Download Common Core Math I, II, and III – Summary of Concepts

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
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 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
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 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)
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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) = 𝑎𝑏 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
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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
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