Michigan State Standards for Mathematics: Grade 9

MI.L. Quantitative Literacy and Logic (L)

L1: Reasoning About Numbers, Systems, and Quantitative Situations: 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

L1.1.1 Know the different properties that hold in different number systems and recognize that the applicable properties change in the transition from the positive integers to all integers, to the rational numbers, and to the real numbers.

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.

L1.1.3 Explain how the properties of associativity, commutativity, and distributivity, as well as identity and inverse elements, are used in arithmetic and algebraic calculations.

L1.1.4 Describe the reasons for the different effects of multiplication by, or exponentiation of, a positive number by a number less than 0, a number between 0 and 1, and a number greater than 1.

L1.1.5 Justify numerical relationships

L1.1.6 Explain the importance of the irrational numbers square root of 2 and square root of 3 in basic right triangle trigonometry, and the importance of pi because of its role in circle relationships.

L1.2 Representations and Relationships

L1.2.1 Use mathematical symbols to represent quantitative relationships and situations.

L1.2.2 Interpret representations that reflect absolute value relationships in such contexts as error tolerance.

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.

L1.2.5 Read and interpret representations from various technological sources, such as contour or isobar diagrams. (Recommended)

L1.3 Counting and Probabilistic Reasoning

L1.3.1 Describe, explain, and apply various counting techniques; relate combinations to Pascal's triangle; know when to use each technique.

L1.3.2 Define and interpret commonly used expressions of probability.

L1.3.3 Recognize and explain common probability misconceptions such as ''hot streaks'' and ''being due.''

L2: Calculation, Algorithms, and Estimation: 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.

L2.1.2 Calculate fluently with numerical expressions involving exponents; use the rules of exponents; evaluate numerical expressions involving rational and negative exponents; transition easily between roots and exponents.

L2.1.3 Explain the exponential relationship between a number and its base 10 logarithm and use it to relate rules of logarithms to those of exponents in expressions involving numbers.

L2.1.4 Know that the complex number i is one of two solutions to x^2 = -1.

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

L2.1.7 Understand the mathematical bases for the differences among voting procedures. (Recommended)

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.

L2.2.4 Compute sums of infinite geometric sequences. (Recommended)

L2.3 Measurement Units, Calculations, and Scales

L2.3.1 Convert units of measurement within and between systems; explain how arithmetic operations on measurements affect units, and carry units through calculations correctly.

L2.3.2 Describe and interpret logarithmic relationships in such contexts as the Richter scale, the pH scale, or decibel measurements; solve applied problems.

L2.4 Understanding Error

L2.4.1 Determine what degree of accuracy is reasonable for measurements in a given situation; express accuracy through use of significant digits, error tolerance, or percent of error; describe how errors in measurements are magnified by computation; recognize accumulated error in applied situations.

L2.4.2 Describe and explain round-off error, rounding, and truncating.

L2.4.3 Know the meaning of and interpret statistical significance, margin of error, and confidence level.

L3: Mathematical Reasoning, Logic, and Proof: 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.

L3.1 Mathematical Reasoning

L3.1.1 Distinguish between inductive and deductive reasoning, identifying and providing examples of each.

L3.1.2 Differentiate between statistical arguments (statements verified empirically using examples or data) and logical arguments based on the rules of logic.

L3.1.3 Define and explain the roles of axioms (postulates), definitions, theorems, counterexamples, and proofs in the logical structure of mathematics. Identify and give examples of each.

L3.2 Language and Laws of Logic

L3.2.1 Know and use the terms of basic logic.

L3.2.2 Use the connectives ''not,'' ''and,'' ''or,'' and ''if..., then,'' in mathematical and everyday settings. Know the truth table of each connective and how to logically negate statements involving these connectives.

L3.2.3 Use the quantifiers ''there exists'' and ''all'' in mathematical and everyday settings and know how to logically negate statements involving them.

L3.2.4 Write the converse, inverse, and contrapositive of an ''if..., then...'' statement. Use the fact, in mathematical and everyday settings, that the contrapositive is logically equivalent to the original, while the inverse and converse are not.

L3.3 Proof

L3.3.1 Know the basic structure for the proof of an ''if..., then...'' statement (assuming the hypothesis and ending with the conclusion) and that proving the contrapositive is equivalent.

L3.3.2 Construct proofs by contradiction. Use counterexamples, when appropriate, to disprove a statement.

L3.3.3 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.

MI.A. Algebra and Functions (A)

A1: Expressions, Equations, and Inequalities: 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

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.

A1.1.2 Know the definitions and properties of exponents and roots transition fluently between them, and apply them in algebraic expressions.

A1.1.3 Factor algebraic expressions using, for example, greatest common factor, grouping, and the special product identities.

A1.1.4 Add, subtract, multiply, and simplify polynomials and rational expressions.

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.

A1.1.7 Transform trigonometric expressions into equivalent forms using basic identities such as sin^2 theta + cos^2 theta = 1 and tan^2 theta + 1 = sec^2 theta (Recommended)

A1.2 Solutions of Equations and Inequalities

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 linear and quadratic equations and inequalities including systems of up to three linear equations with three unknowns. Justify steps in the solution, and apply the quadratic formula appropriately.

A1.2.4 Solve absolute value equations and inequalities, and justify steps in the solution.

A1.2.5 Solve polynomial equations and equations involving rational expressions, and justify steps in the solution.

A1.2.6 Solve power equations and equations including radical expressions, justify steps in the solution, and explain how extraneous solutions may arise.

A1.2.7 Solve exponential and logarithmic equations, and justify steps in the solution.

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

A1.2.9 Know common formulas and apply appropriately in contextual situations.

A1.2.10 Use special values of the inverse trigonometric functions to solve trigonometric equations over specific intervals.

A2: Functions: 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. 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.

A2.1 Definitions, Representations, and Attributes of Functions

A2.1.1 Determine 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.

A2.1.5 Recognize that functions may be defined recursively. Compute values of and graph simple recursively defined functions.

A2.1.6 Identify the zeros of a function, the intervals where the values of a function are positive or negative, and describe the behavior of a function as x approaches positive 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).

A2.2 Operations and Transformations

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

A2.2.2 Apply given transformations 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.

A2.2.4 If a function has an inverse, find the expression(s) for the inverse. (Recommended)

A2.2.5 Write an expression for the composition of one function with another; recognize component functions when a function is a composition of other functions. (Recommended)

A2.2.6 Know and interpret the function notation for inverses and verify that two functions are inverses using composition. (Recommended)

A2.3 Representations of Functions

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.

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

A2.3.3 Write the general symbolic forms that characterize each family of functions.

A2.4 Models of Real-world Situations Using Families of Functions

A2.4.1 Identify the family of function best suited for modeling a given real-world situation.

A2.4.2 Adapt the general symbolic form of a function to one that fits the specification of a given situation by using the information to replace arbitrary constants with numbers.

A2.4.3 Using the adapted general symbolic form, draw reasonable conclusions about the situation being modeled.

A2.4.4 Use methods of linear programming to represent and solve simple real-life problems. (Recommended)

A3: Families of Functions: Students study the symbolic and graphical forms of each function family. By recognizing the unique characteristics of each family, they can use them as tools for solving problems or for modeling real-world situations.

A3.1 Lines and Linear Functions

A3.1.1 Write the symbolic forms of linear functions (standard, point-slope, and slope-intercept) given appropriate information, and convert between forms.

A3.1.2 Graph lines (including those of the form x = h and y = k) given appropriate information.

A3.1.3 Relate the coefficients in a linear function to the slope and x- and y-intercepts of its graph.

A3.1.4 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.

A3.2 Exponential and Logarithmic Functions

A3.2.1 Write the symbolic form and sketch the graph of an exponential function given appropriate information.

A3.2.2 Interpret the symbolic forms and recognize the graphs of exponential and logarithmic functions; recognize the logarithmic function as the inverse of the exponential function.

A3.2.3 Apply properties of exponential and logarithmic functions.

A3.2.4 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.

A.3.2.5 Relate exponential and logarithmic functions to real phenomena, including half-life and doubling time.

A3.3 Quadratic Functions

A3.3.1 Write the symbolic form and sketch the graph of a quadratic function given appropriate information.

A3.3.2 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.

A3.3.3 Convert quadratic functions from standard to vertex form by completing the square.

A3.3.4 Relate the number of real solutions of a quadratic equation to the graph of the associated quadratic function.

A3.3.5 Express quadratic functions in vertex form to identify their maxima or minima, and in factored form to identify their zeros.

A3.4 Power Functions

A3.4.1 Write the symbolic form and sketch the graph of power functions.

A3.4.2 Express direct and inverse relationships as functions and recognize their characteristics.

A3.4.3 Analyze the graphs of power functions, noting reflectional or rotational symmetry.

A3.5 Polynomial Functions

A3.5.1 Write the symbolic form and sketch the graph of simple polynomial functions.

A3.5.2 Understand the effects of degree, leading coefficient, and number of real zeros on the graphs of polynomial functions of degree greater than 2.

A3.5.3 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.

A3.6 Rational Functions

A3.6.1 Write the symbolic form and sketch the graph of simple rational functions.

A3.6.2 Analyze graphs of simple rational functions and understand the relationship between the zeros of the numerator and denominator and the function's intercepts, asymptotes, and domain.

A3.7 Trigonometric Functions

A3.7.1 Use the unit circle to define sine and cosine; approximate values of sine and cosine; use sine and cosine to define the remaining trigonometric functions; explain why the trigonometric functions are periodic.

A3.7.2 Use the relationship between degree and radian measures to solve problems.

A3.7.3 Use the unit circle to determine the exact values of sine and cosine, for integer multiples of pi/6 and pi/4.

A3.7.4 Graph the sine and cosine functions; analyze graphs by noting domain, range, period, amplitude, and location of maxima and minima.

A3.7.5 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.

MI.G. Geometry and Trigonometry (G)

G1: Figures and Their Properties: 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 multistep problems and construct proofs involving vertical angles, linear pairs of angles supplementary angles, complementary angles, and right angles.

G1.1.2 Solve multistep 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.

G1.1.5 Given a line segment in terms of its endpoints in the coordinate plane, determine its length and midpoint.

G1.1.6 Recognize Euclidean geometry as an axiom system. Know the key axioms and understand the meaning of and distinguish between undefined terms, axioms, definitions, and theorems.

G1.2 Triangles and Their Properties

G1.2.1 Prove that the angle sum of a triangle is 180 degrees 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 multistep 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 multistep problems.

G1.2.4 Prove and use the relationships among the side lengths and the angles of 30 degrees - 60 degrees - 90 degrees triangles and 45 degrees - 45 degrees - 90 degrees triangles.

G1.2.5 Solve multistep problems and construct proofs about the properties of medians, altitudes, 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.

G1.3.2 Know and use the Law of Sines and the Law of Cosines and use them to solve problems. Find the area of a triangle with sides a and b and included angle theta using the formula Area = (1/2) absin theta.

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

G1.4 Quadrilaterals and Their Properties

G1.4.1 Solve multistep 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 multistep problems and construct proofs involving quadrilaterals using Euclidean methods or coordinate geometry.

G1.4.3 Describe and justify hierarchical relationships among quadrilaterals.

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

G1.4.5 Understand the definition of a cyclic quadrilateral and know and use the basic properties of cyclic quadrilaterals. (Recommended)

G1.5 Other Polygons and Their Properties

G1.5.1 Know and use subdivision or circumscription methods to find areas of polygons.

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 multistep problems involving circumference and area of circles.

G1.6.2 Solve problems and justify arguments about chords and lines tangent to circles.

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

G1.6.4 Know and use properties of arcs and sectors and find lengths of arcs and areas of sectors.

G1.7 Conic Sections and Their Properties

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.

G1.7.2 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.

G1.7.4 Know and use the relationship between the vertices and foci in and ellipse, the vertices and foci in a hyperbola, and the directrix and focus in a parabola, interpret these relationships in applied contexts. (Recommended)

G1.8 Three- Dimensional Figures

G1.8.1 Solve multistep 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: Relationships Between Figures: 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.

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 three-dimensional figure, given two-dimensional views. Create a two-dimensional representation of a three-dimensional figure.

G2.2.2 Identify or sketch cross sections of three-dimensional figures. Identify or sketch solids formed by revolving two-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 that right triangles, are congruent using 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, k^2, and k^3, respectively.

G3: Transformations of Figures in the Plane: 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 magniture of the dilation.

G3.2.3 Find the image of a figure under the composition of a dilation and an isometry. (Recommended)

MI.S. Statistics and Probability (S)

S1: Univariate Data - Examining Distributions: 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

S1.3.1 Explain the concept of distribution and the relationship between summary statistics for a data set and parameters of a distribution.

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

S1.3.3 Know and use the fact that about 68%, 95%, and 99.7% of the data lie within one, two, and three standard deviations of the mean, respectively in a normal distribution.

S1.3.4 Calculate z-scores, use z-scores to recognize outliers, and use z-scores to make informed decisions.

S2: Bivariate Data - Examining Relationships: 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 that 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: Samples, Surveys, and Experiments: 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

S3.1.1 Know the meanings of a sample from a population and a census of a population, and distinguish between sample statistics and population parameters.

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.

S3.1.4 Design simple experiments or investigations to collect data to answer questions of interest; interpret and present results. (Recommended)

S3.1.5 Understand methods of sampling, including random sampling, stratified sampling, and convenience samples, and be able to determine, in context, the advantages and disadvantages of each. (Recommended)

S3.1.6 Explain the importance of randomization, double-blind protocols, replication, and the placebo effect in designing experiments and interpreting the results of studies. (Recommended)

S3.1.7. Explain the basic ideas of statistical process control, including recording data from a process over time. (Recommended)

S3.1.8. Read and interpret basic control charts; detect patterns and departures from patterns. (Recommended)

S4: Probability Models and Probability Calculation: 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.

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.

S4.1.3 Design and carry out an appropriate simulation using random digits to estimate answers to questions about probability; estimate probabilities using results of a simulation; compare results of simulations to theoretical probabilities. (Recommended)

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.

MI.P. Precalculus

P1: Functions

P1.1 Know and use a definition of a function to decide if a given relation is a function.

P1.2 Perform algebraic operations (including compositions) on functions and apply transformations (translations, reflections, and rescalings).

P1.3 Write an expression for the composition of one given function with another and find the domain, range, and graph of the composite function. Recognize components when a function is composed of two or more elementary functions.

P1.4 Determine whether a function (given symbolically or graphically) has an inverse and express the inverse (symbolically, if the function is given symbolically, or graphically, if given graphically) if it exists. Know and interpret the function notation for inverses.

P1.5 Determine whether two given functions are inverses, using composition.

P1.6 Identify and describe discontinuities of a function (e.g., greatest integer function, 1/x) and how these relate to the graph.

P1.8 Explain how the rates of change of functions in different families (e.g., linear functions, exponential functions, etc.) differ, referring to graphical representations.

P1.7 Understand the concept of limit of a function as x approaches a number or infinity. Use the idea of limit to analyze a graph as it approaches an asymptote. Compute limits of simple functions (e.g., find the limit as x approaches 0 of f(x) = 1/x) informally.

P2: Exponential and Logarithmic Functions

P2.1 Use the inverse relationship between exponential and logarithmic functions to solve equations and problems.

P2.2 Graph logarithmic functions. Graph translations and reflections of these functions.

P2.3 Compare the large-scale behavior of exponential and logarithmic functions with different bases and recognize that different growth rates are visible in the graphs of the functions

P2.4 Solve exponential and logarithmic equations when possible, (e.g. 5^x=3^(x+1)). For those that cannot be solved analytically, use graphical methods to find approximate solutions.

P2.5 Explain how the parameters of an exponential or logarithmic model relate to the data set or situation being modeled. Find an exponential or logarithmic function to model a given data set or situation. Solve problems involving exponential growth and decay.

P3: Quadratic Functions

P3.1 Solve quadratic-type equations (e.g. e^(2x) - 4 e^(x+4)=0) by substitution.

P3.2 Apply quadratic functions and their graphs in the context of motion under gravity and simple optimization problems.

P3.3 Explain how the parameters of an exponential or logarithmic model relate to the data set or situation being modeled. Find a quadratic function to model a given data set or situation.

P4: Polynomial Functions

P4.1 Given a polynomial function whose roots are known or can be calculated, find the intervals on which the function's values are positive and those where it is negative.

P4.2 Solve polynomial equations and inequalities of degree greater than or equal to three. Graph polynomial functions given in factored form using zeros and their multiplicities, testing the sign-on intervals and analyzing the function's large-scale behavior.

P4.3 Know and apply fundamental facts about polynomials: the Remainder Theorem, the Factor Theorem, and the Fundamental Theorem of Algebra.

P5: Rational Functions and Difference Quotients

P5.1 Solve equations and inequalities involving rational functions. Graph rational functions given in factored form using zeros, identifying asymptotes, analyzing their behavior for large x values, and testing intervals.

P5.2 Given vertical and horizontal asymptotes, find an expression for a rational function with these features.

P5.3 Know and apply the definition and geometric interpretation of difference quotient. Simplify difference quotients and interpret difference quotients as rates of change and slopes of secant lines.

P6: Trigonometric Functions

P6.1 Define (using the unit circle), graph, and use all trigonometric functions of any angle. Convert between radian and degree measure. Calculate arc lengths in given circles.

P6.2 Graph transformations of the sine and cosine functions (involving changes in amplitude, period, midline, and phase) and explain the relationship between constants in the formula and transformed graph.

P6.3 Know basic properties of the inverse trigonometric functions sin^-1 x, cos^-1 x, tan^-1 x, including their domains and ranges. Recognize their graphs.

P6.4 Know the basic trigonometric identities for sine, cosine, and tangent (e.g., the Pythagorean identities, sum and difference formulas, co-functions relationships, double-angle and half-angle formulas).

P6.5 Solve trigonometric equations using basic identities and inverse trigonometric functions.

P6.6 Prove trigonometric identities and derive some of the basic ones (e.g., double-angle formula from sum and difference formulas, half-angle formula from double-angle formula, etc.).

P6.7 Find a sinusoidal function to model a given data set or situation and explain how the parameters of the model relate to the data set or situation.

P7: Vectors, Matrices, and Systems of Equations

P7.1 Perform operations (addition, subtraction, and multiplication by scalars) on vectors in the plane. Solve applied problems using vectors.

P7.2 Know and apply the algebraic and geometric definitions of the dot product of vectors.

P7.3 Know the definitions of matrix addition and multiplication. Add, subtract, and multiply matrices. Multiply a vector by a matrix.

P7.4 Represent rotations of the plane as matrices and apply to find the equations of rotated conics.

P7.5 Define the inverse of a matrix and compute the inverse of two-by-two and three-by-three matrices when they exist.

P7.6 Explain the role of determinants in solving systems of linear equations using matrices and compute determinants of two-by-two and three-by-three matrices.

P7.7 Write systems of two and three linear equations in matrix form. Solve such systems using Gaussian elimination or inverse matrices.

P7.8 Represent and solve systems of inequalities in two variables and apply these methods in linear programming situations to solve problems.

P8: Sequences, Series, and Mathematical Induction

P8.1 Know, explain, and use sigma and factorial notation.

P8.2 Given an arithmetic, geometric, or recursively defined sequence, write an expression for the nth term when possible. Write a particular term of a sequence when given the nth term.

P8.3 Understand, explain, and use the formulas for the sums of finite arithmetic and geometric sequences.

P8.4 Compute the sums of infinite geometric series. Understand and apply the convergence criterion for geometric series.

P8.5 Understand and explain the principle of mathematical induction and prove statements using mathematical induction.

P8.6 Prove the binomial theorem using mathematical induction. Show its relationships to Pascal's triangle and to combinations. Use the binomial theorem to find terms in the expansion of a binomial to a power greater than 3.

P9: Polar Coordinates, Parameterizations, and Conic Sections

P9.1 Convert between polar and rectangular coordinates. Graph functions given in polar coordinates.

P9.2 Write complex numbers in polar form. Know and use De Moivre's Theorem.

P9.3 Evaluate parametric equations for given values of the parameter.

P9.4 Convert between parametric and rectangular forms of equations.

P9.5 Graph curves described by parametric equations and find parametric equations for a given graph.

P9.6 Use parametric equations in applied contexts (e.g., orbits and projectiles) to model situations and solve problems.

P9.7 Know, explain, and apply the locus definitions of parabolas, ellipses, and hyperbolas and recognize these conic sections in applied situations.

P9.8 Identify parabolas, ellipses, and hyperbolas from equations, write the equations in standard form, and sketch an appropriate graph of the conic section.

P9.9 Derive the equation for a conic section from given geometric information (e.g., find the equation of an ellipse given its two axes). Identify key characteristics (e.g. foci and asymptotes) of a conic section from its equation or graph.

P9.10 Identify conic sections whose equations are in polar or parametric form.

S1.4 Identify influential points in a bivariate data set and predict and verify the effect of their removal on the least-squares line.

S1.5 When applicable, use logarithmic and power transformations to achieve linearity and use the transformed data to make predictions.

S1.6 Explore categorical data via contingency tables, computing and interpreting marginal, joint, and conditional relative frequencies and examining measures of association.

S2.3 Know and recognize in context the concepts of treatment group, control group, and experimental unit and demonstrate the importance of double-blind protocol, random assignment, experimental unit, and replication.

S2.4 Understand and describe how to implement completely randomized and randomized block designs (including matched-pair designs), recognizing when and how blocking can lower variability.

S3.2 Use basic probability rules such as the addition rule, law of total probability, and complement rule to compute probabilities in a variety of models.

S3.3 Use Bayes' Theorem to solve conditional probability problems, with emphasis on the interpretation of results.

S3.4 Know the definition of random variable and be able to derive a discrete probability distribution based on the probability model of the original sample space.

S3.5 Compute the expected value and standard deviation of discrete random variables and know the effect of a linear transformation of a random variable on its mean and standard deviation.

S3.6 Apply standard discrete distributions, including the binomial, geometric, and hypergeometric.

S3.7 Know the definition of independence of two discrete random variables and use the joint distribution to determine whether two discrete random variables are independent.

S3.8 Use tables and technology to determine probabilities and percentiles of normal distributions.

S3.9 Use simulation methods to answer questions about probability models that are too complex for analytical treatment at this level, e.g., interacting particle system models.

S4.3 Assuming a normal model or the applicability of the Central Limit Theorem, compute probabilities for the sample mean, including probabilities that are needed to compute p-values.

S4.4 Apply the (large sample) distribution of the sample proportion to compute probabilities for the sample proportion and know and use rules of thumb for the applicability of the large sample distribution.

S4.5 Assuming a normal model or the applicability of the Central Limit Theorem, derive a P% confidence interval for the mean under the assumption that the population standard deviation is known.

S4.6 Compute control limits for commonly used control charts and use these to assess whether a process is out of control.

S5: Point and Interval Estimation

S5.1 Compute bias, variance, and mean squared error of estimators of the mean and proportion.

S5.2 Know the logic of confidence intervals, the meaning of confidence level, and the effect of changing sample size, confidence level, and variability on the width of the interval.

S5.3 Compute and interpret confidence intervals for one mean and for the difference between two means (in both the paired and unpaired setting) when the standard deviation is unknown, using the t distribution.

S5.4 Compute and interpret (large sample) confidence intervals for one proportion and the difference between two proportions using the normal distribution.

S5.5 Compute the sample size required for a fixed confidence level and interval width for confidence intervals for means and proportions.

S6: Significance Testing

S6.1 Know the terminology and logic of significance testing, including null and alternative hypotheses, p-value, Type I and Type II errors, and power.

S6.2 Assuming a normal model and known standard deviation, carry out a significance test for a single mean, with emphasis on understanding the computation and interpretation of the p-value, and compute the power curve of a test.

S6.3 Carry out (large sample) significance tests for one proportion and the difference of two proportions, with emphasis on proper interpretation of results.

S6.4 Carry out significance tests for one mean and the difference of two means (paired and unpaired) using the t distribution, with emphasis on proper interpretation of results.

S6.5 Carry out chi-squared significance tests of homogeneity, independence, and goodness-of-fit, with emphasis on proper interpretation of results.

S6.6 Assuming a normal model and known standard deviation, compute the sample size necessary to achieve a pre-specified power at a pre-specified value of the population mean.

S6.7 Demonstrate, in the context of specific studies, the understanding that a result can be statistically significant while of insignificant practical importance and that a failure to reject a null hypothesis may be due to low power and does not necessarily imply the null hypothesis is true.

S7: Inference for Regression

S7.1 Know the statistical model for regression, including linearity, normality of errors, and constancy of error variance.

S7.2 Compute and interpret a confidence interval for the slope of a regression line using the t distribution.

S7.3 Test hypotheses about the slope of a regression line, with emphasis on interpretation of results.

S8: Assessing Assumptions of Statistical Models

S8.1 Demonstrate knowledge of the assumptions required for all of the inferential procedures (confidence intervals and significance tests).

S8.2 In the context of specific studies, recognize aspects of study design that either support or offer evidence against required assumptions.

S8.3 Demonstrate knowledge of the possible effects of incorrect assumptions (i.e., improperly specified models) on inferential procedures and of the robustness of inferential procedures to departures from specified assumptions.

S8.4 Show in context an understanding that statistical models are approximations to reality and that care should be exercised in assigning too much precision to measures such as confidence levels or p-values.

MI.AI. Algebra I

L1.1. Number Systems and Number Sense

L1.1.1. Know the different properties that hold in different number systems and recognize that the applicable properties change in the transition from the positive integers to all integers, to the rational numbers, and to the real numbers.

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.

L1.1.3. Explain how the properties of associativity, commutativity, and distributivity, as well as identity and inverse elements, are used in arithmetic and algebraic calculations.

L1.1.4. Describe the reasons for the different effects of multiplication by, or exponentiation of, a positive number by a number less than 0, a number between 0 and 1, and a number greater than 1.

L1.1.5. Justify numerical relationships.

L1.2. Representations and Relationships

L1.2.2. Interpret representations that reflect absolute value relationships.

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.

L2.1. Calculation Using Real and Complex Numbers

L2.1.1. Explain the meaning and uses of weighted averages.

L2.1.2. Calculate fluently with numerical expressions involving exponents; use the rules of exponents; evaluate numerical expressions involving rational and negative exponents; transition easily between roots and exponents.

L2.1.4. Know that the imaginary number i is one of two solutions to x^2 = -1.

A1.1. Construction, Interpretation, and Manipulation of Expressions

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.

A1.1.2. Know the properties of exponents and roots and apply them in algebraic expressions.

A1.1.3. Factor algebraic expressions using, for example, greatest common factor, grouping, and the special product identities.

A1.2. Solutions of Equations and Inequalities

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 linear and quadratic equations and inequalities including systems of up to three linear equations with three unknowns. Justify steps in the solution, and apply the quadratic formula appropriately.

A1.2.4. Solve absolute value equations and inequalities and justify steps in the solution.

A1.2.6. Solve power equations and equations including radical expressions; 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. Justify steps in the solution.

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

A2.1.1. Determine 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.

A2.1.5. Recognize that functions may be defined recursively. Compute values of and graph simple recursively defined functions.

A2.1.6. Identify the zeros of a function, the intervals where the values of a function are positive or negative, and describe the behavior of a function as x approaches positive 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(s).

A2.2. Operations and Transformations with Functions

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

A2.2.2. Apply given transformations to parent functions and represent symbolically.

A2.2.3. Determine whether a function (given in tabular or graphical form) has an inverse and recognize simple inverse pairs.

A2.3. Representations of Functions

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.

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

A2.3.3. Write the general symbolic forms that characterize each family of functions.

A2.4. Models of Real-World Situations Using Families of Functions

A2.4.1. Identify the family of function best suited for modeling a given real-world situation.

A2.4.2. 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.

A2.4.3. Using the adapted general symbolic form, draw reasonable conclusions about the situation being modeled.

A3.1. Lines and Linear Functions

A3.1.1. Write the symbolic forms of linear functions (standard, point-slope, and slope-intercept) given appropriate information and convert between forms.

A3.1.2. Graph lines (including those of the form x = h and y = k) given appropriate information.

A3.1.3. Relate the coefficients in a linear function to the slope and x- and y- intercepts of its graph.

A3.1.4. 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.

A3.2. Exponential and Logarithmic Functions

A3.2.1. Write the symbolic form and sketch the graph of an exponential function given appropriate information.

A3.2.4. Understand and use the fact that the base of an exponential function determines whether the function increases or decreases and how base affects the rate of growth or decay.

A3.2.5. Relate exponential functions to real phenomena, including half-life and doubling time.

A3.3. Quadratic Functions

A3.3.1. Write the symbolic form and sketch the graph of a quadratic function given appropriate information.

A3.3.2. 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.

A3.3.3. Convert quadratic functions from standard to vertex form by completing the square.

A3.3.4. Relate the number of real solutions of a quadratic equation to the graph of the associated quadratic function.

A3.3.5. Express quadratic functions in vertex form to identify their maxima or minima and in factored form to identify their zeros.

A3.4. Power Functions

A3.4.1. Write the symbolic form and sketch the graph of power functions.

A3.4.2. Express directly and inversely proportional relationships as functions and recognize their characteristics.

A3.4.3. Analyze the graphs of power functions, noting reflectional or rotational symmetry.

A3.5. Polynomial Functions

A3.5.1. Write the symbolic form and sketch the graph of simple polynomial functions.

A3.5.2. Understand the effects of degree, leading coefficient, and number of real zeros on the graphs of polynomial functions of degree greater than 2.

A3.5.3. Determine the maximum possible number of zeroes of a polynomial function and understand the relationship between the x-intercepts of the graph and the factored form of the function.

S2.1. Scatter plots and Correlation

S2.1.1. Construct a scatter plot for a bivariate data set with appropriate labels and scales.

S2.1.2. Given a scatter plot, 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 scatter plot 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 that 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.

MI.AII. Algebra II

L1.2.1. Use mathematical symbols to represent quantitative relationships and situations.

L1.3. Counting and Probabilistic Reasoning

L1.3.1. Describe, explain, and apply various counting techniques; relate combinations to Pascal's triangle; know when to use each technique.

L1.3.2. Define and interpret commonly used expressions of probability.

L1.3.3. Recognize and explain common probability misconceptions such as ''hot streaks'' and ''being due.''

L2.1.3. Explain the exponential relationship between a number and its base 10 logarithm, and use it to relate rules of logarithms to those of exponents in expressions involving numbers.

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.

L2.3. Measurement Units, Calculations, and Scales

L2.3.2. Describe and interpret logarithmic relationships in such contexts as the Richter scale, the pH scale, or decibel measurements; solve applied problems.

L2.4. Understanding Error

L2.4.1. Determine what degree of accuracy is reasonable for measurements in a given situation; express accuracy through use of significant digits, error tolerance, or percent of error; describe how errors in measurements are magnified by computation; recognize accumulated error in applied situations.

L2.4.2. Describe and explain round-off error, rounding, and truncating.

L2.4.3. Know the meaning of and interpret statistical significance, margin of error, and confidence level.

A1.1.4. Add, subtract, multiply, and simplify polynomials and rational expressions.

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.

A1.2.5. Solve polynomial equations and equations involving rational expressions and justify steps in the solution.

A1.2.7. Solve exponential and logarithmic equations and justify steps in the solution.

A1.2.9. Know common formulas and apply appropriately in contextual situations.

A1.2.10. Use special values of the inverse trigonometric functions to solve trigonometric equations over specific intervals.

A3.2.2. Interpret the symbolic forms and recognize the graphs of exponential and logarithmic functions.

A3.2.3. Apply properties of exponential and logarithmic functions.

A3.6. Rational Functions

A3.6.1. Write the symbolic form and sketch the graph of simple rational functions.

A3.6.2. Analyze graphs of simple rational functions and understand the relationship between the zeros of the numerator and denominator, and the function's intercepts, asymptotes, and domain.

A3.7. Trigonometric Functions

A3.7.1. Use the unit circle to define sine and cosine; approximate values of sine and cosine; use sine and cosine to define the remaining trigonometric functions; explain why the trigonometric functions are periodic.

A3.7.2. Use the relationship between degree and radian measures to solve problems.

A3.7.3. Use the unit circle to determine the exact values of sine and cosine, for integer multiples of pi/6 and pi/4.

A3.7.4. Graph the sine and cosine functions; analyze graphs by noting domain, range, period, amplitude, and location of maxima and minima.

A3.7.5. Graph transformations of basic trigonometric functions (involving changes in period, amplitude, phase, and midline) and understand the relationship between constants in the formula and the transformed graph.

G1.7. Conic Sections and Their Properties

G1.7.1. Find an equation of a circle given its center and radius; given the equation of a circle, find its center and radius.

G1.7.2. 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.

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

S1.3.1. Explain the concept of distribution and the relationship between summary statistics for a data set and parameters of a distribution.

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

S1.3.3. Know and use the fact that about 68%, 95%, and 99.7% of the data lie within one, two, and three standard deviations of the mean, respectively in a normal distribution.

S1.3.4. Calculate z-scores, use z-scores to recognize outliers, and use z-scores to make informed decisions.

S3.1. Data Collection and Analysis

S3.1.1. Know the meanings of a sample from a population and a census of a population, and distinguish between sample statistics and population parameters.

S3.1.2. Identify possible sources of bias in data collection, 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.

S4.1. Probability

S4.1.1. Understand and construct sample spaces in simple situations

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.

L1.1.6. Explain the importance of the irrational numbers square root of 2 an square root of 3 in basic right triangle trigonometry, and the importance of pi because of its role in circle 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.

L2.3.1. Convert units of measurement within and between systems; explain how arithmetic operations on measurements affect units, and carry units through calculations correctly.

L3.1. Mathematical Reasoning

L3.1.1. Distinguish between inductive and deductive reasoning, identifying and providing examples of each.

L3.1.2. Differentiate between statistical arguments (statements verified empirically using examples or data) and logical arguments based on the rules of logic.

L3.1.3. Define and explain the roles of axioms (postulates), definitions, theorems, counterexamples, and proofs in the logical structure of mathematics. Identify and give examples of each.

L3.2. Language and Laws of Logic

L3.2.1. Know and use the terms of basic logic.

L3.2.2. Use the connectives ''not,'' ''and,'' ''or,'' and ''if..., then,'' in mathematical and everyday settings. Know the truth table of each connective and how to logically negate statements involving these connectives.

L3.2.3. Use the quantifiers ''there exists'' and ''all'' in mathematical and everyday settings and know how to logically negate statements involving them.

L3.2.4. Write the converse, inverse, and contrapositive of an ''If..., then...'' statement. Use the fact, in mathematical and everyday settings, that the contrapositive is logically equivalent to the original while the inverse and converse are not.

L3.3. Proof

L3.3.1. Know the basic structure for the proof of an ''If..., then...'' statement (assuming the hypothesis and ending with the conclusion) and that proving the contrapositive is equivalent.

L3.3.2. Construct proofs by contradiction. Use counter-examples, when appropriate, to disprove a statement.

L3.3.3. 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.

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

G1.1.1. Solve multistep problems and construct proofs involving vertical angles, linear pairs of angles, supplementary angles, complementary angles, and right angles.

G1.1.2. Solve multistep 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.

G1.1.5. Given a line segment in terms of its endpoints in the coordinate plane, determine its length and midpoint.

G1.1.6. Recognize Euclidean geometry as an axiom system. Know the key axioms and understand the meaning of and distinguish between undefined terms, axioms, definitions, and theorems.

G1.2. Triangles and Their Properties

G1.2.1. Prove that the angle sum of a triangle is 180 degrees 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 multistep 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.

G1.2.4. Prove and use the relationships among the side lengths and the angles of 30 degree - 60 degree - 90 degrees triangles and 45 degree - 45 degree - 90 degree triangles.

G1.2.5. Solve multistep 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.

G1.3.2. Know and use the Law of Sines and the Law of Cosines and use them to solve problems. Find the area of a triangle with sides a and b and included angle theta using the formula Area = (1/2) absin theta.

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

G1.4. Quadrilaterals and Their Properties

G1.4.1. Solve multistep 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 multistep problems and construct proofs involving quadrilaterals using Euclidean methods or coordinate geometry.

G1.4.3. Describe and justify hierarchical relationships among quadrilaterals.

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.

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 multistep problems involving circumference and area of circles.

G1.6.2. Solve problems and justify arguments about chords and lines tangent to circles.

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

G1.6.4. Know and use properties of arcs and sectors, and find lengths of arcs and areas of sectors.

G1.8. Three-dimensional Figures

G1.8.1. Solve multistep 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.

G2.1. 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.

G2.1.3. Know and use the relationship between the volumes of pyramids and prisms.

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

G2.2.1. Identify or sketch a possible three-dimensional figure, given two-dimensional views. Create a two-dimensional representation of a three-dimensional figure.

G2.2.2. Identify or sketch cross sections of three-dimensional figures. Identify or sketch solids formed by revolving two-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 that right triangles are congruent using 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, k^2, and k^3, respectively.

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.