### PA: Pre Algebra

#### PA.I: Students will expand number sense to understand, perform operations, and solve problems with rational numbers.

PA.I.1: Compute fluently with understanding and make reasonable estimates with rational numbers.

PA.I.1.a: Compute fluently using all four operations with integers, and explain why the corresponding algorithms work.

PA.I.1.b: Compute fluently using all four operations with rational numbers, including negative fractions and decimals, and explain why the corresponding algorithms work.

PA.I.2: Analyze relationships among rational numbers, including negative rational numbers, and operations involving these numbers.

PA.I.2.a: Order rational numbers in various forms, including scientific notation (positive and negative exponents), and place numbers on a number line.

PA.I.2.b: Predict the effect of operating with fractions, decimals, percents, and integers as an increase or a decrease of the original value.

PA.I.2.d: Recognize and use the inverse operations of adding and subtracting a fixed number, multiplying and dividing by a fixed number, and computing squares of whole numbers and taking square roots of perfect squares.

PA.I.3: Solve problems involving rational numbers using addition, subtraction, multiplication, and division.

PA.I.3.a: Recognize the absolute value of a rational number as its distance from zero.

PA.I.3.b: Simplify numerical expressions, including those with whole number exponents and absolute values, using the order of operations.

PA.I.3.c: Solve problems involving rational numbers, percents, and proportions.

#### PA.II: Students will use proportion and similarity to solve problems.

PA.II.1: Model and illustrate meanings of ratios, percents, and decimals.

PA.II.1.a: Compare ratios to determine if they are equivalent.

PA.II.1.b: Compare ratios using the unit rate.

PA.II.1.c: Represent percents as ratios based on 100 and decimals as ratios based on powers of ten.

PA.II.1.d: Graph proportional relationships and identify the unit rate as the slope of the related line.

PA.II.2: Solve a wide variety of problems using ratios and proportional reasoning.

PA.II.2.a: Set up and solve problems involving proportional reasoning using variables.

PA.II.2.b: Solve percent problems, including problems involving discounts, interest, taxes, tips, and percent increase or decrease.

PA.II.2.c: Solve ratio and rate problems using informal methods.

PA.II.3: Recognize similar polygons and use properties of similar triangles to solve problems and define the slope of a line.

PA.II.3.a: Define similar polygons as polygons with corresponding angles congruent and corresponding sides that are proportional.

PA.II.3.b: Identify pairs of similar triangles using two pairs of congruent angles, or two pairs of proportional sides with congruent included angles.

PA.II.3.c: Find missing lengths of similar triangles, including inaccessible lengths, using proportions.

PA.II.3.d: Define the slope of a line as the ratio of the vertical change to the horizontal change between two points, and show that the slope is constant using similarity of right triangles.

#### PA.III: Students will develop fluency with the language and operations of algebra to analyze and represent relationships.

PA.III.1: Generalize and express patterns using algebraic expressions.

PA.III.1.a: Compare representations of a relation using tables, graphs, algebraic symbols, and mathematical rules.

PA.III.1.b: Describe simple patterns using a mathematical rule or algebraic expression.

PA.III.1.c: Create and extend simple numerical and visual patterns.

PA.III.2: Evaluate, simplify, and solve algebraic expressions, equations, and inequalities.

PA.III.2.a: Evaluate algebraic expressions, including those with whole number exponents, when given values for the variable(s).

PA.III.2.b: Simplify algebraic expressions using the order of operations, algebraic properties, and exponent rules.

PA.III.2.c: Solve single-variable linear equations and inequalities, including those that must be simplified on one side or those with variables on both sides of an equation.

PA.III.3: Represent relationships using graphs, tables, and other models.

PA.III.3.a: Identify approximate rational coordinates when given the graph of a point on a rectangular coordinate system.

PA.III.3.b: Graph ordered pairs of rational numbers on a rectangular coordinate system.

PA.III.3.c: Graph linear equations using ordered pairs or tables.

PA.III.3.d: Recognize that all first order equations produce linear graphs.

PA.III.3.e: Model real-world problems using graphs, tables, equations, manipulatives, and pictures, and identify extraneous information.

#### PA.IV: Students will use algebraic, spatial, and logical reasoning to solve geometry and measurement problems.

PA.IV.1: Apply the properties of proportionality of different units of measure.

PA.IV.1.b: Create and interpret scale drawings and approximate distance on maps using scale factors.

PA.IV.1.c: Solve problems using scale factors.

PA.IV.2: Derive formulas for surface areas and volume of three-dimensional figures.

PA.IV.2.a: Derive formulas for and calculate surface area and volume of right prisms and cylinders using appropriate units.

PA.IV.2.b: Explain that if a scale factor describes how corresponding lengths in two similar objects are related, then the square of the scale factor describes how corresponding areas are related and the cube of the scale factor describes how corresponding volumes are related.

PA.IV.2.c: Find lengths, areas, and volumes of similar figures, using the scale factor.

PA.IV.2.d: Select appropriate two- and three-dimensional figures to model real-world objects, and solve a variety of problems involving surface areas and volumes of cylinders and prisms.

#### PA.V: Students will understand concepts from probability and statistics and apply statistical methods to solve problems.

PA.V.1: Calculate probabilities of events and compare theoretical and experimental probability.

PA.V.1.b: Calculate the probability of an event or sequence of events with and without replacement using models.

PA.V.1.d: Make approximate predictions using theoretical probability and proportions.

PA.V.1.e: Collect and interpret data to show that as the number of trials increases, experimental probability approaches the theoretical probability.

PA.V.2: Formulate questions and answer the questions by organizing and analyzing data.

PA.V.2.a: Formulate questions that can be answered through data collection and analysis.

PA.V.2.b: Determine the 25th and 75th percentiles (first and third quartiles) to obtain information about the spread of data.

PA.V.2.c: Graphically summarize data of a single variable using histograms and box-andwhisker plots.

PA.V.2.d: Compute the mean and median of a numerical characteristic and relate these values to the histogram of the data.

PA.V.2.e: Use graphical representations and numerical summaries to answer questions and interpret data.

### A1: Algebra 1

#### A1.I: Students will expand number sense to understand, perform operations, and solve problems with real numbers.

A1.I.1: Represent real numbers as points on the number line and distinguish rational numbers from irrational numbers.

A1.I.1.a: Define a rational number as a point on the number line that can be expressed as the ratio of two integers, and points that cannot be so expressed as irrational.

A1.I.1.c: Classify pi and square roots of non-perfect square numbers as irrational.

A1.I.1.d: Place rational and irrational numbers on a number line between two integers.

A1.I.2: Compute fluently and make reasonable estimates with rational and irrational numbers.

A1.I.2.a: Simplify, add, subtract, multiply, and divide expressions with square roots.

A1.I.2.b: Evaluate and simplify numerical expressions containing rational numbers and square roots using the order of operations.

A1.I.2.d: Calculate the measures of the sides of a right triangle using the Pythagorean Theorem.

#### A1.II: Students will extend concepts of proportion to represent and analyze linear relations.

A1.II.1: Represent and analyze the slope of a line.

A1.II.1.a: Identify the slope of a line when given points, a graph, or an equation.

A1.II.1.b: Identify horizontal and vertical lines given the equations or slopes.

A1.II.1.c: Determine the effect of changes in slope or y-intercept in y = mx + b.

A1.II.1.d: Determine and explain the meaning of slopes and intercepts using real-world examples.

A1.II.2: Model and interpret problems having a constant rate of change using linear functions.

A1.II.2.a: Write algebraic expressions or equations to generalize visual patterns, numerical patterns, relations, data sets, or scatter plots.

A1.II.2.b: Represent linear equations in slope-intercept form, y = mx + b, and standard form, Ax+ By =C.

A1.II.2.c: Distinguish between linear and non-linear functions by examining a table, equation, or graph.

A1.II.2.d: Interpret the slope of a linear function as a rate of change in real-world situations.

A1.II.3: Represent and analyze linear relationships using algebraic equations, expressions, and graphs.

A1.II.3.a: Write the equation of a line when given two points or the slope and a point on the line.

A1.II.3.b: Approximate the equation of a line given the graph of a line.

A1.II.3.c: Identify the x- and y-intercepts from an equation or graph of a line or a table of values.

A1.II.3.d: Graph linear relations and inequalities by plotting points, by finding x- and y-intercepts, or by using the slope and any point on the line.

#### A1.III: Students will develop fluency with the language and operations of algebra to analyze and represent relationships.

A1.III.1: Simplify polynomials and the quotient of monomials.

A1.III.1.a: Simplify and evaluate monomial expressions and formulas.

A1.III.1.c: Multiply monomials by a polynomial.

A1.III.1.d: Multiply binomials.

A1.III.1.e: Simplify the quotient of monomials using positive exponents.

A1.III.2: Solve and interpret linear equations and inequalities in various situations including real-world problems.

A1.III.2.a: Solve single-variable linear equations and inequalities algebraically and graphically.

A1.III.2.b: Solve real-world problems involving constant rates of change.

A1.III.2.c: Solve equations for a specified variable.

A1.III.2.d: Solve proportions that include algebraic first-degree expressions.

A1.III.3: Solve and interpret pairs of linear equations and inequalities.

A1.III.3.a: Solve systems of two linear equations graphically and algebraically with and without technology.

A1.III.3.b: Determine the number of possible solutions for a system of two linear equations.

A1.III.3.c: Graph a system of linear inequalities and identify the solution.

A1.III.4: Factor polynomials with common monomial factors and factor simple quadratic expressions.

A1.III.4.a: Find the greatest common monomial factor of a polynomial.

A1.III.4.b: Factor trinomials with integer coefficients of the form x2 + bx + c.

A1.III.4.c: Factor the difference of two squares and perfect square trinomials.

A1.III.5: Solve quadratic equations using factoring or by taking square roots.

A1.III.5.a: Solve quadratic equations that can be simplified to the form x2 = a where a ¡Ý 0 by taking square roots.

A1.III.5.b: Solve quadratic equations using factoring.

A1.III.5.c: Write a quadratic equation when given the solutions.

#### A1.IV: Students will understand concepts from statistics and apply statistical methods to solve problems.

A1.IV.1: Summarize, display, and analyze bivariate data.

A1.IV.1.a: Collect, record, organize, and display a set of data with at least two variables.

A1.IV.1.b: Determine whether the relationship between two variables is approximately linear or non-linear by examination of a scatter plot.

A1.IV.1.c: Characterize the relationship between two linear related variables as having positive, negative, or approximately zero correlation.

A1.IV.2: Estimate, interpret, and use lines fit to bivariate data.

A1.IV.2.a: Estimate the equation of a line of best fit to make and test conjectures.

A1.IV.2.b: Interpret the slope and y-intercept of a line through data.

### G: Geometry

#### G.I: Students will use algebraic, spatial, and logical reasoning to solve geometry problems.

G.I.1: Use inductive and deductive reasoning to develop mathematical arguments.

G.I.1.a: Write conditional statements, converses, and inverses, and determine the truth value of these statements.

G.I.2: Analyze characteristics and properties of angles.

G.I.2.a: Use accepted geometric notation for lines, segments, rays, angles, similarity, and congruence.

G.I.2.b: Identify and determine relationships in adjacent, complementary, supplementary, or vertical angles and linear pairs.

G.I.2.c: Classify angle pairs formed by two lines and a transversal.

G.I.2.e: Prove lines parallel or perpendicular using slope or angle relationships.

G.I.3: Analyze characteristics and properties of triangles.

G.I.3.a: Prove congruency and similarity of triangles using postulates and theorems.

G.I.3.b: Prove the Pythagorean Theorem in multiple ways, find missing sides of right triangles using the Pythagorean Theorem, and determine whether a triangle is a right triangle using the converse of the Pythagorean Theorem.

G.I.3.c: Prove and apply theorems involving isosceles triangles.

G.I.3.d: Apply triangle inequality theorems.

G.I.3.e: Identify medians, altitudes, and angle bisectors of a triangle, and the perpendicular bisectors of the sides of a triangle, and justify the concurrency theorems.

G.I.4: Analyze characteristics and properties of polygons and circles.

G.I.4.a: Use examples and counterexamples to classify subsets of quadrilaterals.

G.I.4.b: Prove properties of quadrilaterals using triangle congruence relationships, postulates, and theorems.

G.I.4.c: Derive, justify, and use formulas for the number of diagonals, lines of symmetry, angle measures, perimeter, and area of regular polygons.

G.I.4.d: Define radius, diameter, chord, secant, arc, sector, central angle, inscribed angle, and tangent of a circle, and solve problems using their properties.

G.I.4.e: Show the relationship between intercepted arcs and inscribed or central angles, and find their measures.

G.I.5: Perform basic geometric constructions, describing and justifying the procedures used.

G.I.5.a: Investigate geometric relationships using constructions.

G.I.5.c: Construct perpendicular and parallel lines.

G.I.5.d: Justify procedures used to construct geometric figures.

G.I.5.e: Discover and investigate conjectures about geometric properties using constructions.

G.I.6: Analyze characteristics and properties of three-dimensional figures.

G.I.6.a: Identify and classify prisms, pyramids, cylinders and cones based on the shape of their base(s).

G.I.6.b: Identify three-dimensional objects from different perspectives using nets, crosssections, and two-dimensional views.

G.I.6.c: Describe the symmetries of three-dimensional figures.

#### G.II: Students will use the language and operations of algebra to explore geometric relationships with coordinate geometry.

G.II.1: Describe the properties and attributes of lines and line segments using coordinate geometry.

G.II.1.a: Verify the classifications of geometric figures using coordinate geometry to find lengths and slopes.

G.II.1.b: Find the distance between two given points and find the coordinates of the midpoint.

G.II.1.c: Write an equation of a line perpendicular or a line parallel to a line through a given point.

G.II.a: Graph a circle given the equation in the form(x − h)2 + (y − k)2 = r 2 , and write the equation when given the graph.

#### G.III: Students will extend concepts of proportion and similarity to trigonometric ratios.

G.III.1: Use triangle relationships to solve problems.

G.III.1.b: Identify the trigonometric relationships of sine, cosine, and tangent with the appropriate ratio of sides of a right triangle.

G.III.1.c: Express trigonometric relationships using exact values and approximations.

G.III.2: Use the trigonometric ratios of sine, cosine, and tangent to represent and solve for missing parts of triangles.

G.III.2.a: Find the angle measure in degrees when given the trigonometric ratio.

G.III.2.b: Find the trigonometric ratio given the angle measure in degrees, using a calculator.

G.III.2.c: Find unknown measures of right triangles using sine, cosine, and tangent functions and inverse trigonometric functions.

#### G.IV: Students will use algebraic, spatial, and logical reasoning to solve measurement problems.

G.IV.1: Find measurements of plane and solid figures.

G.IV.1.b: Develop surface area and volume formulas for polyhedra, cones, and cylinders.

G.IV.1.c: Determine perimeter, area, surface area, lateral area, and volume for prisms, cylinders, pyramids, cones, and spheres when given the formulas.

G.IV.2: Solve real-world problems using visualization and spatial reasoning.

G.IV.2.a: Solve problems using the Pythagorean Theorem and its converse.

G.IV.2.b: Solve problems using the distance formula.

G.IV.2.c: Solve problems involving trigonometric ratios.

G.IV.2.d: Solve problems involving geometric probability.

### AII: Algebra 2

#### AII.I: Students will use the language and operations of algebra to evaluate, analyze and solve problems.

AII.I.1: Evaluate, analyze, and solve mathematical situations using algebraic properties and symbols.

AII.I.1.a: Solve and graph first-degree absolute value equations of a single variable.

AII.I.1.c: Solve absolute value and compound inequalities of a single variable.

AII.I.2: Solve systems of equations and inequalities.

AII.I.2.a: Solve systems of linear, absolute value, and quadratic equations algebraically and graphically.

AII.I.2.b: Graph the solutions of systems of linear, absolute value, and quadratic inequalities on the coordinate plane.

AII.I.2.c: Solve application problems involving systems of equations and inequalities.

AII.I.3: Represent and compute fluently with complex numbers.

AII.I.3.b: Simplify expressions involving complex numbers and express them in standard form, a + bi.

AII.I.4: Model and solve quadratic equations and inequalities.

AII.I.4.b: Approximate the real solutions of quadratic equations graphically.

AII.I.4.c: Solve quadratic equations of a single variable over the set of complex numbers by factoring, completing the square, and using the quadratic formula.

AII.I.4.d: Solve quadratic inequalities of a single variable.

AII.I.4.e: Write a quadratic equation when given the solutions of the equation.

#### AII.II: Students will understand and represent functions and analyze function behavior.

AII.II.1: Represent mathematical situations using relations.

AII.II.1.b: Describe a pattern using function notation.

AII.II.1.c: Determine when a relation is a function.

AII.II.1.d: Determine the domain and range of relations.

AII.II.2: Evaluate and analyze functions.

AII.II.2.a: Find the value of a function at a given point.

AII.II.2.c: Add, subtract, multiply, and divide functions.

AII.II.2.d: Determine whether or not a function has an inverse, and find the inverse when it exists.

AII.II.2.e: Identify the domain and range of a function resulting from the combination or composition of functions.

AII.II.3: Define and graph exponential functions and use them to model problems in mathematical and real-world contexts.

AII.II.3.a: Define exponential functions as functions of the form y = abx ,b > 0,b ?? 1.

AII.II.3.b: Model problems of growth and decay using exponential functions.

AII.II.3.c: Graph exponential functions.

AII.II.4: Define and graph logarithmic functions and use them to solve problems in mathematics and real-world contexts.

AII.II.4.a: Relate logarithmic and exponential functions.

AII.II.4.e: Graph logarithmic functions.

AII.II.4.f: Solve problems involving growth and decay.

#### AII.III: Students will use algebraic, spatial, and logical reasoning to solve geometry and measurement problems.

AII.III.1: Examine the behavior of functions using coordinate geometry.

AII.III.1.a: Identify the domain and range of the absolute value, quadratic, radical, sine, and cosine functions.

AII.III.1.c: Graph functions using transformations of parent functions.

AII.III.1.d: Write an equation of a parabola in the form y = a(x − h)2 + k when given a graph or an equation..

AII.III.3: Determine trigonometric measurements using appropriate techniques, tools, and formulas.

AII.III.3.a: Define the sine, cosine, and tangent functions using the unit circle.

AII.III.3.b: Determine the exact values of the sine, cosine, and tangent functions for the special angles of the unit circle using reference angles.

AII.III.3.c: Find the length of an arc using radian measure.

AII.III.3.d: Find the area of a sector in a circle using radian measure.

#### AII.IV: Students will understand concepts from probability and statistics and apply statistical methods to solve problems.

AII.IV.1: Apply basic concepts of probability.

AII.IV.1.a: Distinguish between permutations and combinations and identify situations in which each is appropriate.

AII.IV.1.b: Calculate probabilities using permutations and combinations to count events.

AII.IV.1.c: Compute conditional and unconditional probabilities in various ways, including by definitions, the general multiplication rule, and probability trees.

AII.IV.2: Use percentiles and measures of variability to analyze data.

AII.IV.2.a: Compute different measures of spread, including the range, standard deviation, and interquartile range.

AII.IV.2.b: Compare the effectiveness of different measures of spread, including the range, standard deviation, and interquartile range in specific situations.

AII.IV.2.d: Use histograms to obtain percentiles.

### PC: Pre-Calculus

#### PC.I: Students will use the language and operations of algebra to evaluate, analyze and solve problems.

PC.I.1: Compute with matrices and use matrices to solve problems.

PC.I.1.e: Solve systems of linear equations with up to three variables using matrices.

PC.I.2: Analyze the behavior of sequences and series.

PC.I.2.a: Describe a sequence as a function where the domain is the set of natural numbers.

PC.I.2.b: Represent sequences and series using various notations.

PC.I.2.c: Identify arithmetic and geometric sequences and series.

PC.I.2.d: Discover and justify the formula for a finite arithmetic series.

PC.I.2.e: Discover and justify the formulas for finite and infinite geometric series.

#### PC.II: Students will understand and represent functions and analyze function behavior.

PC.II.1: Analyze and solve problems using polynomial functions.

PC.II.1.a: Raise a binomial to a power using the Binomial Theorem and Pascal?s Triangle.

PC.II.1.b: Determine the number and nature of solutions to polynomial equations with real coefficients over the complex numbers.

PC.II.1.c: Factor polynomials to solve equations and real-world applications.

PC.II.1.d: Understand the relationships among the solutions of a polynomial equation, the zeros of a function, the x-intercepts of a graph, and the factors of a polynomial.

PC.II.1.e: Write an equation with given solutions.

PC.II.2: Model and graph functions and transformations of functions.

PC.II.2.b: Graph rational, piece-wise, power, exponential, and logarithmic functions.

PC.II.2.c: Identify the effects of changing the parameter a in y = af (x), y = f (ax), y = f (x − a) , and y = f (x) + a , given the graph of y = f (x).

PC.II.3: Analyze the behavior of functions.

PC.II.3.a: Identify the domain, range, and other attributes of families of functions and their inverses.

PC.II.3.b: Approximate instantaneous rates of change and find average rates of change using graphs and numerical data.

PC.II.3.c: Identify and analyze continuity, end behavior, asymptotes, symmetry (odd and even functions), and limits, and connect these concepts to graphs of functions.

PC.II.3.e: Relate the graphical representation of discontinuities and end behavior to the concept of limit.

#### PC.III: Students will use algebraic, spatial, and logical reasoning to solve geometry and measurement problems.

PC.III.1: Solve problems using trigonometry.

PC.III.1.a: Define the six trigonometric functions using the unit circle.

PC.III.1.b: Prove trigonometric identities using definitions, the Pythagorean Theorem, or other relationships.

PC.III.1.c: Simplify trigonometric expressions and solve trigonometric equations using identities.

PC.III.1.e: Construct the graphs of the trigonometric functions and their inverses, and describe their behavior, including periodicity and amplitude.

PC.III.2: Graph curves using polar and parametric equations.

PC.III.2.a: Define and use polar coordinates and relate them to Cartesian coordinates.

PC.III.2.b: Represent complex numbers in rectangular and polar form, and convert between rectangular and polar form.

PC.III.2.c: Translate equations in Cartesian coordinates into polar coordinates and graph them in the polar coordinate plane.

PC.III.3: Solve problems involving the geometric properties of conic sections.

PC.III.3.a: Write equations of conic sections in standard form.

PC.III.3.b: Identify the geometric properties of conic sections (i.e., vertex, foci, lines of symmetry, directrix, major and minor axes, and asymptotes).

PC.III.3.c: Solve real-world applications of conic sections.

#### PC.IV: Students will understand concepts from probability and statistics and apply statistical methods to solve problems.

PC.IV.1: Compute probabilities for discrete distributions and use sampling distributions to calculate approximate probabilities.

PC.IV.1.a: Obtain sample spaces and probability distributions for simple discrete random variables.

PC.IV.1.b: Compute binomial probabilities using Pascal?s Triangle and the Binomial Theorem.

PC.IV.1.c: Compute means and variances of discrete random variables.

PC.IV.1.d: Compute probabilities using areas under the Normal Curve.

PC.IV.1.e: Calculate parameters of sampling distributions for the sample average, sum, and proportion.

PC.IV.1.f: Calculate probabilities in real problems using sampling distributions.

PC.IV.2: Analyze bivariate data using linear regression methods.

PC.IV.2.a: Fit regression lines to pairs of numeric variables and calculate the means and standard deviations of the two variables and the correlation coefficient, using technology.

Correlation last revised: 3/29/2010

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