Arithmetic Properties: Identification and use of arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable.
Properties of Numbers: Use of properties of numbers to demonstrate whether assertions are true or false.
Roots and Powers: Understanding and use of operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power.
Simplification and Solving Equations: Simplifying and solving equations and inequalities involving absolute values.
Word Problems: Multistep problems, including word problems, involving linear equations and linear inequalities in one variable
Plotting Graphs: Plotting graphs of a linear equation and computing the x- and y- intercepts
Gradient: Verification of the fact that a point lies on a line, given an equation of the line. Derivation of linear equations by using the point-slope formula.
Parallel and Perpendicular Lines: Concepts of parallel lines and perpendicular lines and how those slopes are related.
Equation in Two Variables: Solving system of two linear equations in two variables algebraically and interpretating the answer graphically.
Word Problems: Addition, subtraction, multiplication, and division of monomials and polynomials. Multistep problems, including word problems, by using these techniques.
Solving Equations Using Factors: Application of basic factoring techniques to second-and simple third-degree polynomials.
Quadratic Equations: Solving a quadratic equation by factoring or completing the square.
Word Problems: Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems.
Functions: Students understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions.
Domain and Range: Students determine the domain of independent variables and the range of dependent variables defined by a graph, a set of ordered pairs, or a symbolic expression.
Quadratic Equations: Students know the quadratic formula and are familiar with its proof by completing the square.
Quadratic Equation: Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations.
Graph of Quadratic Function: Students graph quadratic functions and know that their roots are the x- intercepts.
Word Problems on Quadratic Equations: Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity.
Logical Arguments: Students use and know simple aspects of a logical argument.
Inductive and Deductive reasoning: Students explain the difference between inductive and deductive reasoning and identify and provide examples of each.
Hypothesis and Conclusion: Students identify the hypothesis and conclusion in logical deduction.
Justifying Results: Students use properties of the number system to judge the validity of results, to justify each step of a procedure, and to prove or disprove statements:
Construction of Arguments: Students use properties of numbers to construct simple, valid arguments (direct and indirect) for, or formulate counterexamples to, claimed assertions.
Geometrical Proofs: Students write geometric proofs, including proofs by contradiction.
Students prove basic theorems involving congruence and similarity.
Congruency: Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles.
Triangle Inequality Theorem: Students know and are able to use the triangle inequality theorem.
Theorems: Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles.
Mensuration: Students know, derive, and solve problems involving the perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures.
Change in Dimensions: Students determine how changes in dimensions affect the perimeter, area, and volume of common geometric figures and solids.
Exterior and Interior Angles: Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems.
Angles: Students prove relationships between angles in polygons by using properties of complementary, supplementary, vertical, and exterior angles.
Pythagoras Theorem: Students prove the Pythagorean theorem.
Use of Pythagoras Theorem: Students use the Pythagorean theorem to determine distance and find missing lengths of sides of right triangles.
Construction: Students perform basic constructions with a straightedge and compass, such as angle bisectors, perpendicular bisectors, and the line parallel to a given line through a point off the line.
Theorems of Coordinate Geometry: Students prove theorems by using coordinate geometry, including the midpoint of a line segment, the distance formula, and various forms of equations of lines and circles.
Trigonometric Functions: Students know the definitions of the basic trigonometric functions defined by the angles of a right triangle. They also know and are able to use elementary relationships between them. For example, tan( x ) = sin( x )/cos( x ), (sin( x )) 2 + (cos( x )) 2 = 1.
Angle and Side Relationship: Students know and are able to use angle and side relationships in problems with special right triangles, such as 30°, 60°, and 90° triangles and 45°, 45°, and 90° triangles.
Rigid Motion of Figures: Students know the effect of rigid motions on figures in the coordinate plane and space, including rotations, translations, and reflections.
Solving Equations: Students solve equations and inequalities involving absolute value.
Linear Equations and Inequalities: Students solve systems of linear equations and inequalities (in two or three variables) by substitution, with graphs, or with matrices.
Operations on Polynomials: Students are adept at operations on polynomials, including long division.
Factorization of Polynomials: Students factor polynomials representing the difference of squares, perfect square trinomials, and the sum and difference of two cubes.
Graphical Representation of Complex Numbers: Students demonstrate knowledge of how real and complex numbers are related both arithmetically and graphically. In particular, they can plot complex numbers as points in the plane.
Operations on Complex Numbers: Students add, subtract, multiply, and divide complex numbers.
Quadratic Equations: Students solve and graph quadratic equations by factoring, completing the square, or using the quadratic formula. Students apply these techniques in solving word problems. They also solve quadratic equations in the complex number system.
Maxima and Minima: Students graph quadratic functions and determine the maxima, minima, and zeros of the function.
Laws of Logarithms
Relationship Between Exponents and Logarithms: Students understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
Exponential Growth and Decay: Students know the laws of fractional exponents, understand exponential functions, and use these functions in problems involving exponential growth and decay.
Logrithms: Students use the definition of logarithms to translate between logarithms in any base.
Graph of a Conic Section: Students demonstrate and explain how the geometry of the graph of a conic section (e.g., asymptotes, foci, eccentricity) depends on the coefficients of the quadratic equation representing it.
Graph of a Quadratic Equation: Given a quadratic equation of the form ax2 + by2 + cx + dy + e = 0, students can use the method for completing the square to put the equation into standard form and can recognize whether the graph of the equation is a circle, ellipse, parabola, or hyperbola. Students can then graph the equation.
Fundamental Counting Principle: Students use fundamental counting principles to compute combinations and permutations.
Combination and Permutation: Students use combinations and permutations to compute probabilities.
Binomial Theorem: Students know the binomial theorem and use it to expand binomial expressions that are raised to positive integer powers.
Mathematical Induction: Students apply the method of mathematical induction to prove general statements about the positive integers.
Series: Students find the general term and the sums of arithmetic series and of both finite and infinite geometric series.
Relationship between Degrees and Radians: Students understand the notion of angle and how to measure it, in both degrees and radians. They can convert between degrees and radians.
Graph of Trignometric Functions: Students know the definition of sine and cosine as y- and x- coordinates of points on the unit circle and are familiar with the graphs of the sine and cosine functions.
Trigonometric Identities: Students know and can prove the identity cos2 (x) + sin2 (x) = 1:
Graph of Complex Trigonometric Functions: Students graph functions of the form f(t) = A sin ( Bt + C ) or f(t) = A cos ( Bt + C) and interpret A, B, and C in terms of amplitude, frequency, period, and phase shift.
Slope of a Line: Students know that the tangent of the angle that a line makes with the x- axis is equal to the slope of the line.
Graph of Inverse Trigonometric Functions: Students know the definitions of the inverse trigonometric functions and can graph the functions.
Half-angle and Double-angle Formulae for sines and cosines: Students demonstrate an understanding of half-angle and double-angle formulas for sines and cosines and can use those formulas to prove and/ or simplify other trigonometric identities.
Area of a Triangle Students determine the area of a triangle, given one angle and the two adjacent sides.
Polar Coordinates: Students are familiar with polar coordinates. In particular, they can determine polar coordinates of a point given in rectangular coordinates and vice versa.
Complex Numbers: Students are familiar with complex numbers. They can represent a complex number in polar form and know how to multiply complex numbers in their polar form.
DeMoivre's theorem: Students know DeMoivre's theorem and can give n th roots of a complex number given in polar form.
Word Problems: Students are adept at using trigonometry in a variety of applications and word problems.
Polar and Rectangular Coordinates Students are familiar with, and can apply, polar coordinates and vectors in the plane. In particular, they can translate between polar and rectangular coordinates and can interpret polar coordinates and vectors graphically.
Complex Number and DeMoivre's Theorem: Students are adept at the arithmetic of complex numbers. They can use the trigonometric form of complex numbers and understand that a function of a complex variable can be viewed as a function of two real variables. They know the proof of DeMoivre's theorem.
Mathematical Induction Students can give proofs of various formulas by using the technique of mathematical induction.
Fundamental Theorem of Algebra: Students know the statement of, and can apply, the fundamental theorem of algebra.
Conic Sections: Students are familiar with conic sections, both analytically and geometrically:
Asymptotes: Students find the roots and poles of a rational function and can graph the function and locate its asymptotes.
Functions: Students demonstrate an understanding of functions and equations defined parametrically and can graph them.
Limit of Sequence: Students are familiar with the notion of the limit of a sequence and the limit of a function as the independent variable approaches a number or infinity.
Gauss-Jordan elimination. Method: Students solve linear equations in any number of variables by using Gauss-Jordan elimination.
The Gauss-Jordan method as Row Operations on the Coefficient Matrix.
Row Echelon Form of Matrices
Addition on Matrices and Vectors.
Solution of Linear Systems: Students demonstrate an understanding that linear systems are inconsistent (have no solutions), have exactly one solution, or have infinitely many solutions.
Geometric Interpretation of Vectors
Inverse of a Matrix
Determinamts: Students compute the determinants of 2 x 2 and 3 x 3 matrices and are familiar with their geometric interpretations as the area and volume of the parallelepipeds spanned by the images under the matrices of the standard basis vectors in two-dimensional and three-dimensional spaces.
Row Reduction Method and Cramer's Rule: Students know that a square matrix is invertible if, and only if, its determinant is nonzero. They can compute the inverse to 2 x 2 and 3 x 3 matrices using row reduction methods or Cramer's rule.
Dot Product: Students compute the scalar (dot) product of two vectors in n- dimensional space and know that perpendicular vectors have zero dot product.
Probability and Statistics
Independent Events: Students know the definition of the notion of independent events and can use the rules for addition, multiplication, and complementation to solve for probabilities of particular events in finite sample spaces.
Conditional Probability: Students know the definition of conditional probability and use it to solve for probabilities in finite sample spaces.
Discrete Random Variables: Students demonstrate an understanding of the notion of discrete random variables by using them to solve for the probabilities of outcomes, such as the probability of the occurrence of five heads in 14 coin tosses.
Standard Distribution: Students are familiar with the standard distributions (normal, binomial, and exponential) and can use them to solve for events in problems in which the distribution belongs to those families.
Standard Deviation: Students determine the mean and the standard deviation of a normally distributed random variable.
Mean, Median and Mode: Students know the definitions of the mean, median, and mode of a distribution of data and can compute each in particular situations.
Variance and Standard Deviation of Distributed Data: Students compute the variance and the standard deviation of a distribution of data.
Methods to Describe Distribution: Students organize and describe distributions of data by using a number of different methods, including frequency tables, histograms, standard line and bar graphs, stem-and-leaf displays, scatter plots, and box-and-whisker plots.
Advanced Placement Probability and Statistics
Independent Events: Students solve probability problems with finite sample spaces by using the rules for addition, multiplication, and complementation for probability distributions and understand the simplifications that arise with independent events.
Conditional Probability: Students know the definition of conditional probability and use it to solve for probabilities in finite sample spaces.
Discrete Random Variables: Students demonstrate an understanding of the notion of discrete random variables by using this concept to solve for the probabilities of outcomes, such as the probability of the occurrence of five or fewer heads in 14 coin tosses.
Continuous Random Variable: Students understand the notion of a continuous random variable and can interpret the probability of an outcome as the area of a region under the graph of the probability density function associated with the random variable.
Mean of Discrete Random Variable: Students know the definition of the mean of a discrete random variable and can determine the mean for a particular discrete random variable.
Variance: Students know the definition of the variance of a discrete random variable and can determine the variance for a particular discrete random variable.
Standard Distribution: Students demonstrate an understanding of the standard distributions (normal, binomial, and exponential) and can use the distributions to solve for events in problems in which the distribution belongs to those families.
Mean and the Standard Deviation: Students determine the mean and the standard deviation of a normally distributed random variable.
Central Limit Theorem: Students know the central limit theorem and can use it to obtain approximations for probabilities in problems of finite sample spaces in which the probabilities are distributed binomially.
Mean, Median and Mode: Students know the definitions of the mean, median, and mode of distribution of data and can compute each of them in particular situations.
Regression: Students find the line of best fit to a given distribution of data by using least squares regression.
Correlation: Students know what the correlation coefficient of two variables means and are familiar with the coefficient's properties.
Methods to Describe Distribution: Students organize and describe distributions of data by using a number of different methods, including frequency tables, histograms, standard line graphs and bar graphs, stem-and-leaf displays, scatter plots, and box-and-whisker plots.
Population Distribution: Students know basic facts concerning the relation between the mean and the standard deviation of a sampling distribution and the mean and the standard deviation of the population distribution.
Confidence Intervals: Students determine confidence intervals for a simple random sample from a normal distribution of data and determine the sample size required for a desired margin of error.
P- value: Students determine the P- value for a statistic for a simple random sample from a normal distribution.
Chi - square: Students are familiar with the chi- square distribution and chi- square test and understand their uses.
Limits: formal definition and the graphical interpretation of limit of values of functions. One-sided limits, infinite limits, and limits at infinity. Definition of convergence and divergence of a function as the domain variable approaches either a number or infinity:
Theorems: Proof and use of theorems evaluating the limits of sums, products, quotients, and composition of functions.
Use of Graphical Calculators: Use of graphical calculators to verify and estimate limits.
Special Limits: Proofs and use of special limits, such as the limits of (sin(x))/x and (1-cos(x))/x as x tends to 0.
Continuity of Function: Knowledge of both the formal definition and the graphical interpretation of continuity of a function.
Intermediate and the Extreme Value Theorem: An Understanding and the application of the intermediate value theorem and the extreme value theorem.
Differntiation: Formal definition of the derivative of a function at a point and the notion of differentiability, derivative of a function as the slope of the tangent line to the graph of the function.
Slope of a Line: Students demonstrate an understanding of the derivative of a function as the slope of the tangent line to the graph of the function.
Rate of Change: Students demonstrate an understanding of the interpretation of the derivative as an instantaneous rate of change. Students can use derivatives to solve a variety of problems from physics, chemistry, economics, and so forth that involve the rate of change of a function.
Diffentiation and Continuity: Students understand the relation between differentiability and continuity.
Functions: Students derive derivative formulas and use them to find the derivatives of algebraic, trigonometric, inverse trigonometric, exponential, and logarithmic functions.
Chain Rule: Students know the chain rule and its proof and applications to the calculation of the derivative of a variety of composite functions.
Rolle's Theorem and L'Hopital's Rule: Students know and can apply Rolle's theorem, the mean value theorem, and L'Hopital's rule.
Maxima and Minima: Students use differentiation to sketch, by hand, graphs of functions. They can identify maxima, minima, inflection points, and intervals in which the function is increasing and decreasing.
Newton's Method of Estimation: Students know Newton's method for approximating the zeros of a function.
Word problems: Students use differentiation to solve related rate problems in a variety of pure and applied contexts.
Riemann Sums: Students know the definition of the definite integral by using Riemann sums.
Word Problems on Integration: Students apply the definition of the integral to model problems in physics, economics, and so forth, obtaining results in terms of integrals.
Fundamental Theorem of Calculus: Students demonstrate knowledge and proof of the fundamental theorem of calculus and use it to interpret integrals as antiderivatives.
Definite Integrals: Students use definite integrals in problems involving area, velocity, acceleration, volume of a solid, area of a surface of revolution, length of a curve, and work.
Indefinite Integrals: Students know the definitions and properties of inverse trigonometric functions and the expression of these functions as indefinite integrals.
Simpson's Rule and Newton's Method: Students understand the algorithms involved in Simpson's rule and Newton's method. They use calculators or computers or both to approximate integrals numerically.
Test for Convergence: Students demonstrate an understanding of the definitions of convergence and divergence of sequences and series of real numbers. By using such tests as the comparison test, ratio test, and alternate series test, they can determine whether a series converges.
Power Series: Students differentiate and integrate the terms of a power series in order to form new series from known ones.
Taylor Series: Students calculate Taylor polynomials and Taylor series of basic functions, including the remainder term.
Growth and Decay Problems: Students know the techniques of solution of selected elementary differential equations and their applications to a wide variety of situations, including growth-and-decay problems.