra to represent situations and solve problems, especially those that involve linear relationships; solve linear equations, systems of linear equations and inequalities; use equations, tables and graphs to analyze and interpret linear functions; use and understand set theory and simple counting techniques; determine the theoretical probability of simple events; and make inferences from statistical data, particularly data that can be modeled by linear functions.  Instruction and assessment should include the appropriate use of manipulatives and technology. Topics should be represented in multiple ways, such as concrete/pictorial, verbal/written, numeric/data-based, graphical, and symbolic. Concepts should be introduced and used, where appropriate, in the context of realistic phenomena.

ALGEBRA  Students will use linear algebra to represent, analyze and solve problems. They will use equations, tables, and graphs to investigate linear relations and functions, paying particular attention to slope as a rate of change.

M8A3. Students will understand relations and linear functions.
(1)    a. Recognize a relation as a correspondence between varying quantities.
(1)    b. Recognize a function as a correspondence between inputs and outputs where the output for each input must be unique.
(1)    c. Distinguish between relations that are functions and those that are not functions.
(1)    d. Recognize functions in a variety of representations and a variety of contexts.
(1)    e. Use tables to describe sequences recursively and with a formula in closed form.

(Also at Session 1:  number line,  irrationality of sqrt(2),  decimal expansions of rational numbers)

(2)    f. Understand and recognize arithmetic sequences as linear functions with whole number input values.
(2)    g. Interpret the constant difference in an arithmetic sequence as the slope of the associated linear function.
(2)    h. Identify relations and functions as linear or nonlinear.
(1-2)  i. Translate among verbal, tabular, graphic, and algebraic representations

DATA ANALYSIS AND PROBABILITY Students will use and understand set theory and simple counting techniques; determine the theoretical probability of simple events; and make inferences from data, particularly data that can be modeled by linear functions.

M8D4. Students will organize, interpret, and make inferences from statistical data

a. Gather data that can be modeled with a linear function.
b. Estimate and determine a line of best fit from a scatter plot.

MATH I

ALGEBRA  Students will explore functions and solve simple equations. Students will simplify and operate with radical, polynomial, and rational expressions.

MM1A1. Students will explore and interpret the characteristics of functions, using graphs, tables, and simple algebraic techniques.
(1)    a. Represent functions using function notation.
(3)    b. Graph the basic functions f(x) = x^n, where n = 1 to 3,  f(x) = sqrt(x),  f(x) = |x|, and f(x) = 1/x.

(5)    c. Graph transformations of basic functions including vertical shifts, stretches, and shrinks, as well as reflections across the         x- and y-axes.
(4)    d. Investigate and explain the characteristics of a function: domain, range, zeros, intercepts, intervals of increase and                     decrease, maximum and minimum values, and end behavior.
(4)    e. Relate to a given context the characteristics of a function, and use graphs and tables to investigate its behavior.
(4)    f. Recognize sequences as functions with domains that are whole numbers.
(4)    g. Explore rates of change, comparing constant rates of change (i.e., slope) versus variable rates of change. Compare rates of         change of linear, quadratic, square root, and other function families.
(4)    h. Determine graphically and algebraically whether a function has symmetry and whether it is even, odd, or neither.
(2)    i. Understand that any equation in x can be interpreted as the equation f(x) = g(x), and interpret the solutions of the                     equation as the x-value(s) of the intersection point(s) of the graphs of y = f(x) and y = g(x).

Mathematics 2  This is the second course in a sequence of courses designed to provide students with arigorous program of study in mathematics. It includes complex numbers; quadratic, piecewise, and exponential functions; right triangles, and right triangular trigonometry; properties of circles; and statistical inference. (Prerequisite: Successful completion of Math 1).  Instruction and assessment should include the appropriate use of manipulatives and technology. Topics should be represented in multiple ways, such as concrete/pictorial, verbal/written, numeric/data-based, graphical, and symbolic. Concepts should be introduced and used, where appropriate, in the context of realistic phenomena.

ALGEBRA  Students will investigate piecewise, exponential, and quadratic functions, using numerical, analytical, and graphical approaches, focusing on the use of these functions in problem-solving situations. Students will solve equations and inequalities and explore inverses of functions.

MM2A1. Students will investigate step and piecewise functions, including greatest integer and absolute value functions.
(3)    a. Write absolute value functions as piecewise functions.
(3)    b. Investigate and explain characteristics of a variety of piecewise functions including domain, range, vertex, axis of                     symmetry, zeros, intercepts, extrema, points of discontinuity, intervals over which the function is constant, intervals of increase and decrease, and rates of change.
(3)    c. Solve absolute value equations and inequalities analytically, graphically, and by using appropriate technology.

MM2A2. Students will explore exponential functions.
(6)   a. Extend properties of exponents to include all integer exponents.
(7)   b. Investigate and explain characteristics of exponential functions, including domain and range, asymptotes, zeros,                         intercepts, intervals of increase and decrease, rates of change, and end behavior.
(7)    c. Graph functions as transformations of f(x) = ax.
(8)    d. Solve simple exponential equations and inequalities analytically, graphically, and by using appropriate technology.
(7)    e. Understand and use basic exponential functions as models of real phenomena.

(7)    f. Understand and recognize geometric sequences as exponential functions with domains that are whole numbers.

(7)    g. Interpret the constant ratio in a geometric sequence as the base of the associated exponential function.

MM2A3. Students will analyze quadratic functions in the forms f(x) = ax2 + bx + c and f(x) = a( x - b/(2a) )2 + k.
(4)    a. Convert between standard and vertex form.
(4)    b. Graph quadratic functions as transformations of the function f(x) = x2.
(4)    c. Investigate and explain characteristics of quadratic functions, including domain, range, vertex, axis of symmetry, zeros,             intercepts, extrema, intervals of increase and decrease, and rates of change.
(5)    d. Explore arithmetic series and various ways of computing their sums.
(5)    e. Explore sequences of partial sums of arithmetic series as examples of quadratic functions.

MM2A4. Students will solve quadratic equations and inequalities in one variable.
(5)    a. Solve equations graphically using appropriate technology.
(5)    b. Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula.
(5)    c. Analyze the nature of roots using technology and using the discriminant.
d. Solve quadratic inequalities both graphically and algebraically, and describe the solutions using linear inequalities.

MM2A5. Students will explore inverses of functions.
(8)
a. Discuss the characteristics of functions and their inverses, including one-to-oneness, domain, and range.
(8)    b. Determine inverses of linear, quadratic, and power functions and functions of the form f(x) = a/x, including the use of         restricted domains.
(8)    c. Explore the graphs of functions and their inverses.
(8)    d. Use composition to verify that functions are inverses of each other.

MM2D2. Students will determine an algebraic model to quantify the association between two quantitative variables.

a. Gather and plot data that can be modeled with linear and quadratic functions.
b. Examine the issues of curve fitting by finding good linear fits to data using simple methods such as the median-median line and "eyeballing".
c. Understand and apply the processes of linear and quadratic regression for curve fitting using appropriate technology.
d. Investigate issues that arise when using data to explore the relationship between two variables, including confusion                     between correlation and causation.

MM3A1. Students will analyze graphs of polynomial functions of higher degree.
a. Graph simple polynomial functions as translations of the function f(x) = axn.
b. Understand the effects of the following on the graph of a polynomial function: degree, lead coefficient, and multiplicity of real zeros.
c. Determine whether a polynomial function has symmetry and whether it is even, odd, or neither.
d. Investigate and explain characteristics of polynomial functions, including domain and range, intercepts, zeros, relative and absolute extrema, intervals of increase and decrease, and end behavior.

MM3A2. Students will explore logarithmic functions as inverses of exponential functions.
(8)    a. Define and understand the properties of nth roots.
(8)    b. Extend properties of exponents to include rational exponents.
(8)    c. Define logarithmic functions as inverses of exponential functions.
(8)    d. Understand and use properties of logarithms by extending laws of exponents.
(8)    e. Investigate and explain characteristics of exponential and logarithmic functions including domain and range, asymptotes, zeros, intercepts, intervals of increase and decrease, and rate of change.
f. Graph functions as transformations of f(x) = ax, f(x) = logax, f(x) = ex, f(x) = ln x.
(7)    g. Explore real phenomena related to exponential and logarithmic functions including half-life and doubling time.