Organization: SAVVAS
Product Name: envision Integrated Mathematics II 2019 Common Core
Product Version: 1




Source:     IMS Online Validator
Profile:    1.2.0
Identifier: realize-1f907e5c-6191-3440-badd-064bc01c7c83
Timestamp:  Thursday, December 17, 2020 10:45 AM EST
Status:     VALID!
Conformant: true


-----    VALID!    -----
Resource Validation Results
The document is valid.


-----    VALID!    -----
Schema Location Results
Schema locations are valid.


-----    VALID!    -----
Schema Validation Results
The document is valid.


-----    VALID!    -----
Schematron Validation Results
The document is valid.


Curriculum Standards:


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
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
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
Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. - CCSS.Math.Content.HSS-ID.B.6a
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. - G-SRT.5
Understand similarity in terms of transformations. - NC.M2.G-SRT.2
Informally assess the fit of a function by plotting and analyzing residuals. - CCSS.Math.Content.HSS-ID.B.6b
Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions, or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. (A2, M3) - S.ID.6a
Verify experimentally the properties of dilations with given center and scale factor. - NC.M2.G-SRT.1
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
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
Emphasize the selection of a type of function for a model based on behavior of data and context. (A2, M3) - F.IF.5c
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
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
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
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
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
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
Extend to include more complicated function situations with the option to solve with technology. (A2, M3) - A.CED.1c
Focus on applying linear and simple exponential expressions. (A1, M1) - A.CED.1a
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 volume formulas for cylinders, pyramids, cones, and spheres to solve problems. - MAFS.912.G-GMD.1.3
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
Use permutations and combinations to compute probabilities of compound events and solve problems. ★ - S.CP.9 (+)
Apply probability concepts to real-world situations to make informed decisions. - 9.4.3.8
Use the relationship between conditional probabilities and relative frequencies in contingency tables. - 9.4.3.9
Define congruence of two figures in terms of rigid motions (a sequence of translations, rotations, and reflections); show that two figures are congruent by finding a sequence of rigid motions that maps one figure to the other. - GM.24
Verify criteria for showing triangles are congruent using a sequence of rigid motions that map one triangle to another. - GM.25
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Example: For example, rearrange Ohm’s law 𝘝 = 𝘝𝘭𝘝𝘭𝘙 to highlight resistance 𝘝𝘭𝘙𝘙. - CCSS.Math.Content.HSA-CED.A.4
Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. Example: For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. - CCSS.Math.Content.HSA-CED.A.3
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. - CCSS.Math.Content.HSA-CED.A.2
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. - CCSS.Math.Content.HSA-CED.A.1
Graph linear functions and indicate intercepts. (A1, M1) - F.IF.7a
Graph quadratic functions and indicate intercepts, maxima, and minima. (A1, M2) - F.IF.7b
Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. (A2, M3) - F.IF.7c
Graph simple exponential functions, indicating intercepts and end behavior. (A1, M1) - F.IF.7e
Derive the formula for the area of a sector, and use it to solve problems. - G.C.5b
Classify two-dimensional figures in a hierarchy based on properties. - G.CO.14
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. ★ (A2, M3) - F.IF.6
Apply similarity to relate the length of an arc intercepted by a central angle to the radius. Use the relationship to solve problems. - G.C.5a
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
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
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. - G.CO.13
Prove and apply theorems about triangles. Theorems include but are not restricted to the following: 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
Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. - F.IF.2
Prove and apply theorems about parallelograms. Theorems include but are not restricted to the following: 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
Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). - F.IF.1
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
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 the following: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. ★ (A2, M3) - F.IF.4
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
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. - CCSS.Math.Content.HSG-CO.C.9
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
Construct logical arguments to prove triangle congruence (SSS, SAS, ASA, AAS and HL) and triangle similarity (AA, SSS, SAS). - G.2D.1.8
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
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
Use congruence and similarity criteria for triangles to solve problems in real-world contexts. - GM.34
Analyze the structure of binomials, trinomials, and other polynomials in order to rewrite equivalent expressions. - A2.ASE.2
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. - CCSS.Math.Content.HSA-APR.A.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
Identify congruent right triangles. - HSM.G.4.5
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems if one of the two acute angles and a side length is given. (G, M2) - G.SRT.8a
Apply theorems about isosceles and equilateral triangles to solve problems. - HSM.G.4.2
Use a composition of rigid motions to show that two objects are congruent. - HSM.G.4.1
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. - MAFS.912.G-SRT.2.5
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. - MP.3
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. ★ (A2, M3) - G.SRT.8b (+)
Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. ★ - S.ID.7
Construct a tangent line from a point outside a given circle to the circle. - G.C.4 (+)
Interpret parts of an expression, such as terms, factors, and coefficients. - A.SSE.1a
Interpret parts of an expression, such as terms, factors, and coefficients. - CCSS.Math.Content.HSA-SSE.A.1a
ordering the angles by degree measure, given side lengths; - T.G.5.b
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. - CCSS.Math.Content.HSG-GMD.A.3
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
Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. - CCSS.Math.Content.HSA-APR.B.3
Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. - CCSS.Math.Content.HSS-ID.B.5
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
Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. - CCSS.Math.Content.HSG-GMD.A.2
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. - CCSS.Math.Content.HSG-GMD.A.1
ordering the sides by length, given angle measures; - T.G.5.a
Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. - CCSS.Math.Content.HSN-CN.A.3
Know there is a complex number 𝘝𝘭𝘙𝘙𝘪 such that 𝘝𝘭𝘙𝘙𝘪𝘪² = –1, and every complex number has the form 𝘝𝘭𝘙𝘙𝘪𝘪𝘢 + 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪 with 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢 and 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣 real. - CCSS.Math.Content.HSN-CN.A.1
Use the relation 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. - CCSS.Math.Content.HSN-CN.A.2
Use theorems to compare the sides and angles of a triangle. - HSM.G.5.4
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. ★ - G.GPE.7
Graph polynomials identifying zeros when suitable factorizations are available and indicating end behavior. Write a polynomial function of least degree corresponding to a given graph. (Limit to polynomials with degrees 3 or less.) - A2.AAPR.3
Find the point on a directed line segment between two given points that partitions the segment in a given ratio. - G.GPE.6
Justify 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
Use coordinates to prove simple geometric theorems algebraically and to verify geometric relationships algebraically, including properties of special triangles, quadrilaterals, and circles. For example, determine if a figure defined by four given points in the coordinate plane is a rectangle; determine if a specific point lies on a given circle. (G, M2) - G.GPE.4
Add, subtract, and multiply polynomials and understand that polynomials are closed under these operations. - A2.AAPR.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.1
Identify figures that have rotational symmetry; determine the angle of rotation, and use rotational symmetry to analyze properties of shapes - G.CO.3b
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
Identify figures that have line symmetry; draw and use lines of symmetry to analyze properties of shapes. - G.CO.3a
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. - CCSS.Math.Content.HSG-CO.A.1
Determine the maximum or minimum value of a quadratic function by completing the square. - A2.ASE.3b
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. - CCSS.Math.Content.HSG-CO.A.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. - CCSS.Math.Content.HSG-CO.A.5
Limit to pairs of linear equations in two variables. (A1, M1) - A.REI.6a
Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. - G.GGPE.1
Extend to include solving systems of linear equations in three variables, but only algebraically. (A2, M3) - A.REI.6b
Identify and describe relationships among angles, radii, chords, tangents, and arcs and use them to solve problems. Include the relationship between central, inscribed, and circumscribed angles and their intercepted arcs; 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
Construct the inscribed and circumscribed circles of a triangle; prove and apply the property that opposite angles are supplementary for a quadrilateral inscribed in a circle. - G.C.3
Prove that all circles are similar using transformational arguments. - G.C.1
Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. - G.SRT.3
Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. - CCSS.Math.Content.HSG-GMD.B.4
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
Use congruence and similarity criteria for triangles to solve problems and to justify relationships in geometric figures that can be decomposed into triangles. - G.SRT.5
Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. - CCSS.Math.Content.HSS-ID.A.2
Prove and apply theorems about triangles. Theorems include but are not restricted to the following: 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
Explain and use the relationship between the sine and cosine of complementary angles - G.SRT.7
Know and apply the Binomial Theorem for the expansion of (𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹 + 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺)ⁿ in powers of 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹 and y for a positive integer 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯, where 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹 and 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺 are any numbers, with coefficients determined for example by Pascal’s Triangle. The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument. - CCSS.Math.Content.HSA-APR.C.5
Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. - CCSS.Math.Content.HSS-ID.A.4
Prove polynomial identities and use them to describe numerical relationships. Example: For example, the polynomial identity (𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹² + 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺²)² = (𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹² – 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺²)² + (2𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺)² can be used to generate Pythagorean triples. - CCSS.Math.Content.HSA-APR.C.4
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
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. - SPT.M.GHS.18
Use coordinates to prove simple geometric theorems algebraically. - G.GGPE.4
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 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
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 structure of an expression to identify ways to rewrite it. Example: For example, see 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹⁴ – 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺⁴ as (𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹²)² – (𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺²)², thus recognizing it as a difference of squares that can be factored as (𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹² – 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺²)(𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹² + 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺²). - CCSS.Math.Content.HSA-SSE.A.2
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
Focus on linear, quadratic, and exponential functions. (A1, M2) - F.IF.4b
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). - CCSS.Math.Content.HSG-SRT.D.11
Combine standard function types using arithmetic operations. Example: For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. - CCSS.Math.Content.HSF-BF.A.1b
Prove the Laws of Sines and Cosines and use them to solve problems. - CCSS.Math.Content.HSG-SRT.D.10
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 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎+𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖 for real numbers 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎 and 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏. - A2.AREI.4b
The student, given information in the form of a figure or statement, will prove two triangles are congruent. - T.G.6
Define probability distributions to represent experiments and solve problems. - HSM.G.12.4
Calculate, interpret, and apply expected value. - HSM.G.12.5
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
Investigate, prove, and apply theorems about triangles, including but not limited to: the sum of the measures of the interior angles of a triangle is 180˚; the base angles of isosceles triangles are congruent; the segment joining the midpoints of two sides of a triangle is parallel to the third side and half the length; a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem using triangle similarity. - GM.31.b
Identify the effect on the graph of replacing 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧(𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹) by 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧(𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹) + 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬, 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧(𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹), 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧(𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹), and 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧(𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹 + 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬) for specific values of 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬 (both positive and negative); find the value of 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬 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. - CCSS.Math.Content.HSF-BF.B.3
For exponential models, express as a logarithm the solution to 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣 to the 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵 power = 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥 where 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢, 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤, and 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥 are numbers and the base 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣 is 2, 10, or 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦; evaluate the logarithm using technology. - CCSS.Math.Content.HSF-LE.A.4
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. - CCSS.Math.Content.HSF-LE.A.3
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). - CCSS.Math.Content.HSF-LE.A.2
Develop the definition of congruence in terms of rigid motions. - G.CO.B.6
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
Use the properties of exponents to interpret expressions for exponential functions. Example: For example, identify percent rate of change in functions such as y = (1.02) to the 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵 power, 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺 = (0.97) to the 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵 power, 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺 = (1.01) to the 12𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵 power, 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺 = (1.2) to the 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵/10 power, and classify them as representing exponential growth or decay. - CCSS.Math.Content.HSF-IF.C.8b
Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. - CCSS.Math.Content.HSF-IF.C.8a
Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. - CCSS.Math.Content.HSA-REI.D.12
Explain why the 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹-coordinates of the points where the graphs of the equations 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺 = 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧(𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹) and 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺 = 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔(𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹) intersect are the solutions of the equation 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧(𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹) = 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔(𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧(𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹) and/or 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔(𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. - CCSS.Math.Content.HSA-REI.D.11
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. - CCSS.Math.Content.HSG-CO.D.13
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. - CCSS.Math.Content.HSF-IF.C.9
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. - CCSS.Math.Content.HSG-CO.D.12
Use dilation and rigid motion to establish triangle similarity theorems. - HSM.G.7.3
Determine whether figures are similar. - HSM.G.7.2
Use similarity and the geometric mean to solve problems involving right triangles. - HSM.G.7.4
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. - CCSS.Math.Content.HSG-SRT.C.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. - CCSS.Math.Content.HSG-SRT.C.6
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
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. - G-CO.7
Explain and use the relationship between the sine and cosine of complementary angles. - CCSS.Math.Content.HSG-SRT.C.7
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
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
Use the definition of similarity to decide if figures are similar and to solve problems involving similar figures. - G.SRT.A.2
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 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
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. - CCSS.Math.Content.HSF-TF.C.9
Graph linear and quadratic functions and show intercepts, maxima, and minima. - CCSS.Math.Content.HSF-IF.C.7a
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = −3x and the circle x² + y² = 3. - A.REI.7
Solve systems of linear equations algebraically and graphically - A.REI.6
Verify 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 - A.REI.5
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
Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. - CCSS.Math.Content.HSF-IF.C.7e
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
Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. - CCSS.Math.Content.HSF-IF.C.7d
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
Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. - CCSS.Math.Content.HSF-IF.C.7c
Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. - CCSS.Math.Content.HSF-IF.C.7b
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
Extend polynomial identities to the complex numbers. Example: For example, rewrite 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹² + 4 as (𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹 + 2𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪)(𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹 – 2𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪). - CCSS.Math.Content.HSN-CN.C.8
Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. - CCSS.Math.Content.HSA-REI.A.1
comparing ratios between lengths, perimeters, areas, and volumes of similar figures; - TDF.G.14.a
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
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. - G.GSRT.5
Interpret the parameters in a linear or exponential function in terms of a context. - CCSS.Math.Content.HSF-LE.B.5
Compare properties of two functions given in different representations such as algebraic, graphical, tabular, or verbal. - A2.FIF.9
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
Use the properties of prisms and cylinders to calculate their volumes. - HSM.G.11.2
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
Apply the Addition Rule, 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗(𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈 or 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉) = 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗(𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈) + 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗(𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉) – 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗(𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈 and 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉), and interpret the answer in terms of the model. - CCSS.Math.Content.HSS-CP.B.7
Given a function in graphical, symbolic, or tabular form, determine the average rate of change of the function over a specified interval. Interpret the meaning of the average rate of change in a given context. - A2.FIF.6
Use permutations and combinations to compute probabilities of compound events and solve problems. - CCSS.Math.Content.HSS-CP.B.9
Relate the domain and range of a function to its graph and, where applicable, to the quantitative relationship it describes. - A2.FIF.5
Interpret key features of a function that models the relationship between two quantities when given in graphical or tabular form. Sketch the graph of a function from a verbal description showing key features. Key features include intercepts; intervals where the function is increasing, decreasing, constant, positive, or negative; relative maximums and minimums; symmetries; end behavior and periodicity. - A2.FIF.4
Evaluate and compare strategies on the basis of expected values. Example: For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident. - CCSS.Math.Content.HSS-MD.B.5b
Find the expected payoff for a game of chance. Example: For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant. - CCSS.Math.Content.HSS-MD.B.5a
Find the conditional probability of 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈 given 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉 as the fraction of 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉’s outcomes that also belong to 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈, and interpret the answer in terms of the model. - CCSS.Math.Content.HSS-CP.B.6
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. - A.REI.3
Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. - A.REI.1
Find the point on a directed line segment between two given points that partitions the segment in a given ratio. - CCSS.Math.Content.HSG-GPE.B.6
Identify different sets of properties necessary to define and construct figures. - GM.29.b
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. - CCSS.Math.Content.HSG-GPE.B.7
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. - CCSS.Math.Content.HSG-C.B.5
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, √3) lies on the circle centered at the origin and containing the point (0, 2). - CCSS.Math.Content.HSG-GPE.B.4
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). - CCSS.Math.Content.HSG-GPE.B.5
Construct a tangent line from a point outside a given circle to the circle. - CCSS.Math.Content.HSG-C.A.4
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. - CCSS.Math.Content.HSG-C.A.2
Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. - CCSS.Math.Content.HSG-C.A.3
determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. - RLT.G.3.d
Prove that all circles are similar. - CCSS.Math.Content.HSG-C.A.1
investigating and using formulas for determining distance, midpoint, and slope; - RLT.G.3.a
The student will use surface area and volume of three-dimensional objects to solve practical problems. - TDF.G.13
use text evidence to support an appropriate response - SPAN_TEKS_K_6_C_i
Prove that all circles are similar. - G.GCI.1
Extend to polynomial expressions beyond those expressions that simplify to forms that are linear or quadratic. (A2, M3) - A.APR.1b
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
Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. - CCSS.Math.Content.HSS-ID.C.7
Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. - G.GCI.5
Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. - G.GCI.4
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. - CCSS.Math.Content.HSN-RN.B.3
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
Factor a quadratic expression to reveal the zeros of the function it defines. - CCSS.Math.Content.HSA-SSE.B.3a
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
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 properties of exponents to transform expressions for exponential functions. Example: For example the expression 1.15 to the 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵 power can be rewritten as ((1.15 to the 1/12 power) to the 12𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵 power) is approximately equal to (1.012 to the 12𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵 power) to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. - CCSS.Math.Content.HSA-SSE.B.3c
Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. - CCSS.Math.Content.HSA-SSE.B.3b
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
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. - CCSS.Math.Content.HSG-SRT.B.4
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. - CCSS.Math.Content.HSA-REI.B.3
Define probability distributions to represent experiments and solve problems. - HSM.A2.12.4
Calculate, interpret, and apply expected value. - HSM.A2.12.5
Focus on formulas in which the variable of interest is linear or square. For example, rearrange Ohm's law V = IR to highlight resistance R, or rearrange the formula for the area of a circle A = (π)r² to highlight radius r. (A1) - A.CED.4a
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
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. - G.SRT.1b
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
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. - CCSS.Math.Content.HSG-GPE.A.1
Derive the equation of a parabola given a focus and directrix. - CCSS.Math.Content.HSG-GPE.A.2
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
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. Example: For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. - CCSS.Math.Content.HSS-CP.A.4
Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. - G.GMD.4
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. ★ - G.GMD.3
Understand that two events 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈 and 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉 are independent if the probability of 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈 and 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉 occurring together is the product of their probabilities, and use this characterization to determine if they are independent. - CCSS.Math.Content.HSS-CP.A.2
When figures are similar, understand and apply the fact that when a figure is scaled by a factor of k, the effect on lengths, areas, and volumes is that they are multiplied by k, k², and k³, respectively. - G.GMD.6
Understand the conditional probability of 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈 given 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉 as 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗(𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈 and 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉)/𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗(𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉), and interpret independence of 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈 and 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉 as saying that the conditional probability of 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈 given 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉 is the same as the probability of 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈, and the conditional probability of 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉 given 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈 is the same as the probability of 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉. - CCSS.Math.Content.HSS-CP.A.3
Understand how and when changes to the measures of a figure (lengths or angles) result in similar and non-similar figures. - G.GMD.5
Give an informal argument for the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments - G.GMD.1
Interpret the parameters in a linear or exponential function in terms of the context. - A2.FLQE.5
Extend polynomial identities to the complex numbers. For example, rewrite x² + 4 as (x + 2 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑖)(x − 2 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑖𝑖). - N.CN.8 (+)
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
the perpendicular bisector of a line segment; - RLT.G.4.b
Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. - CCSS.Math.Content.HSG-SRT.A.3
Use geometric shapes, their measures, and their properties to describe real-world objects. - G.GM.1
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
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
Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. (A2, M3) - F.BF.1b
Rewrite expressions involving radicals and rational exponents using the properties of exponents. - CCSS.Math.Content.HSN-RN.A.2
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. - CCSS.Math.Content.HSN-RN.A.1
Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. ★ - F.BF.2
Identify the effect on the graph of replacing f(x) by f(x) + k, kf(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. (A2, M3) - F.BF.3
Use the volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems. - NC.M3.G-GMD.3
Know and apply properties of congruent and similar figures to solve problems and logically justify results. - 9.3.3.6
Describe the effect of the transformations 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑖𝑖𝑘𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑖𝑖𝑘𝑓(𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑖𝑖𝑘𝑓𝑥), 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑖𝑖𝑘𝑓𝑥𝑓(𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑖𝑖𝑘𝑓𝑥𝑓𝑥)+𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑖𝑖𝑘𝑓𝑥𝑓𝑥𝑘, 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑖𝑖𝑘𝑓𝑥𝑓𝑥𝑘𝑓(𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑖𝑖𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥+𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑖𝑖𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘), and combinations of such transformations on the graph of 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑖𝑖𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦=𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑖𝑖𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓(𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑖𝑖𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥) for any real number 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑖𝑖𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘. Find the value of 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑖𝑖𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘 given the graphs and write the equation of a transformed parent function given its graph. - A2.FBF.3
Know and apply properties of equilateral, isosceles and scalene triangles to solve problems and logically justify results. - 9.3.3.3
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
Prove and apply theorems about lines and angles. Theorems include but are not restricted to the following: 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
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
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
Calculate the expected value of a random variable; interpret it as the mean of the probability distribution. - CCSS.Math.Content.HSS-MD.A.2
Solve an equation of the form 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑖𝑖𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧(𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑖𝑖𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹) = 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑖𝑖𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘤 for a simple function 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑖𝑖𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘤𝘧 that has an inverse and write an expression for the inverse. Example: For example, 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑖𝑖𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘤𝘧𝘧(𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑖𝑖𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘤𝘧𝘧𝘹) =2 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑖𝑖𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘤𝘧𝘧𝘹𝘹³ or 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑖𝑖𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧(𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑖𝑖𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹) = (𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑖𝑖𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹+1)/(𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑖𝑖𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹–1) for 𝘝𝘭𝘙𝘙𝘪𝘪𝘢𝘣𝘪𝘢𝘣𝘪𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝑎𝑏𝑖𝑎𝑏𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘢𝘣𝘤𝘵𝘥𝘢𝘤𝘥𝘣𝘦𝘵𝘺𝘵𝘺𝘵𝘺𝘵𝘹𝘺𝘧𝘹𝘺𝑔𝘹𝘧𝘹𝑔𝘹𝘧𝘹𝑔𝘹𝘹𝘹𝘪𝘹𝘪𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝘵𝘵𝘵𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝑖𝑖𝑘𝑓𝑥𝑓𝑥𝑘𝑓𝑥𝑘𝑦𝑓𝑥𝑘𝑘𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹 ≠ 1. - CCSS.Math.Content.HSF-BF.B.4a
Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only