9-12.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.
9-12.L1.1 Number Systems and Number Sense
9-12.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.
9-12.L1.1.5 Justify numerical relationships (e.g., show that the sum of even integers is even; that every integer can be written as 3m+k, where k is 0, 1, or 2, and m is an integer; or that the sum of the first n positive integers is n (n+1)/2).
9-12.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 pi because of its role in circle relationships; and the role of e in applications such as continuously compounded interest.
9-12.L1.2.5 Read and interpret representations from various technological sources, such as contour or isobar diagrams.
9-12.L1.3 Counting and Probabilistic Reasoning
9-12.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.
9-12.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.
9-12.L2.1 Calculation Using Real and Complex Numbers
9-12.L2.1.1 Explain the meaning and uses of weighted averages (e.g., GNP, consumer price index, grade point average).
9-12.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.
9-12.L3.1.2 Describe and interpret logarithmic relationships in such contexts as the Richter scale, the pH scale, or decibel measurements (e.g., explain why a small change in the scale can represent a large change in intensity); solve applied problems.
9-12.L3.2 Understanding Error
9-12.L3.2.1 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.
9-12.L3.2.3 Know the meaning of and interpret statistical significance, margin of error, and confidence level.
9-12.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.
9-12.L4.1 Mathematical Reasoning
9-12.L4.1.1 Distinguish between inductive and deductive reasoning, identifying and providing examples of each.
9-12.L4.2.2 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.
9-12.L4.2.3 Use the quantifiers "THERE EXISTS" and "ALL" in mathematical and everyday settings and know how to logically negate statements involving them.
9-12.L4.2.4 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.
9-12.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.
9-12.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.
9-12.A1.1 Construction, Interpretation, and Manipulation of Expressions (linear, quadratic, polynomial, rational, power, exponential, logarithmic, and trigonometric)
9-12.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.
9-12.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.
9-12.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.
9-12.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.
9-12.A1.2.10 Use special values of the inverse trigonometric functions to solve trigonometric equations over specific intervals (e.g., 2sin x - 1 = 0 for 0 "is less than or equal to" x "is less than or equal to" 2 pi).
9-12.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.
9-12.A2.1 Definitions, Representations, and Attributes of Functions
9-12.A2.1.1 Recognize whether a relationship (given in contextual, symbolic, tabular, or graphical form) is a function; and identify its domain and range.
9-12.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).
9-12.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 positive or negative infinity, given the symbolic and graphical representations.
9-12.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).
9-12.A2.2.6 Know and interpret the function notation for inverses and verify that two functions are inverses using composition.
9-12.A2.3 Families of Functions (linear, quadratic, polynomial, rational, power, exponential, logarithmic, and trigonometric)
9-12.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.
9-12.A2.4.1 Write the symbolic forms of linear functions (standard [i.e., Ax + By = C, where B "is not equal to" 0], point-slope, and slope-intercept) given appropriate information, and convert between forms.
9-12.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.
9-12.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).
9-12.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.
9-12.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.
9-12.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).
9-12.A2.9.1 Write the symbolic form and sketch the graph of simple rational functions.
9-12.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.
9-12.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.
9-12.A2.10.4 Graph the sine and cosine functions; analyze graphs by noting domain, range, period, amplitude, and location of maxima and minima.
9-12.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.
9-12.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.
9-12.A3.1 Models of Real-world Situations Using Families of Functions.
9-12.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 = Pto the base 0 of a to the t power)
9-12.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 Pto the base 0 = 300 and a = 1.02 to obtain P = 300(1.02) to the t power.
9-12.A3.1.3 Using the adapted general symbolic form, draw reasonable conclusions about the situation being modeled. In the example above, the exact solution is 365.698, but for this problem an appropriate approximation is 365.
9-12.A3.1.4 Use methods of linear programming to represent and solve simple real-life problems.
9-12.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.
9-12.G1.1 Lines and Angles; Basic Euclidean and Coordinate Geometry
9-12.G1.1.1 Solve multi-step problems and construct proofs involving vertical angles, linear pairs of angles supplementary angles, complementary angles, and right angles.
9-12.G1.1.3 Perform and justify constructions, including midpoint of a line segment and bisector of an angle, using straightedge and compass.
9-12.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.
9-12.G1.1.5 Given a line segment in terms of its endpoints in the coordinate plane, determine its length and midpoint.
9-12.G1.1.6 Recognize Euclidean Geometry as an axiom system; know the key axioms and understand the meaning of and distinguish between undefined terms (e.g., point, line, plane), axioms, definitions, and theorems.
9-12.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.
9-12.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.
9-12.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) a b sin theta.
9-12.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).
9-12.G1.7.3 Graph ellipses and hyperbolas with axes parallel to the x- and y-axes, given equations.
9-12.G1.7.4 Know and use the relationship between the vertices and foci in an ellipse, the vertices and foci in a hyperbola, and the directrix and focus in a parabola; interpret these relationships in applied contexts.
9-12.G1.8.2 Identify symmetries of pyramids, prisms, cones, cylinders, hemispheres, and spheres.
9-12.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.
9-12.G2.1 Relationships Between Area and Volume Formulas
9-12.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.
9-12.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).
9-12.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.
9-12.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.
9-12.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.
9-12.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.
9-12.S1.1 Producing and Interpreting Plots
9-12.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.
9-12.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.
9-12.S1.3.1 Explain the concept of distribution and the relationship between summary statistics for a data set and parameters of a distribution.
9-12.S1.3.2 Describe characteristics of the normal distribution, including its shape and the relationships among its mean, median, and mode.
9-12.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.
9-12.S1.3.4 Calculate z-scores, use z-scores to recognize outliers, and use z-scores to make informed decisions.
9-12.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.
9-12.S2.1 Scatterplots and Correlation
9-12.S2.1.1 Construct a scatterplot for a bivariate data set with appropriate labels and scales.
9-12.S2.1.2 Given a scatterplot, identify patterns, clusters, and outliers; recognize no correlation, weak correlation, and strong correlation.
9-12.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.
9-12.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.
9-12.S2.2 Linear Regression
9-12.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.
9-12.S2.2.2 Use the equation of the least squares regression line to make appropriate predictions.
9-12.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.
9-12.S3.1 Data Collection and Analysis
9-12.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.
9-12.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.
9-12.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.
9-12.S3.2.1 Explain the basic ideas of statistical process control, including recording data from a process over time.
9-12.S3.2.2 Read and interpret basic control charts; detect patterns and departures from patterns.
9-12.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.
9-12.S4.1.1 Understand and construct sample spaces in simple situations (e.g., tossing two coins, rolling two number cubes and summing the results).
9-12.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.
9-12.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.
9-12.S4.2 Application and Representation
9-12.S4.2.1 Compute probabilities of events using tree diagrams, formulas for combinations and permutations, Venn diagrams, or other counting techniques.