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Curriculum Standards: Use trigonometric ratios and the Pythagorean Theorem to solve problems involving right triangles in terms of a context - NC.M2.G-SRT.8 Analyze functions that include absolute value expressions. - HSM.A1.5.1 Graph and apply piecewise-defined functions. - HSM.A1.5.2 Explain and use the relationship between the sine and cosine of complementary angles. - G-SRT.7 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. - G-SRT.8 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 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. - G-SRT.3 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 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. - G-SRT.4 Understand similarity in terms of transformations. - NC.M2.G-SRT.2 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. - G-SRT.5 Verify experimentally the properties of dilations with given center and scale factor. - NC.M2.G-SRT.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. - G-SRT.6 Graph and apply step functions. - HSM.A1.5.3 Use similarity to solve problems and to prove theorems about triangles. Use theorems about triangles to prove relationships in geometric figures: a line parallel to one side of a triangle divides the other two sides proportionally and its converse; the Pythagorean Theorem. - NC.M2.G-SRT.4 Use transformations (rigid motions and dilations) to justify the AA criterion for triangle similarity. - NC.M2.G-SRT.3 Graph and analyze transformations of the absolute value function. - HSM.A1.5.4 Verify experimentally that the side ratios in similar right triangles are properties of the angle measures in the triangle, due to the preservation of angle measure in similarity. Use this discovery to develop definitions of the trigonometric ratios for acute angles. - NC.M2.G-SRT.6 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. - G-SRT.2 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 Predict the behavior of polynomial functions. - HSM.A2.3.1 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. - G-GPE.A.1 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 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 Interpret key features of graphs, tables, and verbal descriptions in context to describe functions that arise in applications relating two quantities, including: intercepts; intervals where the function is increasing, decreasing, positive, or negative; and maximums and minimums. - NC.M1.F-IF.4 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. - MAFS.912.G-SRT.1.3 Build an understanding 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 by recognizing that: 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). - NC.M1.F-IF.1 Use function notation to evaluate linear, quadratic, and exponential functions for inputs in their domains, and interpret statements that use function notation in terms of a context. - NC.M1.F-IF.2 Analyze linear, exponential, and quadratic functions by generating different representations, by hand in simple cases and using technology for more complicated cases, to show key features, including: domain and range; rate of change; intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; and end behavior. - NC.M1.F-IF.7 intercepts; - F.AII.7.e 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). - F-LE.2 Use equivalent expressions to reveal and explain different properties of a function. - NC.M1.F-IF.8 connections between and among multiple representations of functions using verbal descriptions, tables, equations, and graphs; - F.AII.7.g Interpret a function in terms of the context by relating its domain and range to its graph and, where applicable, to the quantitative relationship it describes. - NC.M1.F-IF.5 Calculate and interpret the average rate of change over a specified interval for a function presented numerically, graphically, and/or symbolically. - NC.M1.F-IF.6 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 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 Identify the effect on the graph of replacing 𝘧(𝘧𝘹) by 𝘧𝘹𝘧(𝘧𝘹𝘧𝘹) + 𝘧𝘹𝘧𝘹𝘬, 𝘧𝘹𝘧𝘹𝘬𝘬 𝘧𝘹𝘧𝘹𝘬𝘬𝘧(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹), 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹), and 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧(𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹 + 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬) for specific values of 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬 (both positive and negative); find the value of 𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬 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. - F-BF.3 Use graphs to find approximate solutions to systems of equations. - HSM.A1.4.1 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 Graph solutions to linear inequalities in two variables. - HSM.A1.4.4 Graph and solve a system of linear inequalities. - HSM.A1.4.5 Use congruence in terms of rigid motion. Justify the ASA, SAS, and SSS criteria for triangle congruence. Use criteria for triangle congruence (ASA, SAS, SSS, HL) to determine whether two triangles are congruent. - NC.M2.G-CO.8 Prove theorems about lines and angles and use them to prove relationships in geometric figures including: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent; when a transversal crosses parallel lines, corresponding angles are congruent; points are on a perpendicular bisector of a line segment if and only if they are equidistant from the endpoints of the segment; use congruent triangles to justify why the bisector of an angle is equidistant from the sides of the angle. - NC.M2.G-CO.9 Determine whether two figures are congruent by specifying a rigid motion or sequence of rigid motions that will transform one figure onto the other. - NC.M2.G-CO.6 Find the zeros of quadratic functions. - HSM.A2.2.3 Use the properties 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. - NC.M2.G-CO.7 Solve problems with complex numbers. - HSM.A2.2.4 Verify experimentally properties of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. - NC.M2.G-CO.4 Identify key features of quadratic functions. - HSM.A2.2.1 Given a geometric figure and a rigid motion, find the image of the figure. Given a geometric figure and its image, specify a rigid motion or sequence of rigid motions that will transform the pre-image to its image. - NC.M2.G-CO.5 Write and graph quadratic functions in standard form. - HSM.A2.2.2 Experiment with transformations in the plane: represent transformations in the plane; compare rigid motions that preserve distance and angle measure (translations, reflections, rotations) to transformations that do not preserve both distance and angle measure (e.g. stretches, dilations); understand that rigid motions produce congruent figures while dilations produce similar figures. - NC.M2.G-CO.2 Solve linear-quadratic systems. - HSM.A2.2.7 Given a triangle, quadrilateral, or regular polygon, describe any reflection or rotation symmetry i.e., actions that carry the figure onto itself. Identify center and angle(s) of rotation symmetry. Identify line(s) of reflection symmetry. - NC.M2.G-CO.3 Solve quadratic equations by completing the square. - HSM.A2.2.5 Solve quadratic equations using the Quadratic Formula. - HSM.A2.2.6 Graph a rational function and identify the x- and y-intercepts, vertical and horizontal asymptotes, using various methods and tools that may include a graphing calculator or other appropriate technology. (Excluding slant or oblique asymptotes and holes.) - A2.F.1.6 Prove theorems about parallelograms: opposite sides of a parallelogram are congruent; opposite angles of a parallelogram are congruent; diagonals of a parallelogram bisect each other; if the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. - NC.M3.G-CO.11 Use simulation to determine whether the experimental probability generated by sample data is consistent with the theoretical probability based on known information about the population. - NC.M2.S-IC.2 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 Apply properties, definitions, and theorems of two-dimensional figures to prove geometric theorems and solve problems. - NC.M3.G-CO.14 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 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. - MAFS.912.G-SRT.2.5 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 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 Use patterns to multiply binomials. - HSM.A1.7.3 Factor a polynomial. - HSM.A1.7.4 Understand the concept of an asymptote and identify asymptotes for exponential functions and reciprocals of linear functions, using symbolic and graphical methods. - 9.2.1.7 evaluate algebraic expressions for given replacement values of the variables. - EO.A.1.b 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 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 Graph and transform radical functions. - HSM.A2.5.3 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 Represent the inverse of a relation using tables, graphs, and equations. - HSM.A2.5.6 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 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 Build an understanding that operations with polynomials are comparable to operations with integers by adding and subtracting quadratic expressions and by adding, subtracting, and multiplying linear expressions. - NC.M1.A-APR.1 Understand the relationships among the factors of a quadratic expression, the solutions of a quadratic equation, and the zeros of a quadratic function. - NC.M1.A-APR.3 Identify and interpret key characteristics of functions represented graphically, with tables and with algebraic symbolism to solve problems. - A2.IF.A.1 Describe and graph exponential functions. - HSM.A1.6.2 Interpret expressions composed of multiple parts by viewing one or more of their parts as a single entity to give meaning in terms of a context. - NC.M3.A-SSE.1b 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 Justify a chosen solution method and each step of the solving process for quadratic, square root and inverse variation equations using mathematical reasoning. - NC.M2.A-REI.1 Use exponential functions to model situations and make predictions. - HSM.A1.6.3 Identify and describe geometric sequences. - HSM.A1.6.4 Perform, analyze, and use transformations of exponential functions. - HSM.A1.6.5 Describe the effects of transformations algebraically and graphically, creating vertical and horizontal translations, vertical and horizontal reflections and dilations (expansions/compressions) for linear, quadratic, cubic, square and cube root, absolute value, exponential and logarithmic functions. - A2.BF.A.3 Use inverse variation and graph translations of the reciprocal function. - HSM.A2.4.1 Graph rational functions. - HSM.A2.4.2 Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. - G.GGPE.1 Find the sum or difference of rational expressions. - HSM.A2.4.4 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 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 Solve for all solutions of quadratic equations in one variable. - NC.M2.A-REI.4 Use tables, graphs, and algebraic methods to approximate or find exact solutions of systems of linear and quadratic equations, and interpret the solutions in terms of a context. - NC.M2.A-REI.7 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 Write and solve equations with a variable on both sides to solve problems. - HSM.A1.1.3 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 Rewrite and use literal equations to solve problems. - HSM.A1.1.4 Use permutations and combinations to find the number of outcomes in a probability experiment. - HSM.G.12.3 Define probability distributions to represent experiments and solve problems. - HSM.G.12.4 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 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 Use probability to make decisions. - HSM.G.12.6 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 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 Write and solve absolute-value equations and inequalities - HSM.A1.1.7 Use relationships among events to find probabilities. - HSM.G.12.1 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 Use dilation and rigid motion to establish triangle similarity theorems. - HSM.G.7.3 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 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 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. - G-C.2 Prove that all circles are similar. - G-C.1 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. - G-C.3 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 Use the properties of prisms and cylinders to calculate their volumes. - HSM.G.11.2 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). - G-MG.1 The student will use surface area and volume of three-dimensional objects to solve practical problems. - TDF.G.13 Create equations and inequalities in one variable that represent linear, exponential, and quadratic relationships and use them to solve problems. - NC.M1.A-CED.1 Use trigonometry to solve problems. - HSM.G.8.5 Create and graph equations in two variables to represent linear, exponential, and quadratic relationships between quantities. - NC.M1.A-CED.2 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 Use trigonometric ratios to find lengths and angle measures of right triangles. - HSM.G.8.2 Prove the Pythagorean Theorem using similarity and establish the relationships in special right triangles. - HSM.G.8.1 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 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 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 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 Determine whether two figures are similar by specifying a sequence of transformations that will transform one figure into the other. - NC.M2.G-SRT.2a 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 Use the properties of dilations to show that two triangles are similar when all corresponding pairs of sides are proportional and all corresponding pairs of angles are congruent. - NC.M2.G-SRT.2b Extend the understanding that operations with polynomials are comparable to operations with integers by adding, subtracting, and multiplying polynomials. - NC.M2.A-APR.1 The length of the image of a line segment is equal to the length of the line segment multiplied by the scale factor. - NC.M2.G-SRT.1b Identify and interpret parts of a linear, exponential, or quadratic expression, including terms, factors, coefficients, and exponents. - NC.M1.A-SSE.1a The distance between the center of a dilation and any point on the image is equal to the scale factor multiplied by the distance between the dilation center and the corresponding point on the pre-image. - NC.M2.G-SRT.1c Interpret a linear, exponential, or quadratic expression made of multiple parts as a combination of entities to give meaning to an expression. - NC.M1.A-SSE.1b Interpret key features of linear, quadratic, and absolute value functions given an equation or a graph. - HSM.A2.1.1 Dilations preserve angle measure. - NC.M2.G-SRT.1d Solve for the real solutions of quadratic equations in one variable by taking square roots and factoring. - NC.M1.A-REI.4 Use a 2-way table to develop understanding of the conditional probability of A given B (written P(A|B)) as the likelihood that A will occur given that B has occurred. That is, P(A|B) is the fraction of event B;™s outcomes that also belong to event A. - NC.M2.S-CP.3a Write a function that describes a relationship between two quantities. - NC.M3.F-BF.1 Apply transformations to graph functions and write equations. - HSM.A2.1.2 Extend an understanding of the effects on the graphical and tabular representations of a function when replacing f(x) with k*f(x), f(x) + k, f(x + k) to include f(k*x) for specific values of k (both positive and negative). - NC.M3.F-BF.3 Justify a chosen solution method and each step of the solving process for linear and quadratic equations using mathematical reasoning. - NC.M1.A-REI.1 Graph and interpret piecewise-defined functions. - HSM.A2.1.3 Understand that event A is independent from event B if the probability of event A does not change in response to the occurrence of event B. That is P(A|B)=P(A). - NC.M2.S-CP.3b Use a variety of tools to solve systems of linear equations and inequalities. - HSM.A2.1.6 The student will verify and use properties of quadrilaterals to solve problems, including practical problems. - PC.G.9 Use geometric shapes, their measures, and their properties to describe real-world objects. - G.GM.1 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. - G-CO.10 Prove that all circles are similar. - CWC.M.GHS.34 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. - G-CO.12 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. - G-CO.11 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 Describe the effect of the transformations 99𝑘99𝑘𝑓(99𝑘𝑓𝑥), 99𝑘𝑓𝑥𝑓(99𝑘𝑓𝑥𝑓𝑥)+99𝑘𝑓𝑥𝑓𝑥𝑘, 99𝑘𝑓𝑥𝑓𝑥𝑘𝑓(99𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥+99𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘), and combinations of such transformations on the graph of 99𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦=99𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓(99𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥) for any real number 99𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘. Find the value of 99𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘 given the graphs and write the equation of a transformed parent function given its graph. - A2.FBF.3 When a line segment passes through the center of dilation, the line segment and its image lie on the same line. When a line segment does not pass through the center of dilation, the line segment and its image are parallel. - NC.M2.G-SRT.1a Use the equations and graphs of parabolas to solve problems. - HSM.G.9.4 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 Develop properties of special right triangles (45-45-90 and 30-60-90) and use them to solve problems. - NC.M2.G-SRT.12 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 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 standard form. - HSM.A1.2.3 Write equations of parallel lines and perpendicular lines. - HSM.A1.2.4 Write and graph linear equations using slope-intercept form. - HSM.A1.2.1 Use volume formulas for cylinders, pyramids, cones, spheres and composite figures to solve problems. - G.GMD.A.2 Know there is a complex number i such that i^2 = -1, and every complex number has the form a + bi where a and b are real numbers. - NC.M2.N-CN.1 Identify the effect on the graph of replacing 99𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧(99𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹) by 99𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧(99𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹) + 99𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬, 99𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬 99𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧(99𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹), 99𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧(99𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬99𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹), and 99𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧(99𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹 + 99𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬) for specific values of 99𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬 (both positive and negative); find the value of 99𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬 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 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 Interpret expressions that represent a quantity in terms of its context. - NC.M3.A-SSE.1 trigonometric ratios. - T.G.8.c Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. - G-GMD.3 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 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 Create and graph linear, quadratic and exponential equations in two variables. - A1.CED.A.2 Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. - MAFS.912.F-IF.3.7.d 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 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 Interpret expressions that represent a quantity in terms of its context. - NC.M2.A-SSE.1 Identify the function family when given an equation or graph. - HSM.A1.10.3 Write an equivalent form of a quadratic expression by completing the square, where a is an integer of a quadratic expression, ax^2 + bx + c, to reveal the maximum or minimum value of the function the expression defines. - NC.M2.A-SSE.3 Identify the key features of the cube root function. - HSM.A1.10.2 Describe the key features of the square root function. - HSM.A1.10.1 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. - MAFS.912.G-GMD.1.3 Analyze piecewise, absolute value, polynomials, exponential, rational, and trigonometric functions (sine and cosine) using different representations to show key features of the graph, by hand in simple cases and using technology for more complicated cases, including: domain and range; intercepts; intervals where the function is increasing, decreasing, positive, or negative; rate of change; relative maximums and minimums; symmetries; end behavior; period; and discontinuities. - NC.M3.F-IF.7 Interpret key features of graphs, tables, and verbal descriptions in context to describe functions that arise in applications relating two quantities to include periodicity and discontinuities. - NC.M3.F-IF.4 Create equations and inequalities in one variable that represent quadratic, square root, inverse variation, and right triangle trigonometric relationships and use them to solve problems. - NC.M2.A-CED.1 Create and graph equations in two variables to represent quadratic, square root and inverse variation relationships between quantities. - NC.M2.A-CED.2 Create systems of linear, quadratic, square root, and inverse variation equations to model situations in context. - NC.M2.A-CED.3 Draw and describe the rotation of a figure about a point of rotation for a given angle of rotation. - HSM.G.3.3 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 Use the relationship between conditional probabilities and relative frequencies in contingency tables. - 9.4.3.9 Identify different rigid motions used to transform two-dimensional shapes. - HSM.G.3.4 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 Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression 99𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑥² + 999𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑥𝑥 + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(99𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑥𝑥𝑥 – 99𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑥𝑥𝑥𝑦)² as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers 99𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑥𝑥𝑥𝑦𝑥 and 99𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝑥𝑥𝑥𝑦𝑥𝑦. - MAFS.K12.MP.7.1.a 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 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x). - NC.M3.A-APR.6 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 90ก, 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 Organize data in two-way frequency tables and use them to make inferences and generalizations. - HSM.A1.11.5 Use the properties of rational and irrational numbers to explain why: the sum or product of two rational numbers is rational; the sum of a rational number and an irrational number is irrational; the product of a nonzero rational number and an irrational number is irrational. - NC.M2.N-RN.3 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 Quantify and analyze the spread of data. - HSM.A1.11.4 Use measures of center and spread to compare data sets. - HSM.A1.11.2 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 Prove theorems about polygons. - G.CO.C.10 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 Use triangle congruence to solve problems with overlapping triangles. - HSM.G.4.6 Identify congruent right triangles. - HSM.G.4.5 The student will represent and solve problems, including practical problems, involving inverse variation, joint variation, and a combination of direct and inverse variations. - S.AII.10 Explain how expressions with rational exponents can be rewritten as radical expressions. - NC.M2.N-RN.1 Prove that all circles are similar. - MAFS.912.G-C.1.1 Apply theorems about isosceles and equilateral triangles to solve problems. - HSM.G.4.2 Understand that the quadratic formula is the generalization of solving ax^2 + bx + c by using the process of completing the square. - NC.M2.A-REI.4a Rewrite expressions with radicals and rational exponents into equivalent expressions using the properties of exponents. - NC.M2.N-RN.2 Use a composition of rigid motions to show that two objects are congruent. - HSM.G.4.1 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 Explain when quadratic equations will have non-real solutions and express complex solutions as a ± bi for real numbers a and b. - NC.M2.A-REI.4b 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 Use relationships between circles, angles, and arcs. - HSM.G.10.4 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. - SPT.M.GHS.21 Explain and use the relationship between the sine and cosine of complementary angles. - SPT.M.GHS.20 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 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 Construct geometric figures using various tools and methods. - G.CO.D.11 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. - G-GPE.6 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. - G-GPE.7 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 Relate the length of a chord to the central angle it subtends and the arc it intercepts. - HSM.G.10.3 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). - G-GPE.5 Write a function that describes a relationship between two quantities by building quadratic functions with real solution(s) and inverse variation functions given a graph, a description of a relationship, or ordered pairs (include reading these from a table). - NC.M2.F-BF.1 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in context. - NC.M2.S-CP.7 Apply the general Multiplication Rule P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in context. Include the case where A and B are independent: P(A and B) = P(A) P(B). - NC.M2.S-CP.8 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 Prove theorems about triangles and use them to prove relationships in geometric figures including: the sum of the measures of the interior angles of a triangle is 180 degrees; an exterior angle of a triangle is equal to the sum of its remote interior angles; the base angles of an isosceles triangle are congruent: the segment joining the midpoints of two sides of a triangle is parallel to the third side and half the length. - NC.M2.G-CO.10 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 Use geometric shapes, their measures and their properties to describe objects. - G.MG.A.1 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.2.6 Build a function that models a relationship between two quantities by combining linear, exponential, or quadratic functions with addition and subtraction or two linear functions with multiplication. - NC.M1.F-BF.1b Build linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two ordered pairs (include reading these from a table). - NC.M1.F-BF.1a Rewrite a quadratic function to reveal and explain different key features of the function. - NC.M1.F-IF.8a Describe events as subsets of the outcomes in a sample space using characteristics of the outcomes or as unions, intersections and complements of other events. - NC.M2.S-CP.1 Develop and understand independence and conditional probability. - NC.M2.S-CP.3 Verify and apply properties of right triangles, including properties of 45-45-90 and 30-60-90 triangles, to solve problems using algebraic and logical reasoning. - G.RT.1.2 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: Use transformations of functions to find models as students consider increasingly more complex situations. Observe the effect of multiple transformations on a single graph and the common effect of each transformation across function types. - MF.M.A2HS.34 Represent data on two categorical variables by constructing a two-way frequency table of data. Interpret the two-way table as a sample space to calculate conditional, joint and marginal probabilities. Use the table to decide if events are independent. - NC.M2.S-CP.4 Use the definition of the trigonometric functions to determine the sine, cosine, and tangent ratio of an acute angle in a right triangle. Apply the inverse trigonometric functions as ratios to find the measure of an acute angle in right triangles. - G.RT.1.3 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. - NC.M2.S-CP.5 Apply the trigonometric functions as ratios (sine, cosine, and tangent) to find side lengths in right triangles in real-world and mathematical problems. - G.RT.1.4 Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in context. - NC.M2.S-CP.6 Understand the effects of the graphical and tabular representations of a linear, quadratic, square root, and inverse variation function f with k*f(x), f(x) + k, f(x + k) for specific values of k (both positive and negative). - NC.M2.F-BF.3 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. - SPT.M.GHS.18 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. - SPT.M.GHS.17 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. - SPT.M.GHS.16 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. - SPT.M.GHS.15 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. - SPT.M.GHS.19 Construct linear, quadratic and exponential equations given graphs, verbal descriptions or tables. - A1.LQE.A.3 Use properties of sides, angles, and diagonals to identify a parallelogram. - HSM.G.6.4 Use the properties of parallel lines, diagonals, and triangles to investigate parallelograms. - HSM.G.6.3 Identify rhombuses, rectangles, and squares by the characteristics of their diagonals. - HSM.G.6.6 Use the properties of rhombuses, rectangles, and squares to solve problems. - HSM.G.6.5 Prove theorems about lines and angles; use theorems about lines and angles to solve problems. 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. - MAFS.912.G-CO.3.9 Use triangle congruence to understand kites and trapezoids. - HSM.G.6.2 Apply the distance formula and the Pythagorean Theorem and its converse to solve real-world and mathematical problems, as approximate and exact values, using algebraic and logical reasoning (include Pythagorean Triples). - G.RT.1.1 Find the sums of the measures of the exterior angles and interior angles of polygons. - HSM.G.6.1 The student, given information in the form of a figure or statement, will prove two triangles are congruent. - T.G.6 The student, given information in the form of a figure or statement, will prove two triangles are similar. - T.G.7 Develop the criteria for triangle congruence from the definition of congruence in terms of rigid motions. - G.CO.B.7 Write equivalent radical expressions. - HSM.A1.9.3 Solve quadratic equations by taking square roots. - HSM.A1.9.4 (HONORS ONLY) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). - MAFS.912.G-SRT.4.11 Use completing the square to solve quadratic equations. - HSM.A1.9.5 (HONORS ONLY) Prove the Laws of Sines and Cosines and use them to solve problems. - MAFS.912.G-SRT.4.10 Use the quadratic formula to solve quadratic equations. - HSM.A1.9.6 Use tables and graphs to find solutions of quadratic equations. - HSM.A1.9.1 Find the solution of a quadratic equation by factoring. - HSM.A1.9.2 Solve a system with linear and quadratic equations. - HSM.A1.9.7 Develop the definition of congruence in terms of rigid motions. - G.CO.B.6 Derive the equation of a circle. - G.GPE.A.1 Evaluate polynomial and rational expressions and expressions containing radicals and absolute values at specified points in their domains. - 9.2.3.1 Derive the equation of a parabola given a focus and directrix. - G.GPE.A.2 Add, subtract and multiply polynomials; divide a polynomial by a polynomial of equal or lower degree. - 9.2.3.2 Understand and apply the fact that the effect of a scale factor k on length, area and volume is to multiply each by k, k2 and k3, respectively. - 9.3.1.4 Determine the surface area and volume of pyramids, cones and spheres. Use measuring devices or formulas as appropriate. - 9.3.1.1 Compose and decompose two- and three-dimensional figures; use decomposition to determine the perimeter, area, surface area and volume of various figures. - 9.3.1.2 solve problems, including practical problems, involving angles formed when parallel lines are intersected by a transversal. - RLT.G.2.b prove two or more lines are parallel; and - RLT.G.2.a Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. - G.C.A.3 Identify and describe relationships among inscribed angles, radii and chords of circles. - G.C.A.2 Prove that all circles are similar using similarity transformations. - G.C.A.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. - G-CO.1 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. - G-CO.3 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. - G-CO.5 Apply the distance and midpoint formula, where appropriate, to develop the equation of a circle in standard form. - G.C.1.4 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. - G-CO.6 Recognize and write the radius r, center (h, k), and standard form of the equation of a circle (x _ h)_ + (y _ k)_ = r_ with and without graphs. - G.C.1.3 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. - G-CO.9 Use the definition of similarity to decide if figures are similar and to solve problems involving similar figures. - G.SRT.A.2 Apply the properties of circles and relationships among angles; arcs; and distances in a circle among radii, chords, secants and tangents to solve problems using algebraic and logical reasoning. - G.C.1.2 Construct and analyze scale changes of geometric figures. - G.SRT.A.1 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. - G-CO.8 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. - G.SRT.A.3 solving problems, including practical problems, about similar geometric figures. - TDF.G.14.d Prove that two triangles are similar using the Angle-Angle criterion and apply the proportionality of corresponding sides to solve problems and justify results. - G.GSRT.3 Use quadratic functions to model real-world situations. - HSM.A1.8.4 determining how changes in one or more dimensions of a figure affect area and/or volume of the figure; - TDF.G.14.b Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. - G.GSRT.4 Understand 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. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. - G.GSRT.1 comparing ratios between lengths, perimeters, areas, and volumes of similar figures; - TDF.G.14.a Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. - G.GSRT.2 Explain and use the relationship between the sine and cosine of complementary angles. - G.GSRT.7 Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships. - G.GCO.11 Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. - G.GSRT.8 Identify key features of the graph of the quadratic parent function. - HSM.A1.8.1 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. - G.GSRT.5 Graph quadratic functions using the vertex form. - HSM.A1.8.2 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. - G.GSRT.6 Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. - G.GCO.10 Graph quadratic functions using standard form. - HSM.A1.8.3 Demonstrate the ability to rotate, reflect or translate a figure, and determine a possible sequence of transformations between two congruent figures. - G.CO.A.5 Graph logarithmic functions and find equations of the inverses of exponential and logarithmic functions. - HSM.A2.6.4 Recognize the key features of exponential functions. - HSM.A2.6.1 Write exponential models in different ways to solve problems. - HSM.A2.6.2 Define angle, circle, perpendicular line, parallel line, line segment and ray based on the undefined notions of point, line, distance along a line and distance around a circular arc. - G.CO.A.1 Represent and solve problems in various contexts using exponential functions, such as investment growth, depreciation and population growth. - 9.2.2.2 Factor common monomial factors from polynomials, factor quadratic polynomials, and factor the difference of two squares. - 9.2.2.3 Describe the rotational symmetry and lines of symmetry of two- dimensional figures. - G.CO.A.3 Solve exponential and logarithmic equations. - HSM.A2.6.6 investigating symmetry and determining whether a figure is symmetric with respect to a line or a point; and - RLT.G.3.c applying slope to verify and determine whether lines are parallel or perpendicular; - RLT.G.3.b determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. - RLT.G.3.d investigating and using formulas for determining distance, midpoint, and slope; - RLT.G.3.a The dilation of a line segment is longer or shorter in the ratio given by the scale factor. - G-SRT.1b Prove that all circles are similar. - G.GCI.1 Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. - G.GCI.3 Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. - G.GCI.2 Write an equivalent form of a quadratic expression ax^2+bx+c, where a is an integer, by factoring to reveal the solutions of the equation or the zeros of the function the expression defines. - NC.M1.A-SSE.3 Use coordinates to solve geometric problems involving polygons algebraically: use coordinates to compute perimeters of polygons and areas of triangles and rectangles; use coordinates to verify algebraically that a given set of points produces a particular type of triangle or quadrilateral. - NC.M1.G-GPE.4 Interpret key features of graphs, tables, and verbal descriptions in context to describe functions that arise in applications relating two quantities, including: domain and range, rate of change, symmetries, and end behavior. - NC.M2.F-IF.4 Extend the use of function notation to express the image of a geometric figure in the plane resulting from a translation, rotation by multiples of 90 degrees about the origin, reflection across an axis, or dilation as a function of its pre-image. - NC.M2.F-IF.2 Compare key features of two functions (linear, quadratic, square root, or inverse variation functions) each with a different representation (symbolically, graphically, numerically in tables, or by verbal descriptions). - NC.M2.F-IF.9 Use equivalent expressions to reveal and explain different properties of a function by developing and using the process of completing the square to identify the zeros, extreme values, and symmetry in graphs and tables representing quadratic functions, and interpret these in terms of a context. - NC.M2.F-IF.8 Analyze quadratic, square root, and inverse variation functions by generating different representations, by hand in simple cases and using technology for more complicated cases, to show key features, including: domain and range; intercepts; intervals where the function is increasing, decreasing, positive, or negative; rate of change; maximums and minimums; symmetries; and end behavior. - NC.M2.F-IF.7 Use coordinates to find the midpoint or endpoint of a line segment. - NC.M1.G-GPE.6 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. - G-SRT.1a Use coordinates to prove the slope criteria for parallel and perpendicular lines and use them to solve problems: determine if two lines are parallel, perpendicular, or neither; find the equation of a line parallel or perpendicular to a given line that passes through a given point. - NC.M1.G-GPE.5 Understand that the graph of a two variable equation represents the set of all solutions to the equation. - NC.M1.A-REI.10 Build an understanding of why the x-coordinates of the points where the graphs of two linear, exponential, and/or quadratic equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x) and approximate solutions using graphing technology or successive approximations with a table of values. - NC.M1.A-REI.11 Extend the concept of a function to include geometric transformations in the plane by recognizing that: the domain and range of a transformation function f are sets of points in the plane; the image of a transformation is a function of its pre-image. - NC.M2.F-IF.1 The dilation of a line segment is longer or shorter in the ratio given by the scale factor. - SPT.M.GHS.14.b 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. - SPT.M.GHS.14.a Use relationships among events to find probabilities. - HSM.A2.12.1 number of sides of a regular polygon. - PC.G.10.c measure of an interior and/or exterior angle; and - PC.G.10.b Find the probability of an event given that another event has occurred. - HSM.A2.12.2 sum of the interior and/or exterior angles; - PC.G.10.a Use permutations and combinations to find the number of outcomes in a probability experiment. - HSM.A2.12.3 Create and graph equations in two variables to represent absolute value, polynomial, exponential and rational relationships between quantities. - NC.M3.A-CED.2 Define probability distributions to represent experiments and solve problems. - HSM.A2.12.4 Calculate, interpret, and apply expected value. - HSM.A2.12.5 Use probability to make decisions. - HSM.A2.12.6 Solve real-world and mathematical problems using the surface area and volume of prisms, cylinders, pyramids, cones, spheres, and composites of these figures. Use nets, measuring devices, or formulas as appropriate. - G.3D.1.1 Interpret the parameters a and b in a linear function f(x) = ax+b or an exponential function g(x) = ab^x in terms of a context. - NC.M1.F-LE.5 Write a function that describes a relationship between two quantities. - NC.M1.F-BF.1 Identify situations that can be modeled with linear and exponential functions, and justify the most appropriate model for a situation based on the rate of change over equal intervals. - NC.M1.F-LE.1 Understand and apply the geometric properties of a parabola. - HSM.A2.9.1 Understand and apply the geometric properties of a hyperbola. - HSM.A2.9.4 Write, graph, and apply the equation of a circle. - HSM.A2.9.2 the perpendicular bisector of a line segment; - RLT.G.4.b Know and apply properties of parallel and perpendicular lines, including properties of angles formed by a transversal, to solve problems and logically justify results. - 9.3.3.1 (HONORS ONLY) Derive the equation of a parabola given a focus and directrix. - MAFS.912.G-GPE.1.2 Know and apply properties of angles, including corresponding, exterior, interior, vertical, complementary and supplementary angles, to solve problems and logically justify results. - 9.3.3.2 a line segment congruent to a given line segment; - RLT.G.4.a Identify and interpret parts of a quadratic, square root, inverse variation, or right triangle trigonometric expression, including terms, factors, coefficients, radicands, and exponents. - NC.M2.A-SSE.1a Interpret quadratic and square root expressions made of multiple parts as a combination of single entities to give meaning in terms of a context. - NC.M2.A-SSE.1b 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.1.1 Use properties of polygons—including quadrilaterals and regular polygons—to define them, classify them, solve problems and logically justify results. - 9.3.3.7 Know and apply properties of a circle to solve problems and logically justify results. - 9.3.3.8 Know and apply properties of congruent and similar figures to solve problems and logically justify results. - 9.3.3.6 Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. - 9.3.3.3 Apply the Pythagorean Theorem and its converse to solve problems and logically justify results. - 9.3.3.4 Describe a data set using data displays, including box-and-whisker plots; describe and compare data sets using summary statistics, including measures of center, location and spread. Measures of center and location include mean, median, quartile and percentile. Measures of spread include standard deviation, range and inter-quartile range. Know how to use calculators, spreadsheets or other technology to display data and calculate summary statistics. - 9.4.1.1 Analyze the effects on summary statistics of changes in data sets. - 9.4.1.2 Explain and use the relationship between the sine and cosine of complementary angles. - G.SRT.C.6 Understand that side ratios in right triangles define the trigonometric ratios for acute angles. - G.SRT.C.5 Use scatterplots to analyze patterns and describe relationships between two variables. Using technology, determine regression lines (line of best fit) and correlation coefficients; use regression lines to make predictions and correlation coefficients to assess the reliability of those predictions. - 9.4.1.3 Identify and describe transformations of two-dimensional figures. - HSM.G.3 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles. - G.SRT.C.7 Use the relationships between sides, segments, and angles of triangles to solve problems. - HSM.G.5 Write conditionals and biconditionals and find their truth values. - HSM.G.1.5 Use inductive reasoning to make conjectures about mathematical relationships. - HSM.G.1.4 Use deductive reasoning to prove theorems. - HSM.G.1.7 Use deductive reasoning to draw conclusions. - HSM.G.1.6 Use properties of segments and angles to find their measures. - HSM.G.1.1 an angle congruent to a given angle; - RLT.G.4.f the bisector of a given angle, - RLT.G.4.e Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). - MAFS.912.G-MG.1.1 Use the midpoint and distance formulas to solve problems. - HSM.G.1.3 Use a straightedge and compass to construct basic figures. - HSM.G.1.2 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles. - G.GPE.B.6 arc length; and - PC.G.11.c Find the point on a directed line segment between two given points that partitions the segment in a given ratio. - G.GPE.B.5 lengths of segments formed by intersecting chords, secants, and/or tangents; - PC.G.11.b angle measures formed by intersecting chords, secants, and/or tangents; - PC.G.11.a Prove theorems about triangles. - G.CO.C.9 Prove theorems about lines and angles. - G.CO.C.8 Represent relationships in various contexts using systems of linear inequalities; solve them graphically. Indicate which parts of the boundary are included in and excluded from the solution set using solid and dotted lines. - 9.2.4.4 Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context. - 9.2.4.7 Create equations in two variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [Note this standard appears in future courses with a slight variation in the standard language.] - A-CED.2 Use the Law of Sines and the Law of Cosines. - HSM.A2.8.2 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). - MG.M.GHS.53 Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find non-real complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities. - 9.2.4.1 Prove the slope criteria for parallel and perpendicular lines and use them to solve problems. - G.GPE.B.4 Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations. - 9.2.4.2 Accurately interpret and use words and phrases such as "if…then," "if and only if," "all," and "not." Recognize the logical relationships between an "if…then" statement and its inverse, converse and contrapositive. - 9.3.2.2 Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. - 9.3.2.1 Calculate conditional probabilities of events. - G.CP.A.3 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. - G.CP.A.4 Recognize and explain the concepts of conditional probability and independence in a context. - G.CP.A.5 Apply and interpret the Addition Rule for calculating probabilities. - G.CP.A.6 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. 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. Implementation of this standard may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for M.GHS.36. - CPC.M.GHS.10 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. 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.11 Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations. - 9.3.2.4 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. Instructional Note: Build on prior student experience with simple constructions. Emphasize the ability to formalize and explain how these constructions result in the desired objects. Some of these constructions are closely related to previous standards and can be introduced in conjunction with them. - CPC.M.GHS.12 Understand the definition of independent events and use it to solve problems. - G.CP.A.2 Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets. - 9.3.2.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. - G.SRT.B.4 Use permutations and combinations to solve problems. - G.CP.A.8 Use slope to solve problems about parallel and perpendicular lines. - HSM.G.2.4 Solve problems using the measures of interior and exterior angles of triangles. - HSM.G.2.3 Use angle relationships to prove that lines are parallel. - HSM.G.2.2 Determine the measures of the angles formed when parallel lines are intersected by a transversal. - HSM.G.2.1 List of all Files Validated: imsmanifest.xml I_00395277-428d-390b-bda0-b5a297737f6e_1_R/BasicLTI.xml I_0069cb7d-0fa0-3bff-be79-7a6a3ba8e5b6_1_R/BasicLTI.xml I_0075eff4-7bcd-3705-8929-928171810bcd_1_R/BasicLTI.xml I_00828abf-4c08-388b-b386-3b1d551eb5dd_1_R/BasicLTI.xml I_0087f9f8-588d-318e-9272-ac37b2188d21_R/BasicLTI.xml 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