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PRECALCULUS OVERVIEW
Number and Quantity
The Complex Number System Mathematical Practices
● Perform arithmetic operations with complex numbers.
● Represent complex numbers and operations on the complex 1. Make sense of problems
plane. and persevere in solving
Vector and Matrix Quantities them.
● Represent and model with vector quantities. 2. Reason abstractly and
● Perform operations on vectors. quantitatively.
● Perform operations on matrices and use matrices in 3. Construct viable
applications. arguments and critique
Algebra the reasoning of others.
Seeing Structure in Expressions 4. Model with mathematics.
● Interpret the structure of expressions. 5. Use appropriate tools
Arithmetic with Polynomials and Rational Expressions strategically.
● Rewrite rational expressions. 6. Attend to precision.
Creating Equations 7. Look for and make use of
● Create equations that describe numbers or relationships. structure.
Reasoning with Equations and Inequalities 8. Look for and express
● Solve systems of equations regularity in repeated
reasoning.
Functions
Interpreting Functions
● Build new functions from existing functions.
Trigonometric Functions
● Expand the domain of trigonometric functions using a unit circle.
● Model periodic phenomena with trigonometric functions.
● Prove and apply trigonometric identities.
Geometry
Similarity, Right Triangles, and Trigonometry
● Apply trigonometry to general triangles.
Expressing Geometric Properties with Equations
● Translate between the geometric description and the equation for a conic section.
Semesters at a Glance
Semester 1 Semester 2
● Functions & Their Graphs/ linear systems ● Trigonometric Functions (4 wks)
(3 wks) ● Analytic Trigonometry (4 wks)
● Polynomial & Rational Functions (4 wks) ● Vectors & Trigonometry (4 wks)
● Conics (3 wks) ● Polar Functions (2 wks)
● Exponential/Logarithmic Functions (3 wks) ● Parametric (2 wks)
● Matrices (2 wks) ● Complex Functions ( 1 wks)
CRITICAL AREAS
For the Pre-Calculus course, instructional time should focus on four critical areas:
(1) Functions (2) Trigonometry (3) Analytic Geometry (4) Linear Systems using Matrices
(1) While many of the standards for functions appeared in previous courses, students now apply them in
cases of polynomials of degree greater than two, more complicated rational functions, and exponential or
logarithmic functions. Students examine end behavior of these functions and learn to find asymptotes.
In addition, students will analyze functions using different representations.
(2) Students will expand their understanding of trigonometric functions using the unit circle. They will
model periodic phenomena with trigonometric functions, prove and apply trigonometric idenitties, and
apply trigonometry to triangles (law of sines/cosines, vectors, trigonometric form of complex numbers).
(3) Students derive the equations of conics (circles, parabolas, ellipses, and hyperbolas) and translate
between their graphs and equations. Students work with parametrics, converting to Cartesian form.
They understand polar coordinates and the graphs of polar functions (circles, cardioids, limacons, roses).
(4) Students expand their knowledge of linear systems by solving application problems using matricies.
Mathematical Explanation and Examples
Practice
MP.1 Students expand their repertoire of expressions and functions that can used
Make sense of problems and to solve problems. They grapple with understanding the connection between
persevere in solving them. complex numbers, polar coordinates, and vectors, and reason about them.
MP.2 Reason Abstractly and Students understand the connection between transformations and matrices, seeing a
quantitatively matrix as an algebraic representation of a transformation of the plane
MP.3 Construct viable Students continue to reason through the solution of an equation and justify their
arguments and critique the reasoning to their peers. Students defend their choice of a function to model a real-
reasoning of others world situation.
MP.4 Model with mathematics Students apply their new mathematical understanding to real-world problems.
Students also discover mathematics through experimentation and examining
patterns in data from real-world contexts.
MP.5 Use appropriate tools Students continue to use graphing technology to deepen their understanding of the
strategically behavior of polynomial, rational, square root, and trigonometric functions.
MP.6 Attend to precision Students make note of the precise definition of complex number, understanding that
real numbers are a subset of the complex numbers. They pay attention to units in
real-world problems and use unit analysis as a method for verifying their answers.
MP.7 Look for and make Students understand that matrices form an algebraic system in which the order of
use of structure multiplication matters, especially when solving linear systems using them. They see
that complex numbers can be represented by polar coordinates, and that the structure
of the plane yields a geometric interpretation of complex multiplication
MP.8 Look for and express Students multiply several vectors by matrices and observe that some matrices give
regularity in repeated rotations or reflections. They compute with complex numbers and generalize the
reasoning results to understand the geometric nature of their operations.
Unit #1: Functions & Their Graphs (3 Weeks)
Goal: Describe, analyze, and interpret graphs of functions.
- Analyze graphs to determine domain and range, zeros, local maxima and minima.
- Recognize graphs and transformations of common functions.
- Sketch the graph of a transformation.
- Use knowledge of graphical symmetry to determine if a function is even, odd or neither
- Identify and graph linear, absolute value, square root, quadratic, cubic and piecewise functions.
- Perform combinations and compositions of multiple functions.
- Find the inverse of a function algebraically and graphically.
- Solving multivariable linear systems analytically and/or graphically.
I) General Forms of Linear Equations Section 1.1
II) Function notation and Domain/Range Section 1.2
III) Analyzing Graphs of Functions Section 1.3
IV) Piecewise Functions Section 1.3
V) Transformations Section 1.4
VI) Combinations & Compositions Section 1.5
VII) Inverse Functions Section 1-6
Content Standards: F-IF 4 F-IF 5 F-BF 3 F-BF 4
**Common Task is Illustrative Math Medieval Archer
Unit #2: Polynomial & Rational Functions (4 Weeks)
Goal: Investigate polynomial functions and equations (with or without technology).
- Determine domain/range, zeros, local max/min, intervals of increasing/decreasing, and end behavior.
- Use common characteristics of a polynomial function to sketch the graph.
- Analyze a function numerically and graphically to determine if the function is odd, even, or neither.
- Use the Fundamental Theorem of Algebra to determine the number of zeros of a polynomial.
- Find all rational, irrational, and complex zeros of a polynomial using algebraic methods.
- Use polynomial and rational functions to model and to solve real-world problems.
- Algebraically identify intercepts, holes, and asymptotes in order to sketch graphs of rational functions.
- Use graphical and algebraic methods to solve rational equations.
I) Quadratic Functions Section 2.1
II) Polynomial Functions of Higher Degree Section 2.2
III) Real Zeros of Polynomial Functions Section 2.3
IV) Complex Numbers Section 2.4
V) The Fundamental Theorem of Algebra Section 2.5
VI) Rational Functions & Asymptotes Section 2.6
Content Standards F-IF 7 F-IF 7d N-CN 3
**common formative assessment/ task on “polynomial box application problem”
Unit #3: Conics (2.5 Weeks)
Goal: Demonstrate the ability to define and analyze conic sections algebraically and graphically.
To use a problem-solving approach to investigate conic sections.
- Define and write the equations of parabolas, circles, ellipses, and hyperbolas in standard form.
- Analyze and sketch parabolas, circles, ellipses, and hyperbolas.
- Given a quadratic equation in general form complete the square to write it in standard form.
- Use conic sections to model and solve real-world problems.
I) Circles & Parabolas Section 9.1
II) Ellipses Section 9.2
III) Hyperbolas Section 9.3
IV) Application problems
Content Standards: G-GPE 3 G-GPE 3.1
** Formative Assessment on Conics designed by Torres (SAMOHI)
Unit #4: Exponential & Logarithmic Functions (3 Weeks)
Goal: Investigate exponential and logarithmic functions and solve real-world problems.
- Sketch and analyze exponential and logarithmic functions and their transformations.
- Understand the inverse relationship between exponents and logarithms.
- Define the natural base.
- Evaluate logarithms to any base with and without a calculator.
- Use and apply the laws of logarithms and the change of base formula.
- Solve exponential and logarithmic equations.
I) Exponential Functions & Their Graphs Section 3.1
II) Logarithmic Functions & Their Graphs Section 3.2
III) Properties of Logarithms Section 3.3
IV) Solving Exponential / Logarithmic Equations Section 3.4
V) Exponential & Logarithmic Models Section 3.5
Content Standards: F-IF 7e
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