### AI: Algebra I

AI.N.3: Calculate and apply ratios, proportions, rates, and percentages to solve a range of consumer and practical problems.

AI.N.4: Use estimation to judge the reasonableness of results of computations and of solutions to problems involving real numbers, including approximate error in measurement and the approximate value of square roots. (Reminder: This is without the use of calculators.)

AI.P.1: Recognize, describe, and extend patterns governed by a linear, quadratic, or exponential functional relationship or by a simple iterative process (e.g., the Fibonacci sequence).

AI.P.3: Demonstrate an understanding of relations and functions. Identify the domain, range, and dependent and independent variables of functions.

AI.P.4: Translate between different representations of functions and relations: graphs, equations, sets of ordered pairs (scatter plots), verbal, and tabular.

AI.P.5: Demonstrate an understanding of the relationship between various representations of a line. Determine a line’s slope and x- and y-intercepts from its graph or from a linear equation that represents the line.

AI.P.6: Find a linear function describing a line from a graph or a geometric description of the line (e.g., by using the point-slope or slope y-intercept formulas). Explain the significance of a positive, negative, zero, or undefined slope.

AI.P.7: Find linear functions that represent lines either perpendicular or parallel to a given line and through a point (e.g., by using the point-slope form of the equation).

AI.P.8: Add, subtract, and multiply polynomials with emphasis on 1st- and 2nd-degree polynomials.

AI.P.9: Demonstrate facility in symbolic manipulation of polynomial and rational expressions by rearranging and collecting terms, factoring [e.g., a² – b² = (a + b)(a – b), x² + 10x + 21 = (x + 3) (x + 7), 5x the the 4th power + 10x³ – 5x² = 5x² (x² + 2x – 1)], identifying and canceling common factors in rational expressions, and applying the properties of positive integer exponents.

AI.P.10: Divide polynomials by monomials with emphasis on 1st- and 2nd-degree polynomials.

AI.P.11: Perform basic arithmetic operations with rational expressions and functions.

AI.P.12: Find solutions to quadratic equations (with real roots) by factoring, completing the square, or using the quadratic formula. Demonstrate an understanding of the equivalence of the methods.

AI.P.13: Solve equations and inequalities, including those involving absolute value of linear expressions (e.g., |x – 2| > 5), and apply to the solution of problems.

AI.P.14: Solve everyday problems (e.g., compound interest and direct and inverse variation problems) that can be modeled using linear or quadratic functions. Apply appropriate graphical or symbolic methods to the solution.

AI.P.15: Solve everyday problems (e.g., mixture, rate, and work problems) that can be modeled using systems of linear equations or inequalities. Apply algebraic and graphical methods to the solution.

AI.D.1: Select, create, and interpret an appropriate graphical representation (e.g., scatter plot, table, stem-and-leaf plots, circle graph, line graph, and line plot) for a set of data, and use appropriate statistics (e.g., mean, median, range, and mode) to communicate information about the data. Use these notions to compare different sets of data.

### AII: Algebra II

AII.P.1: Describe, complete, extend, analyze, generalize, and create a wide variety of patterns, including iterative and recursive patterns such as Fibonacci Numbers and Pascal’s Triangle.

AII.P.2: Identify arithmetic and geometric sequences and finite arithmetic and geometric series. Use the properties of such sequences and series to solve problems, including finding the formula for the general term and the sum, recursively and explicitly.

AII.P.3: Understand functional notation, evaluate a function at a specified point in its domain, and perform operations on functions with emphasis on the domain and range.

AII.P.4: Understand exponential and logarithmic functions and their basic arithmetic properties, including change of base and formulas for exponential of a sum and logarithm of a product.

AII.P.5: Given algebraic, numeric, and/or graphical representations, recognize functions as polynomial, rational, logarithmic, or exponential, and describe their behavior.

AII.P.6: Find solutions to radical equations; find solutions to quadratic equations (with real coefficients and real or complex roots) graphically, by factoring, by completing the square, or by using the quadratic formula.

AII.P.7: Solve a variety of equations and inequalities using algebraic, graphical, and numerical methods, including the quadratic formula. Include polynomial, exponential, and logarithmic functions, expressions involving the absolute values, and simple rational expressions.

AII.P.9: Use symbolic, numeric, and graphical methods to solve systems of equations and/or inequalities involving algebraic, exponential, and logarithmic expressions. Describe the relationships among the methods.

AII.P.10: Solve everyday problems that can be modeled using polynomial, rational, exponential, logarithmic, and step functions; absolute values; and square roots. Apply appropriate graphical, tabular, or symbolic methods to the solution. Include compound interest, exponential growth and decay, and direct and inverse variation problems.

AII.P.11: Recognize translations and scale changes of a given function f(x) resulting from substitutions for the various parameters a, b, c, and d in y = af(b(x + c/b)) + d. In particular, describe qualitatively the effect of such changes on polynomial, rational, exponential, and logarithmic functions.

AII.P.12: Simplify rational expressions. Solve rational equations and inequalities.

AII.G.1: Define the sine, cosine, and tangent of an acute angle. Apply to the solution of problems.

AII.G.2: Explain the identity sin²q + cos²q = 1. Relate the identity to the Pythagorean theorem.

AII.G.3: Relate geometric and algebraic representations of lines and simple curves.

AII.D.1: Select an appropriate graphical representation for a set of data and use appropriate statistics (e.g., quartile or percentile distribution) to communicate information about the data, including box plots.

AII.D.2: Use combinatorics (e.g., fundamental counting principle, permutations, and combinations) to solve problems, including computing geometric probabilities and probabilities of compound events.

### G: Geometry

G.G.1: Know correct geometric notation, including the notation for line segment (AB) and angle (

G.G.2: Recognize special types of polygons (e.g., isosceles triangles, parallelograms, and rhombuses).

G.G.3: Apply properties of sides, diagonals, and angles in special polygons; identify their parts and special segments (e.g., altitudes, midsegments); determine interior angles for regular polygons.

G.G.5: Detect symmetries of geometric figures.

G.G.6: Apply the triangle inequality and other inequalities associated with triangles (e.g., the longest side is opposite the greatest angle) to prove theorems and to solve problems.

G.G.7: Use properties and theorems about congruent and similar figures and about perpendicular and parallel lines to solve problems.

G.G.8: Write simple proofs of theorems in geometric situations, such as theorems about triangles, congruent and similar figures, and perpendicular and parallel lines (e.g., the longest side is opposite the greatest angle, two lines parallel to a third are parallel to each other; perpendicular bisectors of line segments are the set of all points equidistant from the two end points).

G.G.9: Distinguish between postulates and theorems. Use inductive and deductive reasoning, as well as proof by contradiction. Given a conditional statement, write its inverse, converse, and contrapositive.

G.G.11: Draw congruent and similar figures using a compass, straightedge, or protractor. Justify the constructions by logical argument.

G.G.12: Apply congruence and similarity correspondences (e.g., DABC @ DXYZ) and properties of the figures to find missing parts of geometric figures, and provide logical justification.

G.G.13: Apply properties of angles, parallel lines, arcs, radii, chords, tangents, and secants to solve problems.

G.G.14: Solve simple triangle problems using the triangle angle sum property and/or the Pythagorean theorem; study and understand more than one proof of this theorem.

G.G.15: Use the properties of special triangles (e.g., isosceles, equilateral, 30º-60º-90º, 45º-45º-90º) to solve problems.

G.G.16: Define the sine, cosine, and tangent of an acute angle. Apply to the solution of problems.

G.G.17: Demonstrate an understanding of the relationship between various representations of a line. Determine a line's slope and x- and y-intercepts from its graph or from a linear equation that represents the line. Find a linear equation describing a line from a graph or a geometric description of the line (e.g., by using the point-slope or slope y-intercept formulas). Explain the significance of a positive, negative, zero, or undefined slope.

G.G.18: Using rectangular coordinates, calculate midpoints of segments, slopes of lines and segments, and distances between two points, and apply the results to the solutions of problems.

G.G.19: Find linear equations that represent lines either perpendicular or parallel to a given line and through a point (e.g., by using the point-slope form of the equation).

G.G.20: Draw the results and interpret transformations on figures in the coordinate plane such as translations, reflections, rotations, scale factors, and the results of successive transformations. Apply transformations to the solution of problems.

G.G.21: Demonstrate the ability to visualize solid objects and recognize their projections, cross sections, and graph points in 3-D.

G.G.22: Find and use measures of perimeter, circumference, and area of common geometric figures such as parallelograms, trapezoids, circles, and triangles.

G.G.23: Find and use measures of lateral areas, surface areas, and volumes of prisms, pyramids, spheres, cylinders, and cones, and relate these measures to each other using formulas.

G.G.24: Relate changes in the measurement (including units) of one attribute of an object to changes in other attributes.

### PCT: Precalculus

PCT.N.2: Plot complex numbers using both rectangular and polar coordinates systems. Represent complex numbers using polar coordinates, i.e., a + bi = r (cos theta + i sin theta).

PCT.P.1: Relate the number of roots of a polynomial to its degree. Solve quadratic equations with complex coefficients, including use of completing the square.

PCT.P.2: Demonstrate an understanding of the trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent). Relate the functions to their geometric definitions.

PCT.P.4: Given algebraic, numeric, and/or graphical representations, recognize functions as polynomial, rational, logarithmic, or exponential.

PCT.P.5: Combine functions by composition, as well as by addition, subtraction, multiplication, and division.

PCT.P.6: Identify whether a function has an inverse and when functions are inverses of each other; explain why the graph of a function and its inverse are reflections of one another over the line y = x.

PCT.P.7: Identify maximum and minimum values of functions. Apply to the solution of problems.

PCT.P.8: Describe the translations and scale changes of a given function f(x) resulting from substitutions for the various parameters a, b, c, and d in y = a f(b (x + c/b)) + d. In particular, describe the effect of such changes on polynomial, rational, exponential, and logarithmic functions.

PCT.P.9: Derive and apply basic trigonometric identities i.e., sin²theta + cos²theta = 1, tan²theta + 1 = sec²theta and the laws of sines and cosines.

PCT.P.10: Demonstrate an understanding of the formulas for the sine and cosine of the sum or the difference of two angles. Relate the formulas to DeMoivre's theorem and use them to prove other trigonometric identities. Apply to the solution of problems.

PCT.P.11: Understand, predict, and interpret the effects of the parameters a, w, b, and c on the graph of y = asin (infinity (x — b)) + c; do the same for the cosine and tangent. Use to model periodic processes.

PCT.P.16: Identify maximum and minimum values of functions in simple situations. Apply to the solution of problems.

PCT.G.2: Use vectors to solve problems. Describe addition of vectors, multiplication of a vector by a scalar, and the dot product of two vectors, both symbolically and geometrically. Use vector methods to obtain geometric results.

PCT.G.3: Apply properties of angles, parallel lines, arcs, radii, chords, tangents, and secants to solve problems.

PCT.D.3: Compare the results of simulations (e.g., random number tables, random functions, and area models) with predicted probabilities.

### PS: Probability and Statistics

#### PS.10: Approximate a line of best fit (trend line) given a set of data (e.g., scatter plot).

Correlation last revised: 12/2/2009

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