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Curriculum Standards: Analyze functions that include absolute value expressions. - HSM.A1.5.1 Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. - G.GGMD.4 Graph and apply piecewise-defined functions. - HSM.A1.5.2 Explain the derivation of the formulas for the volume of a sphere and other solid figures using Cavalieri’s principle. - G.GGMD.2 Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. - G.GGMD.3 Prove the slope criteria for parallel and perpendicular lines and uses them to solve geometric problems. (e.g., Find the equation of a line parallel or perpendicular to a given line that passes through a given point.) Instructional Note: Relate work on parallel lines to work in High School Algebra I involving systems of equations having no solution or infinitely many solutions. - CAG.M.GHS.30 Explain the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Apply these formulas to solve mathematical and real-world problems. - G.GGMD.1 Express linear equations in slope-intercept, point-slope, and standard forms and convert between these forms. Given sufficient information (slope and y-intercept, slope and one-point on the line, two points on the line, x- and y-intercept, or a set of data points), write the equation of a line. - A1.A.4.3 Rewrite expressions involving radicals and rational exponents using the properties of exponents. Instructional Note: Address this standard before discussing exponential functions with continuous domains. - LER.M.A1HS.12 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. - LER.M.A1HS.13 Calculate and interpret slope and the x- and y-intercepts of a line using a graph, an equation, two points, or a set of data points to solve real-world and mathematical problems. - A1.A.4.1 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. (e.g., We define 5¹/³ to be the cube root of 5 because we want (5¹/³)³ = 5(¹/³)³ to hold, so (5¹/³)³ must equal 5.) Instructional Note: Address this standard before discussing exponential functions with continuous domains. - LER.M.A1HS.11 Graph and apply step functions. - HSM.A1.5.3 Graph and analyze transformations of the absolute value function. - HSM.A1.5.4 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. - MAFS.912.F-IF.3.9 Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). - CAG.M.GHS.29 Add, subtract, and multiply polynomials. - HSM.A2.3.2 Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and-if there is a flaw in an argument-explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. - MAFS.K12.MP.3.1.a Prove and use polynomial identities. - HSM.A2.3.3 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.1.2 Predict the behavior of polynomial functions. - HSM.A2.3.1 Model and solve problems using the zeros of a polynomial function. - HSM.A2.3.5 Find and graph the inverse of a function, if it exists, in real-world and mathematical situations. Know that the domain of a function f is the range of the inverse function f-_, and the range of the function f is the domain of the inverse function f-_. - A2.F.2.3 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. - ETD.M.GHS.25 Apply the inverse relationship between exponential and logarithmic functions to convert from one form to another. - A2.F.2.4 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Instructional Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k² times the area of the first. Similarly, volumes of solid figures scale by k³ under a similarity transformation with scale factor k. - ETD.M.GHS.26 Add, subtract, multiply, and divide functions using function notation and recognize domain restrictions. - A2.F.2.1 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. - ETD.M.GHS.27 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Instructional Note: Focus on situations that require relating two- and three-dimensional objects, determining and using volume, and the trigonometry of general triangles. - ETD.M.GHS.28 vertical and horizontal asymptotes; - F.AII.7.i 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.1.2 end behavior; - F.AII.7.h determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; - EI.A.6.a composition of functions algebraically and graphically. - F.AII.7.k Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. - MAFS.912.G-SRT.1.3 write the equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line; and - EI.A.6.b intercepts; - F.AII.7.e graph linear equations in two variables. - EI.A.6.c zeros; - F.AII.7.d connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; - F.AII.7.g values of a function for elements in its domain; - F.AII.7.f domain, range, and continuity; - F.AII.7.a Find the point on a directed line segment between two given points that partitions the segment in a given ratio. - CAG.M.GHS.31 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem. - CAG.M.GHS.32 extrema; - F.AII.7.c Derive the equation of a parabola given a focus and directrix. Instructional Note: The directrix should be parallel to a coordinate axis. - CAG.M.GHS.33 intervals in which a function is increasing or decreasing; - F.AII.7.b Use graphs to find approximate solutions to systems of equations. - HSM.A1.4.1 Solve an equation of the form ??(????) = ?????? for a simple function ???????? that has an inverse and write an expression for the inverse. Example: For example, ??????????(????????????) =2 ??????????????³ or ????????????????(??????????????????) = (????????????????????+1)/(??????????????????????–1) for ???????????????????????? ? 1. - MAFS.912.F-BF.2.4.a Solve systems of linear equations using the substitution method. - HSM.A1.4.2 Solve systems of linear equations using the elimination method. - HSM.A1.4.3 Describe the effect of the transformations ??????????????????????????????????????????????????????(??????????????????????????????), ????????????????????????????????(??????????????????????????????????)+????????????????????????????????????, ??????????????????????????????????????(????????????????????????????????????????+??????????????????????????????????????????), and combinations of such transformations on the graph of ????????????????????????????????????????????=??????????????????????????????????????????????(????????????????????????????????????????????????) for any real number ??????????????????????????????????????????????????. Find the value of ???????????????????????????????????????????????????? given the graphs and write the equation of a transformed parent function given its graph. (Limit to linear; quadratic; exponential with integer exponents; vertical shift and vertical stretch.) - A1.FBF.3 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Example: For example, if the function ??????????????????????????????????????????????????????(????????????????????????????????????????????????????????) gives the number of person-hours it takes to assemble ?????????????????????????????????????????????????????????? engines in a factory, then the positive integers would be an appropriate domain for the function. - MAFS.912.F-IF.2.5 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.2.6 Graph solutions to linear inequalities in two variables. - HSM.A1.4.4 Graph and solve a system of linear inequalities. - HSM.A1.4.5 Find the zeros of quadratic functions. - HSM.A2.2.3 Solve problems with complex numbers. - HSM.A2.2.4 Identify key features of quadratic functions. - HSM.A2.2.1 Write and graph quadratic functions in standard form. - HSM.A2.2.2 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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. - MAFS.912.F-IF.2.4 Solve linear-quadratic systems. - HSM.A2.2.7 Solve quadratic equations by completing the square. - HSM.A2.2.5 Solve quadratic equations using the Quadratic Formula. - HSM.A2.2.6 Graph exponential and logarithmic functions. Identify asymptotes and x- and y-intercepts using various methods and tools that may include graphing calculators or other appropriate technology. Recognize exponential decay and growth graphically and algebraically. - A2.F.1.4 Recognize the graphs of exponential, radical (square root and cube root only), quadratic, and logarithmic functions. Predict the effects of transformations [f(x + c), f(x) + c, f(cx), and cf(x), where c is a positive or negative real-valued constant] algebraically and graphically, using various methods and tools that may include graphing calculators or other appropriate technology. - A2.F.1.2 Graph a quadratic function. Identify the x- and y-intercepts, maximum or minimum value, axis of symmetry, and vertex using various methods and tools that may include a graphing calculator or appropriate technology. - A2.F.1.3 Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. - A2.ASE.1 Graph piecewise functions with no more than three branches (including linear, quadratic, or exponential branches) and analyze the function by identifying the domain, range, intercepts, and intervals for which it is increasing, decreasing, and constant. - A2.F.1.8 Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. - A2.ASE.2 solve multistep linear inequalities in one variable algebraically and represent the solution graphically; - EI.A.5.a solve practical problems involving inequalities; and - EI.A.5.c Create and solve equations and inequalities in one variable that model real-world problems involving linear, quadratic, simple rational, and exponential relationships. Interpret the solutions and determine whether they are reasonable. (Limit to linear; quadratic; exponential with integer exponents.) - A1.ACE.1 Use the structure of an expression to identify ways to rewrite it. Instructional Note: Extend to polynomial and rational expressions. Example:: For example, see x4 – y4 as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²). - PRR.M.A2HS.7 Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. (Limit to linear; quadratic; exponential with integer exponents; direct and indirect variation.) - A1.ACE.2 Solve literal equations and formulas for a specified variable including equations and formulas that arise in a variety of disciplines. - A1.ACE.4 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. Instructional Note: Extend beyond the quadratic polynomials found in Algebra I. - PRR.M.A2HS.9 Prove 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. - MAFS.912.G-SRT.2.4 Extend polynomial identities to the complex numbers. Instructional Note: Limit to polynomials with real coefficients. Example:: For example, rewrite x² + 4 as (x + 2i)(x – 2i). - PRR.M.A2HS.4 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. - MAFS.912.G-SRT.2.5 intercepts; - F.A.7.d values of a function for elements in its domain; and - F.A.7.e connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs. - F.A.7.f Factor a quadratic trinomial. - HSM.A1.7.5 Factor a quadratic trinomial when a ? 1. - HSM.A1.7.6 determining whether a relation is a function; - F.A.7.a domain and range; - F.A.7.b Factor special trinomials. - HSM.A1.7.7 Combine like terms to simplify polynomials. - HSM.A1.7.1 Multiply two polynomials. - HSM.A1.7.2 Represent relationships in various contexts with compound and absolute value inequalities and solve the resulting inequalities by graphing and interpreting the solutions on a number line. - A1.A.2.2 Use patterns to multiply binomials. - HSM.A1.7.3 Factor a polynomial. - HSM.A1.7.4 Identify intercepts, zeros, maxima, minima and intervals of increase and decrease from the graph of a function. - 9.2.1.6 evaluate algebraic expressions for given replacement values of the variables. - EO.A.1.b Make qualitative statements about the rate of change of a function, based on its graph or table of values. - 9.2.1.8 Determine how translations affect the symbolic and graphical forms of a function. Know how to use graphing technology to examine translations - 9.2.1.9 The student will perform operations on complex numbers and express the results in simplest form using patterns of the powers of i. - EO.AII.2 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 ???????????????????????????????????????????????????????????? is a function and ?????????????????????????????????????????????????????????????? is an element of its domain, then ????????????????????????????????????????????????????????????????(??????????????????????????????????????????????????????????????????) denotes the output of ???????????????????????????????????????????????????????????????????? corresponding to the input ??????????????????????????????????????????????????????????????????????. The graph of ???????????????????????????????????????????????????????????????????????? is the graph of the equation ?????????????????????????????????????????????????????????????????????????? = ????????????????????????????????????????????????????????????????????????????(??????????????????????????????????????????????????????????????????????????????). - MAFS.912.F-IF.1.1 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.1.2 Relate roots and rational exponents and use them to simplify expressions and solve equations. - HSM.A2.5.1 represent verbal quantitative situations algebraically; and - EO.A.1.a Perform operations on functions to answer real-world questions. - HSM.A2.5.5 Using technology, create scatterplots and analyze those plots to compare the fit of linear, quadratic, or exponential models to a given data set. Select the appropriate model, fit a function to the data set, and use the function to solve problems in the context of the data. - A1.SPID.6 Graph and transform radical functions. - HSM.A2.5.3 Create a linear function to graphically model data from a real-world problem and interpret the meaning of the slope and intercept(s) in the context of the given problem. - A1.SPID.7 Understand the definition of a function. Use functional notation and evaluate a function at a given point in its domain - 9.2.1.1 Create symbolic representations of linear and exponential functions, including arithmetic and geometric sequences, given graphs, verbal descriptions, and tables. (Limit to linear; exponential.) - A1.FLQE.2 Distinguish between functions and other relations defined symbolically, graphically or in tabular form. - 9.2.1.2 Find the domain of a function defined symbolically, graphically or in a real-world context. - 9.2.1.3 Represent the inverse of a relation using tables, graphs, and equations. - HSM.A2.5.6 Obtain information and draw conclusions from graphs of functions and other relations. - 9.2.1.4 Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.) - A1.FLQE.5 Identify the vertex, line of symmetry and intercepts of the parabola corresponding to a quadratic function, using symbolic and graphical methods, when the function is expressed in the form f (x) = ax2 + bx + c, in the form - 9.2.1.5 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. - CCSS.Math.Content.HSG-GMD.A.3 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. - AP.M.GHS.48 The student will solve problems involving equations of circles. - PC.G.12 Explain and use the relationship between the sine and cosine of complementary angles. - MAFS.912.G-SRT.3.7 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. - MAFS.912.G-SRT.3.8 Use permutations and combinations to compute probabilities of compound events and solve problems. - AP.M.GHS.50 Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. - A2.AAPR.1 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.3.6 Use properties of exponents to solve equations with rational exponents. - HSM.A1.6.1 Interpret the parameters in a linear or exponential function in terms of a context. Instructional Note: Limit exponential functions to those of the form f(x) = b? + k. - LER.M.A1HS.32 Describe and graph exponential functions. - HSM.A1.6.2 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). Instructional Note: In constructing linear functions, draw on and consolidate previous work in Grade 8 on finding equations for lines and linear functions. - LER.M.A1HS.30 Use exponential functions to model situations and make predictions. - HSM.A1.6.3 Solve equations involving several variables for one variable in terms of the others. - A1.A.3.1 Identify and describe geometric sequences. - HSM.A1.6.4 Perform, analyze, and use transformations of exponential functions. - HSM.A1.6.5 Understand and justify that the steps taken when solving simple equations in one variable create new equations that have the same solution as the original. - A1.AREI.1 applying the laws of exponents to perform operations on expressions; - EO.A.2.a Determine the maximum or minimum value of a quadratic function by completing the square. - A2.ASE.3b add, subtract, multiply, divide, and simplify rational algebraic expressions; - EO.AII.1.a add, subtract, multiply, divide, and simplify radical expressions containing rational numbers and variables, and expressions containing rational exponents; and - EO.AII.1.b Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Instructional Note: Limit to linear and exponential functions. Connect arithmetic sequences to linear functions and geometric sequences to exponential functions. - LER.M.A1HS.27 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. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Instructional Note: Focus on vertical translations of graphs of linear and exponential functions. Relate the vertical translation of a linear function to its y-intercept. While applying other transformations to a linear graph is appropriate at this level, it may be difficult for students to identify or distinguish between the effects of the other transformations included in this standard. - LER.M.A1HS.28 factor polynomials completely in one or two variables. - EO.AII.1.c Recognize that geometric sequences are exponential using equations, tables, graphs and verbal descriptions. Given the formula f(x) = a(r)x, find the next term and define the meaning of a and r within the context of the problem. - A1.A.3.6 Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. - G.GGPE.1 Solve systems of linear equations using the substitution method. - A1.AREI.6 Justify that the solution to a system of linear equations is not changed when one of the equations is replaced by a linear combination of the other equation. - A1.AREI.5 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. - A1.AREI.3 identifying the converse, inverse, and contrapositive of a conditional statement; - RLT.G.1.a determining the validity of a logical argument. - RLT.G.1.c translating a short verbal argument into symbolic form; and - RLT.G.1.b 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. Instructional Note: Focus on linear functions and exponential functions whose domain is a subset of the integers. The Unit on Quadratic Functions and Modeling in this course and the Algebra II course address other types of functions. - LER.M.A1HS.23 Use coordinates to prove simple geometric theorems algebraically. - G.GGPE.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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Instructional Note: Focus on linear and exponential functions. - LER.M.A1HS.21 Analyze slopes of lines to determine whether lines are parallel, perpendicular, or neither. Write the equation of a line passing through a given point that is parallel or perpendicular to a given line. Solve geometric and real-world problems involving lines and slope. - G.GGPE.5 Given two points, find the point on the line segment between the two points that divides the segment into a given ratio. - G.GGPE.6 Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. - G.GGPE.7 Recognize 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). Instructional Note: Students should experience a variety of types of situations modeled by functions. Detailed analysis of any particular class of function at this stage is not advised. Students should apply these concepts throughout their future mathematics courses. Draw examples from linear functions and exponential functions having integral domains. - LER.M.A1HS.18 Use function notation, evaluate functions for inputs in their domains and interpret statements that use function notation in terms of a context. Instructional Note: Students should experience a variety of types of situations modeled by functions. Detailed analysis of any particular class of function at this stage is not advised. Students should apply these concepts throughout their future mathematics courses. Draw examples from linear functions and exponential functions having integral domains. - LER.M.A1HS.19 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Instructional Note: Build on student experiences graphing and solving systems of linear equations from middle school to focus on justification of the methods used. Include cases where the two equations describe the same line (yielding infinitely many solutions) and cases where two equations describe parallel lines (yielding no solution); connect to standards in Geometry which require students to prove the slope criteria for parallel lines. - LER.M.A1HS.14 use knowledge of transformations to convert between equations and the corresponding graphs of functions. - F.AII.6.b recognize the general shape of function families; and - F.AII.6.a Find the probability of an event given that another event has occurred. - HSM.G.12.2 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. - MAFS.912.G-CO.1.3 Solve quadratic equations by inspection, 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 ????????????????????????????????????????????????????????????????????????????????+?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? for real numbers ?????????????????????????????????????????????????????????????????????????????????????? and ????????????????????????????????????????????????????????????????????????????????????????. - A2.AREI.4b Rewrite and use literal equations to solve problems. - HSM.A1.1.4 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.1.2 Use permutations and combinations to find the number of outcomes in a probability experiment. - HSM.G.12.3 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.1.1 Define probability distributions to represent experiments and solve problems. - HSM.G.12.4 Solve and graph inequalities. - HSM.A1.1.5 Calculate, interpret, and apply expected value. - HSM.G.12.5 Write and solve compound inequalities. - HSM.A1.1.6 Reason about operations with real numbers. - HSM.A1.1.1 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.1.5 Create and solve linear equations with one variable. - HSM.A1.1.2 Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. - MAFS.912.A-SSE.2.3.b Factor a quadratic expression to reveal the zeros of the function it defines. - MAFS.912.A-SSE.2.3.a 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.1.2 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Example: For example, rearrange Ohm’s law ?????????????????????????????????????????????????????????????????????????????????????????? = ?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? to highlight resistance ????????????????????????????????????????????????????????????????????????????????????????????????. - MAFS.912.A-CED.1.4 Write and solve absolute-value equations and inequalities - HSM.A1.1.7 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.1.1 Interpret parts of an expression, such as terms, factors, and coefficients. - RQ.M.A1HS.4.a Use relationships among events to find probabilities. - HSM.G.12.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Instructional Note: Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. - RQ.M.A1HS.5 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Limit to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. - RQ.M.A1HS.6 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (e.g., Rearrange Ohm’s law V = IR to highlight resistance R.) Instructional Note: Limit to formulas with a linear focus. - RQ.M.A1HS.8 Solve common and natural logarithmic equations using the properties of logarithms. - A2.A.1.6 Represent real-world or mathematical problems using quadratic equations and solve using various methods (including graphing calculator or other appropriate technology), factoring, completing the square, and the quadratic formula. Find non-real roots when they exist. - A2.A.1.1 Represent real-world or mathematical problems using exponential equations, such as compound interest, depreciation, and population growth, and solve these equations graphically (including graphing calculator or other appropriate technology) or algebraically. - A2.A.1.2 Use dilation and rigid motion to establish triangle similarity theorems. - HSM.G.7.3 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (e.g., Rearrange Ohm’s law V = IR to highlight resistance R. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. Extend this standard to formulas involving squared variables. - EE.M.A1HS.47 Determine whether figures are similar. - HSM.G.7.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. - EE.M.A1HS.46 Find the lengths of segments using proportional relationships in triangles resulting from parallel lines. - HSM.G.7.5 Create 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. Instructional Note: Extend work on linear and exponential equations in the Relationships between Quantities and Reasoning with Equations unit to quadratic equations. - EE.M.A1HS.45 Use similarity and the geometric mean to solve problems involving right triangles. - HSM.G.7.4 Dilate figures and identify characteristics of dilations. - HSM.G.7.1 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.2.8 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Instructional Note: Include systems consisting of one linear and one quadratic equation. Include systems that lead to work with fractions. Example:: For example, find the points of intersection between the line y = –3x and the circle x² + y² = 3. Example:: For example, finding the intersections between x² + y² = 1 and y = (x+1)/2 leads to the point (3/5, 4/5) on the unit circle, corresponding to the Pythagorean triple 3² + 4² = 5². - EE.M.A1HS.49 Use 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. - MAFS.912.G-CO.2.7 Represent real-world or mathematical problems using systems of linear equations with a maximum of three variables and solve using various methods that may include substitution, elimination, and graphing (may include graphing calculators or other appropriate technology). - A2.A.1.8 Solve systems of equations containing one linear equation and one quadratic equation using tools that may include graphing calculators or other appropriate technology. - A2.A.1.9 Use the volumes of right and oblique pyramids and cones to solve problems. - HSM.G.11.3 Calculate the volume of a sphere and solve problems involving the volumes of spheres. - HSM.G.11.4 Solve absolute value equations and interpret the solutions in the original context. - A1.A.1.2 Analyze and solve real-world and mathematical problems involving systems of linear equations with a maximum of two variables by graphing (may include graphing calculator or other appropriate technology), substitution, and elimination. Interpret the solutions in the original context. - A1.A.1.3 Identify space figures and their relationships with polygons to solve problems. - HSM.G.11.1 Use the properties of prisms and cylinders to calculate their volumes. - HSM.G.11.2 Use knowledge of solving equations with rational values to represent and solve mathematical and real-world problems (e.g., angle measures, geometric formulas, science, or statistics) and interpret the solutions in the original context. - A1.A.1.1 Add, subtract, multiply, divide, and simplify polynomial and rational expressions. - A2.A.2.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. - A2.A.2.4 The student will use surface area and volume of three-dimensional objects to solve practical problems. - TDF.G.13 Interpret parts of an expression, such as terms, factors, and coefficients. - PRR.M.A2HS.6.a Factor polynomial expressions including but not limited to trinomials, differences of squares, sum and difference of cubes, and factoring by grouping using a variety of tools and strategies. - A2.A.2.1 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.1.1 Use trigonometry to solve problems. - HSM.G.8.5 Prove 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 segment’s endpoints. Instructional Note: Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. - CPC.M.GHS.9 Interpret the parameters in a linear or exponential function in terms of the context. (Limit to linear.) - HSF-LE.B.5 Use trigonometric ratios to find lengths and angle measures of right triangles. - HSM.G.8.2 Use 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. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. - CPC.M.GHS.7 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. - CPC.M.GHS.8 Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. - HSM.G.8.1 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, for example, graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Instructional Note: Build on student experience with rigid motions from earlier grades. Point out the basis of rigid motions in geometric concepts, (e.g., translations move points a specified distance along a line parallel to a specified line; rotations move objects along a circular arc with a specified center through a specified angle) - CPC.M.GHS.5 Use the Law of Cosines to solve problems. - HSM.G.8.4 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. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. - CPC.M.GHS.6 Use the Law of Sines to solve problems. - HSM.G.8.3 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. - CCSS.Math.Content.HSG-SRT.B.5 Use statistics appropriate to the shape of the data distribution to compare center and spread of two or more different data sets that include all real numbers. - G.SPID.2 A 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. - MAFS.912.G-SRT.1.1.a Determine whether a function is a relation. - HSM.A1.3.1 Identify, evaluate, and graph linear functions. - HSM.A1.3.2 Transform linear equations - HSM.A1.3.3 Identify and describe arithmetic sequences. - HSM.A1.3.4 Interpret the parameters in a linear or exponential function in terms of a context. - MAFS.912.F-LE.2.5 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. Instructional Note: Build on student experience with rigid motions from earlier grades. Point out the basis of rigid motions in geometric concepts, (e.g., translations move points a specified distance along a line parallel to a specified line; rotations move objects along a circular arc with a specified center through a specified angle). - CPC.M.GHS.3 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. - CPC.M.GHS.1 The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve practical problems, using mathematical models of quadratic and exponential functions. - S.AII.9 Represent transformations in the plane using, for example, 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). Instructional Note: Build on student experience with rigid motions from earlier grades. Point out the basis of rigid motions in geometric concepts, (e.g., translations move points a specified distance along a line parallel to a specified line; rotations move objects along a circular arc with a specified center through a specified angle). - CPC.M.GHS.2 The dilation of a line segment is longer or shorter in the ratio given by the scale factor. - MAFS.912.G-SRT.1.1.b Interpret the meanings of coefficients, factors, terms, and expressions based on their real-world contexts. Interpret complicated expressions as being composed of simpler expressions. (Limit to linear; quadratic; exponential.) - A1.ASE.1 Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. - HSM.A2.1.1 Use graphs and tables to approximate solutions to algebraic equations and inequalities. - HSM.A2.1.5 Apply transformations to graph functions and write equations. - HSM.A2.1.2 Graph and interpret piecewise-defined functions. - HSM.A2.1.3 Use a variety of tools to solve systems of linear equations and inequalities. - HSM.A2.1.6 Solve systems of equations using matrices. - HSM.A2.1.7 The student will verify and use properties of quadrilaterals to solve problems, including practical problems. - PC.G.9 Interpret the parameters in a linear or exponential function in terms of the context. - A2.FLQE.5 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. - CCSS.Math.Content.HSG-SRT.A.2 Use geometric shapes, their measures, and their properties to describe real-world objects. - G.GM.1 Prove that all circles are similar. - CWC.M.GHS.34 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. - CWC.M.GHS.36 Identify 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. - CWC.M.GHS.35 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. - MAFS.912.A-REI.2.3 Describe the effect of the transformations ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????(??????????????????????????????????????????????????????????????????????????????????????????????????????), ????????????????????????????????????????????????????????????????????????????????????????????????????????(??????????????????????????????????????????????????????????????????????????????????????????????????????????)+????????????????????????????????????????????????????????????????????????????????????????????????????????????, ??????????????????????????????????????????????????????????????????????????????????????????????????????????????(????????????????????????????????????????????????????????????????????????????????????????????????????????????????+??????????????????????????????????????????????????????????????????????????????????????????????????????????????????), and combinations of such transformations on the graph of ????????????????????????????????????????????????????????????????????????????????????????????????????????????????????=??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????(????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????) for any real number ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????. Find the value of ???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? given the graphs and write the equation of a transformed parent function given its graph. - A2.FBF.3 Use the equations and graphs of parabolas to solve problems. - HSM.G.9.4 Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. Example: 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? - CCSS.Math.Content.HSS-MD.A.4 Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. Example: 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. - CCSS.Math.Content.HSS-MD.A.3 Calculate the expected value of a random variable; interpret it as the mean of the probability distribution. - CCSS.Math.Content.HSS-MD.A.2 Use the coordinate plane to analyze geometric figures. - HSM.G.9.1 Use the equations and graphs of circles to solve problems. - HSM.G.9.3 Prove geometric theorems using algebra and the coordinate plane. - HSM.G.9.2 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Instructional Note: Focus on situations in which the analysis of circles is required. - CWC.M.GHS.41 Write and graph linear equations using point-slope form. - HSM.A1.2.2 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.1.2 Use coordinates to prove simple geometric theorems algebraically. (e.g., Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2).) Instructional Note: Include simple proofs involving circles. - CWC.M.GHS.40 Write and graph linear equations using standard form. - HSM.A1.2.3 Write equations of parallel lines and perpendicular lines. - HSM.A1.2.4 For exponential models, express as a logarithm the solution to ?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? to the ?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? power = ?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? where ????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????, ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????, and ???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? are numbers and the base ?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? is 2, 10, or ????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????; evaluate the logarithm using technology. - MAFS.912.F-LE.1.4 Write and graph linear equations using slope-intercept form. - HSM.A1.2.1 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. Example: For example, we define 5 to the 1/3 power to be the cube root of 5 because we want (5 to the 1/3 power)³ = (5 to the 1/3 power)³ to hold, so (5 to the 1/3 power)³ must equal 5. - MAFS.912.N-RN.1.1 Rewrite expressions involving radicals and rational exponents using the properties of exponents. - MAFS.912.N-RN.1.2 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) - A2.AREI.7 Identify the effect on the graph of replacing ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????(????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????) by ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????(????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????) + ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????, ???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????(????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????), ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????(??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????), and ????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????(?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? + ????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????) for specific values of ?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? (both positive and negative); find the value of ???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? 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. - MAFS.912.F-BF.2.3 Prove 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. - CCSS.Math.Content.HSG-CO.C.10 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. Instructional Note: Emphasize the similarity of all circles. Reason that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. - CWC.M.GHS.38 Construct a tangent line from a point outside a given circle to the circle. - CWC.M.GHS.37 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. - CWC.M.GHS.39 the Pythagorean Theorem and its converse; - T.G.8.a properties of special right triangles; and - T.G.8.b trigonometric ratios. - T.G.8.c 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.3.5 Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. - G.GCO.1 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line ?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? = –3???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? and the circle ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????² + ????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????² = 3. - MAFS.912.A-REI.3.7 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.3.6 Describe rotations and reflections that carry a regular polygon onto itself and identify types of symmetry of polygons, including line, point, rotational, and self-congruence, and use symmetry to analyze mathematical situations. - G.GCO.3 Represent translations, reflections, rotations, and dilations of objects in the plane by using paper folding, sketches, coordinates, function notation, and dynamic geometry software, and use various representations to help understand the effects of simple transformations and their compositions. - G.GCO.2 Predict and describe the results of transformations on a given figure using geometric terminology from the definitions of the transformations, and describe a sequence of transformations that maps a figure onto its image. - G.GCO.5 Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence c - G.GCO.7 Demonstrate that triangles and quadrilaterals are congruent by identifying a combination of translations, rotations, and reflections in various representations that move one figure onto the other. - G.GCO.6 Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. - G.GCO.9 Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. - G.GCO.8 Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; 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. - MAFS.912.G-CO.3.10 Prove theorems about parallelograms; use theorems about parallelograms to solve problems. 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. - MAFS.912.G-CO.3.11 Identify the dependent and independent variables as well as the domain and range given a function, equation, or graph. Identify restrictions on the domain and range in real-world contexts. - A1.F.1.2 Explain 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. - MAFS.912.N-RN.2.3 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.3.7 (HONORS ONLY) Extend polynomial identities to the complex numbers. Example: For example, rewrite ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????² + 4 as (???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? + 2??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????)(???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? – 2??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????). - MAFS.912.N-CN.3.8 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. - MAFS.912.G-GMD.2.4 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.3.7.e 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. - CCSS.Math.Content.HSG-CO.B.6 Use 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. - CCSS.Math.Content.HSG-CO.B.7 Graph linear and quadratic functions and show intercepts, maxima, and minima. - MAFS.912.F-IF.3.7.a Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. - CCSS.Math.Content.HSG-CO.B.8 Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. - MAFS.912.F-IF.3.7.b Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. - MAFS.912.G-CO.4.13 Make 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. - MAFS.912.G-CO.4.12 Create equations in two or more variables to represent relationships between quantities. Graph the equations on coordinate axes using appropriate labels, units, and scales. - A2.ACE.2 Write linear functions, using function notation, to model real-world and mathematical situations. - A1.F.1.3 Use systems of equations and inequalities to represent constraints arising in real-world situations. Solve such systems using graphical and analytical methods, including linear programing. Interpret the solution within the context of the situation. (Limit to linear programming.) - A2.ACE.3 Interpret parts of an expression, such as terms, factors, and coefficients. - MAFS.912.A-SSE.1.1.a Create and solve equations and inequalities in one variable that model real-world problems involving linear, quadratic, simple rational, and exponential relationships. Interpret the solutions and determine whether they are reasonable. - A2.ACE.1 (HONORS ONLY) Use permutations and combinations to compute probabilities of compound events and solve problems. - MAFS.912.S-CP.2.9 (HONORS ONLY) Apply the Addition Rule, ????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????(?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? or ????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????) = ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????(????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????) + ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????(????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????) – ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????(???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? and ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????), and interpret the answer in terms of the model. - MAFS.912.S-CP.2.7 Find the conditional probability of ???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? given ?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? as the fraction of ????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????’s outcomes that also belong to ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????, and interpret the answer in terms of the model. - MAFS.912.S-CP.2.6 Use inverse functions to solve problems. - HSM.A1.10.7 Add, subtract, and multiply functions. - HSM.A1.10.6 Change functions to compress or stretch their graphs. - HSM.A1.10.5 Graph and analyze transformations of functions. - HSM.A1.10.4 Identify the function family when given an equation or graph. - HSM.A1.10.3 Identify the key features of the cube root function. - HSM.A1.10.2 The student will investigate and describe the relationships among solutions of an equation, zeros of a function, x-intercepts of a graph, and factors of a polynomial expression. - F.AII.8 Describe the key features of the square root function. - HSM.A1.10.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. - MAFS.912.G-GMD.1.1 (HONORS ONLY) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. - MAFS.912.G-GMD.1.2 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. - MAFS.912.G-GMD.1.3 Describe the concepts of intersections, unions and complements using Venn diagrams. Understand the relationships between these concepts and the words AND, OR, NOT, as used in computerized searches and spreadsheets. - 9.4.3.6 Understand and use simple probability formulas involving intersections, unions and complements of events. - 9.4.3.7 Describe the properties of a figure before and after translation. - HSM.G.3.2 Apply probability concepts to real-world situations to make informed decisions. - 9.4.3.8 Identify different types of symmetry in two-dimensional figures. - HSM.G.3.5 Identify different rigid motions used to transform two-dimensional shapes. - HSM.G.3.4 Use the relationship between conditional probabilities and relative frequencies in contingency tables. - 9.4.3.9 Calculate experimental probabilities by performing simulations or experiments involving a probability model and using relative frequencies of outcomes. - 9.4.3.2 Draw and describe the reflection of a figure across a line of reflection. - HSM.G.3.1 Apply probability concepts such as intersections, unions and complements of events, and conditional probability and independence, to calculate probabilities and solve problems. - 9.4.3.5 Apply the properties of angles, including corresponding, exterior, interior, vertical, complementary, and supplementary angles to solve real-world and mathematical problems using algebraic reasoning and proofs. - G.2D.1.2 Apply theorems involving the interior and exterior angle sums of polygons and use them to solve real-world and mathematical problems using algebraic reasoning and proofs. - G.2D.1.3 Apply the properties of special quadrilaterals (square, rectangle, trapezoid, isosceles trapezoid, rhombus, kite, parallelogram) and use them to solve real-world and mathematical problems involving angle measures and segment lengths using algebraic reasoning and proofs. - G.2D.1.4 Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints, and slopes of line segments. - G.2D.1.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). - MAFS.912.G-GPE.2.5 Apply the properties of polygons to solve real-world and mathematical problems involving perimeter and area (e.g., triangles, special quadrilaterals, regular polygons up to 12 sides, composite figures). - G.2D.1.6 Apply the properties of congruent or similar polygons to solve real-world and mathematical problems using algebraic and logical reasoning. - G.2D.1.7 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. - MAFS.912.G-GPE.2.6 Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). - G.2D.1.8 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. - MAFS.912.G-GPE.2.7 Use numeric, graphic and algebraic representations of transformations in two dimensions, such as reflections, translations, dilations, and rotations about the origin by multiples of 90?, to solve problems involving figures on a coordinate plane and identify types of symmetry. - G.2D.1.9 Understand how the properties of similar right triangles allow the trigonometric ratios to be defined, and determine the sine, cosine and tangent of an acute angle in a right triangle. - 9.3.4.1 Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. - MF.M.A2HS.30.b Use coordinates to prove simple geometric theorems algebraically. Example: For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, v3) lies on the circle centered at the origin and containing the point (0, 2). - MAFS.912.G-GPE.2.4 Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. - MF.M.A2HS.30.a Organize data in two-way frequency tables and use them to make inferences and generalizations. - HSM.A1.11.5 Apply the properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve real-world and mathematical problems and determine if two lines are parallel, using algebraic reasoning and proofs. - G.2D.1.1 Use numeric, graphic and symbolic representations of transformations in two dimensions, such as reflections, translations, scale changes and rotations about the origin by multiples of 90°, to solve problems involving figures on a coordinate grid. - 9.3.4.6 Use algebra to solve geometric problems unrelated to coordinate geometry, such as solving for an unknown length in a figure involving similar triangles, or using the Pythagorean Theorem to obtain a quadratic equation for a length in a geometric figure. - 9.3.4.7 Use coordinate geometry to represent and analyze line segments and polygons, including determining lengths, midpoints and slopes of line segments. - 9.3.4.4 Know the equation for the graph of a circle with radius r and center (h, k), (x – h)2 + (y – k)2 = r2, and justify this equation using the Pythagorean Theorem and properties of translations. - 9.3.4.5 Apply the trigonometric ratios sine, cosine and tangent to solve problems, such as determining lengths and areas in right triangles and in figures that can be decomposed into right triangles. Know how to use calculators, tables or other technology to evaluate trigonometric ratios. - 9.3.4.2 Use calculators, tables or other technologies in connection with the trigonometric ratios to find angle measures in right triangles in various contexts. - 9.3.4.3 Evaluate reports based on data published in the media by identifying the source of the data, the design of the study, and the way the data are analyzed and displayed. Show how graphs and data can be distorted to support different points of view. Know how to use spreadsheet tables and graphs or graphing technology to recognize and analyze distortions in data displays. - 9.4.2.1 Identify and explain misleading uses of data; recognize when arguments based on data confuse correlation and causation. - 9.4.2.2 Use triangle congruence to solve problems with overlapping triangles. - HSM.G.4.6 Identify congruent right triangles. - HSM.G.4.5 Apply theorems about isosceles and equilateral triangles to solve problems. - HSM.G.4.2 Prove that all circles are similar. - MAFS.912.G-C.1.1 The student will compute and distinguish between permutations and combinations. - S.AII.12 Identify 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. - MAFS.912.G-C.1.2 Use a composition of rigid motions to show that two objects are congruent. - HSM.G.4.1 Determine congruent triangles by comparing two angles and one side. - HSM.G.4.4 Use SAS and SSS to determine whether triangles are congruent. - HSM.G.4.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. - MAFS.912.G-C.1.3 (HONORS ONLY) Construct a tangent line from a point outside a given circle to the circle. - MAFS.912.G-C.1.4 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. - SPT.M.GHS.21 Use relationships between circles, angles, and arcs. - HSM.G.10.4 Explain and use the relationship between the sine and cosine of complementary angles. - SPT.M.GHS.20 Use angle measures and segment lengths formed by intersecting lines and circles to solve problems. - HSM.G.10.5 Understand the use of undefined terms, definitions, postulates, and theorems in logical arguments/proofs. - G.RL.1.1 Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles. Instructional Note: With respect to the general case of the Laws of Sines and Cosines, the definitions of sine and cosine must be extended to obtuse angles. - SPT.M.GHS.24 Solve quadratic equations by inspection (e.g., for ????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????² = 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 ?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? ± ?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? for real numbers ???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? and ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????. - MAFS.912.A-REI.2.4.b Prove the Laws of Sines and Cosines and use them to solve problems. Instructional Note: With respect to the general case of the Laws of Sines and Cosines, the definitions of sine and cosine must be extended to obtuse angles. - SPT.M.GHS.23 Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. - SPT.M.GHS.22 Use the relation ????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. - MAFS.912.N-CN.1.2 Use the method of completing the square to transform any quadratic equation in ?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? into an equation of the form (???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? – ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????)² = ???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? that has the same solutions. Derive the quadratic formula from this form. - MAFS.912.A-REI.2.4.a Analyze and draw conclusions based on a set of conditions using inductive and deductive reasoning. Recognize the logical relationships between a conditional statement and its inverse, converse, and contrapositive. - G.RL.1.2 Find arc length and sector area of a circle and use them to solve problems. - HSM.G.10.1 Assess the validity of a logical argument and give counterexamples to disprove a statement. - G.RL.1.3 Use properties of tangent lines to solve problems. - HSM.G.10.2 Relate the length of a chord to the central angle it subtends and the arc it intercepts. - HSM.G.10.3 Use function notation; evaluate a function, including nonlinear, at a given point in its domain algebraically and graphically. Interpret the results in terms of real-world and mathematical problems. - A1.F.3.2 Add, subtract, and multiply functions using function notation. - A1.F.3.3 ordering the angles by degree measure, given side lengths; - T.G.5.b determining whether a triangle exists; and - T.G.5.c determining the range in which the length of the third side must lie. - T.G.5.d ordering the sides by length, given angle measures; - T.G.5.a Compare a pair of sides of two triangles when the remaining pairs of sides are congruent. - HSM.G.5.5 Use theorems to compare the sides and angles of a triangle. - HSM.G.5.4 Use perpendicular and angle bisectors to solve problems. - HSM.G.5.1 Find the points of concurrency for the medians of a triangle and the altitudes of a triangle. - HSM.G.5.3 Use triangle bisectors to solve problems. - HSM.G.5.2 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-C.2.5 Solve an equation of the form ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????(????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????)=??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????(????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????) graphically by identifying the ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????-coordinate(s) of the point(s) of intersection of the graphs of ?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????