1. Mathematical Reasoning-

     A. Construct Proofs Based On Deductive Reasoning

• Euclidean and analytic direct proofs.

     B. Construct Indirect Proofs

• Euclidean indirect proofs.

2. Number and Numeration-

     A. Understand and Use Rational and Irrational Numbers

• Determine from the discriminant of a quadratic equation whether the roots are rational or irrational.
• Rationalize denominators.
• Simplifying of algebraic fractions with polynomial denominators.
• Simplify complex fractions.

     B. Recognize the Order of the Real Numbers

• Give rational approximations of irrational numbers to a specific degree of accuracy.

     C. Apply the Properties of the Real Numbers to Various Subsets of Numbers

• Use the properties of real numbers in the development of algebraic skills.

     D. Recognize the Hierarchy of the Complex Number System

• Subsets of complex numbers.

     E. Model the Structure of the Complex Number System

• Imaginary unit of complex numbers.
• Standard form of complex numbers.

3. Operations-

     A. Use Addition, Subtraction, Multiplication, Division, and Exponentiation with Real Numbers and Algebraic Expressions

• Operations with fractions with polynomial denominators.
• Add and subtract rational fractions with monomial and binomial denominators.

     B. Develop an Understanding Of and Use the Composition of Functions and Transformations

• Understand the general concept and symbolism of the composition of transformations.
• Apply the composition of transformations (line reflections, rotations, translations, glide reflections).
• Identify graphs that are symmetric with respect to the axes or origin.
• Isometries (direct and opposite).
• Applications to graphing (inverse functions, symmetry).
• Define and compute compositions of functions and transformations.

     C. Use Transformations on Figures and Functions in the Coordinate Plane

• Apply transformations (line reflection, point reflection, rotation, translation, and dilation) on figures and functions in the coordinate plane.
• Use slope and midpoint to demonstrate transformations.
• Use the ideas of transformations to investigate relationships of two circles.
• Use translation and reflection to investigate the parabola.

     D. Use Rational Exponents on Real Numbers and All Operations on Complex Numbers

• Absolute value of complex numbers.
• Evaluate expressions with fractional exponents.
• Basic arithmetic operations with complex numbers.
• Simplify square roots with negative radicands.
• Use the product of a complex number and its conjugate to express the quotient of two complex numbers.
• Cyclic nature of powers of i.
• Solving quadratic equations.
• Laws of rational exponents.

     E. Combine Functions, Using the Basic Operations and the Composition of Two Functions

• Determine the value of compound functions.
• Pairs of equations.

4. Modeling/Multiple Representation

     A. Represent Problem Situations Symbolically by Using Algebraic Expressions, Sequences, Tree Diagrams, Geometric Figures and Graphs

• Express quadratic, circular, exponential, and logarithmic functions in problem situations algebraically.
• Use symbolic form to represent an explicit rule for a sequence.
• Definition and graph of an inverse variation (hyperbola).

     B. Manipulate Symbolic Representations to Explore Concepts at an Abstract Level.

• Use positive, negative, and zero exponents and be familiar with the laws used in working with expressions containing exponents.
• In the development of the use of exponents, the students should review scientific notation and its use in expressing very large or very small numbers.
• Rewrite the equality log(b)a = c as a = b^c.
• Solve equations, using logarithmic expressions.
• Rewrite expressions involving exponents and logarithms.
• Compound functions.

     C. Choose Appropriate Representations to Facilitate the Solving of a Problem

• Select exponential or logarithmic process to solve an equation.
• Recognize that a variety of phenomena can be modeled by the same type of function.

     D. Develop Meaning for Basic Conic Sections

• Circles.
• Parabloas.
• Using the intercepts, recognize the ellipse and non-rectangular hyperbola.

     E. Model Real-World Problems With Systems of Equations and Inequalities

• Solve systems of equations: linear, quadratic, and linear-quadratic systems.

     F. Model Vector Quantities Both Algebraically and Geometrically

• The LAw of Sines and the Law of Cosines can be used with a wide variety of problems involving triangles, parallelograms and other geometric figures in applications involving resolution of forces both algebraically and geometrically.

     G. Represent Graphically the Sum and Difference of Two Complex Numbers

• Represent the basic operations of addition and subtraction.

     H. Model Quadratic Inequalities Both Algebraically and Graphically

• Use multiple representation to show inequalities algebraically and graphically to find the possible solutions.

     I. Model the Composition of Transformations

• The composition of two line reflections when the two lines are parallel.
• The composition of two rotations about the same point.
• The composition of two translations.
• The composition of a line reflection and a translation in a direction parallel to the line of reflection (glide reflection).

     J. Determine the Effects of Changing Parameters of the Graphs of Functions

• Be able to sketch the effects of changing the value of a in the function y = a^x. Characteristics to be emphasized are:
  the domain of an exponential function is the set of real numbers
  • the range of an exponential function is the set of positive numbers
  the graph of any exponential function will contain the point (0, 1)
  • the exponential function is one-to-one
• If a > 1, the graph rises, but if 0 < a < 1, the graph falls.
• The graphs of y = a^x and y = a^-x, a > 0, and a is not equal to 1, are reflections of each other in the y-axis.
• The logarithmic function is the inverse of the exponential function with the following characteristics:
  since the exponential function is one-to-one, its inverse, the logarithmic function, exists
  the domain of the logarithmic function is the set of positive real numbers
  • the range of the logarithmic function is the set of all real numbers
  the graph of any logarithmic function will contain the point (1, 0).
• The graphs of y = a^x and x = a^y, a > 0, and a is not equal to 1, are reflections of each other in the line y = x.

     K. Use Polynomial, Trigonometric, and Exponential Functions to Model Real-World Relationships

• Recognize when a real-world relationship can be represented by a linear, quadratic, trigonometric, or exponential function.
• Solve real-world problems by using linear, quadratic, trigonometric, and exponential functions.

     L. Use Algenraic Relationships to Analyze the Conic Sections

• Write the equation of a circle with a given center and radius and determine the radius and center of a circle whose equation is in the form  (x - h)^2 + (y - k)^2 = r^2.
• Recognize an equation in the form y = ax^2 + bx + c, a is not equal to 0 as an equation of a parabola, and
  • be able to form a table of values in order to sketch its graph
  • find the axis of symmetry
  • determine the abscissa of the vertex to provide a point of reference for choosing the x-coordinates to be plotted
  • find the y-interceot of the parabola.
• Turning point.
• Maximum or minimum.

     M. Use Circular Functions to Study and Model Periodic Real-World Phenomena

• Use the concept of the unit circle to solve real-world problems involving:
  • radian measure
  • sine
  • cosine
  • tangent
  • reciprocal trigonometric functions.
• Relate reference angles, amplitude, period, and translations to the solution of real-world problems.

     N. Use Graphing Utilities to Create and Explore Geometric and Algebraic Models

• Graph quadratic equations and observe where the graph crosses the x-axis, or note that it does not.

5. Measurement

      A. Use Trigonometry as a Method to Measure Indirectly

• Triangle solutions.
• Right triangle trigonometry.
• Unit circle.
• Angle rotation- the measure of an angle can be a real number.

     B. Understand Error in Measurement and Its Consequence on Subsequent Calculations

• Error of measurement of angles and length of the sides of a triangle and its consequence to the solution of trigonometric problems.

     C. Derive and Apply Formulas Relating Angle Measure and Arc Degree Measure in a Circle

• Express angle measures in terms of radians and degrees.
• Reference and coterminal angles.
• Understand the derivation and apply formulas for sine, cosine, tangent, and their reciprocal trigonometric function.
• Sum and difference of two angles.
• Double and half angles for sine and cosine.
• Vectors.
• Angles formed by arcs, chords, tangents, and secants.

     D. Prove and Apply Theorems Related to Lengths of Segments in a Circle

• Prove and apply theorems related to arcs, chords, tangents, secants, and angles.
• Prove theorems related to congruence and similarity including right triangle proportions.

     E. Define the Trigonometric Functions in Terms of the Unit Circle

• Sine, cosine, tangent, and their reciprocal functions on the unit circle.
• Radian measure.
• Coordinates of a point on the unit circle expressed as (cos A, sin A).
• Special angles 30 degrees, 45 degrees, 60 degrees.
• Reference angles.
• Amplitude and period.
• Reflections in the line y = x.
• Inverse functions.

     F. Relate Trigonometric Relationships to the Area of a Triangle and to General Solutions of Triangles

• Application of the sine function in the solution of the area of a triangle.
• Law of Sines:
  finding a side given ASA or AAS.
  • the ambiguous case (SSA).
  • finding a side given SSA.
• Law of Cosines:
  • finding a side given SAS.
  • finding an angle given SSS.
• Solutions of Triangles.

     Apply the Normal Curve and its Properties to Familiar Contexts

• Intuitive use of the normal curve in real-world situations.
• Mean on the bell curve.
• Standard deviation.

     H. Derive Formulas to Find Measures Such as Length, Area, and Volume in Real-World Context

• Includes Pythagorean Theorem, perimeter of polygons, circumference of circles, area of polygons and circles, and volume of solids.

     I. Design a Statistical Experiment to Study a Problem and Communicate the Outcome, Including Dispersion

• Bias.
• Random sample.
• Choose appropriate statistical measures.

     J. Use Statistical Methods, Including Scatter Plots and Lines of Best Fit, to Make Predictions

• Given data, produce scatter plots and lines of best fit.
• Make predictions.
• Discuss probability of error in predictions.

6. Uncertainty

     A. Judge the Reasonableness of Results Obtained From Applications in Algebra, Geometry, Trigonometry, Probability, and Statistics

• Use substitution as a check for solutions to equations and inequalities.
• Using proof as a check on the validity of geometric constructions.
• Compare histograms with formula-derived solutions for mean, median, variation, and standard deviation.

     B. Judge the Reasonableness of a Graph Produced by a Calculator or Computer

• Determine the effects of changing the parameters of graphs of linear, quadratic, trigonometric, exponential, and circular functions.

     C. Interpret Probabilities in Real-World Situations

• Applications of the probability of exactly, at least, or at most r successes in n trials of a Bernoulli experiment.
• Simple applications of the binomial theorem.

     D. Use a Bernoulli Experiment to Determine Probabilities for Experiments With Exactly Two Outcomes

• Definition of a Bernoulli experiment.
• Case where r successes are assumed to occur first.
• General case.

     E. Use Curve Fitting to Fit Data

• Linear, logarithmic, exponential, and power regressions from scatter plots.
• Linear correlation coefficient.

     F. Create and Interpret Applications of Discrete and Continuous Probability Distributions

• Measures of central tendency.
• Use of the summation notation. ( ? )
• Measures of dispersion.
• Range.
• Mean absolute deviation.
• Variance using the calculator.
• Standard deviation using the calculator.
• Binomial theorem.
• Normal approximation for the binomial distribution.

     G. Make Predictions Based On Interpolations and Extrapolations From Data

• Domain and range.
• Interpolate and extrapolate from graphs of linear, quadratic, trigonometric, circular, exponential, and logarithmic function.

7. Patterns/Functions

     A. Use Function Vocabulary and Notation

• Definition of a relation.
• Determining if a relation is a function.
• Definition of inverse function.
• Notation for absolute value, composite functions.
• Expressing exponential functions as logs.
• Functions (inverse, exponential, logarithmic).

     B. Represent and Analyze Functions, Using Verbal Descriptions, Tables, Equations, and Graphs

• Represent and analyze exponential, logarithmic, quadratic, and trigonometric functions.

     C. Translate Among the Verbal Descriptions, Tables, Equations, and Graphic Forms of Functions

• Relate algebraic expressions to the graphs of functions.

     D. Analyze the Effect of Parametric Changes on the Graphs of Functions

• Use graphing calculators or sketchers to analyze the effects of changing parameters of functions.

     E. Apply Linear, Exponential, and Quadratic Functions in the Solution of Problems

• Solve real-world problems by using linear, exponential, and quadratic functions.

     F. Apply and Interpret Transformations to Functions

• Use ideas of transformations to investigate the relationships between functions.

     G. Model Real-World Situations With the Appropriate Function

• Characteristics of linear, quadratic, trigonometric, circular, exponential, and logarithmic functions.

     H. Apply Axiomatic Structure to Algebra and Geometry

• Algebraic and geometric proof.
• Find the solution of a quadratic equation both algebraically and graphically as a check.
• Use the quotient identities, reciprocal identities, and the Pythagorean identities.

     I. Solve Equations With Complex Roots, Using a Variety of Algebraic and Graphical Methods With Appropriate Tools

• Determine from the discriminant of a quadratic equation whether the roots are imaginary, rational, or irrational.

     J. Evaluate and Form the Composition of Functions

• Evaluate composite functions.
• Use composite functions in problem-solving situations.

     K. Solve Equations, Using Fractions, Absolute Values, and Radicals

• Fractional equations.
• Equations with radicals.
• Linear inequalities.
• Absolute value inequalities.
• Quadratic inequalities.

     L. Use Basic Transformations to Demonstrate Similarity and Congruence of Figures

• Transformations that provide congruence.
• Direct isometries.
• Opposite isometries.
• Transformations that provide similarity.
• Dilation.

     M. Identify and Differentiate Between Direct and Indirect Isometries

• Transformations that provide congruence.

     N. Analyze Inverse Functions, Using Transformations

• Identify inverse functions which are reflections in the line y = x.

     O. Apply the Ideas of Symmetries in Sketching and Analyzing Graphs of Functions

• Simplify the graphing of functions by using symmetries with respect to an axis, the origin, or some other point.

     P. Use the Normal Curve to Answer Questions About Data

• Standard deviation for grouped data.
• Measures of central tendency.

     Q. Develop Methods to Solve Trigonometric Equations and Verify Trigonometric Functions

• Solve first-degree trigonometric equations.
• Solve quadratic trigonometric equations.