Course Title:Advanced Algebra with Trigonometry (Level 2 )

Course Title:Advanced Algebra With Trigonometry (Level 2 )

Grade:12

Length of Course:One Year (5 credits)

Prerequisite:Algebra 2, or special permission of the Math Supervisor

Description:

The Math 12 course, although consisting primarily of Precalculus concepts, integrates geometry, discrete mathematics and statistics together with advanced algebra concepts. Pure and applied mathematics is also integrated throughout the course. These unifying strands are employed for a specific purpose – to motivate, justify, extend, and enhance critical mathematical skills and concepts. A real-world orientation is also emphasized in guiding the approaches that allow students in working out exercises and problems.

In addition to marinating a real-world orientation and integrating up-to-date technology (graphing calculators and computers), the course emphasizes facility with more advanced algebraic expressions and functions – especially quadratic and trigonometric relations – and other functions based on these concepts. Students also will begin to develop an understanding of less traditional topics, such as Sampling and Surveys – necessary in solidifying connections between the abstractions of mathematics and the real world.

This course is consistent with the district K-12 Mathematics Program, as well as with the New Jersey Core Curriculum Content Standards, as students continue to build on the previously studied content standards in addition to the process standards of Problem Solving, Communication, Reasoning, Connections, and Representation.

This Twelfth Grade course aims at preparing students for success in college-level Mathematics – Pre - Calculus and eventually Calculus. To succeed in these areas of mathematics, students must acquire an understanding of functions – particularly the properties, behavior and manipulation of important functions such as polynomial, exponential, logarithmic and trigonometric functions. Beyond functions, students must also then have a firm understanding of analytic trigonometry, of sequences and series and of introductory limits. In addition, units on Probability and Counting, and Elementary Statistics (in lieu of the more rigorous treatment of limits prescribed in the curriculum) are included to prepare students for college-level Statistics.

Please note that the scope and sequence for this twelfth grade course may vary slightly from year to year depending on the needs of the students. In order to succeed in many of the topics above, pupils need an on-going, thorough review of the fundamental foundations of the understanding of this material. Thus, included and built into the curriculum are many essential topics of review.

Evaluation:

Student performance will be measured using a variety of instructor-specific quizzes and chapter tests as well as a common departmental Quarterly, Midterm and Final Exam. Assessments will equally emphasize measurement of the degree to which required skills have been mastered as well as how well key concepts have been understood.

Scope and Sequence:

A pacing guide for Level 2 is attached.

Texts:

Advanced Algebra, Scott, Foresman, and Company (1996)

Functions, Statistics and Trigonometry [FST], Scott, Foresman and Company 1996

Advanced Mathematics, Richard G. Brown, Houghton Mifflin Company 1992

Reference Texts:

Precalculus with Trigonometry, Concepts & Applications, Paul A. Foerster, Key Curriculum Press (2003)

Precalculus with Limits: Houghton Mifflin (2001)

Unit 1: Fundamentals

Learning Objectives
The student will … / Content Outline
Key Definitions, Skills and Concepts / Instructional Materials
1.a Develop, apply, and explain methods for solving problems involving rational and negative exponents;
They will perform basic algebraic operations on exponential expressions, radical expressions, and polynomial expressions.
(4.1-B2, B4,
4.3-A3, D1, D3) / What is an exponent? What is a radical? What is a rational exponent? What is a polynomial?
Skills check, ability to:
Simplify expressions with positive, negative and fractional exponents
Simplify expressions with roots of degree 2 or higher
Switch back and forth from radical notation to fractional exponent notation
Express numbers in scientific notation
Add, subtract and multiply polynomials
Factor polynomials using a variety of techniques, such as factoring formulas, trial and error or factoring a common monomial
Concept check:
How can our understanding of exponent notation be used to prove each of the “exponent rules”? / [FST] – Chapter 2
Advanced Algebra – Chapter 1, 7, 8
Brown – Chapter 1, 2, 5
1.b Recognize and extend previous knowledge of rational numbers to rational expressions. Perform basic algebraic operations on rational expressions.
(4.1-B1
4.3-A3, D1, D2, D3) / What is a rational expression?
Skills check, ability to:
Simplify rational expressions by canceling common factors from both numerator and denominator
Multiply and divide rational expressions
Rationalize numerator or denominator using conjugate radical
Concept check:
How is simple fractional arithmetic similar to manipulating rational expressions?
Are the following statements correct? If not, why not?

/ Advanced Algebra – Chapter 7
1.c Solve linear and quadratic equations; Connect and explain important aspects of these equations and their application to real life situations.
(4.1-A3, B1
4.3-A3, D2, D3) / What is a linear equation? What is a quadratic equation?
Skills check, ability to:
Solve a variety of linear equations
Solve multivariate equations for a given variable
Solve quadratic equations by factoring, and the quadratic formula
Solve equations radicals
Concept check:
How do you know when something is a solution to an equation?
Why is the discriminant important? / Brown – Chapter 1
Learning Objectives
The student will … / Content Outline
Key Definitions, Skills and Concepts / Instructional Materials
1.d Compare and contrast the methodologies used in the solving of linear equations and absolute value equations. Evaluate Algebraic Expressions
(4.1-A3, B1
4.3-A3, D2, D3) / Expressions and Formulas
Expressions vs. Equations: Differences and commonalities
Variables as unknowns, varying quantities, and in formulas
Evaluating Expressions & Using Formulas
Using formulas by substituting for the independent variable and simplifying
Solving Equations
Solving linear equations with one variable
Translating verbal & algebraic expressions
Reverse order of operations to solve equation
Properties of Equality: Reflexive, Symmetric, Transitive, Substitution
Solving for a particular variable in a formula
Solving Absolute Value Equations
Solve for variables inside abs value brackets
Separate into 2 equations, find 2 solutions / Brown : Chapters 1 and 3
1.e Illustrate methods of solving Quadratic Formula and relate the solutions to the graphic representation of quadratics. Interpret the significance of the discriminant and illustrate how it helps in understanding the nature of the roots of the quadratic equation.
(4.1-A3, B1
4.3-A3, D2, D3) / The Quadratic Formula and the Discriminant
The four stages of the Quadratic Formula:
Standardize: Put quadratic equation into standard form equal to zero. Id.values of a, b,c
Substitute. Simplify
Split: Separate the plus/minus expression to create (up to) 2 solutions
The Discriminant
Use discriminant to determine the nature of the solutions to the equation, and number of x- intercepts on the graph / Brown Chapter 1
FST – Chapter 2
Advanced Algebra – Chapter 6

Unit 2: Functions

Learning Objectives
The student will … / Content Outline
Key Definitions, Skills and Concepts / Instructional Materials
2.a Use functions to model real - world phenomena and solve problems that involve varying quantities.
(4.3-A3, B1, B2, D3) / What is a function? What is the domain and range of a function? When is a variable independent? When is it dependent?
Skills check, ability to:
Evaluate functions (including piecewise defined functions)
Concept check:
What does it mean to say that f is a function of x?
True or false: the same as. Explain why.
Give examples of functions in real life. Explain why your examples are functions.
Represent functions using machine diagrams and arrow diagrams.
Represent a given function verbally, algebraically, graphically (visually) and numerically (i.e. using a table of values). / [FST] 2-1 (The Language of Functions)
Advanced Algebra – Chapter 1

Unit 2: Functions – continued

2.b Combine functions to create new functions and identify their resulting domains.
(4.3-B4) / What are the algebraic properties of functions? What is a composite function?
Skills check, ability to:
Perform addition, subtraction, division and multiplication of functions (algebraically)
Find composite functions and their corresponding domains / [FST] 3-7
Advanced Algebra – Chapter 8
Brown – Chapter 4
2.c Identify one-to-one functions, determining and interpreting the meaning and significance of their corresponding inverses.
(4.1-B4) / What is a one-to-one function? What is the definition of an inverse of a function? What is the inverse function property?
Skills check, ability to:
Test for whether a given function is one-to-one and the existence of an inverse function (horizontal line test)
Verify whether two functions are inverses
Concept check:
Why does a function that is not one-to-one not have an inverse?
Explain how one finds an inverse. / Brown – Chapter 4
[FST] 3-8 (Inverse Functions)
Advanced Algebra – Chapter 8

Unit 3: Polynomial and Rational Functions

Learning Objectives
The student will … / Content Outline
Key Definitions, Skills and Concepts / Instructional Materials
3.a Perform basic algebraic operations on imaginary numbers; explain their meaning and significance. / Imaginary numbers
Factor radicals to extract a negative & replace w/ i
Properties of imaginary numbers
Operations on imaginary numbers (+ - x ÷ )
Solve quadratic equations w/imaginary solutions
Complex Numbers
Definition of Complex Number in form.
Arithmetic with complex numbers (add / subtract)
Multiplying complex numbers (using FOIL)
Complex conjugates
Divide complex numbers
Divide a complex number by a constant:
Separate fractions
Divide a complex number by imaginary number
Divide 2 complex numbers -use complex conjugate
Complex Numbers
Perform operations with complex numbers
Find complex roots for quadratic equations
Rationalize complex fractions by using the complex conjugate / [FST] 9.6 (Complex Numbers)
Advanced Algebra: Chapter 6
Brown – Chapter 11
Unit 3: Polynomial and Rational Functions - continued
3.b Connect the procedures in solving basic algebraic operations to those necessary in work with complex numbers; find complex solutions to quadratic equations.
(4.3-A3) / What is a complex number?
Skills check, ability to:
Recognize complex numbers and their parts
Add, subtract, multiply and divide complex numbers
Simplify expressions with square roots of negative numbers
Find complex solutions to quadratic equations
Concept Check:
What is the value of? What is the value of?
How do you determine whether a quadratic equation has complex solutions? / [FST] 9.6 (Complex Numbers)
Advanced Algebra: Chapter 6
Brown – Chapter 11

Unit 4: Exponential and Log Functions

Learning Objectives
The student will … / Content Outline
Key Definitions, Skills and Concepts / Instructional Materials
4.a Recognize and evaluate exponential functions and relate insights to graphical interpretations
(4.3-B5) / What is an exponential function? What is a natural exponential function?
Skills check, ability to:
Express an exponential function in standard form
Evaluate exponential functions (including natural exponential functions)
Graph exponential functions (including natural exponential functions)
Identify and distinguish graphs of exponential functions.
Concept check:
What distinguishes an exponential function from a linear function?
Give a verbal representation of an exponential function.
What is the number e and when is it used? (Or, what is so natural about the number e?) / [FST] 4.3 (Exponential Functions, note: natural exponentials covered in FST 4.6)
Advanced Algebra – Chapter 9
Brown – Chapter 5
4.b Recognize, evaluate, graph and apply transformations to logarithmic functions and convert logarithmic functions to exponential functions (and vice versa).
(4.3-B5) / What is a logarithmic function? What is a common logarithm? What is a natural logarithm?
Skills check, ability to:
Switch back and forth from logarithmic to exponential expressions.
Evaluate logarithms using basic properties of logarithms
Graph logarithmic functions
Evaluate common logarithms
Evaluate natural logarithms
Find the domain of a logarithmic function
Concept check:
How are logarithmic functions related to exponential functions?
Why is the domain of a logarithmic function restricted? / [FST] 4.5 (Logarithmic Functions)
[FST] 4.6 (e and Natural Logarithms)
Advanced Algebra – Chapter 9
Brown – Chapter 5
4.c Manipulate (i.e. expand or combine) and evaluate logarithmic expressions using the laws of logarithms. Recognize, explain and apply insights to real -world models.
(4.3-D1) / What are the laws of logarithms?
Skills check, ability to:
Use the laws of logarithms to evaluate logarithmic expressions
Expand and combine logarithmic expressions
Concept check:
How do the laws of exponents give rise to the laws of logarithms? / [FST] 4.7
Advanced Algebra – Chapter 9
Brown – Chapter 5
Unit 4: Exponential and Log Functions - continued
4.d Solve exponential and logarithmic equations. Compare and contrast the various methodologies.
(4.3-A3, B1, B4, D3) / What is an exponential equation? What is a logarithmic equation?
Skills check, ability to:
Solve equations that involve variables in the exponent (algebraically)
Solve equations that involve logarithms of a variable (algebraically)
Solve more complicated compound interest problems (e.g. finding the term for an investment to double)
Concept check:
Why are logarithms useful in solving exponential equations?
Describe the steps involved in solving a typical logarithmic equation. / [FST] 4.8 (Solving Exponential Equations)
Advanced Algebra – Chapter 9
Brown – Chapter 5
4.e Apply exponential and logarithmic functions to real-life situations.
(4.3-A3, B1, B2, B4, D3) / What is an exponential growth model? What is an exponential decay model? What are logarithmic scales?
Skills check, ability to:
Apply exponential growth models to real life situations: e.g. predicting the future (and past) size of a population growing exponentially
Apply exponential decay models to real life situations: e.g. calculating the amount of mass remaining of a radioactively decaying substance after t units of time.
Convert relative magnitudes measured in logarithmic scales to relative magnitudes measured in linear scales
Concept check:
What does it mean for something to grow or decay exponentially?
How do we know we can use an exponential growth or decay function to model physical phenomena?
Can a half-life decay model be alternatively expressed using a different decay factor? Give an example.
Why are logarithmic scales useful? / [FST] 4.4 (Finding Exponential Models)
[FST] 4.9 (Exponential and Logarithmic Modeling)
Advanced Algebra – Chapter 9
Brown – Chapter 5

Units 5 & 6: Trigonometric Ratios and Functions

Learning Objectives
The student will … / Content Outline
Key Definitions, Skills and Concepts / Instructional Materials
5.a Distinguish between the use of radian measure and that of degrees. They will use radian measure to calculate the size of an angle (or amount of rotation) and convert between degree and radian measure. They will recognize the significance of the differences in use.
(4.3-B4, D1, D2) / What are radian and degree measures for angles? What are coterminal angles? What are arc length and sector areas?
Skills check, ability to:
Convert between degree and radian measure
Find coterminal angles
Find arc length and sector areas
Concept check:
Using a piece of string, demonstrate how to create an angle of measure 1, 2 and 3 radians on a circle. Hint: how is the radius of circle related to its circumference?
In your own words, explain the concept of radian measure. Hint: Think about a circle of radius 1. / [FST] 5.1 (Measures of Angles and Rotations),
[FST] 5.2 (Lengths of Arcs and Areas of Sectors)
5.b Investigate the origins and procedures used with the trigonometric ratio of an acute angle inside a right triangle.
(4.3-B4, D1, D2) / What are the six right triangle trigonometric ratios? What are special triangles?
Skills check, ability to:
Find exact values of the trigonometric ratios when given two lengths of a right triangle
Use the trigonometric ratios to solve right triangles
Find the trigonometric values of special right triangles (45-45-90 and 30-60-90) without the use of a calculator
Use the inverse function on the calculator to solve for angles in applications problems
Concept check:
Justify why trigonometric ratios within a right triangle makes sense using the geometric theorem of similarity. / [FST] 5.3 (Trigonometric Ratios of Acute Angles)
5.c Illustrate techniques in finding the value of the trigonometric function of an angle (of any size).
(4.3-B4, D1, D2) / What are the six trigonometric ratios as defined when the angle is placed in standard position? What is a reference angle?
Skills check, ability to:
Find reference angle for any angle in standard position
Find the exact value of any special angle, including nonacute special angles
Determine in what quadrant an angle must lie given the signs of the trigonometric functions
Find the exact values of the trig functions when given one of the values
Find the area of a triangle using the SAS formula
Concept check:
To determine sin 150, sin 210, sin 330 and sin 570, I only need to know the value of sin 30. Is this true or false and why? / [FST] 5.4 (The Sine, Cosine and Tangent Functions),
[FST] 5.5 (Exact Values of Trigonometric Functions)
Unit 5: Trigonometric Ratios and Functions - continued
5.d Illustrate the use of the unit circle to find the trigonometric ratios of a given angle (of any size) and find the terminal point of a given rotation around the unit circle.
(4.3-B4, D1, D2) / What is the unit circle? What are the even/odd properties?
Skills check, ability to:
Use the unit circle to find the values of the six trig functions for special angles
Find a terminal point on the unit circle when given a variety of information
Concept Check:
How are the trig functions defined for the unit circle and how is this consistent with the definitions we have seen so far?
If you are told to compare cos 77 and cos 82 and say which one is bigger without using a calculator. How would you do it? / [FST] 5.5 (Exact Values of Trigonometric Functions),
[FST] 5.7 (Properties of Sines, Cosines and Tangents)
5.e Recognize and sketch graphs of trigonometric functions (comparing their structures) and identify their key attributes (e.g. amplitude, period).
(4.3-B4, D1, D2) / What is a periodic graph? What is meant by amplitude and period?
Skills check, ability to:
Recognize the graphs of the three major trigonometric functions
State the amplitude and period of a given trigonometric function
Write the trigonometric function for a given graph
Concept Check:
Why is the graph of any trignometric function periodic?
How do the period and amplitude relate to our earlier studies of transforming functions? / [FST] 5.6 (Graphs of the Sine, Cosine and Tangent Functions),
5.f Explain the use of the Law of Sines and the Law of Cosines to solve real - world problems involving triangles.
(4.3-B4, D1, D2) / What is the Law of Sines? What is the Law of Cosines?
Skills check, ability to:
Use the Law of Sines and Law of Cosines to solve for all possible triangles when given a set of conditions
Use Law of Sines and Cosines to solve problems involving bearing and direction
Concept Check:
How is the Law of Cosines related to the Pythagorean theorem? / [FST] 5.8 (The Law of Cosines),
[FST] 5.9 (The Law of Sines)

Unit 7: Sequences and Series