# Georgia Standards of Excellence Curriculum Map. Mathematics. GSE Analytic Geometry

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1 Georgia Standards of Excellence Curriculum Map Mathematics GSE Analytic Geometry These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement.

2 GSE Analytic Geometry Curriculum Map 1 st Semester 2 nd Semester Unit 1 (6 7 weeks) Similarity, Congruence, and Proofs MGSE9-12.G.SRT.1 MGSE9-12.G.SRT.2 MGSE9-12.G.SRT.3 MGSE9-12.G.SRT.4 MGSE9-12.G.SRT.5 MGSE9-12.G.CO.6 MGSE9-12.G.CO.7 MGSE9-12.G.CO.8 MGSE9-12.G.CO.9 MGSE9-12.G.CO.10 MGSE9-12.G.CO.11 MGSE9-12.G.CO.12 MGSE9-12.G.CO.13 MGSE9-12.G.GPE.4 Unit 2 (2 3 weeks) Right Triangle Trigonometry MGSE9-12.G.SRT.6 MGSE9-12.G.SRT.7 MGSE9-12.G.SRT.8 Unit 3 (4 5 weeks) Circles and Volume MGSE9-12.G.C.1 MGSE9-12.G.C.2 MGSE9-12.G.C.3 MGSE9-12.G.C.4 MGSE9-12.G.C.5 MGSE9-12.G.GMD.1 MGSE9-12.G.GMD.2 MGSE9-12.G.GMD.3 MGSE9-12.G.GMD.4 Unit 4 (2 3 weeks) Extending the Number System MGSE9-12.N.RN.2 MGSE9-12.N.RN.3 MGSE9-12.A.APR.1 Unit 5 (7 8 weeks) Quadratic Functions MGSE9-12.A.SSE.1 MGSE9-12.A.SSE.1a MGSE9-12.A.SSE.1b MGSE9-12.A.SSE.2 MGSE9-12.A.SSE.3 MGSE9-12.A.SSE.3a MGSE9-12.A.SSE.3b MGSE9-12.A.CED.1 MGSE9-12.A.CED.2 MGSE9-12.A.CED.4 MGSE9-12.A.REI.4 MGSE9-12.A.REI.4a MGSE9-12.A.REI.4b MGSE9-12.F.IF.4 MGSE9-12.F.IF.5 MGSE9-12.F.IF.6 MGSE9-12.F.IF.7 MGSE9-12.F.IF.7a MGSE9-12.F.IF.8 MGSE9-12.F.IF.8a MGSE9-12.F.IF.9 MGSE9-12.F.BF.1 MGSE9-12.F.BF.1a MGSE9-12.F.BF.3 MGSE9-12.F.LE.3 MGSE9-12.S.ID.6 MGSE9-12.S.ID.6a Unit 6 (4 5 weeks) Geometric and Algebraic Connections MGSE9-12.G.GPE.1 MGSE9-12.G.GPE.4 MGSE9-12.G.MG.1 MGSE9-12.G.MG.2 MGSE9-12.G.MG.3 Unit 7 (4 5 weeks) Applications of Probability MGSE9-12.S.CP.1 MGSE9-12.S.CP.2 MGSE9-12.S.CP.3 MGSE9-12.S.CP.4 MGSE9-12.S.CP.5 MGSE9-12.S.CP.6 MGSE9-12.S.CP.7 These units were written to build upon concepts from prior units, so later units contain tasks that depend upon the concepts addressed in earlier units. All units will include the Mathematical Practices and indicate skills to maintain. NOTE: Mathematical standards are interwoven and should be addressed throughout the year in as many different units and tasks as possible in order to stress the natural connections that exist among mathematical topics. Grade 9-12 Key: Number and Quantity Strand: RN = The Real Number System, Q = Quantities, CN = Complex Number System, VM = Vector and Matrix Quantities Algebra Strand: SSE = Seeing Structure in Expressions, APR = Arithmetic with Polynomial and Rational Expressions, CED = Creating Equations, REI = Reasoning with Equations and Inequalities Functions Strand: IF = Interpreting Functions, LE = Linear and Exponential Models, BF = Building Functions, TF = Trigonometric Functions Geometry Strand: CO = Congruence, SRT = Similarity, Right Triangles, and Trigonometry, C = Circles, GPE = Expressing Geometric Properties with Equations, GMD = Geometric Measurement and Dimension, MG = Modeling with Geometry Statistics and Probability Strand: ID = Interpreting Categorical and Quantitative Data, IC = Making Inferences and Justifying Conclusions, CP = Conditional Probability and the Rules of Probability, MD = Using Probability to Make Decisions July 2016 Page 2 of 9

3 Georgia Standards of Excellence Analytic Geometry Curriculum Map Rationale Unit 1: Building on standards from Coordinate Algebra and from middle school, students will use transformations and proportional reasoning to develop a formal understanding of similarity and congruence. Students will identify criteria for similarity and congruence of triangles, develop facility with geometric proofs (variety of formats), and use the concepts of similarity and congruence to prove theorems involving lines, angles, triangles, and other polygons. Unit 2: Students will apply similarity in right triangles to understand right triangle trigonometry. Students will use the Pythagorean Theorem and the relationship between the sine and cosine of complementary angles to solve problems involving right triangles. Unit 3: Students will understand and apply theorems about circles, find arc lengths of circles, and find areas of sectors of circles. Students will develop and explain formulas related to circles and the volume of solid figures and use the formulas to solve problems. Building on standards from middle school, students will extend the study of identifying cross-sections of three-dimensional shapes to identifying three-dimensional objects generated by rotations of two-dimensional objects. Unit 4: Students will address properties of rational and irrational numbers, rewrite expressions involving radicals, and perform operations on polynomials in preparation for working with quadratic functions later in the course. Unit 5: Students will analyze quadratic functions. Students will investigate key features of graphs, solve quadratic equations by taking the square roots, factoring (x 2 + bx + c AND ax 2 + bx + c), completing the square, and using the quadratic formula. Students will compare and contrast graphs in standard, vertex, and intercept forms. Students will only work with real number solutions. Unit 6: Students will verify algebraically geometric relationships of circles in the coordinate plane. Students will derive the equation of a circle and model real-world objects using geometric shapes and concepts. Unit 7: Students will understand independence and conditional probability and use them to interpret data. Building on standards from middle school, students will formalize the rules of probability and use the rules to compute probabilities of compound events in a uniform probability model. July 2016 Page 3 of 9

5 given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. MGSE9-12.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. MGSE9-12.G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. (Extend to include HL and AAS.) Prove geometric theorems MGSE9-12.G.CO.9 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. MGSE9-12.G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; 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. MGSE9-12.G.CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Make geometric constructions MGSE9-12.G.CO.12 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. MGSE9-12.G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon, each inscribed in a circle. cone using Cavalieri s principle. MGSE9-12.G.GMD.2 Give an informal argument using Cavalieri s principle for the formulas for the volume of a sphere and other solid figures. MGSE9-12.G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Visualize relationships between twodimensional and three-dimensional objects MGSE9-12.G.GMD.4 Identify the shapes of two-dimensional cross-sections of threedimensional objects, and identify threedimensional objects generated by rotations of two-dimensional objects. July 2016 Page 5 of 9

6 Use coordinates to prove simple geometric theorems algebraically MGSE9-12.G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. 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). (Focus on quadrilaterals, right triangles, and circles.) July 2016 Page 6 of 9

8 Solve equations and inequalities in one variable MGSE9-12.A.REI.4 Solve quadratic equations in one variable. MGSE9-12.A.REI.4a Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x p) 2 = q that has the same solutions. Derive the quadratic formula from ax 2 + bx + c = 0. MGSE9-12.A.REI.4b Solve quadratic equations by inspection (e.g., for x 2 = 49), taking square roots, factoring, completing the square, and the quadratic formula, as appropriate to the initial form of the equation (limit to real number solutions). Interpret functions that arise in applications in terms of the context MGSE9-12.F.IF.4 Using tables, graphs, and verbal descriptions, interpret the key characteristics of a function which models the relationship between two quantities. Sketch a graph showing key features including: intercepts; interval where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. MGSE9-12.F.IF.5 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. MGSE9-12.F.IF.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. Analyze functions using different representations MGSE9-12.F.IF.7 Graph functions expressed algebraically and show key features of the graph both by hand and by using technology. MGSE9-12.F.IF.7a Graph linear and quadratic functions and show intercepts, maxima, and minima (as determined by the function or by context). MGSE9-12.F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. MGSE9-12.F.IF.8a 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. For example, compare and contrast quadratic functions in standard, vertex, and intercept forms. MGSE9-12.F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one function and an algebraic expression for July 2016 Page 8 of 9 P(A) + P(B) P(A and B), and interpret the answers in context.

9 another, say which has the larger maximum. Build a function that models a relationship between two quantities MGSE9-12.F.BF.1 Write a function that describes a relationship between two quantities. MGSE9-12.F.BF.1a Determine an explicit expression and the recursive process (steps for calculation) from context. For example, if Jimmy starts out with \$15 and earns \$2 a day, the explicit expression 2x+15 can be described recursively (either in writing or verbally) as to find out how much money Jimmy will have tomorrow, you add \$2 to his total today. Jn = Jn 1+ 2, J0 = 15 Build new functions from existing functions MGSE9-12.F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Construct and compare linear, quadratic, and exponential models and solve problems MGSE9-12.F.LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. Summarize, represent, and interpret data on two categorical and quantitative variables MGSE9-12.S.ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. MGSE9-12.S.ID.6a Decide which type of function is most appropriate by observing graphed data, charted data, or by analysis of context to generate a viable (rough) function of best fit. Use this function to solve problems in context. Emphasize linear, quadratic and exponential models. July 2016 Page 9 of 9

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