PRECALCULUS CURRICULUM UNITS
Unit 1: Trigonometry and Trigonometric Graphs
Goal: To acquire the knowledge to demonstrate the ability to define trigonometric ratios and apply trigonometry to solve real-world problems. Students will also demonstrate the ability to sketch and analyze trigonometric graphs and apply trigonometry to solve real-world problems.
Objectives
- To define and evaluate the Six Trigonometric Ratios
- To solve triangles using Trigonometric Ratios
- To define radian measures and convert angle measures between degrees and radians
- To define the Six Trigonometric Functions in terms of the unit circle
- To develop basic Trigonometric Identities
- To use Trigonometric Functions to model and solve real-world problems, including angle-triangle relations, arc length, and speed.
- To graph Sine, Cosine and Tangent functions
- To identify the domain and range of a basic Trigonometric Function
- To graph transformations of the Sine, Cosine, and Tangent graphs
- To identify and sketch the Period, Amplitude, and Phase Shift of the Sine, Cosine, and Tangent Functions
- To use Trigonometric Graphs to model and solve real-world problems
Common Core Standards Clusters
Students will:
- Apply Trigonometry to general triangles
- Extend the domain of trigonometric functions using the unit circle
- Model periodic phenomena and trigonometric functions
Common Core State Standards
Students will be able to:
- Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle
- 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 transversed counterclockwise around the unit circle
- Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency and midline
Unit 2: Trigonometric Equations and Identities
Goal:To acquire the knowledge and skills to demonstrate the ability to solve trigonometric equations, investigate inverse trigonometric functions, and use trigonometric identities.
Objectives
- To solve Trigonometric Equations graphically and algebraically
- To define the domain and range of the Inverse Trigonometric Functions
- To write a Trigonometric Function to model and solve real-world problems
- To apply strategies to prove identities
- To use the addition and subtraction identities for Sine, Cosine, and Tangent functions
- To use the Double-Angle Formula and Half-Angle Identities
- To use identities to solve Trigonometric quations
- To solve triangles using Law of Sines and Law of Cosines
Common Core Standards Clusters
Students will
- Apply Trigonometry to general triangles
- Extend the domain of Trigonometric Functions using the Unit Circle
- Model periodic phenomena and trigonometric functions
- Define Trigonometric Ratios and solve problems involving Right Triangles
Common Core State Standards
Students will be able to
- Prove the Law of Sines and Cosines and use them to solve problems
- Understand and apply the Law of Sines and Cosines to find unknown measurements in right and non-right triangles (e.g. surveying problems, resultant forces)
- Use Trigonometric Ratios and the Pythagorean Theorem to solve Right Triangles in applied problems
- Explain and use the relationship between the Sine and Cosine of Complementary Angles
- Understand that by similarity, side ratios in Right Triangles are properties of the angles in the triangle leading to definitions of trigonometric ratios for acute angles
- Derive the formula - f(x) = ½ ab sin C from the area of a triangle
Unit3: Functions and Graphs
Goal: To acquire the knowledge and skills to demonstrate the ability to solve equations and use function notation. Students will develop skills in constructing and interpreting graphs of functions.
Objectives
- To solve absolute-value, radical, and fractional equations
- To determine whether a relation is a function
- To determine the domain of a function
- To evaluate piecewise-defined and greatest integer functions
- To analyze graphs to determine domain and range, local maxima and minima, and intervals where they are increasing and decreasing
- To find the vertex and intercepts of a quadratic function and sketch its graph
- To transform graphs of parent functions
- To determine whether a graph is symmetric with respect to the x-axis, y-axis, and/or origin
- To perform addition, subtraction, multiplication, division, and composition of functions
- To define inverse relations and functions and determine whether an inverse relation is a function
- To verify inverses using composition
Common Core Standards Clusters
Students will
- Interpret the graphs of equations and analyze functions of graphs
- Interpret functions that arise in applications in terms of a context
- Analyze functions using different representations- algebraically and graphically
- Build a function that models a relationship between two quantities
- Write expressions in equivalent forms to solve problems
- Create equations that describe numbers or relationships
- Solve equations and inequalities in one variable
- Use complex numbers in polynomial identities and equations
- Solve a system of equations algebraically, graphically and using matrices
Common Core State Standards
Students will be able to
- Solve a system of linear equations and a quadratic equation in two variables, algebraically and graphically
- Interpret key features of graphs and tables in terms of quantities, and sketch graphs showing key features given a verbal description of the relationship
- Graph linear and quadratic functions and show intercepts, maxima and minima
- Graph square roots, cube roots and piecewise defined functions, including absolute value functions
- Write a function that describes a relationship between two quantities
- Graph functions expressed symbiotically and show key features of the graph by hand in simple cases and using technology for more complicated cases
Unit 4: Polynomials, Rational Functions, Logarithmic and Exponential Functions
Goal: To acquire the knowledge and skills to demonstrate the ability to solve polynomial equations and sketch and analyze graphs of polynomial and rational functions. And, to acquire the knowledge and skills to use the laws of exponents and logarithms and apply them to real-world situations.
Objectives
- To add, subtract, multiply and divide polynomials and simplify and perform operations on complex number
- To apply the Remainder Theorem and Factor Theorem
- To determine the maximum number of zeros of a polynomials and find all rational zeros of a polynomial
- To solve for the complex zeros of a polynomial
- To analyze and sketch polynomial functions using continuity, end behavior, intercepts, local extrema, and points of inflections, and find the domain of a rational functions
- To use polynomial functions to model and solve real-world problems
- To evaluate and apply properties of logarithms and analyze exponential and logarithmic graphs
- To solve exponential and logarithmic equations
Common Core Standards Clusters
Students will
- Use polynomial identities to solve problems
- Solve equations/systems of equations as a process of reasoning and explain the reasoning
- Represent and solve equations and inequalities graphically
- Analyze functions using different representations – algebraically and graphically
- Use complex numbers in polynomial identities and equations
- Interpret the structure of expression and write expressions in equivalent forms to solve problems
- Perform arithmetic operations on polynomials
- Understand the relationship between zeros and factors of polynomials – Remainder and Factor Theorems
- Interpret functions and analyze polynomial functions using different representations
Common Core State Standards
Students will be able to
- Extend polynomial identities to the complex numbers and
- Solve polynomial, exponential and logarithmic equations
- Interpret expressions that represent a quantity in terms of its context
- Use structures of an expression to identify ways to rewrite functions
- Understand that polynomials form a system of analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication and understand how to add, subtract, multiply and divide polynomials
- Know and apply the Remainder Theorem for a polynomial f(x) and a number, the remainder on division by x – a is f(a), so f(a) = 0 if and only if x –a is a factor of f(x).
- Identify zeros of a polynomials when suitable factorizations are available and use the zeros to construct a graph of the function defined by the polynomial
Unit5: Statistics and Data
Goal: To acquire the knowledge and skills to demonstrate the ability to summarize statistics and relate to different types of data and probability distributions. They will identify different ways of collecting data, including sample surveys, experiments, and simulations and the role that randomness and careful design play in the conclusions that can be drawn.
Objectives
- To be able to collect, create, analyze and summarize data
- To determine normal distributions and make specific estimates
- To build on prior knowledge and understanding of data distributions
- To determine standard deviationsand discover the role of randomness of sample data
- To make conclusions that can be drawn from data
- To understand the concept of statistical significance being developed informally through simulation as meaning a result that is unlikely to have occurred solely as a result of random selection in sampling or random assignment in an experiment
Common Core Standards Clusters
Students will
- Summarize, represent, and interpret data on a single count or measurement variable
- Understand and evaluate random processes underlying statistical experiments
- Make inferences and justify conclusions from sample surveys, experiments, and observational studies.
- Use probability to evaluate outcomes of decisions
Common Core State Standards
Students will be able to
- Use the mean and standard deviation of a data set t fit to a normal distribution and to estimate population percentages
- Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve
- Understand that statistics allows inferences to be made about population parameters based on a random sample from that population
- Decide if a specified model is constraint with results from a given data-generating process using simulation
- Recognize the purposes of and differences among sample surveys, experiments, and observational studies, explain how randomization relates to each
- Use data from a sample survey to estimate a population mean or proportion, develop a margin of error through the use of simulation models for random sampling
- Use data from a randomized experiment to compare two treatments use simulations to decide if differences between parameters are significant
- Use probabilities to make fair decisions and evaluate reports based on data
- Analyze decisions and strategies using probability concepts
Unit 6: Analytic Geometry
Goal: To acquire the knowledge and skills to explore conic sections algebraically and graphically
Objectives
- To define a circle and write the equation
- To analyze and sketch the graph of a circle
- To define an ellipse and write its equation
- To analyze and sketch the graph of an ellipse
- To define a hyperbola and write the equation
- To analyze and sketch the graph of a hyperbola
- To define a parabola and write its equation
- To analyze and sketch the graph of a parabola
- To write the equation of and graph a translated conic section
- To use conic sections to model and solve real-world problems
Common Core Standards Clusters
Students will
- Understand and apply theorems about circles
- Translate between the geometric description and the equation for a conic section
- Connect the equations of circles and parabolas to prior work with quadratic equations. The directrix should be parallel to a coordinate axis.
Common Core State Standards
Students will be able to
- 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
- Derive the equation of a parabola given a focus and directrix
Unit7: Limits and Derivatives
Goal: To acquire the knowledge and skills to demonstrate the ability to calculate limits algebraically and estimate limits from graphs and tables of values.
Objectives - Students will be able to:
- Use the informal definition of limit
- Use and apply the properties of limits to find the limit of various functions
- Find one-sided limits
- Determine if a function is continuous at a point or an interval
- Find the limit as x approaches infinity
Calculus Standards
Standard 1.0
Students demonstrate knowledge of both the formal definition and the graphical interpretation of limit values of functions. This knowledge includes one-sided limits, infinite limits, and limits of infinity. Students know the definition of convergence and divergence of a function as the domain variable approaches either a number or infinity.
1.1Students prove and use theorems evaluating the limits of sums, products, quotients, and composition of functions
1.2Students use graphical calculators to verify and estimate limits
1.3Students prove and use special limits, such as the limits of (sin x) /x and (1 –cos x) /x as x approaches 0
Standard 2.0
Students demonstrate knowledge of both the formal definition and the graphical interpretation of continuity of a function
Standard 4.0
Students demonstrate an understanding of the formal definition of the derivative of a function at a point and the notions of differentiability
4.0Students demonstrate an understanding of the derivative of a function as the slope of the tangent line to the graph of a function
4.1Students demonstrate and understanding of the interpretation of the derivative as an instantaneous rate of change. Students can use derivatives to solve a variety of problems from physics, chemistry, economics, and so forth, that involves the rate of change of a function
4.2Students understand the relation between differentiability and continuity
4.3Students derive derivative formulas and use them to find the derivatives of algebraic, trigonometric, inverse trigonometric, exponential, and logarithmic functions