### L1: Based on their knowledge of the properties of arithmetic, students understand and reason about numbers, number systems, and the relationships between them. They represent quantitative relationships using mathematical symbols, and interpret relationships from those representations.

#### L1.1: Number Systems and Number Sense

L1.1.2: Explain why the multiplicative inverse of a number has the same sign as the number, while the additive inverse of a number has the opposite sign.

L1.1.6: Explain the importance of the irrational numbers square root of 2 and square root of 3 in basic right triangle trigonometry; the importance of π because of its role in circle relationships; and the role of e in applications such as continuously compounded interest.

#### L1.2: Representations and Relationships

L1.2.3: Use vectors to represent quantities that have magnitude and direction; interpret direction and magnitude of a vector numerically, and calculate the sum and difference of two vectors.

L1.2.4: Organize and summarize a data set in a table, plot, chart, or spreadsheet; find patterns in a display of data; understand and critique data displays in the media.

#### L1.3: Counting and Probabilistic Reasoning

L1.3.1: Describe, explain, and apply various counting techniques (e.g., finding the number of different 4-letter passwords; permutations; and combinations); relate combinations to Pascal’s triangle; know when to use each technique.

L1.3.2: Define and interpret commonly used expressions of probability (e.g., chances of an event, likelihood, odds).

L1.3.3: Recognize and explain common probability misconceptions such as “hot streaks” and “being due.”

### L2: Students calculate fluently, estimate proficiently, and describe and use algorithms in appropriate situations (e.g., approximating solutions to equations.) They understand the basic ideas of iteration and algorithms.

#### L2.1: Calculation Using Real and Complex Numbers

L2.1.1: Explain the meaning and uses of weighted averages (e.g., GNP, consumer price index, grade point average).

L2.1.4: Know that the complex number i is one of two solutions to x² = -1.

L2.1.5: Add, subtract, and multiply complex numbers; use conjugates to simplify quotients of complex numbers.

#### L2.2: Sequences and Iteration

L2.2.1: Find the nth term in arithmetic, geometric, or other simple sequences.

L2.2.2: Compute sums of finite arithmetic and geometric sequences.

L2.2.3: Use iterative processes in such examples as computing compound interest or applying approximation procedures.

### L4: Students understand mathematical reasoning as being grounded in logic and proof and can distinguish mathematical arguments from other types of arguments. They can interpret arguments made about quantitative situations in the popular media. Students know the language and laws of logic and can apply them in both mathematical and everyday settings. They write proofs using direct and indirect methods and use counterexamples appropriately to show that statements are false.

#### L4.1: Mathematical Reasoning

L4.1.1: Distinguish between inductive and deductive reasoning, identifying and providing examples of each.

L4.1.2: Differentiate between statistical arguments (statements verified empirically using examples or data) and logical arguments based on the rules of logic.

#### L4.2: Language and Laws of Logic

L4.2.1: Know and use the terms of basic logic (e.g., proposition, negation, truth and falsity, implication, if and only if, contrapositive, and converse).

L4.2.3: Use the quantifiers “THERE EXISTS” and “ALL” in mathematical and everyday settings and know how to logically negate statements involving them.

#### L4.3: Proof

L4.3.3: Explain the difference between a necessary and a sufficient condition within the statement of a theorem; determine the correct conclusions based on interpreting a theorem in which necessary or sufficient conditions in the theorem or hypothesis are satisfied.

### A1: Students recognize, construct, interpret, and evaluate expressions. They fluently transform symbolic expressions into equivalent forms. They determine appropriate techniques for solving each type of equation, inequality, or system of equations, apply the techniques correctly to solve, justify the steps in the solutions, and draw conclusions from the solutions. They know and apply common formulas.

#### A1.1: Construction, Interpretation, and Manipulation of Expressions (linear, quadratic, polynomial, rational, power, exponential, logarithmic, and trigonometric)

A1.1.1: Give a verbal description of an expression that is presented in symbolic form, write an algebraic expression from a verbal description, and evaluate expressions given values of the variables.

A1.1.3: Factor algebraic expressions using, for example, greatest common factor, grouping, and the special product identities (e.g., differences of squares and cubes).

A1.1.4: Add, subtract, multiply, and simplify polynomials and rational expressions (e.g., multiply (x – 1) (1 – x² + 3); simplify (9x - x³)/(x + 3))

A1.1.5: Divide a polynomial by a monomial.

A1.1.6: Transform exponential and logarithmic expressions into equivalent forms using the properties of exponents and logarithms including the inverse relationship between exponents and logarithms.

#### A1.2: Solutions of Equations and Inequalities (linear, quadratic, polynomial, rational, power, exponential, logarithmic, and trigonometric)

A1.2.1: Write equations and inequalities with one or two variables to represent mathematical or applied situations, and solve.

A1.2.2: Associate a given equation with a function whose zeros are the solutions of the equation.

A1.2.3: Solve (and justify steps in the solutions) linear and quadratic equations and inequalities, including systems of up to three linear equations with three unknowns; apply the quadratic formula appropriately.

A1.2.4: Solve absolute value equations and inequalities, (e.g. solve l x - 3 l ≤ 6), and justify steps in the solution.

A1.2.6: Solve power equations (e.g., (x + 1)³ = 8) and equations including radical expressions (e.g., the square root of (3x - 7) = 7), justify steps in the solution, and explain how extraneous solutions may arise.

A1.2.8: Solve an equation involving several variables (with numerical or letter coefficients) for a designated variable, and justify steps in the solution.

A1.2.9: Know common formulas (e.g., slope, distance between two points, quadratic formula, compound interest, distance = velocity x time), and apply appropriately in contextual situations.

### A2: Students understand functions, their representations, and their attributes. They perform transformations, combine and compose functions, and find inverses. Students classify functions and know the characteristics of each family. They work with functions with real coefficients fluently.

#### A2.1: Definitions, Representations, and Attributes of Functions

A2.1.1: Recognize whether a relationship (given in contextual, symbolic, tabular, or graphical form) is a function; and identify its domain and range.

A2.1.2: Read, interpret, and use function notation, and evaluate a function at a value in its domain.

A2.1.3: Represent functions in symbols, graphs, tables, diagrams, or words, and translate among representations.

A2.1.4: Recognize that functions may be defined by different expressions over different intervals of their domains; such functions are piecewise-defined (e.g., absolute value and greatest integer functions).

A2.1.5: Recognize that functions may be defined recursively, and compute values of and graph simple recursively defined functions (e.g., f(0) = 5, and f(n) = f(n-1) + 2).

A2.1.6: Identify the zeros of a function and the intervals where the values of a function are positive or negative, and describe the behavior of a function, as x approaches postive or negative infinity, given the symbolic and graphical representations.

A2.1.7: Identify and interpret the key features of a function from its graph or its formula(e), (e.g. slope, intercept(s), asymptote(s), maximum and minimum value(s), symmetry, average rate of change over an interval, and periodicity).

#### A2.2: Operations and Transformations

A2.2.1: Combine functions by addition, subtraction, multiplication, and division.

A2.2.2: Apply given transformations (e.g., vertical or horizontal shifts, stretching or shrinking, or reflections about the x- and y-axes) to basic functions, and represent symbolically.

A2.2.3: Recognize whether a function (given in tabular or graphical form) has an inverse and recognize simple inverse pairs (e.g., f(x) = x³ and g(x) = x to the 1/3 power).

#### A2.3: Families of Functions (linear, quadratic, polynomial, rational, power, exponential, logarithmic, and trigonometric)

A2.3.1: Identify a function as a member of a family of functions based on its symbolic, or graphical representation; recognize that different families of functions have different asymptotic behavior at infinity, and describe these behaviors.

A2.3.2: Describe the tabular pattern associated with functions having constant rate of change (linear); or variable rates of change.

#### A2.4: Lines and Linear Functions

A2.4.1: Write the symbolic forms of linear functions (standard [i.e., Ax + By = C, where B ≠ 0], point-slope, and slope-intercept) given appropriate information, and convert between forms.

A2.4.2: Graph lines (including those of the form x = h and y = k) given appropriate information.

A2.4.3: Relate the coefficients in a linear function to the slope and x- and y-intercepts of its graph.

A2.4.4: Find an equation of the line parallel or perpendicular to given line, through a given point; understand and use the facts that non-vertical parallel lines have equal slopes, and that non-vertical perpendicular lines have slopes that multiply to give -1.

#### A2.5: Exponential and Logarithmic Functions

A2.5.1: Write the symbolic form and sketch the graph of an exponential function given appropriate information. (e.g., given an initial value of 4 and a rate of growth of 1.5, write f(x) = 4 (1.5) to the x power).

A2.5.2: Interpret the symbolic forms and recognize the graphs of exponential and logarithmic functions (e.g., f(x) = 10 to the x power, f(x) = log x, f(x) = e to the x power, f(x) = ln x); recognize the logarithmic function as the inverse of the exponential function.

A2.5.3: Apply properties of exponential and logarithmic functions (e.g., a ot the x+y power = a to the x power times a to the y power; log(ab)= log a + log b).

A2.5.4: Understand and use the fact that the base of an exponential function determines whether the function increases or decreases and understand how the base affects the rate of growth or decay.

A2.5.5: Relate exponential and logarithmic functions to real phenomena, including half-life and doubling time.

A2.6.1: Write the symbolic form and sketch the graph of a quadratic function given appropriate information (e.g., vertex, intercepts, etc.).

A2.6.2: Identify the elements of a parabola (vertex, axis of symmetry, direction of opening) given its symbolic form or its graph, and relate these elements to the coefficient(s) of the symbolic form of the function.

A2.6.3: Convert quadratic functions from standard to vertex form by completing the square.

A2.6.4: Relate the number of real solutions of a quadratic equation to the graph of the associated quadratic function.

A2.6.5: Express quadratic functions in vertex form to identify their maxima or minima, and in factored form to identify their zeros.

#### A2.7: Power Functions (including roots, cubics, quartics, etc.)

A2.7.2: Express direct and inverse relationships as functions (e.g., y = kx to the n power and y = kx to the -n power, n > 0) and recognize their characteristics (e.g., in y = x³, note that doubling x results in multiplying y by a factor of 8).

A2.7.3: Analyze the graphs of power functions, noting reflectional or rotational symmetry.

#### A2.8: Polynomial Functions

A2.8.1: Write the symbolic form and sketch the graph of simple polynomial functions.

A2.8.2: Understand the effects of degree, leading coefficient, and number of real zeros on the graphs of polynomial functions of degree greater than 2.

A2.8.3: Determine the maximum possible number of zeros of a polynomial function, and understand the relationship between the x-intercepts of the graph and the factored form of the function.

#### A2.9: Rational Functions

A2.9.1: Write the symbolic form and sketch the graph of simple rational functions.

A2.9.2: Analyze graphs of simple rational functions (e.g., f(x) = (2x + 1)/(x - 1); g(x) = x/(x² - 4)) and understand the relationship between the zeros of the numerator and denominator and the function’s intercepts, asymptotes, and domain.

#### A2.10: Trigonometric Functions

A2.10.1: Use the unit circle to define sine and cosine; approximate values of sine and cosine (e.g., sin 3, or cos 0.5); use sine and cosine to define the remaining trigonometric functions; explain why the trigonometric functions are periodic.

A2.10.3: Use the unit circle to determine the exact values of sine and cosine, for integer multiples of pi/6 and pi/4.

A2.10.4: Graph the sine and cosine functions; analyze graphs by noting domain, range, period, amplitude, and location of maxima and minima.

A2.10.5: Graph transformations of basic trigonometric functions (involving changes in period, amplitude, and midline) and understand the relationship between constants in the formula and the transformed graph.

### A3: Students construct or select a function to model a real-world situation in order to solve applied problems. They draw on their knowledge of families of functions to do so.

#### A3.1: Models of Real-world Situations Using Families of Functions.

A3.1.1: Identify the family of function best suited for modeling a given real-world situation (e.g., quadratic functions for motion of an object under the force of gravity; exponential functions for compound interest; trigonometric functions for periodic phenomena. In the example above, recognize that the appropriate general function is exponential (P = P0a to the t power)

A3.1.2: Adapt the general symbolic form of a function to one that fits the specifications of a given situation by using the information to replace arbitrary constants with numbers. In the example above, substitute the given values P0 = 300 and a = 1.02 to obtain P = 300(1.02) to the t power.

### G1: Students represent basic geometric figures, polygons, and conic sections and apply their definitions and properties in solving problems and justifying arguments, including constructions and representations in the coordinate plane. Students represent three-dimensional figures, understand the concepts of volume and surface area, and use them to solve problems. They know and apply properties of common three-dimensional figures.

#### G1.1: Lines and Angles; Basic Euclidean and Coordinate Geometry

G1.1.1: Solve multi-step problems and construct proofs involving vertical angles, linear pairs of angles supplementary angles, complementary angles, and right angles.

G1.1.2: Solve multi-step problems and construct proofs involving corresponding angles, alternate interior angles, alternate exterior angles, and same-side (consecutive) interior angles.

G1.1.3: Perform and justify constructions, including midpoint of a line segment and bisector of an angle, using straightedge and compass.

G1.1.4: Given a line and a point, construct a line through the point that is parallel to the original line using straightedge and compass; given a line and a point, construct a line through the point that is perpendicular to the original line; justify the steps of the constructions.

#### G1.2: Triangles and Their Properties

G1.2.1: Prove that the angle sum of a triangle is 180° and that an exterior angle of a triangle is the sum of the two remote interior angles.

G1.2.2: Construct and justify arguments and solve multi-step problems involving angle measure, side length, perimeter, and area of all types of triangles.

G1.2.3: Know a proof of the Pythagorean Theorem and use the Pythagorean Theorem and its converse to solve multi-step problems.

G1.2.5: Solve multi-step problems and construct proofs about the properties of medians, altitudes, and perpendicular bisectors to the sides of a triangle, and the angle bisectors of a triangle; using a straightedge and compass, construct these lines.

#### G1.3: Triangles and Trigonometry

G1.3.1: Define the sine, cosine, and tangent of acute angles in a right triangle as ratios of sides; solve problems about angles, side lengths, or areas using trigonometric ratios in right triangles.

G1.3.3: Determine the exact values of sine, cosine, and tangent for 0°, 30°, 45°, 60°, and their integer multiples, and apply in various contexts.

#### G1.4: Quadrilaterals and Their Properties

G1.4.1: Solve multi-step problems and construct proofs involving angle measure, side length, diagonal length, perimeter, and area of squares, rectangles, parallelograms, kites, and trapezoids.

G1.4.2: Solve multi-step problems and construct proofs involving quadrilaterals (e.g., prove that the diagonals of a rhombus are perpendicular) using Euclidean methods or coordinate geometry.

G1.4.3: Describe and justify hierarchical relationships among quadrilaterals, (e.g. every rectangle is a parallelogram).

G1.4.4: Prove theorems about the interior and exterior angle sums of a quadrilateral.

#### G1.5: Other Polygons and Their Properties

G1.5.1: Know and use subdivision or circumscription methods to find areas of polygons (e.g., regular octagon, non-regular pentagon).

G1.5.2: Know, justify, and use formulas for the perimeter and area of a regular n-gon and formulas to find interior and exterior angles of a regular n-gon and their sums.

#### G1.6: Circles and Their Properties

G1.6.1: Solve multi-step problems involving circumference and area of circles.

G1.6.2: Solve problems and justify arguments about chords (e.g., if a line through the center of a circle is perpendicular to a chord, it bisects the chord) and lines tangent to circles (e.g., a line tangent to a circle is perpendicular to the radius drawn to the point of tangency).

G1.6.3: Solve problems and justify arguments about central angles, inscribed angles and triangles in circles.

G.1.7.1: Find an equation of a circle given its center and radius; given the equation of a circle, find its center and radius.

#### G1.7: Conic Sections and Their Properties

G1.7.2: Identify and distinguish among geometric representations of parabolas, circles, ellipses, and hyperbolas; describe their symmetries, and explain how they are related to cones.

G1.7.3: Graph ellipses and hyperbolas with axes parallel to the x- and y-axes, given equations.

#### G1.8: Three- Dimensional Figures

G1.8.1: Solve multi-step problems involving surface area and volume of pyramids, prisms, cones, cylinders, hemispheres, and spheres.

G1.8.2: Identify symmetries of pyramids, prisms, cones, cylinders, hemispheres, and spheres.

### G2: Students use and justify relationships between lines, angles, area and volume formulas, and 2- and 3-dimensional representations. They solve problems and provide proofs about congruence and similarity.

#### G2.1: Relationships Between Area and Volume Formulas

G2.1.1: Know and demonstrate the relationships between the area formula of a triangle, the area formula of a parallelogram, and the area formula of a trapezoid.

G2.1.2: Know and demonstrate the relationships between the area formulas of various quadrilaterals (e.g., explain how to find the area of a trapezoid based on the areas of parallelograms and triangles).

G2.1.3: Know and use the relationship between the volumes of pyramids and prisms (of equal base and height) and cones and cylinders (of equal base and height).

#### G2.2: Relationships Between Two-dimensional and Three-dimensional Representations

G2.2.1: Identify or sketch a possible 3-dimensional figure, given 2-dimensional views (e.g., nets, multiple views); create a 2-dimensional representation of a 3-dimensional figure.

G2.2.2: Identify or sketch cross-sections of 3-dimensional figures; identify or sketch solids formed by revolving 2-dimensional figures around lines.

#### G2.3: Congruence and Similarity

G2.3.1: Prove that triangles are congruent using the SSS, SAS, ASA, and AAS criteria, and for right triangles, the hypotenuse-leg criterion.

G2.3.2: Use theorems about congruent triangles to prove additional theorems and solve problems, with and without use of coordinates.

G2.3.3: Prove that triangles are similar by using SSS, SAS, and AA conditions for similarity.

G2.3.4: Use theorems about similar triangles to solve problems with and without use of coordinates.

G2.3.5: Know and apply the theorem stating that the effect of a scale factor of k relating one two dimensional figure to another or one three dimensional figure to another, on the length, area, and volume of the figures is to multiply each by k, k², and k³, respectively.

### G3: Students will solve problems about distance-preserving transformations and shape-preserving transformations. The transformations will be described synthetically and, in simple cases, by analytic expressions in coordinates.

#### G3.1: Distance-preserving Transformations: Isometries

G3.1.1: Define reflection, rotation, translation, and glide reflection and find the image of a figure under a given isometry.

G3.1.2: Given two figures that are images of each other under an isometry, find the isometry and describe it completely.

G3.1.3: Find the image of a figure under the composition of two or more isometries, and determine whether the resulting figure is a reflection, rotation, translation, or glide reflection image of the original figure.

#### G3.2: Shape-preserving Transformations: Dilations and Isometries

G3.2.1: Know the definition of dilation, and find the image of a figure under a given dilation.

G3.2.2: Given two figures that are images of each other under some dilation, identify the center and magnitude of the dilation.

### S1: Students plot and analyze univariate data by considering the shape of distributions and analyzing outliers; they find and interpret commonly-used measures of center and variation; and they explain and use properties of the normal distribution.

#### S1.1: Producing and Interpreting Plots

S1.1.1: Construct and interpret dot plots, histograms, relative frequency histograms, bar graphs, basic control charts, and box plots with appropriate labels and scales; determine which kinds of plots are appropriate for different types of data; compare data sets and interpret differences based on graphs and summary statistics.

S1.1.2: Given a distribution of a variable in a data set, describe its shape, including symmetry or skewness, and state how the shape is related to measures of center (mean and median) and measures of variation (range and standard deviation) with particular attention to the effects of outliers on these measures.

#### S1.2: Measures of Center and Variation

S1.2.1: Calculate and interpret measures of center including: mean, median, and mode; explain uses, advantages and disadvantages of each measure given a particular set of data and its context.

S1.2.2: Estimate the position of the mean, median, and mode in both symmetrical and skewed distributions, and from a frequency distribution or histogram.

S1.2.3: Compute and interpret measures of variation, including percentiles, quartiles, interquartile range, variance, and standard deviation.

#### S1.3: The Normal Distribution

S1.3.2: Describe characteristics of the normal distribution, including its shape and the relationships among its mean, median, and mode.

### S2: Students plot and interpret bivariate data by constructing scatterplots, recognizing linear and nonlinear patterns, and interpreting correlation coefficients; they fit and interpret regression models, using technology as appropriate.

#### S2.1: Scatterplots and Correlation

S2.1.1: Construct a scatterplot for a bivariate data set with appropriate labels and scales.

S2.1.2: Given a scatterplot, identify patterns, clusters, and outliers; recognize no correlation, weak correlation, and strong correlation.

S2.1.3: Estimate and interpret Pearson’s correlation coefficient for a scatterplot of a bivariate data set; recognize that correlation measures the strength of linear association.

S2.1.4: Differentiate between correlation and causation; know that a strong correlation does not imply a cause-and-effect relationship; recognize the role of lurking variables in correlation.

#### S2.2: Linear Regression

S2.2.1: For bivariate data which appear to form a linear pattern, find the least squares regression line by estimating visually and by calculating the equation of the regression line; interpret the slope of the equation for a regression line.

S2.2.2: Use the equation of the least squares regression line to make appropriate predictions.

### S3: Students understand and apply sampling and various sampling methods, examine surveys and experiments, identify bias in methods of conducting surveys, and learn strategies to minimize bias. They understand basic principles of good experimental design.

#### S3.1: Data Collection and Analysis

S3.1.2: Identify possible sources of bias in data collection and sampling methods and simple experiments; describe how such bias can be reduced and controlled by random sampling; explain the impact of such bias on conclusions made from analysis of the data; and know the effect of replication on the precision of estimates.

S3.1.3: Distinguish between an observational study and an experimental study, and identify, in context, the conclusions that can be drawn from each.

### S4: Students understand probability and find probabilities in various situations, including those involving compound events, using diagrams, tables, geometric models and counting strategies; they apply the concepts of probability to make decisions.

#### S4.1: Probability

S4.1.1: Understand and construct sample spaces in simple situations (e.g., tossing two coins, rolling two number cubes and summing the results).

S4.1.2: Define mutually exclusive events, independent events, dependent events, compound events, complementary events and conditional probabilities; and use the definitions to compute probabilities.

#### S4.2: Application and Representation

S4.2.1: Compute probabilities of events using tree diagrams, formulas for combinations and permutations, Venn diagrams, or other counting techniques.

S4.2.2: Apply probability concepts to practical situations, in such settings as finance, health, ecology, or epidemiology, to make informed decisions.

Correlation last revised: 10/24/2008

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