The Pre-Calculus standards present in various courses at the High School are intended to enrich students’ understanding of advanced algebraic and trigonometric themes, use mathematical systems in application, and prepare students for a college-level calculus course. Highlighted topics include: polynomial, rational, advanced trigonometric, logarithmic, exponential, and inverse functions, and advanced work with conic sections. Through investigation students will discover the fundamental properties of conic sections that allow for their construction. Students are introduced to and develop opportunities to integrate matrices in various forms to solve complex algebraic problems. Students investigate and apply vectors, probability models, and random variables and use permutations and combinations to explore complex problems relying on probability. The sequencing of the the High School Pre-Calculus standards are presented within different courses/grades amongst the Regular and Honors levels, allowing students to take a developmentally appropriate, differentiated approach to the discovery of the topics outlined below, but at their core all approaches will be founded on the following essential standards:
- N-Q.1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
- N-CN.3. (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
- N-CN.9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
- N-VM.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).
- N-VM.2. (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
- N-VM.4.a. (+) 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.
- N-VM.6. (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
- N-VM.7. (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
- N-VM.8. (+) Add, subtract, and multiply matrices of appropriate dimensions.
- A-REI.8. (+) Represent a system of linear equations as a single matrix equation in a vector variable.
- A-REI.9. (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).
- F-BF.4.b. Find inverse functions: (+) Verify by composition that one function is the inverse of another.
- F-BF.4.c. Find inverse functions: (+) Read values of an inverse function from a graph or a table, given that the function has an inverse.
- F-BF.4.d. Find inverse functions: (+) Produce an invertible function from a non-invertible function by restricting the domain.
- F-BF.5. (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
- F-TF.6. (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
- F-TF.7. (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.
- G-SRT.11. (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
- G-GPE.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.
- S-CP.9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems.
- S-MD.1. (+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.
- S-MD.2. (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
- S-MD.3. (+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.