Academic standards list

Precalculus — Mathematics

Introduction

Academic standards define the expectations for knowledge and skills that students are to learn in a subject by a certain age or at the end of a school grade level. This page contains a list of standards for a specific content area, grade level, and/or course. The list of standards may be structured using categories and sub-categories.

Number Expressions

TSS.Math.P.N.NE  —  Conceptual Category: Number and Quantity
 
Represent, interpret, compare, and simplify number expressions.
Use the laws of exponents and logarithms to expand or collect terms in expressions; simplify expressions or modify them in order to analyze them or
 
Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
 
Classify real numbers and order real numbers that include transcendental expressions, including roots and fractions of Pi and e.
 
Simplify complex radical and rational expressions; discuss and display understanding that rational numbers are dense in the real numbers and the integers
 
Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by
 

The Complex Number System

TSS.Math.P.N.CN  —  Conceptual Category: Number and Quantity
 
Perform complex number arithmetic and understand the representation on the complex plane.
Perform arithmetic operations with complex numbers expressing answers in the form a + bi.
 
Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
 
Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and
 
Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this
 
Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers
 
Use complex numbers in polynomial identities and equations.
Extend polynomial identities to the complex numbers.
 
Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
 

Vectors and Matrix Quantities

TSS.Math.P.N.VM  —  Conceptual Category: Number and Quantity
 
Represent and model with vector quantities.
Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for
 
Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
 
Solve problems involving velocity and other quantities that can be represented by vectors.
 
Understand the graphic representation of vectors and vector arithmetic.
Add and subtract vectors.
     There are 3 components within this standard.
 
Multiply a vector by a scalar.
     There are 2 components within this standard.
 
Calculate and interpret the dot product of two vectors.
 
Perform operations on matrices and use matrices in applications.
Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
 
Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
 
Add, subtract, and multiply matrices of appropriate dimensions.
 
Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the
 
Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The
 
Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as
 
Work with 2 _ 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.
 

Sequences and Series

TSS.Math.P.A.S  —  Conceptual Category: Algebra
 
Understand and use sequences and series.
Demonstrate an understanding of sequences by representing them recursively and explicitly.
 
Use sigma notation to represent a series; expand and collect expressions in both finite and infinite settings.
 
Derive and use the formulas for the general term and summation of finite or infinite arithmetic and geometric series, if they exist.
     There are 3 components within this standard.
 
Understand that series represent the approximation of a number when truncated; estimate truncation error in specific examples.
 
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
 

Reasoning with Equations and Inequalities

TSS.Math.P.A.REI  —  Conceptual Category: Algebra
 
Solve systems of equations and nonlinear inequalities.
Represent a system of linear equations as a single matrix equation in a vector variable.
 
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).
 
Solve nonlinear inequalities (quadratic, trigonometric, conic, exponential, logarithmic, and rational) by graphing (solutions in interval notation if
 
Solve systems of nonlinear inequalities by graphing.
 

Parametric Equations

TSS.Math.P.A.PE  —  Conceptual Category: Algebra
 
Describe and use parametric equations.
Graph curves parametrically (by hand and with appropriate technology).
 
Eliminate parameters by rewriting parametric equations as a single equation.
 

Conic Sections

TSS.Math.P.A.C  —  Conceptual Category: Algebra
 
Understand the properties of conic sections and model real-world phenomena.
Display all of the conic sections as portions of a cone.
 
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.
 
From an equation in standard form, graph the appropriate conic section: ellipses, hyperbolas, circles, and parabolas. Demonstrate an understanding of the
 
Transform equations of conic sections to convert between general and standard form.
 

Building Functions

TSS.Math.P.F.BF  —  Conceptual Category: Functions
 
Build new functions from existing functions.
Understand how the algebraic properties of an equation transform the geometric properties of its graph. For example, given a function, describe the
 
Develop an understanding of functions as elements that can be operated upon to get new functions: addition, subtraction, multiplication, division, and
 
Compose functions.
 
Construct the difference quotient for a given function and simplify the resulting expressions.
 
Find inverse functions (including exponential, logarithmic, and trigonometric).
     There are 4 components within this standard.
 
Explain why the graph of a function and its inverse are reflections of one another over the line y = x.
 

Interpreting Functions

TSS.Math.P.F.IF  —  Conceptual Category: Functions
 
Analyze functions using different representations.
Determine whether a function is even, odd, or neither.
 
Analyze qualities of exponential, polynomial, logarithmic, trigonometric, and rational functions and solve real-world problems that can be modeled with
 
Identify the real zeros of a function and explain the relationship between the real zeros and the x-intercepts of the graph of a function (exponential,
 
Identify characteristics of graphs based on a set of conditions or on a general equation such as y = ax^2 + c.
 
Visually locate critical points on the graphs of functions and determine if each critical point is a minimum, a maximum, or point of inflection. Describe
 
Graph rational functions, identifying zeros, asymptotes (including slant), and holes (when suitable factorizations are available) and showing end-behavior.
 
Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is
 

Trigonometric Functions

TSS.Math.P.F.TF  —  Conceptual Category: Functions
 
Extend the domain of trigonometric functions using the unit circle.
Convert from radians to degrees and from degrees to radians.
 
Use special triangles to determine geometrically the values of sine, cosine, tangent for Pi/3, Pi/4 and Pi/6, and use the unit circle to express the
 
Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
 
Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
 

Graphing Trigonometric Functions

TSS.Math.P.F.GT  —  Conceptual Category: Functions
 
Model periodic phenomena with trigonometric functions.
Interpret transformations of trigonometric functions.
 
Determine the difference made by choice of units for angle measurement when graphing a trigonometric function.
 
Graph the six trigonometric functions and identify characteristics such as period, amplitude, phase shift, and asymptotes.
 
Find values of inverse trigonometric expressions (including compositions), applying appropriate domain and range restrictions.
 
Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
 
Determine the appropriate domain and corresponding range for each of the inverse trigonometric functions.
 
Graph the inverse trigonometric functions and identify their key characteristics.
 
Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in
 

Applied Trigonometry

TSS.Math.P.G.AT  —  Conceptual Category: Geometry
 
Use trigonometry to solve problems.
Use the definitions of the six trigonometric ratios as ratios of sides in a right triangle to solve problems about lengths of sides and measures of angles.
 
Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
 
Derive and apply the formulas for the area of sector of a circle.
 
Calculate the arc length of a circle subtended by a central angle.
 
Prove the Laws of Sines and Cosines and use them to solve problems.
 
Understand and apply the Law of Sines (including the ambiguous case) and the Law of Cosines to find unknown measurements in right and non-right triangles
 

Trigonometric Identities

TSS.Math.P.G.TI  —  Conceptual Category: Geometry
 
Apply trigonometric identities to rewrite expressions and solve equations.
Apply trigonometric identities to verify identities and solve equations. Identities include: Pythagorean, reciprocal, quotient, sum/difference,
 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
 

Polar Coordinates

TSS.Math.P.G.PC  —  Conceptual Category: Geometry
 
Use polar coordinates.
Graph functions in polar coordinates.
 
Convert between rectangular and polar coordinates.
 
Represent situations and solve problems involving polar coordinates.
 

Model with Data

TSS.Math.P.S.MD  —  Conceptual Category: Statistics and Probability
 
Model data using regressions equations.
Create scatter plots, analyze patterns, and describe relationships for bivariate data (linear, polynomial, trigonometric, or exponential) to model
 
Determine a regression equation to model a set of bivariate data. Justify why this equation best fits the data.
 
Use a regression equation, modeling bivariate data, to make predictions. Identify possible considerations regarding the accuracy of predictions when
 
 
♦  Academic Standards marked with a diamond are "Supporting Content" within the grade level or course.
 
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