NC.M3.A-APR.6 Rewrite simple rational expressions in different forms; write 𝑎(𝑥)/𝑏(𝑥) in the form 𝑞(𝑥) + 𝑟(𝑥)/𝑏(𝑥), where 𝑎(𝑥),𝑏(𝑥),𝑞(𝑥), and 𝑟(𝑥) are polynomials with the degree of 𝑟(𝑥) less than the degree of 𝑏(𝑥).
Write expressions in equivalent forms to solve problems.
NC.M3.A-SSE.3 Write an equivalent form of an exponential expression by using the properties of exponents to transform expressions to reveal rates based on different intervals of the domain.
NC.M3.A-CED Creating Equations
Create equations that describe numbers or relationships.
NC.M3.A-CED.1 Create equations and inequalities in one variable that represent absolute value, polynomial, exponential, and rational relationships and use them to solve problems algebraically and graphically.
NC.M3.A-CED.2 Create and graph equations in two variables to represent absolute value, polynomial, exponential and rational relationships between quantities.
Represent and solve equations and inequalities graphically.
NC.M3.A-REI.11 Extend an understanding that the 𝑥-coordinates of the points where the graphs of two equations 𝑦=𝑓(𝑥) and 𝑦=𝑔(𝑥) intersect are the solutions of the equation 𝑓(𝑥)=𝑔(𝑥) and approximate solutions using a graphing technology or successive approximations with a table of values.
NC.M3.F-IF.2 Use function notation to evaluate piecewise defined functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
Interpret functions that arise in applications in terms of the context.
NC.M3.F-IF.4 Interpret key features of graphs, tables, and verbal descriptions in context to describe functions that arise in applications relating two quantities to include periodicity and discontinuities.
Analyze functions using different representations.
NC.M3.F-IF.7 Analyze piecewise, absolute value, polynomials, exponential, rational, and trigonometric functions (sine and cosine) using different representations to show key features of the graph, by hand in simple cases and using technology for more complicated cases, including: domain and range; intercepts; intervals where the function is increasing, decreasing, positive, or negative; rate of change; relative maximums and minimums; symmetries; end behavior; period; and discontinuities.
NC.M3.F-IF.9 Compare key features of two functions using different representations by comparing properties of two different functions, each with a different representation (symbolically, graphically, numerically in tables, or by verbal descriptions).
NC.M3.F-BF Building Functions
Build a function that models a relationship between two quantities.
NC.M3.F-BF.1 Write a function that describes a relationship between two quantities.
NC.M3.F-BF.1.a Build polynomial and exponential functions with real solution(s) given a graph, a description of a relationship, or ordered pairs (include reading these from a table).
NC.M3.F-BF.3 Extend an understanding of the effects on the graphical and tabular representations of a function when replacing 𝑓(𝑥) with 𝑘 • 𝑓(𝑥), 𝑓(𝑥) + 𝑘, 𝑓(𝑥 + 𝑘) to include 𝑓(𝑘 • 𝑥) for specific values of 𝑘 (both positive and negative).
NC.M3.F-BF.4.a Understand the inverse relationship between exponential and logarithmic, quadratic and square root, and linear to linear functions and use this relationship to solve problems using tables, graphs, and equations.
NC.M3.F-BF.4.c If an inverse function exists for a linear, quadratic and/or exponential function, 𝑓, represent the inverse function, 𝑓⁻¹, with a table, graph, or equation and use it to solve problems in terms of a context.
NC.M3.F-LE Linear, Quadratic, and Exponential Models
Construct and compare linear and exponential models and solve problems.
NC.M3.F-LE.3 Compare the end behavior of functions using their rates of change over intervals of the same length to show that a quantity increasing exponentially eventually exceeds a quantity increasing as a polynomial function.
NC.M3.F-LE.4 Use logarithms to express the solution to 𝘢𝘣 to the 𝘤𝘵 power = 𝑑 where 𝑎, 𝑐, and 𝑑 are numbers and evaluate the logarithm using technology.
Extend the domain of trigonometric functions using the unit circle.
NC.M3.F-TF.1 Understand radian measure of an angle as: The ratio of the length of an arc on a circle subtended by the angle to its radius. A dimensionless measure of length defined by the quotient of arc length and radius that is a real number. The domain for trigonometric functions.
NC.M3.G-CO.11 Prove theorems about parallelograms. Opposite sides of a parallelogram are congruent. Opposite angles of a parallelogram are congruent. Diagonals of a parallelogram bisect each other. If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.
NC.M3.G-C.2 Understand and apply theorems about circles. Understand and apply theorems about relationships with angles and circles, including central, inscribed and circumscribed angles. Understand and apply theorems about relationships with line segments and circles including, radii, diameter, secants, tangents and chords.
NC.M3.G-C.5 Using similarity, demonstrate that the length of an arc, s, for a given central angle is proportional to the radius, r, of the circle. Define radian measure of the central angle as the ratio of the length of the arc to the radius of the circle, s/r. Find arc lengths and areas of sectors of circles.
NC.M3.G-MG.1 Apply geometric concepts in modeling situations: Use geometric and algebraic concepts to solve problems in modeling situations. Use geometric shapes, their measures, and their properties, to model real-life objects. Use geometric formulas and algebraic functions to model relationships. Apply concepts of density based on area and volume. Apply geometric concepts to solve design and optimization problems.
NC.M3.S-IC.5 Use simulation to determine whether observed differences between samples from two distinct populations indicate that the two populations are actually different in terms of a parameter of interest.