NQ: Number and Quantity

NQ.1.PC: Students will use complex numbers and determine how polar and rectangular coordinates are related.

NQ.1.PC.1: Find the conjugate of a complex number. Use conjugates to find quotients of complex numbers. Use conjugates to find moduli.

Points in the Complex Plane
Roots of a Quadratic

NQ.1.PC.2: Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers). Explain why the rectangular and polar forms of a given complex number represent the same number.

Points in the Complex Plane

NQ.1.PC.3: Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; Use properties of geometrical representation for computation.

Points in the Complex Plane

NQ.2.PC: Students will perform operations with vectors and use those skills to solve problems.

NQ.2.PC.1: Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., 𝙫, |𝙫|, ||𝙫||, 𝘷).

Vectors

NQ.2.PC.2: Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

Vectors

NQ.2.PC.3: Solve problems involving velocity and other quantities that can be represented by vectors.

2D Collisions
Golf Range

NQ.2.PC.4: Add and subtract vectors. Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. Understand vector subtraction 𝙫 – 𝙬 as 𝙫 + (–𝙬), where –𝙬 is the additive inverse of 𝙬, with the same magnitude as 𝙬 and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order. Perform vector subtraction component-wise.

Adding Vectors
Vectors

NQ.2.PC.5: Multiply a vector by a scalar. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; Perform scalar multiplication component-wise, e.g., as 𝘤(𝘷ₓ, 𝘷 subscript 𝘺) = (𝘤𝘷ₓ, 𝘤𝘷 subscript 𝘺). Compute the magnitude of a scalar multiple 𝘤𝙫 using ||𝘤𝙫|| = |𝘤|𝙫. Compute the direction of 𝘤𝙫 knowing that when |𝘤|𝙫 ≠ 0, the direction of 𝘤𝙫 is either along 𝙫 (for 𝘤 > 0) or against 𝙫 (for 𝘤 < 0).

Dilations
Vectors

T: Trigonometry

T.3.PC: Students will develop and apply the definitions of the six trigonometric functions and use the definitions to solve problems and verify identities.

T.3.PC.2: Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed around the unit circle.

Cosine Function
Sine Function
Tangent Function

T.3.PC.3: Use special right triangles to determine geometrically the exact values of sine, cosine, tangent for π/3, π/4, π/6, and π/2. Use the unit circle to express the values of sine, cosine, and tangent for π–𝑥, π+𝑥, and 2π–𝑥 in terms of their exact values for 𝑥, where 𝑥 is any real number.

Cosine Function
Sine Function
Sum and Difference Identities for Sine and Cosine
Tangent Function
Translating and Scaling Sine and Cosine Functions

T.3.PC.4: Develop the Pythagorean identity, sin²(θ) + cos²(θ) = 1. Given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle, use the Pythagorean identity to find the remaining trigonometric functions.

Simplifying Trigonometric Expressions
Sine, Cosine, and Tangent Ratios

T.3.PC.5: Develop the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Simplifying Trigonometric Expressions
Sum and Difference Identities for Sine and Cosine

T.4.PC: Students will solve trigonometric equations and sketch the graph of periodic trigonometric functions.

T.4.PC.2: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

Sound Beats and Sine Waves

CS: Conic Sections

CS.5.PC: Students will identify, analyze, and sketch the graphs of the conic sections and relate their equations and graphs.

CS.5.PC.1: Derive the equation of a circle of given center and radius using the Pythagorean Theorem. Complete the square to find the center and radius of a circle given by an equation.

Circles
Distance Formula
Pythagorean Theorem
Pythagorean Theorem with a Geoboard

CS.5.PC.2: Derive the equation of a parabola given a focus and directrix.

Parabolas

CS.5.PC.3: Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

Ellipses
Hyperbolas

CS.5.PC.4: Find the equations for the asymptotes of a hyperbola.

Hyperbolas
Rational Functions

CS.5.PC.5: Complete the square in order to generate an equivalent form of an equation for a conic section; use that equivalent form to identify key characteristics of the conic section.

Circles

CS.5.PC.6: Identify, graph, write, and analyze equations of each type of conic section, using properties such as symmetry, intercepts, foci, asymptotes, and eccentricity, and using technology when appropriate.

Addition and Subtraction of Functions
Circles
Ellipses
Hyperbolas
Parabolas
Rational Functions

CS.5.PC.7: Solve systems of equations and inequalities involving conics and other types of equations, with and without appropriate technology.

Linear Programming
Solving Equations by Graphing Each Side
Solving Linear Systems (Matrices and Special Solutions)
Solving Linear Systems (Standard Form)
Systems of Linear Inequalities (Slope-intercept form)

F: Functions

F.6.PC: Students will be able to find the inverse of functions and use composition of functions to prove that two functions are inverses.

F.6.PC.1: Write a function that describes a relationship between two quantities. From a context, determine an explicit expression, a recursive process, or steps for calculation. Combine standard function types using arithmetic operations. (e.g., given that f(x) and g(x) are functions developed from a context, find (f + g)(x), (f – g)(x), (fg)(x), (f/g)(x), and any combination thereof, given 𝑔(𝑥) ≠ 0.) Compose functions.

Addition and Subtraction of Functions
Arithmetic Sequences
Arithmetic and Geometric Sequences
Function Machines 1 (Functions and Tables)
Geometric Sequences

F.6.PC.3: Understand the inverse relationship between exponents and logarithms. Use the inverse relationship between exponents and logarithms to solve problems.

Logarithmic Functions

F.7.PC: Students will be able to interpret different types of functions and their key characteristics including polynomial, exponential, logarithmic, power, trigonometric, rational, and other types of functions.

F.7.PC.3: Know and apply the Binomial Theorem for the expansion of (𝘹 + 𝘺)ⁿ in powers of 𝘹 and y for a positive integer 𝘯, where 𝘹 and 𝘺 are any numbers with coefficients determined for example by Pascal's Triangle.

Binomial Probabilities

F.7.PC.4: For a function that models a relationship between two quantities: Interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.

Absolute Value with Linear Functions
Exponential Functions
Function Machines 3 (Functions and Problem Solving)
General Form of a Rational Function
Graphs of Polynomial Functions
Logarithmic Functions
Points, Lines, and Equations
Quadratics in Factored Form
Quadratics in Polynomial Form
Quadratics in Vertex Form
Radical Functions

F.7.PC.5: Calculate and interpret the average rate of change of a function (presented algebraically or as a table) over a specified interval. Estimate the rate of change from a graph.

Cat and Mouse (Modeling with Linear Systems)
Slope

F.7.PC.6: Graph functions expressed algebraically and show key features of the graph, with and without technology. Graph linear and quadratic functions and, when applicable, show intercepts, maxima, and minima. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Graph power and polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Graph exponential and logarithmic functions, showing intercepts and end behavior. Graph trigonometric functions, showing period, midline, and amplitude.

Absolute Value with Linear Functions
Cat and Mouse (Modeling with Linear Systems)
Cosine Function
Exponential Functions
Linear Functions
Point-Slope Form of a Line
Points, Lines, and Equations
Quadratics in Factored Form
Quadratics in Polynomial Form
Quadratics in Vertex Form
Radical Functions
Roots of a Quadratic
Sine Function
Slope-Intercept Form of a Line
Standard Form of a Line
Tangent Function
Translating and Scaling Functions
Translating and Scaling Sine and Cosine Functions
Zap It! Game