### AI: Algebra I

AI.3.0: Students solve equations and inequalities involving absolute values.

AI.4.0: Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x - 5) + 4(x - 2) = 12.

AI.5.0: Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step.

AI.6.0: Students graph a linear equation and compute the x-and y-intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4).

AI.7.0: Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the point-slope formula.

AI.8.0: Students understand the concepts of parallel lines and perpendicular lines and how those slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point.

AI.9.0: Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets.

AI.10.0: Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques.

AI.11.0: Students apply basic factoring techniques to second- and simple third-degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials.

AI.12.0: Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms.

AI.14.0: Students solve a quadratic equation by factoring or completing the square.

AI.16.0: 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.

AI.17.0: 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.

AI.18.0: Students determine whether a relation defined by a graph, a set of ordered pairs, or a symbolic expression is a function and justify the conclusion.

AI.19.0: Students know the quadratic formula and are familiar with its proof by completing the square.

AI.20.0: Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations.

AI.21.0: Students graph quadratic functions and know that their roots are the x-intercepts.

AI.22.0: Students use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points.

AI.24.1: Students explain the difference between inductive and deductive reasoning and identify and provide examples of each.

AI.25.2: Students judge the validity of an argument according to whether the properties of the real number system and the order of operations have been applied correctly at each step.

AI.25.3: Given a specific algebraic statement involving linear, quadratic, or absolute value expressions or equations or inequalities, students determine whether the statement is true sometimes, always, or never.

### G: Geometry

G.1.0: Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning.

G.2.0: Students write geometric proofs, including proofs by contradiction.

G.4.0: Students prove basic theorems involving congruence and similarity.

G.5.0: Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles.

G.6.0: Students know and are able to use the triangle inequality theorem.

G.7.0: Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles.

G.8.0: Students know, derive, and solve problems involving the perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures.

G.9.0: Students compute the volumes and surface areas of prisms, pyramids, cylinders, cones, and spheres; and students commit to memory the formulas for prisms, pyramids, and cylinders.

G.10.0: Students compute areas of polygons, including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids.

G.11.0: Students determine how changes in dimensions affect the perimeter, area, and volume of common geometric figures and solids.

G.12.0: Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems.

G.13.0: Students prove relationships between angles in polygons by using properties of complementary, supplementary, vertical, and exterior angles.

G.14.0: Students prove the Pythagorean theorem.

G.15.0: Students use the Pythagorean theorem to determine distance and find missing lengths of sides of right triangles.

G.16.0: 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.

G.17.0: 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.

G.18.0: 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.

G.19.0: Students use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side.

G.21.0: Students prove and solve problems regarding relationships among chords, secants, tangents, inscribed angles, and inscribed and circumscribed polygons of circles.

G.22.0: Students know the effect of rigid motions on figures in the coordinate plane and space, including rotations, translations, and reflections.

### AII: Algebra II

AII.1.0: Students solve equations and inequalities involving absolute value.

AII.2.0: Students solve systems of linear equations and inequalities (in two or three variables) by substitution, with graphs, or with matrices.

AII.3.0: Students are adept at operations on polynomials, including long division.

AII.4.0: Students factor polynomials representing the difference of squares, perfect square trinomials, and the sum and difference of two cubes.

AII.5.0: 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.

AII.7.0: Students add, subtract, multiply, divide, reduce, and evaluate rational expressions with monomial and polynomial denominators and simplify complicated rational expressions, including those with negative exponents in the denominator.

AII.8.0: 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.

AII.9.0: Students demonstrate and explain the effect that changing a coefficient has on the graph of quadratic functions; that is, students can determine how the graph of a parabola changes as a, b, and c vary in the equation y = a(x - b)² + c.

AII.10.0: Students graph quadratic functions and determine the maxima, minima, and zeros of the function.

AII.11.2: Students judge the validity of an argument according to whether the properties of real numbers, exponents, and logarithms have been applied correctly at each step.

AII.12.0: Students know the laws of fractional exponents, understand exponential functions, and use these functions in problems involving exponential growth and decay.

AII.15.0: Students determine whether a specific algebraic statement involving rational expressions, radical expressions, or logarithmic or exponential functions is sometimes true, always true, or never true.

AII.16.0: 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.

AII.17.0: 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.

AII.18.0: Students use fundamental counting principles to compute combinations and permutations.

AII.19.0: Students use combinations and permutations to compute probabilities.

AII.22.0: Students find the general term and the sums of arithmetic series and of both finite and infinite geometric series.

AII.24.0: Students solve problems involving functional concepts, such as composition, defining the inverse function and performing arithmetic operations on functions.

AII.25.0: Students use properties from number systems to justify steps in combining and simplifying functions.

### T: Trigonometry

T.2.0: 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.

T.3.1: Students prove that this identity is equivalent to the Pythagorean theorem (i.e., students can prove this identity by using the Pythagorean theorem and, conversely, they can prove the Pythagorean theorem as a consequence of this identity).

T.3.2: Students prove other trigonometric identities and simplify others by using the identity cos²(x) + sin²(x) = 1. For example, students use this identity to prove that sec²(x) = tan²(x) + 1.

T.4.0: 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.

T.5.0: Students know the definitions of the tangent and cotangent functions and can graph them.

T.6.0: Students know the definitions of the secant and cosecant functions and can graph them.

T.7.0: Students know that the tangent of the angle that a line makes with the x-axis is equal to the slope of the line.

T.9.0: Students compute, by hand, the values of the trigonometric functions and the inverse trigonometric functions at various standard points.

T.10.0: Students demonstrate an understanding of the addition formulas for sines and cosines and their proofs and can use those formulas to prove and/ or simplify other trigonometric identities.

T.11.0: 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.

T.12.0: Students use trigonometry to determine unknown sides or angles in right triangles.

T.15.0: Students are familiar with polar coordinates. In particular, they can determine polar coordinates of a point given in rectangular coordinates and vice versa.

T.16.0: Students represent equations given in rectangular coordinates in terms of polar coordinates.

T.17.0: 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.

T.18.0: Students know DeMoivre's theorem and can give nth roots of a complex number given in polar form.

### MA: Mathematical Analysis

MA.1.0: 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.

MA.2.0: 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.

MA.5.1: Students can take a quadratic equation in two variables; put it in standard form by completing the square and using rotations and translations, if necessary; determine what type of conic section the equation represents; and determine its geometric components (foci, asymptotes, and so forth).

MA.5.2: Students can take a geometric description of a conic section - for example, the locus of points whose sum of its distances from (1, 0) and (-1, 0) is 6 - and derive a quadratic equation representing it.

MA.6.0: Students find the roots and poles of a rational function and can graph the function and locate its asymptotes.

### LA: Linear Algebra

LA.4.0: Students perform addition on matrices and vectors.

LA.7.0: Students demonstrate an understanding of the geometric interpretation of vectors and vector addition (by means of parallelograms) in the plane and in three-dimensional space.

LA.8.0: Students interpret geometrically the solution sets of systems of equations. For example, the solution set of a single linear equation in two variables is interpreted as a line in the plane, and the solution set of a two-by-two system is interpreted as the intersection of a pair of lines in the plane.

LA.12.0: Students compute the scalar (dot) product of two vectors in n-dimensional space and know that perpendicular vectors have zero dot product.

### PS: Probability and Statistics

PS.1.0: 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.

PS.3.0: 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.

PS.4.0: 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.

PS.6.0: Students know the definitions of the mean, median, and mode of a distribution of data and can compute each in particular situations.

PS.8.0: 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, scatterplots, and box-and-whisker plots.

### APPS: Advanced Placement Probability and Statistics

APPS.1.0: 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.

APPS.3.0: 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.

APPS.5.0: Students know the definition of the mean of a discrete random variable and can determine the mean for a particular discrete random variable.

APPS.7.0: 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.

APPS.9.0: 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.

APPS.10.0: Students know the definitions of the mean, median, and mode of distribution of data and can compute each of them in particular situations.

APPS.12.0: Students find the line of best fit to a given distribution of data by using least squares regression.

APPS.13.0: Students know what the correlation coefficient of two variables means and are familiar with the coefficient's properties.

APPS.14.0: 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, scatterplots, and box-and-whisker plots.

APPS.15.0: Students are familiar with the notions of a statistic of a distribution of values, of the sampling distribution of a statistic, and of the variability of a statistic.

APPS.16.0: 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.

### C: Calculus

C.1.3: Students prove and use special limits, such as the limits of (sin(x))/x and (1 - cos(x))/x as x tends to 0.

C.2.0: Students demonstrate knowledge of both the formal definition and the graphical interpretation of continuity of a function.

C.13.0: Students know the definition of the definite integral by using Riemann sums. They use this definition to approximate integrals.

C.16.0: 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.

Correlation last revised: 11/2/2009

This correlation lists the recommended Gizmos for this state's curriculum standards. Click any Gizmo title below for more information.