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Common Core Math I, II, and III – Summary of Concepts Algebra Concepts One-Variable and Two-Variable Equations and Inequalities: Equivalent expressions (A-SSE.3) Represent solution set as a graph for a given equation or inequality(AREI.10,12) Solve equations by graphing (A-REI.11) Solve equations by making a table (A-REI.2,3,4) Solve equations using algebraic methods (A-REI.2,3,4) Solve a literal equation for a given variable (I,II) (A.CED.4) Create equations & inequalities based on a context (A.CED.1,2) Write equations of lines parallel or perpendicular to a given line, passing through a given point (G-GPE.5) Interpret constants and coefficients in context (A-SSE.1) Systems of Two Equations: Solve a system of linear equations by graphing or making a table (I) (A-REI-6) Solve a system of linear equations using algebraic methods (substitution or elimination) (I) (A-REI.5) Solve a linear-exponential system by graphing or making a table (I) (AREI.6) Solve a linear-quadratic system by graphing, making a table or using algebraic methods (II) (A-REI.7) Solve any system by graphing or making a table (involving linear, polynomial, rational, absolute value, exponential, or logarithmic functions) (III) (A-CED.3) Systems of Inequalities: Graph a system of linear inequalities (I) (A-REI.12) Identify feasible region (I) (A.CED.3) Viable vs. non-viable solutions in context (I) (A-CED.3) Optimization by trial and error given an objective function (II) Formal Linear Programming (III) (A.CED.3-see unpacked version) o Objective Function o Corner Principle Geometry Concepts Transformations: (II) (G-CO.2) Rigid motions – translations, reflections, rotations (G-CO.3,4,5) Non-rigid motions – dilations (G-SRT.1) Properties of motions Function Concepts (also see list of parent functions) Write a function rule for a given relationship o Given verbal description, graph, or table of values/set of data (modeling) (F-BF.1) o Function notation (F-IF.2) o Evaluate function for given input (F-IF.2) o Recursive vs. explicit forms (F-BF.1a,2) NOW-NEXT form (I) Formal notation (III) Interpret key features of functions in context (F-IF.4,7) Theoretical vs. practical domain (F-IF.5) Rate of change (I) o Average rate of change (F-IF.6) Calculate given equation or table of values Estimate from graph Interpret in context o Constant rate of change for linear functions (additive) (FLE.1a,b) o Constant percent rate of change for exponential functions (multiplicative) (F-LE.1a,c) o Exponential growth exceeds growth of other functions (F-LE.3) Compare features of two functions represented in different forms (verbal, algebraic, tabular, graphical) (F-IF.9) Combine functions using arithmetic operations (F-IF.1b) o +, − constant to linear, exponential, or quadratic (I) o +, − linear or linear-quadratic (I) Transformations (F-BF.3) o 𝑓(𝑥 + 𝑘), 𝑓(𝑥) + 𝑘 (I) o Fred Function (II) o 𝑘 ∙ 𝑓(𝑥) (II) o 𝑓(𝑘𝑥) (III) o Find k given graph Find the inverse of a function (F-BF.4) (III) Statistics and Probability Concepts Congruence: (II) Defined by rigid motions (G-CO.6,7) Triangle congruence – ASA, SAS, SSS (G-CO.8) Similarity: Defined by dilation (II) (G-SRT.2) Scale factor (II) (G-SRT.1) Triangles – AA Similarity (III) (G-SRT.3) Prove Geometric Theorems: Triangle Angle Sum Theorem (II) (G-CO.10) Midsegment Theorem (II) (G-CO.10); Side-Splitter Theorem (III) (GSRT.4) Isosceles Triangle Theorem (III) (G-CO.10) Vertical angles are congruent. (III) (G-CO.9) Developed by Wake County Public School System Page 1 If ≠, then alternate interior angles and corresponding angles are congruent. (III) (G-CO.9) Points on perpendicular bisector of segment equidistant from endpoints of segment. (III) (G-CO.9) Parallelograms (III) (G-CO.11) o Opposite sides are congruent. o Opposite angles are congruent. o Diagonals bisect each other. Rectangles (III) (G-CO.11) o Diagonals are congruent. Pythagorean Theorem (prove using similarity) (III) (G-SRT.4) Use CPCTC and similarity of triangles to prove relationships in geometric figures. (III) (G-SRT.5) All circles are similar. (III) (G-C.1) If inscribed in a circle, the opposite angles of a quadrilateral are supplementary. (III) Constructions: Copy a segment or angle (III) (G-CO.12) Bisect a segment or angle (III) (G-CO.12) Perpendicular lines and perpendicular bisector of a segment (III) Parallel lines (III) (G-CO.12) Inscribe an equilateral triangle, square, or regular hexagon in a circle (II) (G-CO.13) Inscribe/circumscribe a circle in/around a triangle (III) (G-C.3) Triangles: Solve right triangles (II) (See also Trigonometric Functions) (G-SRT.8) Solve triangles using the Law of Sines and Law of Cosines (II) (GSRT.10) 1 Area of a triangle: 𝐴 = 𝑎𝑏 sin(𝐶) (II) (G-SRT.9) 2 Applications of triangle relationships: surveying problems, resultant forces (II) (G-SRT.11) Circles: Line, segment, and angle relationships: (III) o Measures of central, inscribed, and circumscribed angles (GC.2) o Inscribed angle that intercepts a diameter at its endpoints is a right angle (G-C.2) o Tangent and radius to point of tangency are perpendicular (G-C.2) Measurement relationships: (III) (G-C.5) o Arc length derive using similarity that 𝐿~𝑟 define measure of angle 𝜃as constant of proportionality, where 𝜃is measured in radians so 𝐿 = 𝜃𝑟 find arc lengths o Area of sector 1 Derive the formula 𝐴 = 𝜃𝑟 2 2 Coordinate Geometry: Use coordinates to prove geometric relationships (I) o Slope criteria for parallel and perpendicular lines (G-GPE.5) o Distance – congruent segments, perimeter, area (G-GPE.7) o Midpoint (G-GPE.6) Developed by Wake County Public School System Page 2 Equation of a circle: (G-GPE.1) o Derive using Pythagorean Theorem given center and radius (II) o Find center and radius by completing the square (III) Derive equation of parabola given focus and directrix (III) (G-GPE.2) Perimeter, Area, and Volume: Informal argument for formulas (I) (G-GMD.1) o Circumference, area of a circle o Volume of cylinder, pyramid, cone Use volume formulas to solve problems – cylinders, pyramids, cones, spheres (I) (G-GMD.3) Cross-sections of 3-D figures; solids generated by rotation of 2-D figures (II) (G-GMD.4) Modeling with geometric figures (including density and design problems) (II) (G-MG.1,2,3) Parent Function Linear Math Tools Properties of Operations & Properties of Equality (I) (A-REI.1) Arithmetic sequences (I) (F-LE.2) Real World Connections Dollar Deals Projectile motion Unit Rate Graph: Equation: 𝑓(𝑥) = 𝑥 Domain: all real numbers Range: all real numbers Key Features Intercepts: (0, 0) Quadratic Power Polynomial Graph: Equation: 𝑓(𝑥) = 𝑥 2 Domain: all real numbers Range: y ≥ 0 Key Features Intercepts: (0, 0) Min: (0, 0) Graph: Equation: 𝑓(𝑥) = 𝑥 𝑝 Domain: Range: Key Features Intercepts: Graph: Equation: Domain: Range: Key Features Intercepts: Developed by Wake County Public School System Square roots (I) (A-REI.4b) Factor to find zeros (I) (A-SSE.3a, F-IF.8a) Factor to solve equations (I, II) (A-REI.4b) Quadratic Formula (II) (A-REI.4b) Complex numbers (III) (N-CN.1,2,7) Complete the square (III) (F-IF.8a) o to find the vertex (A-SSE.3b, FIF.8a) o to solve equations (A-REI.4b) Revenue/Income Finding square roots or cube roots of expressions (A-REI.4b) Properties of rational exponents (NRN.1,2) Operations with polynomials: +, − quadratics (I) (A-APR.1) Multiply 2 linear expressions (I) (A-APR.1) +, − any polynomials (II) (A-APR.1) Multiply up to 3 linear expressions (II) (AAPR.1) Multiply any polynomials (III) (A-APR.1) Remainder Theorem (III) (A-APR.2) Use factoring to find zeros & construct rough graph(A-APR.3) Page 3 Parent Function Rational (Inverse Variation) Graph: 1 Equation: 𝑓(𝑥) = 𝑥 Domain: x≠ 0 Range: y ≠ 0 Key Features Intercepts: none Math Tools Operations with rational expressions (Limit to 2nd degree in numerator and/or denominator) (A-APR.7) Long Division(III) (A-APR.6) CAS (III) (A-APR.6) Real World Connections Extraneous solutions (II) (A-REI.2) Square Root Graph: Equation: 𝑓(𝑥) = √𝑥 Domain: x ≥ 0 Range: y ≥ 0 Key Features Intercepts: (0, 0) Cube Root Graph: 3 Equation: 𝑓(𝑥) = √𝑥 Domain: all real numbers Range: all real numbers Key Features Intercepts: (0, 0) Exponential Graph: Equation: 𝑓(𝑥) = 𝑎𝑏 𝑥 Domain: all real numbers Range: y > 0 Key Features Intercepts: x: none; y: (0, 1) Logarithmic Piecewise Defined/Step Graph: Equation: 𝑓(𝑥) = log(𝑥) Domain: x > 0 Range: all real numbers Key Features Intercepts: x: (1, 0); y: none Graph: Equation: Domain: Range: Key Features Intercepts: Absolute Value Properties of Exponents (I, II) (A.SSE.3c, FIF.8b) Evaluate for integer inputs (I) (A.CED.2) Growth vs. decay (I) (F-LE.1c) Common Logs (II) (A.CED.1, F-LE.4) Natural Logs (III) (F-LE.4) Properties of Logs (II, III) Geometric Sequences (I, II) (F-LE.2) and Series(III) (A-SSE.4) Asymptotes Population growth Postage Tolerance Radioactive decay Compound interest Graph: Equation: 𝑦 = |𝑥| Domain: all real numbers Range: y ≥ 0 Key Features Intercepts: x: (0, 0); y: (0, 1) Developed by Wake County Public School System Page 4 Trigonometric Parent Function Graph: Equation: Domain: Range: Key Features Intercepts: Developed by Wake County Public School System Math Tools Sine, cosine, and tangent ratios (II) (GSRT.6,8) Pythagorean Theorem (II) (G-SRT.8) Sine, cosine, tangent functions in standard position (II) Midline, amplitude (II) (F-IF.7e, F-TF.5) Period, frequency (III) (F-IF.7e, F-TF.5) Radian measure (III) Unit Circle (III) Real World Connections Page 5