Mathematics Content Standards

Represent real numbers as points on the line and as decimals (possibly infinite).

Express rational numbers as fractions, finite or repeating decimals, or percents; convert between these.

Understand the density of rational and irrational numbers in real numbers, i.e., between every two real numbers there are rational and irrational numbers.

Identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable.

Compare and order rational numbers in various forms including integers, percents, and positive and negative fractions and decimals.

Know, understand and use the standard algorithms for addition, subtraction, multiplication and division of whole numbers and decimals.

Add, subtract, multiply, and divide rational numbers in problem situations.

Know and apply the Fundamental Theorem of Arithmetic.

Understand number theoretic concepts of primes, factors, and multiples; be able to calculate least common multiple and greatest common factor.

Understand “absolute value” for real numbers, its interpretation as distance from the zero on the number line, and recognize that Mathematics √(a²) = |a| for both positive and negative numbers a.

Use algebraic techniques to solve rate, ratio and proportion problems, such as problems involving percent mixtures, rates of work, unit rates, rates of change, speed, density, pressure, scale drawings, and similar triangles.

Solve consumer problems involving sales tax, tips, interest, discounts, compound interest, markups, commissions, percent increase, and percent decrease using whole numbers, fractions, decimals and percents.

Understand the associative, distributive and commutative properties and use them to simplify calculations.

Approximate the values of irrational numbers as they arise from problem situations (e.g., Mathematics, √2 and their rational multiples).

Estimate squares and square roots, cubes and cube roots, and understand the concept of the nth root.

Give answers to numerical problems to a specified degree of accuracy.

Express numbers in scientific notation, including negative exponents, in appropriate problem situations and perform computations with numbers in scientific notation.

Understand and use the rules of exponents.

Understand and use the inverse relationship between exponents and logarithms.

Deduce and use simple laws of logarithms.

Use the definition of logarithms to translate between logarithms in different bases. Use the properties of logarithms to simplify logarithmic numeric expressions and to identify their approximate values.

Use addition, subtraction, multiplication and division to solve problems involving monomials, binomials, polynomials, and algebraic fractions and mixed expressions.

Use the field properties of the rational and real number systems to prove or disprove mathematical statements concerning arithmetic and algebraic combinations of numbers.

Know how real and complex numbers are related both arithmetically and graphically. In particular, plot complex numbers as points in the plane.

Add, subtract, multiply, and divide complex numbers.

Understand the concepts of relation and function.

Represent relations and functions through tables, graphs, verbal rules, or symbolic rules.

Know that a function is not defined if its domain is not explicitly given; recognize that functions incompletely defined by formulas or rules alone are generally construed as having as domain all the numbers to which the formula or rule can apply, and be able to determine these in common cases; understand the terms “dependent variable” and “independent variable”.

Determine whether a relation defined by a graph, a set of ordered pairs, or a symbolic expression is a function and justify the conclusion.

Derive and use the formulas for general term and sum of arithmetic series and both finite and infinite geometric series.

Derive the summation formulas for arithmetic series and for both finite and infinite geometric series.

Use and interpret formulas to answer questions about quantities and their relationships, e.g., the relationship between distance, speed, and time.

Find compositions, inverses and algebraic combinations of given functions.

Solve linear equations such as (3/2)(2x-3) - 4(x-2) = 12, and be able to explain how and why the procedures which succeed in this case fail for the equations: 2(2x-3) - 4(x-2) = 12 and 2(2x-3) -4(x-2) = 2.

Given an equation of a line, determine whether a particular point lies on it.

Graph a linear equation in the plane, e.g., 2x+6y-4 = 0; find the intercepts and slope of that line; and explain what sort of line would occur if one or more of the three coefficients were changed to zero.

Sketch the region in the plane defined by one or more linear inequalities (e.g., the region defined by 2x+6y < 4 and x-y < 8).

Find an equation of a line given sufficient geometric information, such as two points on the line, or one point and the slope.

Understand the concepts of parallel lines and perpendicular lines and how their slopes are related and why.

Solve equations and inequalities involving absolute values.

Solve systems of linear equations (in two or three variables) by substitution, with graphs, and with matrices.

Be proficient in the four operations on polynomials, including “long division” with remainder.

Factor polynomials representing the difference of squares, perfect squares of binomials and the sum and difference of two cubes.

Simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms.

Add, subtract, multiply, divide, reduce, and evaluate rational expressions with monomial and polynomial denominators and simplify complicated rational expressions.

Solve quadratic equations by factoring, completing the square, and the quadratic formula.

Apply quadratic equations to problems such as those involving the motion of an object under the force of gravity.

Know the binomial coefficients and use the binomial theorem to expand positive integer powers of binomials.

Graph quadratic functions and determine their maxima, minima, and zeros.

Determine how the graph of a parabola changes as a, b, and c vary in the equation: y = a(x-b) 2 + c.

Given a quadratic equation of the form ax 2 + by 2 + cx + dy + e = 0, use the method of completing the square to put the equation into standard form; recognize whether the graph of the equation is a circle, ellipse, parabola, or hyperbola; and graph the equation.

When a conic section is described by a quadratic equation in standard form, explain how its geometry (e.g., asymptotes, foci, eccentricity) depends on the coefficients of the equation.

Know the laws of fractional exponents and deduce them from the laws of integral exponents.

Understand exponential functions, and use these functions in problems involving exponential growth and decay.

Solve quadratic equations in the complex number system.

Know the statement of the Fundamental Theorem of Algebra and deduce that every polynomial of degree n is a product of n complex linear factors.

Identify symmetry in common objects as examples of point, line, or rotational symmetry.

Classify two or three dimensional solid objects and see relationships among them according to their geometric attributes (e.g., shape, number of corners, number and shape of faces, edges, and vertices).

Represent three-dimensional objects using two-dimensional projections or perspective drawings.

Know, whether as axioms or as deduced from other axioms,

• sufficient conditions for the congruence of triangles

• properties of angles formed by transversals to parallels

• that the angles of a triangle sum to 180 degrees.

Know that when three parallels are cut by two transversals, the ratio of the segments formed on one transversal is equal to the ratio of the corresponding segments formed on the other.

Know the relationship of ratio to similarity for rectilinear figures, and be able to write numerical proportions related to given similar figures.

Know the following terms, and be able to construct illustrations for them by straight-edge and compass constructions:

• adjacent, vertical, complementary and supplementary angles

• angle bisectors

• perpendicular bisector of a segment

• a line through a given point, perpendicular to a given line

• a line through a given point, parallel to a given line

• a partition of a given segment into n equal subsegments, for any whole number n.

Understand what skew lines in space are, and make an illustrative drawing.

Use definitions and theorems to solve problems related to convex polygons:

• determine when they are similar or congruent

• compute the sum of the interior angles

• compute the sum of the exterior angles

• in the case of quadrilaterals of special types, such as parallelograms and trapezoids, determine the relationships of the diagonals, sides and angles.

Know the vocabulary related to special triangles and their properties (right, isosceles, equilateral), and

1. know the meaning of altitude, base, median, interior angle, exterior angle, and the fundamental theorems on their relationships.

2. use the definition of area for a rectangle to deduce by partitions and congruence the area for parallelograms, triangles, and trapezoids.

Know and be able to prove theorems concerning arcs, chords, radii, secants and tangents of a circle, the angles they form and the ratios of certain of their lengths.

Prove the Pythagorean theorem and its converse, making explicit the axioms and preceding theorems used in the proofs.

Obtain the equation of a circle in a coordinate plane, given its center and radius; conversely, given the equation, find the radius and the coordinates of the center.

Be able to prove simple Euclidean theorems using coordinate systems and algebra, e.g., that the diagonals of a parallelogram bisect each other.

Convert between US customary and metric systems of measurement, length, area, volume, weight, time and temperature and convert composite measures within and across systems, e.g., feet per second vs. meters per hour.

Use composite measures, such as grams per cubic centimeters and person-hours.

Compute areas of triangles and other polygons.

Know and derive formulas for and solve problems involving the perimeter, circumference, and area of simple two dimensional figures.

Understand the relationships between linear, square, and cubic measures and determine how changes in dimensions affect the perimeter, area, and volume of common geometric figures and solids.

Know the formulas, and be able to compute the volumes and surface areas of prisms, pyramids, cylinders and cones.

Compute arc lengths and the area of sectors of a circle.

Interpret scale drawings, maps and other diagrams.

Solve problems using:

• the Pythagorean theorem, e.g., find the length of the missing side of a right triangle and the lengths of other line segments

• basic trigonometric ratios, e.g., sine and cosine for angles between 0 and 180°

• angle and side relationships in problems with special right triangles, such as 30°, 60°, and 90° triangles and 45°, 45°, and 90° triangles.

Understand the notion of angle and how to measure it in both degrees and radians, and convert between the two.

Know the law of sines and the law of cosines and apply those laws to solve problems; e.g., determine the area of a triangle, given one angle and the two adjacent sides.

Understand the definitions of sine and cosine as the y and x coordinates of the appropriate point on the unit circle.

Know the identity cos 2 (x) + sin 2 (x) = 1; prove this identity by using the Pythagorean theorem, and conversely prove the Pythagorean theorem as a consequence of this identity.

Prove trigonometric identities and simplify expressions by using the identity
cos 2 (x) + sin 2 (x) = 1.

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. Also, show how the sum a cos mt + b sin mt can be rewritten in the form r cos (mt + e), exhibiting its amplitude and phase.

Know the definitions of the tangent and cotangent functions, secant and cosecant functions, and inverse trigonometric functions, and graph them.

Know that the tangent of the angle that a line makes with the x-axis is equal to the slope of the line.

Compute, by hand, the values of the trigonometric functions and the inverse trigonometric functions at various standard points.

Understand the addition formulas for sines and cosines and their proofs; use those formulas to prove identities and to simplify expressions.

Know how the addition formulas imply the half-angle and double-angle formulas; use the latter to prove identities and to simplify expressions.

Convert between rectangular and polar coordinates of a point; obtain the polar equation of a curve given in rectangular coordinates and vice versa.

Understand complex numbers (e.g., represent a complex number in polar form; know how to multiply complex numbers in polar form).

Know DeMoivre's theorem; use it to find powers and roots of complex numbers. Relate DeMoivre's theorem to the addition formulas for sine and cosine.

Know that probability measures the likelihood that an event will occur by assigning a number between 0 and 1 to that event. If the probability of an event is p, then 1-p is the probability of the event not occurring.

Find the probability in certain simple situations, e.g., multiple coin tosses, drawing cards from a deck, throwing dice.

Use fundamental counting principles to compute combinations and permutations.

Use combinations and permutations to compute probabilities.

Solve probability problems with finite sample spaces by using the rules for addition, multiplication, and complementation for probabilities; understand the simplifications that arise with independent events.

Know the definition of conditional probability and use it to solve probabilities for finite sample spaces.

Know the definitions of mean, median, mode of a data distribution, and compute each of them in particular situations.

Compute the variance and the standard deviation of a distribution of data.

Know the meaning of, and compute, the minimum, the lower quartile, the median, the upper quartile and the maximum of a data set.

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, scatterplots, and box-and-whisker plots.

Understand the concept of normal distribution; estimate probabilities using its graph or tabulated values.

Find the line of best fit to a given distribution of data by using least squares regression.

Reduce rectangular matrices to row echelon forms.

Know both the formal definition and the graphical interpretation of limit of values of functions, including one-sided limits, infinite limits, and limits at infinity.

Prove and use theorems evaluating the limits of sums, products, quotients, and compositions of functions

Prove and use special limits, such as the limits of (sin(x))/x and (1-cos(x))/x as x tends to 0.

Know of both the formal definition and the graphical interpretation of continuity of a function.

Understand and apply the intermediate value theorem and the extreme value theorem.

Know the formal definition of the derivative of a function at a point and the notion of differentiability including:

• the derivative of a function as the slope of the tangent line to the graph of the function

• the interpretation of the derivative as an instantaneous rate of change in order to solve a variety of problems from physics, chemistry, economics, and so forth that involve the rate of change of a function

• the relation between differentiability and continuity

• deriving the formulas for derivatives of sums, products and quotients of differentiable functions, and applying them

Know the chain rule and its proof and applications to the calculation of the derivative of a variety of composite functions.

Find the derivatives of parametrically defined functions, and use “implicit differentiation” in applied problems.

Compute derivatives of higher orders.

Know and apply Rolle's theorem, the mean value theorem, and L'Hospital's rule.

Use differentiation to help sketch by hand the graphs of functions; identify maxima, minima, inflection points, and intervals of increase, decrease, and convexity (up or down).

Know Newton's method for approximating the zeros of a function.

Use differentiation to solve optimization (maximum-minimum problems) in pure and applied contexts.

Use differentiation to solve related rate problems in pure and applied contexts.

Know the definition of the definite integral by using Riemann sums and use this definition to approximate integrals.

Apply the derivative and integral to model problems in physics and economics.

Know the Fundamental Theorem of Calculus and its proof; use this theorem to interpret integrals as antiderivatives and to find integrals in closed form.

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.

Compute, by hand, the integrals of functions by using techniques of integration such as substitution, integration by parts, and trigonometric substitution; combine these techniques when appropriate.

Compute, by hand, the integrals of rational functions by combining the techniques above with the algebraic techniques of partial fractions and completing the square.

Know the definitions and properties of inverse trigonometric functions and the expression of these functions as indefinite integrals.

Compute the integrals of trigonometric functions by using the techniques noted above.

Understand and use Simpson's rule and the trapezoidal rule.

Understand improper integrals as limits of definite integrals .

Know the techniques of solution of selected elementary differential equations and their applications, including growth-and-decay problems and oscillating springs or linear circuits.

Understand the definitions of convergence and divergence of sequences and series of real numbers; determine whether a series converges.

Understand and compute the radius (interval) of convergence of power series.

Differentiate and integrate the terms of a power series in order to form new series.

Calculate Taylor polynomials and Taylor series of basic functions, including the remainder term. Use Taylor polynomials for approximations to a desired degree of accuracy.

Know the chain rule and its proof and applications to the calculation of the derivative of a variety of composite functions.