MAFS.912.A-APR: Arithmetic with Polynomials and Rational Expressions

MAFS.912.A-APR.1: Perform arithmetic operations on polynomials

MAFS.912.A-APR.1.1: Understand 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.

MAFS.912.A-APR.2: Understand the relationship between zeros and factors of polynomials

MAFS.912.A-APR.2.2: Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x).

MAFS.912.A-APR.2.3: Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

MAFS.912.A-APR.3: Use polynomial identities to solve problems

MAFS.912.A-APR.3.5: Know and apply the Binomial Theorem for the expansion of (x + y) to the n power in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal?s Triangle.

MAFS.912.A-CED: Creating Equations

MAFS.912.A-CED.1: Create equations that describe numbers or relationships

MAFS.912.A-CED.1.1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational, absolute, and exponential functions.

MAFS.912.A-CED.1.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

MAFS.912.A-CED.1.3: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.

MAFS.912.A-CED.1.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

MAFS.912.A-REI: Reasoning with Equations and Inequalities

MAFS.912.A-REI.1: Understand solving equations as a process of reasoning and explain the reasoning

MAFS.912.A-REI.1.1: Explain each step in solving a simple 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.

MAFS.912.A-REI.1.2: Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

MAFS.912.A-REI.2: Solve equations and inequalities in one variable

MAFS.912.A-REI.2.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

MAFS.912.A-REI.2.4: Solve quadratic equations in one variable.

MAFS.912.A-REI.2.4.b: Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

MAFS.912.A-REI.3: Solve systems of equations

MAFS.912.A-REI.3.5: Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

MAFS.912.A-REI.3.6: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

MAFS.912.A-REI.3.8: Represent a system of linear equations as a single matrix equation in a vector variable.

MAFS.912.A-REI.4: Represent and solve equations and inequalities graphically

MAFS.912.A-REI.4.10: Understand 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).

MAFS.912.A-REI.4.11: Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

MAFS.912.A-REI.4.12: Graph 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.

MAFS.912.A-SSE: Seeing Structure in Expressions

MAFS.912.A-SSE.1: Interpret the structure of expressions

MAFS.912.A-SSE.1.1: Interpret expressions that represent a quantity in terms of its context.

MAFS.912.A-SSE.1.1.a: Interpret parts of an expression, such as terms, factors, and coefficients.

MAFS.912.A-SSE.1.1.b: Interpret complicated expressions by viewing one or more of their parts as a single entity.

MAFS.912.A-SSE.1.2: Use the structure of an expression to identify ways to rewrite it.

MAFS.912.A-SSE.2: Write expressions in equivalent forms to solve problems

MAFS.912.A-SSE.2.3: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

MAFS.912.A-SSE.2.3.a: Factor a quadratic expression to reveal the zeros of the function it defines.

MAFS.912.A-SSE.2.3.c: Use the properties of exponents to transform expressions for exponential functions.

MAFS.912.F-BF: Building Functions

MAFS.912.F-BF.1: Build a function that models a relationship between two quantities

MAFS.912.F-BF.1.1: Write a function that describes a relationship between two quantities.

MAFS.912.F-BF.1.1.a: Determine an explicit expression, a recursive process, or steps for calculation from a context.

MAFS.912.F-BF.1.2: Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

MAFS.912.F-BF.2: Build new functions from existing functions

MAFS.912.F-BF.2.3: Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.

MAFS.912.F-BF.2.4: Find inverse functions.

MAFS.912.F-BF.2.4.b: Verify by composition that one function is the inverse of another.

MAFS.912.F-BF.2.4.c: Read values of an inverse function from a graph or a table, given that the function has an inverse.

MAFS.912.F-BF.2.5: Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

MAFS.912.F-IF: Interpreting Functions

MAFS.912.F-IF.1: Understand the concept of a function and use function notation

MAFS.912.F-IF.1.1: Understand 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 f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

MAFS.912.F-IF.1.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

MAFS.912.F-IF.2: Interpret functions that arise in applications in terms of the context

MAFS.912.F-IF.2.4: For 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.

MAFS.912.F-IF.2.5: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

MAFS.912.F-IF.2.6: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

MAFS.912.F-IF.3: Analyze functions using different representations

MAFS.912.F-IF.3.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

MAFS.912.F-IF.3.7.a: Graph linear and quadratic functions and show intercepts, maxima, and minima.

MAFS.912.F-IF.3.7.b: Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

MAFS.912.F-IF.3.7.c: Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

MAFS.912.F-IF.3.7.d: Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

MAFS.912.F-IF.3.7.e: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude, and using phase shift.

MAFS.912.F-IF.3.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

MAFS.912.F-IF.3.8.a: 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.

MAFS.912.F-IF.3.8.b: Use the properties of exponents to interpret expressions for exponential functions.

MAFS.912.F-LE: Linear, Quadratic, and Exponential Models

MAFS.912.F-LE.1: Construct and compare linear, quadratic, and exponential models and solve problems

MAFS.912.F-LE.1.1: Distinguish between situations that can be modeled with linear functions and with exponential functions.

MAFS.912.F-LE.1.1.a: Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

MAFS.912.F-LE.1.1.b: Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

MAFS.912.F-LE.1.1.c: Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

MAFS.912.F-LE.1.2: Construct 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).

MAFS.912.F-LE.1.4: For exponential models, express as a logarithm the solution to ab to the ct power = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

MAFS.912.F-LE.2: Interpret expressions for functions in terms of the situation they model

MAFS.912.F-LE.2.5: Interpret the parameters in a linear or exponential function in terms of a context.

MAFS.912.F-TF: Trigonometric Functions

MAFS.912.F-TF.2: Model periodic phenomena with trigonometric functions

MAFS.912.F-TF.2.5: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

MAFS.912.F-TF.3: Prove and apply trigonometric identities

MAFS.912.F-TF.3.9: Prove the addition and subtraction, half-angle, and double-angle formulas for sine, cosine, and tangent and use these formulas to solve problems.

MAFS.912.G-C: Circles

MAFS.912.G-C.1: Understand and apply theorems about circles

MAFS.912.G-C.1.2: Identify and describe relationships among inscribed angles, radii, and chords.

MAFS.912.G-C.2: Find arc lengths and areas of sectors of circles

MAFS.912.G-C.2.5: Derive 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.

MAFS.912.G-CO: Congruence

MAFS.912.G-CO.1: Experiment with transformations in the plane

MAFS.912.G-CO.1.1: Know 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.

MAFS.912.G-CO.1.2: Represent transformations in the plane using, e.g., transparencies and 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 (e.g., translation versus horizontal stretch).

MAFS.912.G-CO.1.4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

MAFS.912.G-CO.1.5: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

MAFS.912.G-CO.2: Understand congruence in terms of rigid motions

MAFS.912.G-CO.2.6: Use 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.

MAFS.912.G-CO.2.8: Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions.

MAFS.912.G-CO.3: Prove geometric theorems

MAFS.912.G-CO.3.9: Prove theorems about lines and angles; use theorems about lines and angles to solve problems.

MAFS.912.G-CO.3.10: Prove theorems about triangles; use theorems about triangles to solve problems.

MAFS.912.G-CO.3.11: Prove theorems about parallelograms; use theorems about parallelograms to solve problems.

MAFS.912.G-CO.4: Make geometric constructions

MAFS.912.G-CO.4.12: Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).

MAFS.912.G-GPE: Expressing Geometric Properties with Equations

MAFS.912.G-GPE.1: Translate between the geometric description and the equation for a conic section

MAFS.912.G-GPE.1.1: Derive 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.

MAFS.912.G-GPE.1.2: Derive the equation of a parabola given a focus and directrix.

MAFS.912.G-GPE.1.3: Derive the equations of ellipses and hyperbolas given the foci and directrices.

MAFS.912.G-GPE.2: Use coordinates to prove simple geometric theorems algebraically

MAFS.912.G-GPE.2.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

MAFS.912.G-GMD: Geometric Measurement and Dimension

MAFS.912.G-GMD.1: Explain volume formulas and use them to solve problems

MAFS.912.G-GMD.1.1: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.

MAFS.912.G-GMD.1.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

MAFS.912.G-SRT: Similarity, Right Triangles, and Trigonometry

MAFS.912.G-SRT.1: Understand similarity in terms of similarity transformations

MAFS.912.G-SRT.1.1: Verify experimentally the properties of dilations given by a center and a scale factor:

MAFS.912.G-SRT.1.1.b: The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

MAFS.912.G-SRT.1.2: Given 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.

MAFS.912.G-SRT.2: Prove theorems involving similarity

MAFS.912.G-SRT.2.4: Prove theorems about triangles.

MAFS.912.G-SRT.2.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

MAFS.912.G-SRT.3: Define trigonometric ratios and solve problems involving right triangles

MAFS.912.G-SRT.3.6: Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

MAFS.912.G-SRT.3.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

MAFS.912.N-CN: The Complex Number System

MAFS.912.N-CN.1: Perform arithmetic operations with complex numbers.

MAFS.912.N-CN.1.1: Know there is a complex number i such that i² = -1, and every complex number has the form a + bi with a and b real.

MAFS.912.N-CN.1.2: Use the relation i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

MAFS.912.N-CN.1.3: Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

MAFS.912.N-CN.2: Represent complex numbers and their operations on the complex plane.

MAFS.912.N-CN.2.4: Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.

MAFS.912.N-CN.3: Use complex numbers in polynomial identities and equations.

MAFS.912.N-CN.3.7: Solve quadratic equations with real coefficients that have complex solutions.

MAFS.912.N-RN: The Real Number System

MAFS.912.N-RN.1: Extend the properties of exponents to rational exponents.

MAFS.912.N-RN.1.1: Explain 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.

MAFS.912.N-VM: Vector and Matrix Quantities

MAFS.912.N-VM.1: Represent and model with vector quantities.

MAFS.912.N-VM.1.1: Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).

MAFS.912.N-VM.1.2: Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

MAFS.912.N-VM.1.3: Solve problems involving velocity and other quantities that can be represented by vectors.

MAFS.912.N-VM.2: Perform operations on vectors.

MAFS.912.N-VM.2.4: Add and subtract vectors.

MAFS.912.N-VM.2.4.a: Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.

MAFS.912.N-VM.2.4.b: Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

MAFS.912.N-VM.2.5: Multiply a vector by a scalar.

MAFS.912.N-VM.2.5.a: Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(v subscript x, v subscript y) = (cv subscript x, cv subscript y).

MAFS.912.S-CP: Conditional Probability and the Rules of Probability

MAFS.912.S-CP.1: Understand independence and conditional probability and use them to interpret data

MAFS.912.S-CP.1.1: Describe 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').

MAFS.912.S-CP.1.2: Understand 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.

MAFS.912.S-CP.1.3: Understand the conditional probability of A given B as P(A and B)/P(B), 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.

MAFS.912.S-CP.2: Use the rules of probability to compute probabilities of compound events in a uniform probability model

MAFS.912.S-CP.2.9: Use permutations and combinations to compute probabilities of compound events and solve problems.

MAFS.912.S-ID: Interpreting Categorical and Quantitative Data

MAFS.912.S-ID.1: Summarize, represent, and interpret data on a single count or measurement variable

MAFS.912.S-ID.1.1: Represent data with plots on the real number line (dot plots, histograms, and box plots).

MAFS.912.S-ID.1.2: Use 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.

MAFS.912.S-ID.1.3: Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

MAFS.912.S-ID.2: Summarize, represent, and interpret data on two categorical and quantitative variables

MAFS.912.S-ID.2.6: Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

MAFS.912.S-ID.2.6.a: Fit a function to the data; use functions fitted to data to solve problems in the context of the data.

MAFS.912.S-ID.2.6.b: Informally assess the fit of a function by plotting and analyzing residuals.

MAFS.912.S-ID.2.6.c: Fit a linear function for a scatter plot that suggests a linear association.

MAFS.912.S-ID.3: Interpret linear models

MAFS.912.S-ID.3.7: Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

MAFS.912.S-ID.3.8: Compute (using technology) and interpret the correlation coefficient of a linear fit.

MAFS.912.S-IC: Making Inferences and Justifying Conclusions

MAFS.912.S-IC.2: Make inferences and justify conclusions from sample surveys, experiments, and observational studies

MAFS.912.S-IC.2.4: Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

MAFS.912.S-IC.2.5: Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

MAFS.912.S-MD: Using Probability to Make Decisions

MAFS.912.S-MD.1: Calculate expected values and use them to solve problems

MAFS.912.S-MD.1.2: Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

MAFS.912.S-MD.1.3: Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.

MAFS.912.S-MD.1.4: Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.

MAFS.912.S-MD.2: Use probability to evaluate outcomes of decisions

MAFS.912.S-MD.2.5: Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

MAFS.912.S-MD.2.5.a: Find the expected payoff for a game of chance.

MAFS.912.S-MD.2.5.b: Evaluate and compare strategies on the basis of expected values.

MAFS.912.S-MD.2.6: Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

MAFS.912.S-MD.2.7: Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

Correlation last revised: 9/16/2020

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