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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 Derive and apply the constant ratios of the sides in special right triangles (45˚-45˚-90˚ and 30˚-60˚-90˚). - GM.35.a Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. - G-SRT.8 Fit a linear function for a scatter plot that suggests a linear association. - CCSS.Math.Content.HSS-ID.B.6c Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. - LER.M.A1HS.13 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 Informally assess the fit of a function by plotting and analyzing residuals. - CCSS.Math.Content.HSS-ID.B.6b 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 Fit a linear function for a scatterplot that suggests a linear association. (A1, M1) - S.ID.6c Use statistics appropriate to the shape of the data distribution to compare center and spread of two or more different data sets. - A1.DS.A.2 Focus on linear, quadratic, and exponential functions. (A1, M2) - F.IF.5b 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 Create and graph linear, quadratic and exponential equations in two variables. - A1.CED.A.2 Solve quadratic equations as appropriate to the initial form of the equation by inspection, e.g., for x² = 49; taking square roots; completing the square; applying the quadratic formula; or utilizing the Zero-Product Property after factoring. - A.REI.4b 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 simple quadratic expressions. (A1, M2) - A.CED.1b Focus on applying linear and simple exponential expressions. (A1, M1) - A.CED.1a determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line; - EI.A.6.a graph linear equations in two variables. - EI.A.6.c 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 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 Demonstrate the converse of the Pythagorean Theorem. - GM.35.d Use graphs to find approximate solutions to systems of equations. - HSM.A1.4.1 Solve systems of linear equations using the elimination method. - HSM.A1.4.3 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 simple exponential functions, indicating intercepts and end behavior. (A1, M1) - F.IF.7e Classify two-dimensional figures in a hierarchy based on properties. - G.CO.14 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. ★ - F.IF.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 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 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 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 Identify key features of quadratic functions. - HSM.A2.2.1 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n + 1) = f(n) + f(n − 1) for n ≥ 1. - F.IF.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. - 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 Use congruence and similarity criteria for triangles to solve problems in real-world contexts. - GM.34 Interpret shapes of data displays representing different types of data distributions. - HSM.A1.11.3 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 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 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 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 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 Use a composition of rigid motions to show that two objects are congruent. - HSM.G.4.1 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. - SPT.M.GHS.21 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 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 x^2 + 9x + 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 complex things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y) 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 x and y. - MP.7 domain and range; - F.A.7.b 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 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. ★ - S.ID.5 evaluate algebraic expressions for given replacement values of the variables. - EO.A.1.b Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. ★ - S.ID.7 Represent data with plots on the real number line (dot plots, histograms, and box plots) in the context of real-world applications using the GAISE model. ★ - S.ID.1 In the context of real-world applications by using the GAISE model, use statistics appropriate to the shape of the data distribution to compare center (median and mean) and spread (mean absolute deviation, interquartile range, and standard deviation) of two or more different data sets. ★ - S.ID.2 represent verbal quantitative situations algebraically; and - EO.A.1.a In the context of real-world applications by using the GAISE model, interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). ★ - S.ID.3 Interpret parts of an expression, such as terms, factors, and coefficients. - A.SSE.1a 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 Interpret parts of an expression, such as terms, factors, and coefficients. - CCSS.Math.Content.HSA-SSE.A.1a 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 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 Compute (using technology) and interpret the correlation coefficient of a linear fit. ★ - S.ID.8 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 Use theorems to compare the sides and angles of a triangle. - HSM.G.5.4 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. - MAFS.912.S-ID.1.2 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. ★ - G.GPE.7 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. - MAFS.912.G-SRT.3.8 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 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 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 𝘝𝘭𝘙𝘙𝑥² + 9𝘝𝘭𝘙𝘙𝑥𝑥 + 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(𝘝𝘭𝘙𝘙𝑥𝑥𝑥 – 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦)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 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥 and 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦. - CCSS.Math.Practice.MP7.a 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 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 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. - CCSS.Math.Content.HSG-CO.A.4 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 Extend to include solving systems of linear equations in three variables, but only algebraically. (A2, M3) - A.REI.6b Solve systems of linear equations using the substitution method. - A1.AREI.6 Justify that the solution to a system of linear equations is not changed when one of the equations is replaced by a linear combination of the other equation. - A1.AREI.5 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 Represent data with plots on the real number line (dot plots, histograms, and box plots). - CCSS.Math.Content.HSS-ID.A.1 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 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 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). - CCSS.Math.Content.HSS-ID.A.3 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 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. - NC.M2.S-CP.5 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 Use the structure of an expression to identify ways to rewrite it. Example: For example, see 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹⁴ – 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺⁴ as (𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹²)² – (𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺²)², thus recognizing it as a difference of squares that can be factored as (𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹² – 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺²)(𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹² + 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺²). - CCSS.Math.Content.HSA-SSE.A.2 Focus on linear, quadratic, and exponential functions. (A1, M2) - F.IF.4b Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Instructional Note: Build on student experiences graphing and solving systems of linear equations from middle school to focus on justification of the methods used. Include cases where the two equations describe the same line (yielding infinitely many solutions) and cases where two equations describe parallel lines (yielding no solution); connect to standards in Geometry which require students to prove the slope criteria for parallel lines. - LER.M.A1HS.14 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 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 The student, given information in the form of a figure or statement, will prove two triangles are congruent. - T.G.6 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 Develop the criteria for triangle congruence from the definition of congruence in terms of rigid motions. - G.CO.B.7 Identify the effect on the graph of replacing 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧(𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹) by 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧(𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹) + 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬, 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧(𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹), 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧(𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹), and 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧(𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹 + 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬) for specific values of 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬 (both positive and negative); find the value of 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬 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 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 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 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 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 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. Instructional Note: In grades 6 – 8, students describe center and spread in a data distribution. Here they choose a summary statistic appropriate to the characteristics of the data distribution, such as the shape of the distribution or the existence of extreme data points. - DS.M.A1HS.34 Develop the definition of congruence in terms of rigid motions. - G.CO.B.6 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 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. - CCSS.Math.Content.HSG-CO.D.13 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. - CCSS.Math.Content.HSG-CO.D.12 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 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 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 Describe a data set using data displays, describe and compare data sets using summary statistics, including measures of central tendency, location, and spread. Know how to use calculators, spreadsheets, or other appropriate technology to display data and calculate summary statistics. - A1.D.1.1 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 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 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 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. - CCSS.Math.Content.HSA-REI.A.2 Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. - CCSS.Math.Content.HSF-IF.C.7b 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 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 Identify key features of the graph of the quadratic parent function. - HSM.A1.8.1 Analyze and solve real-world and mathematical problems involving systems of linear equations with a maximum of two variables by graphing (may include graphing calculator or other appropriate technology), substitution, and elimination. Interpret the solutions in the original context. - A1.A.1.3 Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. - G.GSRT.8 Interpret the parameters in a linear or exponential function in terms of a context. - CCSS.Math.Content.HSF-LE.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. - G.GSRT.5 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 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. - AP.M.GHS.43 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. - A.REI.3 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. - A.REI.2 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 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 determining whether a figure has been translated, reflected, rotated, or dilated, using coordinate methods. - RLT.G.3.d 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. - S-ID.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 Focus on polynomial expressions that simplify to forms that are linear or quadratic. (A1, M2) - A.APR.1a Compute (using technology) and interpret the correlation coefficient of a linear fit. - CCSS.Math.Content.HSS-ID.C.8 Distinguish between correlation and causation. - CCSS.Math.Content.HSS-ID.C.9 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 Use technology to represent data with plots on the real number line (histograms, and box plots). - NC.M1.S-ID.1 Examine the effects of extreme data points (outliers) on shape, center, and/or spread. - NC.M1.S-ID.3 Create systems of linear equations and inequalities to model situations in context. - NC.M1.A-CED.3 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. Interpret differences in shape, center, and spread in the context of the data sets. - NC.M1.S-ID.2 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. Instructional Note: Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. - CPC.M.GHS.7 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 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 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 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. - CCSS.Math.Content.HSG-SRT.B.5 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. - A.REI.12 Use relationships among events to find probabilities. - HSM.A2.12.1 Solve systems of linear equations using linear combination. - A1.AREI.6b Solve systems of linear equations using the substitution method. - A1.AREI.6a Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. - CCSS.Math.Content.HSA-REI.B.3 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 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 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 Understand that two events 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈 and 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉 are independent if the probability of 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈 and 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉 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 Interpret the parameters in a linear or exponential function in terms of the context. - A2.FLQE.5 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 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 Use tables, graphs, or algebraic methods (substitution and elimination) to find approximate or exact solutions to systems of linear equations and interpret solutions in terms of a context. - NC.M1.A-REI.6 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 Explain why replacing one equation in a system of linear equations by the sum of that equation and a multiple of the other produces a system with the same solutions. - NC.M1.A-REI.5 Know and apply properties of congruent and similar figures to solve problems and logically justify results. - 9.3.3.6 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. - A2.FBF.2 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 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 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 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 systems of two linear equations in two variables algebraically and graphically; and - EI.A.4.d 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 Solve an equation of the form 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉𝘧(𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉𝘧𝘹) = 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉𝘧𝘹𝘤 for a simple function 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉𝘧𝘹𝘤𝘧 that has an inverse and write an expression for the inverse. Example: For example, 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉𝘧𝘹𝘤𝘧𝘧(𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉𝘧𝘹𝘤𝘧𝘧𝘹) =2 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉𝘧𝘹𝘤𝘧𝘧𝘹𝘹³ or 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧(𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹) = (𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹+1)/(𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹–1) for 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹 ≠ 1. - CCSS.Math.Content.HSF-BF.B.4a 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 practical problems involving equations and systems of equations. - EI.A.4.e Understand that polynomials form a system analogous to the integers, namely, that they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. - A.APR.1 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles. - G.SRT.C.7 Interpret the parameters in a linear or exponential function in terms of a context. ★ - F.LE.5 Use the relationships between sides, segments, and angles of triangles to solve problems. - HSM.G.5 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically. ★ (A1, M2) - F.LE.3 Know precise definitions of ray, angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and arc length. - G.CO.1 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: For example, the Fibonacci sequence is defined recursively by 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧(0) = 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧(1) = 1, 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧(𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯+1) = 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧(𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯) + 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧(𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯-1) for 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯 greater than or equal to 1. - CCSS.Math.Content.HSF-IF.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). ★ - F.LE.2 Create equations and inequalities in one variable and use them to solve problems. Include equations and inequalities arising from linear, quadratic, simple rational, and exponential functions. ★ - A.CED.1 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. ★ - A.CED.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. - CCSS.Math.Content.HSF-IF.A.2 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. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. ★ (A1, M1) - A.CED.3 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. ★ - A.CED.4 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 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧 is a function and 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹 is an element of its domain, then 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧(𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹) denotes the output of 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧 corresponding to the input 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹. The graph of 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧 is the graph of the equation 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺 = 𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧(𝘝𝘭𝘙𝘙𝑥𝑥𝑥𝑦𝑥𝑦𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘧𝘹𝘧𝘹𝘬𝘬𝘧𝘹𝘧𝘬𝘹𝘧𝘹𝘬𝘬𝘬𝘈𝘉𝘈𝘉𝘧𝘹𝘤𝘧𝘧𝘹𝘹𝘧𝘹𝘹𝘹𝘹𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝘧𝘹𝘧𝘹𝘧𝘹𝘧𝘺𝘧𝘹). - CCSS.Math.Content.HSF-IF.A.1 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 arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. - CCSS.Math.Content.HSF-BF.A.2 Extend to include more complicated function situations with the option to graph with technology. (A2, M3) - A.CED.2c Focus on applying simple quadratic expressions. (A1, M2) - A.CED.2b Focus on applying linear and simple exponential expressions. (A1, M1) - A.CED.2a Prove theorems about triangles. - G.CO.C.9 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. - CCSS.Math.Content.HSA-REI.C.6 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. - CCSS.Math.Content.HSA-REI.C.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. - CCSS.Math.Practice.MP3.a Use the structure of an expression to identify ways to rewrite it. For example, to factor 3x(x − 5) + 2(x − 5), students should recognize that the "x − 5" is common to both expressions being added, so it simplifies to (3x + 2)(x − 5); or see 34𝘹⁴ – 34𝘹𝘺⁴ as (34𝘹𝘺𝘹²)² – (34𝘹𝘺𝘹𝘺²)²,thus recognizing it as a difference of squares that can be factored as (34𝘹𝘺𝘹𝘺𝘹² – 34𝘹𝘺𝘹𝘺𝘹𝘺²)(34𝘹𝘺𝘹𝘺𝘹𝘺𝘹² + 34𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺²). - A.SSE.2 Solve systems of linear equations algebraically, exactly, and graphically while focusing on pairs of linear equations in two variables. - A-REI.6 Given a system of two equations in two variables, show and explain why the sum of equivalent forms of the equations produces the same solution as the original system. - A-REI.5 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. - CCSS.Math.Content.HSG-CO.C.10 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line 34𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘺 = –334𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘺𝘹 and the circle 34𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘺𝘹𝘹² + 34𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘺𝘹𝘹𝘺² = 3. - CCSS.Math.Content.HSA-REI.C.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Understand that such systems may have zero, one, two, or infinitely many solutions. (Limit to linear equations and quadratic functions.) - A2.AREI.7 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 Informally determine the input of a function when the output is known. (A1, M1) - F.BF.4a Solve simple rational and radical equations in one variable and understand how extraneous solutions may arise. - A2.AREI.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 properties of special right triangles; and - T.G.8.b Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments. - 9.3.2.1 Focus on transformations of graphs of quadratic functions, except for f(kx); (A1, M2) - F.BF.3a Understand that two events A and B are independent if and only if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. ★ - S.CP.2 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. - MAFS.912.A-REI.3.5 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 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 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 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. - MAFS.912.A-REI.3.6 Understand the definition of independent events and use it to solve problems. - G.CP.A.2 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 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. - CCSS.Math.Content.HSF-IF.B.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 Solve quadratic equations by inspection (e.g., for 34𝘹² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as 34𝘹𝘢 ± 34𝘹𝘢𝘣34𝘹𝘢𝘣𝘪 for real numbers 34𝘹𝘢𝘣𝘪𝘢 and 34𝘹𝘢𝘣𝘪𝘢𝘣. - CCSS.Math.Content.HSA-REI.B.4b Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Example: For example, if the function 34𝘹𝘢𝘣𝘪𝘢𝘣𝘩(34𝘹𝘢𝘣𝘪𝘢𝘣𝘩𝘯) gives the number of person-hours it takes to assemble 34𝘹𝘢𝘣𝘪𝘢𝘣𝘩𝘯𝘯 engines in a factory, then the positive integers would be an appropriate domain for the function. - CCSS.Math.Content.HSF-IF.B.5 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. - CCSS.Math.Content.HSF-IF.B.4 Solve a system of linear equations algebraically and/or graphically. - A1.REI.B.3 Justify that the technique of linear combination produces an equivalent system of equations. - A1.REI.B.5 List of all Files Validated: imsmanifest.xml I_001f5c91-e21d-366f-80b8-4e88de375b9d_R/BasicLTI.xml I_003215f8-341c-3a2c-be74-b74f15077526_1_R/BasicLTI.xml I_0058846a-67c8-3dea-bb97-2a827a630ab8_1_R/BasicLTI.xml I_00aef186-285b-3493-aa5b-4b98fe9dbcfc_1_R/BasicLTI.xml I_00d81eff-6720-3320-9a87-6026b33bb376_R/BasicLTI.xml I_00d82571-4efb-367b-9cc1-29777d125592_R/BasicLTI.xml I_00e00a30-c32e-3209-bde5-1a98d6ea3750_1_R/BasicLTI.xml I_00e00a30-c32e-3209-bde5-1a98d6ea3750_3_R/BasicLTI.xml I_01290e9f-8e22-39d9-baaa-fd3c130aa0fd_1_R/BasicLTI.xml 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