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 Results for Math, 9-12 (2006):
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 (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). L1.1.6 Explain the importance of the irrational numbers √2 and √3 in basic right triangle trigonometry; the importance of p because of its role in circle relationships; and the role of e in applications such as continuously compounded interest.
L1.2.1 Use mathematical symbols (e.g., interval notation, set notation, summation notation) to represent quantitative relationships and situations. L1.2.2 Interpret representations that reflect absolute value relationships (e.g. l x - a l = b, or a ?? b) 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.
L1.3.1 Describe, explain, and apply various counting techniques (e.g., finding the number of different 4-letter passwords; permutations; and combinations); relate combinations to Pascal's triangle; know when to use each technique. L1.3.2 Define and interpret commonly used expressions of probability (e.g., chances of an event, likelihood, odds). L1.3.3 Recognize and explain common probability misconceptions such as "hot streaks" and "being due."
L2.1.1 Explain the meaning and uses of weighted averages (e.g., GNP, consumer price index, grade point average). 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 x2 = -1. L2.1.5 Add, subtract, and multiply complex numbers; use conjugates to simplify quotients of complex numbers. L2.1.6 Recognize when exact answers aren't always possible or practical; use appropriate algorithms to approximate solutions to equations (e.g., to approximate square roots).
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.
L3.1.1 Convert units of measurement within and between systems; explain how arithmetic operations on measurements affect units, and carry units through calculations correctly. L3.1.2 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.
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. L3.2.2 Describe and explain round-off error, rounding, and truncating. L3.2.3 Know the meaning of and interpret statistical significance, margin of error, and confidence level.
L4.1.1 Distinguish between inductive and deductive reasoning, identifying and providing examples of each. L4.1.2 Differentiate between statistical arguments (statements verified empirically using examples or data) and logical arguments based on the rules of logic. L4.1.3 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.
L4.2.1 Know and use the terms of basic logic (e.g., proposition, negation, truth and falsity, implication, if and only if, contrapositive, and converse). L4.2.2 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. L4.2.3 Use the quantifiers "THERE EXISTS" and "ALL" in mathematical and everyday settings and know how to logically negate statements involving them. L4.2.4 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.
L4.3.1 Know the basic structure for the proof of an "If..., then..." statement (assuming the hypothesis and ending with the conclusion) and know that proving the contrapositive is equivalent. L4.3.2 Construct proofs by contradiction; use counterexamples, when appropriate, to disprove a statement. 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.
L1.2.5 Read and interpret representations from various technological sources, such as contour or isobar diagrams. L2.1.7 Understand the mathematical bases for the differences among voting procedures. L2.2.4 Compute sums of infinite geometric sequences.
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 (e.g., differences of squares and cubes). A1.1.4 Add, subtract, multiply, and simplify polynomials and rational expressions (e.g., multiply (x - 1) (1 - x2 + 3); simplify ( 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.
A1.2.1 Write equations and inequalities with one or two variables to represent mathematical or applied situations, and solve. A1.2.2 Associate a given equation with a function whose zeros are the solutions of the equation. A1.2.3 Solve (and justify steps in the solutions) linear and quadratic equations and inequalities, including systems of up to three linear equations with three unknowns; apply the quadratic formula appropriately. A1.2.4 Solve absolute value equations and inequalities, (e.g. solve l x - 3 l = 6), and justify steps in the solution. A1.2.5 Solve polynomial equations and equations involving rational expressions (e.g. solve -2x(x2 + 4x+3) = 0; solve x - ( A1.2.6 Solve power equations (e.g., (x + 1)3 = 8) and equations including radical expressions (e.g., ( A1.2.7 Solve exponential and logarithmic equations (e.g., 3(2x) = 24), 2 ln(x + 1) = 4), and justify steps in the solution. A1.2.8 Solve an equation involving several variables (with numerical or letter coefficients) for a designated variable, and justify steps in the solution. A1.2.9 Know common formulas (e.g., slope, distance between two points, quadratic formula, compound interest, distance = velocity · time), and apply appropriately in contextual situations. A1.2.10 Use special values of the inverse trigonometric functions to solve trigonometric equations over specific intervals (e.g., 2sin x - I = 0 for 0 = x = 2p).
A2.1.1 Recognize whether a relationship (given in contextual, symbolic, tabular, or graphical form) is a function; and identify its domain and range. A2.1.2 Read, interpret, and use function notation, and evaluate a function at a value in its domain. A2.1.3 Represent functions in symbols, graphs, tables, diagrams, or words, and translate among representations. A2.1.4 Recognize that functions may be defined by different expressions over different intervals of their domains; such functions are piecewise-defined (e.g., absolute value and greatest integer functions). A2.1.5 Recognize that functions may be defined recursively, and compute values of and graph simple recursively defined functions (e.g., f(0) = 5, and f(n) = f(n-1) + 2 ). A2.1.6 Identify the zeros of a function and the intervals where the values of a function are positive or negative, and describe the behavior of a function, as x approaches postive or negative infinity, given the symbolic and graphical representations. A2.1.7 Identify and interpret the key features of a function from its graph or its formula(e), (e.g. slope, intercept(s), asymptote(s), maximum and minimum value(s), symmetry, average rate of change over an interval, and periodicity).
A2.2.1 Combine functions by addition, subtraction, multiplication, and division. A2.2.2 Apply given transformations (e.g., vertical or horizontal shifts, stretching or shrinking, or reflections about the x- and y-axes) to basic functions, and represent symbolically. A2.2.3 Recognize whether a function (given in tabular or graphical form) has an inverse and recognize simple inverse pairs (e.g., f(x) = x3 and g(x) = x?).
A2.3.1 Identify a function as a member of a family of functions based on its symbolic, or graphical representation; recognize that different families of functions have different asymptotic behavior at infinity, and describe these behaviors. A2.3.2 Describe the tabular pattern associated with functions having constant rate of change (linear); or variable rates of change. A2.3.3 Write the general symbolic forms that characterize each family of functions. (e.g., f(x) = A0ax; f(x) = AsinBx )
A2.4.1 Write the symbolic forms of linear functions (standard [i.e., Ax + By = C, where B ≠ 0], point-slope, and slope-intercept) given appropriate information, and convert between forms. A2.4.2 Graph lines (including those of the form x = h and y = k) given appropriate information. A2.4.3 Relate the coefficients in a linear function to the slope and x- and y-intercepts of its graph. A2.4.4 Find an equation of the line parallel or perpendicular to given line, through a given point; understand and use the facts that non-vertical parallel lines have equal slopes, and that non-vertical perpendicular lines have slopes that multiply to give -1.
A2.5.1 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)x). A2.5.2 Interpret the symbolic forms and recognize the graphs of exponential and logarithmic functions (e.g., f(x) = 10x, f(x) = log x, f(x) = ex, f(x) = ln x); recognize the logarithmic function as the inverse of the exponential function. A2.5.3 Apply properties of exponential and logarithmic functions (e.g., ax+y = axa y; log(ab)= log a + log b). A2.5.4 Understand and use the fact that the base of an exponential function determines whether the function increases or decreases and understand how the base affects the rate of growth or decay. A2.5.5 Relate exponential and logarithmic functions to real phenomena, including half-life and doubling time. A2.6.1 Write the symbolic form and sketch the graph of a quadratic function given appropriate information (e.g., vertex, intercepts, etc.). A2.6.2 Identify the elements of a parabola (vertex, axis of symmetry, direction of opening) given its symbolic form or its graph, and relate these elements to the coefficient(s) of the symbolic form of the function. A2.6.3 Convert quadratic functions from standard to vertex form by completing the square. A2.6.4 Relate the number of real solutions of a quadratic equation to the graph of the associated quadratic function. A2.6.5 Express quadratic functions in vertex form to identify their maxima or minima, and in factored form to identify their zeros.
A2.7.1 Write the symbolic form and sketch the graph of power functions. A2.7.2 Express direct and inverse relationships as functions (e.g., y = kxn and y = kx-n, n > 0) and recognize their characteristics (e.g., in y = x3, note that doubling x results in multiplying y by a factor of 8). A2.7.3 Analyze the graphs of power functions, noting reflectional or rotational symmetry.
A2.8.1 Write the symbolic form and sketch the graph of simple polynomial functions. A2.8.2 Understand the effects of degree, leading coefficient, and number of real zeros on the graphs of polynomial functions of degree greater than 2. A2.8.3 Determine the maximum possible number of zeros of a polynomial function, and understand the relationship between the x-intercepts of the graph and the factored form of the function.
A2.9.1 Write the symbolic form and sketch the graph of simple rational functions. A2.9.2 Analyze graphs of simple rational functions (e.g., f(x) = (
A2.10.1 Use the unit circle to define sine and cosine; approximate values of sine and cosine (e.g., sin 3, or cos 0.5); use sine and cosine to define the remaining trigonometric functions; explain why the trigonometric functions are periodic. A2.10.2 Use the relationship between degree and radian measures to solve problems. A2.10.3 Use the unit circle to determine the exact values of sine and cosine, for integer multiples of p/6 and p/4. A2.10.4 Graph the sine and cosine functions; analyze graphs by noting domain, range, period, amplitude, and location of maxima and minima. A2.10.5 Graph transformations of basic trigonometric functions (involving changes in period, amplitude, and midline) and understand the relationship between constants in the formula and the transformed graph.
A3.1.1 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 = P0at) A3.1.2 Adapt the general symbolic form of a function to one that fits the specifications of a given situation by using the information to replace arbitrary constants with numbers. In the example above, substitute the given values P0 = 300 and a = 1.02 to obtain P = 300(1.02)t. 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.
A1.1.7 Transform trigonometric expressions into equivalent forms using basic identities such as sin2θ + cos2θ = 1, tanθ = ( A2.2.4 If a function has an inverse, find the expression(s) for the inverse. 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. A2.2.6 Know and interpret the function notation for inverses and verify that two functions are inverses using composition. A3.1.4 Use methods of linear programming to represent and solve simple real-life problems.
G1.1.1 Solve multi-step problems and construct proofs involving vertical angles, linear pairs of angles supplementary angles, complementary angles, and right angles. G1.1.2 Solve multi-step problems and construct proofs involving corresponding angles, alternate interior angles, alternate exterior angles, and same-side (consecutive) interior angles. G1.1.3 Perform and justify constructions, including midpoint of a line segment and bisector of an angle, using straightedge and compass. G1.1.4 Given a line and a point, construct a line through the point that is parallel to the original line using straightedge and compass; given a line and a point, construct a line through the point that is perpendicular to the original line; justify the steps of the constructions. G1.1.5 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 (e.g., point, line, plane), axioms, definitions, and theorems.
G1.2.1 Prove that the angle sum of a triangle is 180° and that an exterior angle of a triangle is the sum of the two remote interior angles. G1.2.2 Construct and justify arguments and solve multi-step problems involving angle measure, side length, perimeter, and area of all types of triangles. G1.2.3 Know a proof of the Pythagorean Theorem and use the Pythagorean Theorem and its converse to solve multi-step problems. G1.2.4 Prove and use the relationships among the side lengths and the angles of 30°- 60°- 90° triangles and 45°- 45°- 90° triangles. G1.2.5 Solve multi-step problems and construct proofs about the properties of medians, altitudes, and perpendicular bisectors to the sides of a triangle, and the angle bisectors of a triangle; using a straightedge and compass, construct these lines.
G1.3.1 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 θ using the formula Area = (1/2) a b sin θ. G1.3.3 Determine the exact values of sine, cosine, and tangent for 0°, 30°, 45°, 60°, and their integer multiples, and apply in various contexts.
G1.4.1 Solve multi-step problems and construct proofs involving angle measure, side length, diagonal length, perimeter, and area of squares, rectangles, parallelograms, kites, and trapezoids. G1.4.2 Solve multi-step problems and construct proofs involving quadrilaterals (e.g., prove that the diagonals of a rhombus are perpendicular) using Euclidean methods or coordinate geometry. G1.4.3 Describe and justify hierarchical relationships among quadrilaterals, (e.g. every rectangle is a parallelogram). G1.4.4 Prove theorems about the interior and exterior angle sums of a quadrilateral.
G1.5.1 Know and use subdivision or circumscription methods to find areas of polygons (e.g., regular octagon, non-regular pentagon). G1.5.2 Know, justify, and use formulas for the perimeter and area of a regular n-gon and formulas to find interior and exterior angles of a regular n-gon and their sums. G1.6.1 Solve multi-step problems involving circumference and area of circles. G1.6.2 Solve problems and justify arguments about chords (e.g., if a line through the center of a circle is perpendicular to a chord, it bisects the chord) and lines tangent to circles (e.g., a line tangent to a circle is perpendicular to the radius drawn to the point of tangency). G1.6.3 Solve problems and justify arguments about central angles, inscribed angles and triangles in circles. G1.6.4 Know and use properties of arcs and sectors, and find lengths of arcs and areas of sectors. 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.
G1.8.1 Solve multi-step problems involving surface area and volume of pyramids, prisms, cones, cylinders, hemispheres, and spheres. G1.8.2 Identify symmetries of pyramids, prisms, cones, cylinders, hemispheres, and spheres.
G2.1.1 Know and demonstrate the relationships between the area formula of a triangle, the area formula of a parallelogram, and the area formula of a trapezoid. G2.1.2 Know and demonstrate the relationships between the area formulas of various quadrilaterals (e.g., explain how to find the area of a trapezoid based on the areas of parallelograms and triangles). G2.1.3 Know and use the relationship between the volumes of pyramids and prisms (of equal base and height) and cones and cylinders (of equal base and height).
G2.2.1 Identify or sketch a possible 3-dimensional figure, given 2-dimensional views (e.g., nets, multiple views); create a 2-dimensional representation of a 3-dimensional figure. G2.2.2 Identify or sketch cross-sections of 3-dimensional figures; identify or sketch solids formed by revolving 2-dimensional figures around lines.
G2.3.1 Prove that triangles are congruent using the SSS, SAS, ASA, and AAS criteria, and for right triangles, the hypotenuse-leg criterion. G2.3.2 Use theorems about congruent triangles to prove additional theorems and solve problems, with and without use of coordinates. G2.3.3 Prove that triangles are similar by using SSS, SAS, and AA conditions for similarity. G2.3.4 Use theorems about similar triangles to solve problems with and without use of coordinates. G2.3.5 Know and apply the theorem stating that the effect of a scale factor of k relating one two dimensional figure to another or one three dimensional figure to another, on the length, area, and volume of the figures is to multiply each by k, k2, and k3, respectively.
G3.1.1 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.
G3.2.1 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.
G1.4.5 Understand the definition of a cyclic quadrilateral, and know and use the basic properties of cyclic quadrilaterals. 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. G3.2.3 Find the image of a figure under the composition of a dilation and an isometry.
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.
S1.2.1 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.
S1.3.1 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. S2.1.1 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.
S2.2.1 For bivariate data which appear to form a linear pattern, find the least squares regression line by estimating visually and by calculating the equation of the regression line; interpret the slope of the equation for a regression line. S2.2.2 Use the equation of the least squares regression line to make appropriate predictions.
S3.1.1 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.
S4.1.1 Understand and construct sample spaces in simple situations (e.g., tossing two coins, rolling two number cubes and summing the results). S4.1.2 Define mutually exclusive events, independent events, dependent events, compound events, complementary events and conditional probabilities; and use the definitions to compute probabilities.
S4.2.1 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.
S3.1.4 Design simple experiments or investigations to collect data to answer questions of interest; interpret and present results. 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. 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. S3.2.1 Explain the basic ideas of statistical process control, including recording data from a process over time. S3.2.2 Read and interpret basic control charts; detect patterns and departures from patterns. 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.
Students use the definition of function (including domain and range). They combine functions using algebraic operations and composition. They write a given function as a composition of simpler functions. The notion of one-to-one function is introduced, leading to the definition of an inverse function. Students find the symbolic expression for the inverse of a given function and show that two given functions are inverses.
Students graph logarithmic functions as inverses of exponential functions, and solve equations involving exponential and logarithmic functions. They determine the asymptotic behavior of exponential and logarithmic functions with different bases. Students apply these functions in real world situations, such as exponential growth and decay, and compound interest.
Students use the unit circle to define sine, cosine, and the other trigonometric functions. They apply transformations involving changes in amplitude, midline, period, and phase, to trigonometric functions and represent the results graphically and symbolically. The inverse trigonometric functions and their graphs are introduced. They establish and learn sum and difference formulas and other basic trigonometric identities. They use these to simplify trigonometric expressions, to solve trigonometric equations, prove trigonometric identities, and solve applied problems.
Students learn the Fundamental Theorem of Algebra, the Remainder Theorem, and the Factor Theorem. They solve polynomial equations and inequalities by factoring and dividing polynomials. They identify the large-scale behavior of the graph of a polynomial from its leading term. Students solve rational equations and inequalities. They determine the asymptotes of the graph of a rational function.
Students learn the definition and geometric interpretation of difference quotients. Using the definition, they represent and simplify difference quotients and interpret difference quotients as rates of change and slopes of secant lines. They acquire an informal meaning for the limit of a function and relate that meaning to the graph.
Students sketch and perform operations (multiplication by scalars, addition, and subtraction) of vectors in the plane and use vectors in applications. They learn the algebraic and geometric definitions of dot product of vectors, and use them in applications. Students represent rotations of the plane as matrices and apply these in the context of analytic geometry. They multiply matrices and multiply vectors by matrices, compute determinants, and they solve systems of two and three linear equations by matrix methods. Students compute inverses of three-by-three matrices, when they exist.
Students find the nth term in arithmetic sequences, geometric sequences, and recursively defined sequences. They use sigma notation and compute sums of finite arithmetic sequences. Students compute sums of finite and infinite geometric series and apply the convergence criterion for geometric series.
Students use polar coordinates and graph equations in polar form. They write complex numbers in polar form and use DeMoivre's Theorem. Students parameterize segments and curves, and recognize implicitly defined curves. They identify parabolas, ellipses and hyperbolas from their equations, put the equations in standard form, sketch and analyze the graphs and characteristics (e.g. finding foci).
Students prove theorems and use mathematical induction.
Students learn and apply techniques for exploring univariate and bivariate data using both graphical and numerical summaries. Fundamental is fostering students' understanding of variability in data, and learning how to make comparisons between data sets in the presence of variability.
Students learn methods of designing surveys and controlled experiments, design surveys and experiments, and use their knowledge of design to critically assess conclusions. The importance of randomization in minimizing bias, and of methods such as blocking to reduce variability, are stressed.
Students are introduced to commonly-used discrete probability models such as the binomial and hypergeometric models and use these to model real-world phenomena. Conditional probabilities including Bayes' Theorem are used to solve problems from health, public policy, and other areas, and students gain facility with the normal distribution.
Students learn results for sums of random variables, including an informal treatment of the central limit theorem, and apply these results to sampling distributions of common estimators. Statistical process control and the creation and interpretation of control charts are included.
Students study basic properties of point estimators, and then apply their knowledge of sampling distributions to construct confidence interval estimators for means and proportions in one- and two- sample problems for both means and proportions. Correct interpretation of confidence intervals is stressed.
Students learn the terminology and logic of significance testing, including power, learn to perform significance tests for means and proportions in one- and two-sample problems for means and proportions, and are introduced to chi-squared testing. A main focus is on correct interpretation of results.
The statistical model for simple linear regression is introduced, and students learn to construct confidence intervals and perform significance tests for the slope of a regression line.
Throughout the sections on probability and statistical inference, the role of the underlying mathematical model and its assumptions is kept in the forefront. Students learn to assess the validity of the model and to gauge the effect of departures from model assumptions. |
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Strand 1: Quantitative Literacy and Logic
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