Use properties of rational and irrational numbers.
NC.M2.N-RN.3 Use the properties of rational and irrational numbers to explain why: the sum or product of two rational numbers is rational; the sum of a rational number and an irrational number is irrational; the product of a nonzero rational number and an irrational number is irrational.
NC.M2.A-SSE.1 Interpret expressions that represent a quantity in terms of its context.
NC.M2.A-SSE.1.a Identify and interpret parts of a quadratic, square root, inverse variation, or right triangle trigonometric expression, including terms, factors, coefficients, radicands, and exponents.
NC.M2.A-SSE.3 Write an equivalent form of a quadratic expression by completing the square, where 𝑎 is an integer of a quadratic expression, 𝑎𝑥² + 𝑏𝑥 + 𝑐, to reveal the maximum or minimum value of the function the expression defines.
Represent and solve equations and inequalities graphically.
NC.M2.A-REI.11 Extend the understanding that the 𝑥-coordinates of the points where the graphs of two square root and/or inverse variation 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.M2.F-IF Interpreting Functions
Understand the concept of a function and use function notation.
NC.M2.F-IF.1 Extend the concept of a function to include geometric transformations in the plane by recognizing that: the domain and range of a transformation function f are sets of points in the plane; the image of a transformation is a function of its pre-image.
NC.M2.F-IF.2 Extend the use of function notation to express the image of a geometric figure in the plane resulting from a translation, rotation by multiples of 90 degrees about the origin, reflection across an axis, or dilation as a function of its pre-image.
Interpret functions that arise in applications in terms of the context.
NC.M2.F-IF.4 Interpret key features of graphs, tables, and verbal descriptions in context to describe functions that arise in applications relating two quantities, including: domain and range, rate of change, symmetries, and end behavior.
Analyze functions using different representations.
NC.M2.F-IF.7 Analyze quadratic, square root, and inverse variation functions by generating different representations, by hand in simple cases and using technology for more complicated cases, to show key features, including: domain and range; intercepts; intervals where the function is increasing, decreasing, positive, or negative; rate of change; maximums and minimums; symmetries; and end behavior.
NC.M2.F-IF.8 Use equivalent expressions to reveal and explain different properties of a function by developing and using the process of completing the square to identify the zeros, extreme values, and symmetry in graphs and tables representing quadratic functions, and interpret these in terms of a context.
NC.M2.F-IF.9 Compare key features of two functions (linear, quadratic, square root, or inverse variation functions) each with a different representation (symbolically, graphically, numerically in tables, or by verbal descriptions).
Build a function that models a relationship between two quantities.
NC.M2.F-BF.1 Write a function that describes a relationship between two quantities by building quadratic functions with real solution(s) and inverse variation functions given a graph, a description of a relationship, or ordered pairs (include reading these from a table).
NC.M2.F-BF.3 Understand the effects of the graphical and tabular representations of a linear, quadratic, square root, and inverse variation function 𝑓 with 𝑘 • 𝑓(𝑥), 𝑓(𝑥) + 𝑘, 𝑓(𝑥 + 𝑘) for specific values of 𝑘 (both positive and negative).
NC.M2.G-CO.2 Experiment with transformations in the plane. Represent transformations in the plane. Compare rigid motions that preserve distance and angle measure (translations, reflections, rotations) to transformations that do not preserve both distance and angle measure (e.g. stretches, dilations). Understand that rigid motions produce congruent figures while dilations produce similar figures.
NC.M2.G-CO.3 Given a triangle, quadrilateral, or regular polygon, describe any reflection or rotation symmetry i.e., actions that carry the figure onto itself. Identify center and angle(s) of rotation symmetry. Identify line(s) of reflection symmetry.
NC.M2.G-CO.5 Given a geometric figure and a rigid motion, find the image of the figure. Given a geometric figure and its image, specify a rigid motion or sequence of rigid motions that will transform the pre-image to its image.
NC.M2.G-CO.8 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.9 Prove theorems about lines and angles and use them to prove relationships in geometric figures including: Vertical angles are congruent. When a transversal crosses parallel lines, alternate interior angles are congruent. When a transversal crosses parallel lines, corresponding angles are congruent. Points are on a perpendicular bisector of a line segment if and only if they are equidistant from the endpoints of the segment. Use congruent triangles to justify why the bisector of an angle is equidistant from the sides of the angle.
NC.M2.G-CO.10 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 180º. 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-SRT Similarity, Right Triangles, and Trigonometry
Understand similarity in terms of similarity transformations.
NC.M2.G-SRT.1 Verify experimentally the properties of dilations with given center and scale factor:
NC.M2.G-SRT.1.a When a line segment passes through the center of dilation, the line segment and its image lie on the same line. When a line segment does not pass through the center of dilation, the line segment and its image are parallel.
NC.M2.G-SRT.1.c The distance between the center of a dilation and any point on the image is equal to the scale factor multiplied by the distance between the dilation center and the corresponding point on the pre-image.
NC.M2.G-SRT.1.d Dilations preserve angle measure.
NC.M2.G-SRT.2 Understand similarity in terms of transformations.
NC.M2.G-SRT.2.a Determine whether two figures are similar by specifying a sequence of transformations that will transform one figure into the other.
NC.M2.G-SRT.2.b Use the properties of dilations to show that two triangles are similar when all corresponding pairs of sides are proportional and all corresponding pairs of angles are congruent.
NC.M2.G-SRT.3 Use transformations (rigid motions and dilations) to justify the AA criterion for triangle similarity.
Prove theorems involving similarity.
NC.M2.G-SRT.4 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.
Define trigonometric ratios and solve problems involving right triangles.
NC.M2.G-SRT.6 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.S-CP.3 Develop and understand independence and conditional probability.
NC.M2.S-CP.3.a Use a 2-way table to develop understanding of the conditional probability of A given B (written P(A|B)) as the likelihood that A will occur given that B has occurred. That is, P(A|B) is the fraction of event B's outcomes that also belong to event A.
NC.M2.S-CP.4 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.