Algebra
Major Cluster: Represent and solve equations and inequalities graphically.
Generate resourceSupporting Cluster: Write expressions and equations in equivalent forms to solve problems.
Generate resourceSupporting Cluster: Build a function that models a relationship between two quantities.
Generate resourceMajor Cluster: Understand the concept of a function and use function notation.
Generate resourceMajor Cluster: Interpret functions that arise in applications in terms of the context.
Generate resourceSupporting Cluster: Construct and compare linear, quadratic, and exponential models and solve problems.
Generate resourceAdditional Cluster: Summarize, represent, and interpret data on a single count or measurement variable.
Generate resourceSupporting Cluster: Understand and evaluate random processes underlying statistical experiments.
Generate resourceCreate an equation involving one operation with one variable and use it to solve a real-world problem.
Generate resourceIdentify an algebraic expression involving one arithmetic operation to represent a real-world problem.
Generate resourceSolve simple algebraic equations with one variable using multiplication and division.
Generate resourceDetermine the successive term in a geometric sequence given the common ratio.
Generate resourceSelect the appropriate graphical representation (first quadrant) given a situation involving constant rate of change.
Generate resourceDetermine an arithmetic sequence with whole numbers when provided a recursive rule.
Generate resourceConstruct graphs that represent linear functions with different rates of change and interpret which is faster/slower, higher/lower, etc.
Generate resourceModel a simple linear function such as y = mx to show that these functions increase by equal amounts over equal intervals.
Generate resourceModel a simple linear function such as y = mx to show that these functions increase by equal amounts over equal intervals.
Generate resourceModel a simple linear function such as y = mx to show that these functions increase by equal amounts over equal intervals.
Generate resourceUse the commutative, associative, and distributive properties to add, subtract, and multiply whole numbers.
Generate resourceSolve real-world problems involving addition and subtraction of decimals, using models when needed.
Generate resourceSolve real-world problems involving multiplication of decimals and whole numbers, using models when needed.
Generate resourceDetermine the likelihood of an event occurring when the outcomes are equally likely to occur.
Generate resourceDetermine the likelihood of an event occurring when the outcomes are equally likely to occur.
Generate resourceGiven data, construct a simple graph (line, pie, bar, or picture) or table, and interpret the data.
Generate resourceGiven data, construct a simple graph (line, pie, bar, or picture) or table, and interpret the data.
Generate resourceCalculate the mean of a given data set (limit the number of data points to fewer than five).
Generate resourceAlgebra 1
Note: b, c, and d, exist and can be found in A2.
Generate resourceCreate equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
Generate resourceCreate equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
Generate resourceRepresent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
Generate resourceRearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohmβs law ππ = πΌπΌπΌπΌ to highlight resistance π π .
Generate resourceUnderstand solving equations as a process of reasoning and explain the reasoning.
Generate resourceExplain each step-in solving equations as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. (A1 and A2)
Generate resourceSolve rational and radical equations in one variable and give examples showing how extraneous solutions may arise. (A1 and A2)
Generate resourceSolve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Generate resourceUse the method of completing the square to transform any quadratic equation in x into an equation of the form (π₯π₯ β ππ)2 = ππ that has the same solutions.
Generate resourceExplain how the strategy of elimination results in finding solution(s) to a system of equations.
Generate resourceSolve systems of linear equations exactly and approximately. For example, with graphs, focusing on pairs of linear equations in two variables.
Generate resourceSolve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. (A1 and A2)
Generate resourceUnderstand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). (A1 and A2)
Generate resourceExplain why the solution(s) of a system of equations are the point(s) of intersection(s) on a coordinate plane. Find the solutions approximately. For example, using technology to graph the functions, make tables of values, or find successive approximations. Include cases where ππ(π₯π₯) and/or ππ(π₯π₯) are quadratic, exponential, rational, absolute value functions, polynomial, exponential, and logarithmic functions. β (A1 and A2)
Generate resourceGraph and interpret (with the use of technology) the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. (A1 and A2)
Generate resourceInterpret expressions that represent a quantity in terms of its context. Interpret parts of an expression, such as terms, factors, and coefficients.
Generate resourceUse the structure of an expression to identify ways to rewrite it. For example, see π₯π₯4 β π¦π¦4 as (π₯π₯2)2 β (π¦π¦2)2, thus recognizing it as a difference of squares that can be factored as (π₯π₯2β π¦π¦2)(π₯π₯2+ π¦π¦2).
Generate resourceChoose and produce an equivalent form of an expression or equation to reveal and explain properties of the quantity represented by the expression. β (A1 and A2)
Generate resourceFactor a quadratic expression to reveal the zeros of the function it defines.
Generate resourceComplete the square in a quadratic equation to reveal the maximum or minimum value of the function it defines.
Generate resourceUse the properties of exponents to transform expressions for exponential functions. For example, the expression 3π₯π₯ can be rewritten as (1 + 2)π₯π₯ to reveal the growth rate is 200%.
Generate resourceWrite a function that describes a relationship between two quantities. Determine an explicit expression, a recursive process, or steps for calculation from a context.
Generate resourceWrite arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Note: Interpret arithmetic sequences as linear functions and geometric sequences as exponential functions. β (A1 and A2)
Generate resourceIdentify the effect on linear and quadratic graphs of replacing ππ(π₯π₯) by ππ(π₯π₯) + ππ , ππππ(π₯π₯), ππ(ππππ), and ππ(π₯π₯ + ππ) for specific values of ππ (both positive and negative); find the value of ππ given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
Generate resourceUnderstand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If ππ is a function and π₯π₯ is an element of its domain, then ππ(π₯π₯) denotes the output of ππ corresponding to the input π₯π₯. The graph of ππ is the graph of the equation π¦π¦ = ππ(π₯π₯).
Generate resourceUse function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
Generate resourceRecognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by ππ(0) = ππ(1) = 1, ππ(ππ + 1) = ππ(ππ) + ππ(ππ β 1)for ππ β₯ 1.
Generate resourceFor a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features may include intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximum and minimum; and symmetries. β (A1 and A2)
Generate resourceRelate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
Generate resourceCalculate and interpret the average rate of change of a nonlinear function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. β (A1 and A2)
Generate resourceGraph functions expressed symbolically and show key features of the graph by hand in simple cases and using technology for more complicated cases.
Generate resourceGraph linear and quadratic functions expressed symbolically and show key features of the graph by hand in simple cases and using technology for more complicated cases, including intercepts, maxima, and minima if they exist. β (A1 and A2)
Generate resourceWrite a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. For example, use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. (A1 and A2)
Generate resourceConstruct and compare linear, quadratic, and exponential models and solve problems.
Generate resourceDistinguish between situations that can be modeled with linear functions and with exponential functions.
Generate resourceProve that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
Generate resourceRecognize situations in which one quantity changes at a constant rate per unit interval relative to another.
Generate resourceRecognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
Generate resourceConstruct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). (A1 and A2)
Generate resourceUse graphs and tables to show that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
Generate resourceInterpret the parameters in a linear or exponential function in terms of a context. (A1 and A2)
Generate resourceUse units to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. (A1 and A2)
Generate resourceChoose a level of accuracy appropriate to limitations on measurement when reporting quantities.
Generate resourceExplain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
Generate resourceSummarize, represent, and interpret data on a single count or measurement variable.β
Generate resourceRepresent data with plots on the real number line (dot plots, histograms, and box plots) in a modeling context. β
Generate resourceUse statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. β
Generate resourceInterpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). β
Generate resourceSummarize, represent, and interpret data on two categorical and quantitative variables.β
Generate resourceSummarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. β
Generate resourceRepresent data on two quantitative variables on a scatter plot and describe how the variables are related. β
Generate resourceFit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.
Generate resourceInformally assess the fit of a function by plotting and analyzing residuals.
Generate resourceFit a linear function for a scatter plot that suggests a linear association.
Generate resourceInterpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. β
Generate resourceCompute (using technology) and interpret the correlation coefficient of a linear fit. β
Generate resourceAlgebra 2
Note: a exists and can be found in A1.
Generate resourceUnderstand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
Generate resourceKnow and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by π₯π₯ β ππ is ππ(ππ), so ππ(ππ) = 0 if and only if (π₯π₯ β ππ) is a factor of ππ(π₯π₯).
Generate resourceIdentify zeros of polynomials when suitable factorizations are available and use the zeros to construct a rough graph of the function defined by the polynomial.
Generate resourceRewrite simple rational expressions in different forms; write ππ(π₯π₯) ππ(π₯π₯) in the form ππ(π₯π₯) + ππ(π₯π₯) ππ(π₯π₯) , where ππ(π₯π₯), ππ(π₯π₯), ππ(π₯π₯), and ππ(π₯π₯) are polynomials with the degree of r(x) less than the degree of b(x). For example, in the same way one may view 11 7 as (7+4) 7 = 1 + 4 7 , one can view (π₯π₯+7) (π₯π₯+3) as ((π₯π₯+3)+4) (π₯π₯+3) = 1 + 4 π₯π₯+3
Generate resourceUnderstand solving equations as a process of reasoning and explain the reasoning.
Generate resourceExplain each step in solving an equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. (A1 and A2)
Generate resourceSolve rational and radical equations in one variable and give examples showing how extraneous solutions may arise. (A1 and A2)
Generate resourceUse the method of completing the square to transform any quadratic equation in π₯π₯ into an equation of the form (π₯π₯ β ππ)2 = ππ that has the same solutions.
Generate resourceSolve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. (A1 and A2)
Generate resourceUnderstand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). (A1 and A2)
Generate resourceExplain why the solution(s) of a system of equations are the point(s) of intersection(s) on a coordinate plane. Find the solutions approximately. For example, using technology to graph the functions, make tables of values, or find successive approximations. Include cases where ππ(π₯π₯) and/or ππ(π₯π₯) are quadratic, exponential, rational, absolute value functions, polynomial, exponential, and logarithmic functions. β (A1 and A2)
Generate resourceGraph and interpret (with the use of technology) the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. (A1 and A2)
Generate resourceInterpret expressions that represent a quantity in terms of its context. For example, interpret ππ(1 + ππ)ππ as the product of ππ and ππ factor not depending on ππ. β
Generate resourceUse the structure of an expression to identify ways to rewrite it. For example, see π₯π₯4 β π¦π¦4 as (π₯π₯2)2 β (π¦π¦2)2, thus recognizing it as a difference of squares that can be factored as (π₯π₯2 β π¦π¦2)(π₯π₯2 + π¦π¦2).
Generate resourceChoose and produce an equivalent form of an expression or equation to reveal and explain properties of the quantity represented by the expression. (A1 and A2)
Generate resourceFactor a quadratic expression to reveal the zeros of the function it defines.
Generate resourceComplete the square in a quadratic equation to reveal the maximum or minimum value of the function it defines.
Generate resourceUse the properties of exponents to transform expressions for exponential functions. For example, the expression 3π₯π₯ can be rewritten as (1 + 2)π₯π₯ to reveal the growth rate is 200%. β
Generate resourceApply the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. β
Generate resourceWrite a function that describes a relationship between two quantities. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. β
Generate resourceWrite arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. β (A1 and A2)
Generate resourceIdentify the effect on linear and quadratic graphs of replacing ππ(π₯π₯) by ππ(π₯π₯) + ππ , ππππ(π₯π₯), ππ(ππππ), and ππ(π₯π₯ + ππ) for specific values of ππ (both positive and negative); find the value of ππ given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. (A1 and A2)
Generate resourceSolve an equation of the form ππ(π₯π₯) = ππ for a simple function ππ that has an inverse and write an expression for the inverse. For example,ππ(π₯π₯) = 2π₯π₯3 or,ππ(π₯π₯) = (π₯π₯+1) (π₯π₯β1) for π₯π₯ β 1.
Generate resourceFor a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features may include intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximum and minimum; and symmetries. β (A1 and A2)
Generate resourceRelate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. β (A1 and A2)
Generate resourceCalculate and interpret the average rate of change of a nonlinear function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. β (A1 and A2)
Generate resourceGraph functions expressed symbolically and show key features of the graph by hand in simple cases and using technology for more complicated cases. β (A1 and A2)
Generate resourceGraph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
Generate resourceGraph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
Generate resourceGraph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
Generate resourceWrite a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. For example, rewrite rational expressions to show the vertical transformation. (A1 and A2)
Generate resourceCompare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
Generate resourceConstruct and compare linear, quadratic, and exponential models and solve problems.
Generate resourceConstruct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). (A1 and A2)
Generate resourceFor exponential models, express as a logarithm the solution to ππ, ππππππ = ππ where ππ, ππ, ππ, and ππ are numbers and the base ππ is 2, 10, or ππ, evaluate the logarithm using technology.
Generate resourceInterpret the parameters in a linear or exponential function in terms of a context. (A1 and A2)
Generate resourceUnderstand the radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
Generate resourceExplain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
Generate resourceUse special triangles to determine geometrically the values of sine, cosine for ππ 3 , ππ 4 , and ππ 6 , and use the unit circle to express the values of sine and cosine for ππ β π₯π₯, ππ + π₯π₯, and 2ππ β π₯π₯ in terms of their values for π₯π₯, where π₯π₯ is any real number. Note: Does not include tangent.
Generate resourceUse the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
Generate resourceChoose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. β
Generate resourceKnow there is a complex number ππ such that ππ2 = β1, and every complex number has the form ππ + ππππ with ππ and ππ real.
Generate resourceUse the relation ππ 2 = β1and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
Generate resourceSolve quadratic equations with real coefficients that have complex solutions.
Generate resourceUse units to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. (A1 and A2)
Generate resourceChoose a level of accuracy appropriate to limitations on measurement when reporting quantities.
Generate resourceExplain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5 1 3 to be the cube root of 5 because we want (51 3)3 = 5( 1 3) 3 to hold, so 5( 1 3) 3 must equal 5.
Generate resourceRewrite expressions involving radicals and rational exponents using the properties of exponents.
Generate resourceUnderstand and evaluate random processes underlying statistical experiments.β
Generate resourceUnderstand statistics as a process for making inferences about population parameters based on a random sample from that population. β
Generate resourceDecide if a specified model is consistent with results from a given data-generating process. For example, using simulation or a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model? β
Generate resourceRecognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. β
Generate resourceUse data from a sample survey to estimate a population mean or proportion; develop a margin of error using simulation models for random sampling. β
Generate resourceUse data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. β
Generate resourceUse the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. β
Generate resourceMake inferences and justify conclusions from sample surveys, experiments, and observational studies.β
Generate resourceSummarize, represent, and interpret data on a single count or measurement variable. β
Generate resourceCreate equations and inequalities in one variable and use them to solve problems. (A1 and A2)
Generate resourceCreate equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. (A1 and A2)
Generate resourceRepresent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. (A1 and A2)
Generate resourceRearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange the formula for the area of a trapezoid, A =(b1 +b2)/2*h for the length of one of the bases. (A1 and A2)
Generate resourceCalculus
Explore additional topics such as sequences and series, Taylor polynomials, and polar coordinates (optional depending on course length and student readiness).
Generate resourcePrepare for further study in mathematics, science, engineering, and related fields.
Generate resourceDefine the instantaneous rate of change at a point as the limit of average rates of change.
Generate resourceIdentify the derivative of a function using appropriate strategies. For example, rules for sums, differences, products, quotients, and limits of functions.
Generate resourceExplain the relationship between continuity and differentiability at a point.
Generate resourceCompute derivatives of algebraic, trigonometric, inverse, exponential, and logarithmic functions.
Generate resourceApply the rules of differentiation, including the product rule, quotient rule, and chain rule.
Generate resourceSolve first-order differential equations, including linear and exponential growth and decay models.
Generate resourceInterpret derivatives in real-world contexts, such as motion and growth problems.
Generate resourceDetermine analytically and graphically where a function, or derivative function, is positive or negative, increasing or decreasing, and/or concave up or concave down.
Generate resourceExplain the relationships among the behaviors of ππ, ππβ² , ππππππ ππβ²β²; for example, if ππβ²β² is positive, then ππβ² is increasing and ππ is concave up.
Generate resourceAnalyze and sketch graphs of ππ, ππβ² , ππππππ ππβ²β².
Generate resourceUse the first and second derivative tests to find and classify critical points.
Generate resourceState the Fundamental Theorem of Calculus and use it to evaluate definite integrals and construct antiderivatives.
Generate resourceCompute indefinite and definite integrals of algebraic, trigonometric, exponential, inverse, and logarithmic functions.
Generate resourceApply integration techniques, including substitution and integration by parts.
Generate resourceDetermine if an improper integral converges or diverges using limits of definite integrals.
Generate resourceTransform integrands (using substitution and other techniques) to find antiderivatives using a table of integrals.
Generate resourceUse integrals to find the area between curves and volume of solids of revolution.
Generate resourceSolve problems involving continuity and determine the continuity of functions.
Generate resourceGeometry
Not applicable.
Generate resourceSee EE.G-CO.6-8
Generate resourceSee EE.G-CO.6-8
Generate resourceKnow the attributes of perpendicular lines, parallel lines, and line segments; angles; and circles.
Generate resourceGiven a geometric figure and a rotation, reflection, or translation of that figure, identify the components of the two figures that are congruent.
Generate resourceGiven a geometric figure and a rotation, reflection, or translation of that figure, identify the components of the two figures that are congruent.
Generate resourceMake a prediction about the volume of a container, the area of a figure, and the perimeter of a figure, and then test the prediction using formulas or models.
Generate resourceMake a prediction about the volume of a container, the area of a figure, and the perimeter of a figure, and then test the prediction using formulas or models.
Generate resourceMake a prediction about the volume of a container, the area of a figure, and the perimeter of a figure, and then test the prediction using formulas or models.
Generate resourceIdentify the shapes of two-dimensional cross-sections of three-dimensional objects.
Generate resourceFind perimeters and areas of squares and rectangles to solve real-world problems.
Generate resourceConstruct the inscribed and circumscribed circles of a triangle with technology, and investigate properties of a quadrilateral inscribed in a circle.
Generate resourceIdentify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
Generate resourceDerive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
Generate resourceKnow precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
Generate resourceRepresent transformations in the plane using geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not. For example, translation versus horizontal stretch).
Generate resourceGiven a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
Generate resourceDevelop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
Generate resourceGiven a geometric figure and a rotation, reflection, or translation, draw the transformed figure. Specify a sequence of transformations that will carry a given figure onto another. For example, using graph paper, tracing paper, or geometry software.
Generate resourceUse geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
Generate resourceUse the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
Generate resourceExplain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
Generate resourceProve theorems about triangles. Theorems include measures of interior angles of a triangle sum to 180; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
Generate resourceProve theorems about parallelograms. Theorems include opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
Generate resourceProve theorems about lines and angles. Theorems include vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segmentβs endpoints.
Generate resourceMake formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
Generate resourceConstruct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Generate resourceAdditional Cluster: Explain volume formulas and use them to solve problems.
Generate resourceGive an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.
Generate resourceUse volume formulas for cylinders, pyramids, cones, and spheres to solve problems. β
Generate resourceVisualize relationships between two-dimensional and three-dimensional objects.
Generate resourceAdditional Cluster: Visualize relationships between two-dimensional and three-dimensional objects.
Generate resourceIdentify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
Generate resourceTranslate between the geometric description and the equation for a conic section.
Generate resourceAdditional Cluster: Translate between the geometric description and the equation for a conic section.
Generate resourceDerive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
Generate resourceMajor Cluster: Use coordinates to prove simple geometric theorems algebraically.
Generate resourceUse coordinate geometry to prove simple geometric theorems. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle.
Generate resourceProve the slope criteria for parallel and perpendicular lines and use them to solve geometric problems. For example, find the equation of a line parallel or perpendicular to a given line that passes through a given point.
Generate resourceFind the point on a directed line segment between two given points that partitions the segment in a given ratio.
Generate resourceUse coordinates to compute perimeters of polygons and areas of triangles and rectangles. For example, use the distance formula to calculate the distance between the two points. β
Generate resourceUse geometric shapes, their measures, and their properties to describe and explain objects. For example, modeling a tree trunk as a cylinder. β
Generate resourceApply concepts of density based on area and volume in modeling situations. For example, persons per square mile, BTUs per cubic foot. β
Generate resourceApply geometric methods to solve design problems. For example, designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios. β
Generate resourceMajor Cluster: Understand similarity in terms of similarity transformations.
Generate resourceVerify experimentally the properties of dilations given by a center and a scale factor:
Generate resourceA dilation takes a line not passing through the center of the dilation to a parallel line and leaves a line passing through the center unchanged.
Generate resourceThe dilation of a line segment is longer or shorter in the ratio given by the scale factor.
Generate resourceGiven two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
Generate resourceUse the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
Generate resourceProve theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
Generate resourceUse congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Generate resourceDefine trigonometric ratios using similar triangle ratios and the ratios of corresponding side lengths.
Generate resourceExplain and use the relationship between the sine and cosine of complementary angles.
Generate resourceUse trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems, including special right triangles. β
Generate resourceAdditional Cluster: Use independence and conditional probability to interpret data.
Generate resourceDescribe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (βorβ βandβ βnotβ). β
Generate resourceUnderstand that two events π΄π΄ and π΅π΅ are independent if the probability of π΄π΄ and π΅π΅ occurring together is the product of their probabilities and use this characterization to determine if they are independent. β
Generate resourceUnderstand the conditional probability of π΄π΄ given π΅π΅ as ππ(π΄π΄ ππππππ π΅π΅) ππ(π΅π΅) , and interpret independence of π΄π΄ and π΅π΅ as saying that the conditional probability of π΄π΄ given π΅π΅ is the same as the probability of π΄π΄, and the conditional probability of π΅π΅ given π΄π΄ is the same as the probability of π΅π΅.β
Generate resourceConstruct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among mathematics, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. β
Generate resourceRecognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. β
Generate resourceUse the rules of probability to compute probabilities of compound events in a uniform probability model. β
Generate resourceFind the conditional probability of π΄π΄ given π΅π΅ as the fraction of π΅π΅β²π π outcomes that also belong to π΄π΄ and interpret the answer in terms of the model. β
Generate resourceApply the Addition Rule, ππ(π΄π΄ ππππ π΅π΅) = ππ(π΄π΄) + ππ(π΅π΅) β ππ(π΄π΄ ππππππ π΅π΅), and interpret the answer in terms of the model. β
Generate resourceAnalyze decisions and strategies using probability concepts. For example, product testing, medical testing, pulling a hockey goalie at the end of a game. β
Generate resourcePrecalculus
Understand the properties of conic sections, including circles, parabolas, ellipses, and hyperbolas.
Generate resourceSolve problems involving distance, midpoint, slope, and equations of lines and circles.
Generate resourceOperations with Polynomial, Rational, Exponential, and Logarithmic Functions
Generate resourceIdentify and graph various functions, including linear, quadratic, polynomial, rational, exponential, logarithmic, and trigonometric functions.
Generate resourceAnalyze functions by considering domain, range, symmetry, intercepts, and asymptotic behavior.
Generate resourcePerform operations of addition, subtraction, multiplication, division, and composition of functions.
Generate resourceAnalyze, graph, and solve problems using polynomial and rational functions.
Generate resourceUnderstand the behavior of polynomial functions, including end behavior, turning points, and factors.
Generate resourceIdentify and graph rational functions and analyze their asymptotic behavior and discontinuities.
Generate resourceAnalyze, graph, and solve problems using exponential and logarithmic functions.
Generate resourceGraph exponential and logarithmic functions and understand their transformations.
Generate resourceApply exponential and logarithmic functions in realworld contexts such as population growth, compound interest, and exponential decay.
Generate resourceFind the nth term, partial sums, and sums of finite and infinite sequences and series.
Generate resourceApply sequences and series to solve problems in mathematics and other disciplines.
Generate resourceDescribe how quantities change with respect to each other in a parametric function.
Generate resourceGraph conic sections using implicitly defined functions and parametric functions.
Generate resourceGraph trigonometric functions including sine, cosine, tangent, cosecant, secant, and cotangent.
Generate resourceUnderstand the unit circle and use it to define trigonometric values for unique angles.
Generate resourceSolve trigonometric equations and inequalities and apply trigonometric identities.
Generate resourceDescribe how angles and radii change with respect to each other in a polar graph.
Generate resourceUse trigonometric identities to simplify expressions and verify identities.
Generate resourceApply trigonometric identities to solve problems involving triangles, vectors, and periodic phenomena.
Generate resourceStatistics and Probability
Know and apply the Binomial Theorem for the expansion of (π₯π₯ + π¦π¦)ππ in powers of π₯π₯ and π¦π¦ for a positive integer ππ, where π₯π₯ and π¦π¦ are any numbers, with coefficients determined for example by Pascalβs Triangle. For example, constructing a distribution for the number made for a 35% free-throw shooter when given 10 attempts.
Generate resourceUnderstand independence and conditional probability and use them to interpret data.β
Generate resourceDescribe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or" "and" "not.β) β
Generate resourceUnderstand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities and use this characterization to determine if they are independent. β
Generate resourceUnderstand the conditional probability of A given B as ππ(π΄π΄ ππππππ π΅π΅) ππ(π΅π΅) , and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.β
Generate resourceConstruct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among mathematics, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. β
Generate resourceRecognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. β
Generate resourceUse the rules of probability to compute probabilities of compound events in a uniform probability model.
Generate resourceFind the conditional probability of A given B as the fraction of B's outcomes that also belong to A and interpret the answer in terms of the model. β
Generate resourceApply the Addition Rule, P(A or B) = P(A) + P(B) β P(A and B) and interpret the answer in terms of the model. β
Generate resourceApply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B)|A) = P(B)P(A|B), and interpret the answer in terms of the model.β
Generate resourceUse permutations and combinations to compute probabilities of compound events and solve problems. β
Generate resourceUnderstand and evaluate random processes underlying statistical experiments.β
Generate resourceUnderstand statistics as a process for making inferences about population parameters based on a random sample from that population. β
Generate resourceDecide if a specified model is consistent with results from a given data-generating process. For example, using simulation or a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model? β
Generate resourceMake inferences and justify conclusions from sample surveys, experiments, and observational studies.
Generate resourceRecognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. β
Generate resourceUse data from a sample survey to estimate a population mean or proportion; develop a margin of error using simulation models for random sampling. β
Generate resourceUse data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. β
Generate resourceEvaluate reports based on data. β For example, a magazine poll reported on the status of American women. One of the statements in the poll was βIt is better for a family if the father works outside the home and the mother takes care of children.β 51% of the sampled women agreed with the statement while 57% of the sampled men agreed. A note on the polling method says that about 1600 men and 1800 women were randomly sampled in the poll and the margin of error was about two percentage points. What is the margin of error and how is it interpreted in this context?
Generate resourceSummarize, represent, and interpret data on a single count or measurement variable. β
Generate resourceRepresent data with plots on the real number line (dot plots, histograms, and box plots). β
Generate resourceUse statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. β
Generate resourceInterpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). β
Generate resourceUse the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. β
Generate resourceSummarize, represent, and interpret data on two categorical and quantitative variables.β
Generate resourceSummarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. β
Generate resourceRepresent data on two quantitative variables on a scatter plot and describe how the variables are related. β
Generate resourceFit a linear function for a scatter plot that suggests a linear association.
Generate resourceFit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.
Generate resourceInformally assess the fit of a function by plotting and analyzing residuals.
Generate resourceInterpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. β
Generate resourceCompute (using technology) and interpret the correlation coefficient of a linear fit. β
Generate resourceDefine a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions. β
Generate resourceCalculate the expected value of a random variable; interpret it as the mean of the probability distribution. β
Generate resourceDevelop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes. β
Generate resourceDevelop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households? β
Generate resourceWeigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. β
Generate resourceFind the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.
Generate resourceEvaluate and compare strategies based on expected values. For example, compare a high-deductible versus a lowdeductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.
Generate resourceUse probabilities to make fair decisions. For example, drawing by lots, using a random number generator. β
Generate resourceAnalyze decisions and strategies using probability concepts. For example, product testing, medical testing, pulling a hockey goalie at the end of a game. β
Generate resource