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Numbers that share specific characteristics can be "put" into unique **sets**. It's not unlike a set of dishes; each dish shares a certain characteristic. It's the same concept with numbers; a set of numbers are a group of numbers that share a common characteristic.

The main Number Sets that we use today.

Number Set Worksheet ANSWER Videos and ANSWERS

Set Notation

There are many ways to represent a set of numbers. Below are the main two types of set notation.

Set Notation Worksheet ANSWERS and ANSWER Videos

**Sets and Subsets**

Any set "contained" inside another set is called a **subset**. Another way to think about a subset is any set which all the elements of a set are contained in another set.

*Example:* Let A = {1, 3, 5, 7} and let B = {x|x ∈ N}, then **A is a subset of B **(all elements of A are contained in B).

We write subsets like this: **A ⊆ B** (The set A is a subset of set B.)

A **proper** **subset** is a subset that does **not** contain every element of the original set. So, we can also say that the set A is a **proper subset** of set B: A** ⊂ B**.

*Example:* The set of Natural numbers is a **proper subset** of the set of Whole numbers because the set of Natural numbers does not contain the number 0.

**N ⊂ W **(The set of Natural numbers is a **proper subset** of the set of Whole numbers.)

All sets are also subsets of themselves, but not proper subsets of themselves.

*Example:* **N ⊆ N **(The set of Natural numbers is a subset of itself.)

**Unions and Intersections worksheet ANSWER Videos and ANSWERS**

**Venn Diagrams**

Venn Diagrams Worksheet ANSWER Videos and ANSWERS

**Section 1: Set Theory Quiz (Answers under Resources)**

Proof that you are what you are not!

The Cartesian Plane

Vertical and Horizontal Lines on a Graph Worksheet ANSWER Videos ANSWERS

Reflections Worksheet ANSWER Videos and ANSWERS

Intercepts Worksheet Review Video ans answers

Midpoint and Distance between coordinates worksheet ANSWER videos and ANSWERS

**Finding points on a graph with a fixed distance**

Find a point D(x; y) such that the points A(-3; 1), B(4; 0), C(0;-3) and D are the corners of a square.

Circle C passes through the three vertices of triangle T. If the vertices of triangle T are (-1, 2), (1, 6), and (5, 4), then what is the area of cirlce C?

The Cartesian Plane and Coordinate Geometry Worksheet

The Cartesian Plane and Coordinate Geometry Worksheet ANSWER videos and ANSWERS

**Section 2: Cartesian Plane and Coordinate Geometry Quiz (Answers under Resources)**

How did the right triangle get its name?

Relation Graphing Worksheet ANSWER Videos and ANSWERS

Domains and Ranges of Functions Worksheet ANSWER Videos and ANSWERS

Linear Functions: SLOPE

You remember the the slope-intercept equation for a straight line?
It's our old friend ...
y = mx + b

How to algebraically write out a linear equation.

How to graph linear inequalities.

Linear Functions Worksheet ANSWERS and ANSWER VIDEOS

A step-wise function can be thought of as a "multiple" function; which function to use depends on the value of the number (x) being plugged into the step-wise function.

Step-wise Functions Worksheet ANSWER Videos and ANSWERS

Absolute value functions are very common in mathematics. They can be tricky!

Absolute Value Functions Worksheet ANSWER Videos and ANSWERS

How to graph an inequality equation.

Absolute Value Inequalities Worksheet ANSWER Videos and ANSWERS

**Section 3: Functions (General) and Graphing Quiz (Answers under Resources)**

Section 3 - Brain Candy

Graphing Quadratic Functions Worksheet ANSWER Videos and ANSWERS

Graphing Quadratic Functions with x intercepts

Graphing Quadratic Functions with X Intercepts Worksheet ANSWER Videos and ANSWERS

Imaginary and Complex Numbers Worksheet ANSWER Videos and ANSWERS

Graphing Quadratic Equations with No X Intercepts Worksheet ANSWER Videos and ANSWERS

Solving Quadratic Equations by Completing the Square

Solving Quadratic Equations using the Quadratic Formula

Solving Quadratic Equations Worksheet ANSWER Videos and ANSWERS

Graphing Quadratic Inequalities

Graphing Quadratic Inequalities Worksheet ANSWER Videos and ANSWERS

Minimums and maximums of quadratic functions.

Farmer Skippy has 32 linear feet of fencing and he wants to fence in his yard so that it will maximize the area. He decides on a rectangular yard but he only needs three sides because the fourth side will be the wall of his house. Find the dimensions of the sides and the maximum area of the yard.

Minimums and Maximums of Quadratic Functions Worksheet ANSWER Videos and ANSWERS

**Section 4: Quadratic Functions - Quiz (Answers under Resources)**

Section 4 - Brain Candy

Polynomial Function Worksheet ANSWER Videos and ANSWERS

Function Arithmetic Worksheet ANSWER Videos and ANSWERS

Finding a “Zero” of a Function Worksheet ANSWER Videos and ANSWERS

Multiplicities of a Zero Worksheet ANSWER Videos and ANSWERS

Polynomial Division Worksheet ANSWER VIDEOS and ANSWERS

Synthetic Polynomial Division Worksheet

Synthetic Polynomial Division Worksheet ANSWER Videos and ANSWERS

Difference Quotient Function Worksheet ANSWER Videos and ANSWERS

Average Rate of Change Function Worksheet ANSWER Videos and ANSWERS

Function Composition Worksheet ANSWER Videos and ANSWERS

One-to-one and Onto Functions Worksheet ANSWER Videos and ANSWERS

Inverse Functions Worksheet ANSWER Videos and ANSWERS

**Section 5: Polynomial Functions Quiz (Answers under Resources)**

Section 5 - Brain Candy

Rational Exponents Worksheet ANSWER Videos and ANSWERS

Powers of Numbers

Exponential Equations Worksheet ANSWER Videos and ANSWERS

**A few exponential equations for your perusal.**

Solving Log Equations

Solving Log Equations Worksheet ANSWER Videos and ANSWERS

Applications of Exponential and Log Equations (INTEREST problems only)

Some harder log problems worked out

Exponents and Logarithm Formulas Worksheet ANSWER Video and ANSWERS

**Section 6: Exponents and Logarithms Quiz (Answers under Resources)**

Section 6 - Brain Candy

Graphing Systems of Linear Equations Worksheet ANSWER Videos ANSWERS

Solving by Substitution Worksheet ANSWER Video and ANSWERS

Gaussian Elimination Worksheet ANSWER Videos and ANSWERS

Systems with 3 Variables Worksheet ANSWER Videos and ANSWERS

Determinants of a Matrix and Scalars Worksheet ANSWERS

Solving Augmented Matrices Worksheet ANSWER Videos and ANSWERS

How to Use a Matrix to Find a Quadratic Equation Worksheet ANSWER Videos and ANSWERS

Matrix Arithmetic Worksheet Worksheet ANSWER Video and ANSWERS

Matrix Inverses Worksheet ANSWER Videos and ANSWERS

**Section 7: Systems of Linear Equations Test (in pdf format) (Answers under Resources)**

Section 7 - Brain Candy

Sequences: Finding the nth Term Worksheet ANSWER Videos and ANSWERS

Summation of Sequences Worksheet ANSWER Videos and ANSWERS

Principal of Mathematical Induction Worksheet ANSWER Videos and ANSWERS

The Binomial Theorem Worksheet ANSWER Videos and ANSWERS

**Section 8: Sequences Test (in pdf format) (Answers under Resources)**

Section 8 - Brain Candy

*COURSE LEARNING OUTCOMES*

Upon successful completion of this course, students will be able to:

- Write out and describe sets using set-theory notation
- Solve intersections and unions of sets; Venn Diagrams
- Understand the Fundamental Graphing Theory
- Determine the x-intercepts and y-intercepts of polynomial functions
- How to reflect a point and a line across an axis and/or the origin
- Find midpoints and distance between any two given points
- Apply Coordinate Geometry to explore geometric shapes on the Cartesian plane
- Determine if a relation is also a function
- How to graph relations
- How to graph using set-building notation
- Identify the domain and range of a function
- Solve and graph linear equations and inequalities
- Apply step-wise functions
- Solve and graph absolute value equations and inequalities
- Factor polynomials, solve and graph quadratic equations and inequalities
- Calculate with complex numbers, conjugates, discriminants, and real solutions
- Solve quadratic equations by Completing the Square, and with the Quadratic Formula
- Solve application problems including quadratic models and optimization
- Understand and apply the Fundamental Theorem of Algebra
- Determine which functions are polynomial functions and which are not
- Find all of the “zeros” (Real and/or Complex) of a polynomial function
- Define multiplicities of zeros for polynomial functions
- Understand and apply the Factor and Remainder Theorem
- Perform operations on functions, including addition, subtraction, multiplication and division
- Perform synthetic division with polynomial functions
- Understand and apply the Difference Quotient Function
- Find the average rate of change of functions
- Successfully compose functions and define their domains and ranges
- Understand what makes a function one-to-one and/or onto
- Calculate the inverse of a polynomial function, including the domain and range
- Understand and apply the Properties of Exponential Functions
- How to graph exponential functions using MS Excel
- Reduce and calculate with radicals and fractional exponents
- Solve quadratic, radical, polynomial, and rational equations
- Understand and apply the Properties of Logarithmic Functions
- Apply the rules of logarithms to expand logarithms, or to write logarithmic expressions as a single log
- Solve equations and applications involving exponential and logarithmic functions
- How to graph logarithmic functions using MS Excel
- Calculate with Euler’s number (e)
- To solve logarithmic equations with the Change of Base rule
- Apply exponential and logarithmic properties to real-world problems
- Solve systems of linear equations using the Substitution and/or the Gaussian Elimination method
- Translate systems of linear equations into matrices
- Find determinants of 2 x 2 and 3 x 3 matrices
- To convert any augmented matrix into reduced-row echelon form
- Identify if a matrix is consistent and/or dependent
- Understand what the Identity Matrix is and how it is used
- Perform arithmetic with matrices
- Find inverses of matrices
- Find Eigenvalues and Eigenvectors of 3 x 3 matrices
- To calculate information from a sequence (arithmetic and geometric) such as the sum, nth term and other information
- To use the Principal of Mathematical Induction to prove or disprove statements with Natural numbers
- To apply the Binomial Theorem
- Brain Candy

*TARGET AUDIENCE*

This is a robust 11^{th} grade algebra course that was designed for high school students planning on attending, and **graduating** from, college.

*COURSE REQUIREMENTS*

Students taking this course should ALREADY have basic algebra skills, i.e., arithmetic with fractions, decimals, and percents; arithmetic with integers (a negative multiplied by a negative is a positive); how to algebraically move a variable and/or a constant from one side of an equation to the other side; you know - the absolute basics of algebra. A graphing calculator would be nice but not necessary. A calculator that can calculate exponents and logarithms is needed.

*THIS COURSE INCLUDES*

- 140+ video tutorials
- Guided homework worksheets covering every concept (answer sheets with video explanations)
- 8 tests (one for each section)
- 3 Midterms + 3 Practice Midterms (with answers)
- 1 Final Exam + 1 Practice Final Exam (cumulative, with answers)

*COURSE CONCEPTS*

**Basic Algebra Review**

- Basic Algebra Review

**Introductory Set Theory**

- Lesson 1 – Set Theory
- Lesson 2 - Sets of Numbers
- Lesson 3 – Symbols Used in Set Theory
- Lesson 4 - Set Notation
- Lesson 5 - Subsets
- Lesson 6 - Unions and Intersections of Sets
- Lesson 7 - Venn Diagrams

** **

**The Cartesian Plane and Coordinate Geometry**

- Lesson 1 - The Cartesian Plane
- Lesson 2 - The Fundamental Graphing Principle
- Lesson 3 - Vertical and Horizontal Lines on a Graph
- Lesson 4 - Reflections on a Graph
- Lesson 5 - Intercepts
- Lesson 6 - Midpoints
- Lesson 7 - Distance Formula
- Lesson 8 - Finding points on a graph with a fixed distance
- Lesson 9 - Triangle XYZ is a right triangle
- Lesson 10 - Find the third vertex of a 30-60-90-degree triangle
- Lesson 11 - Find the fourth point of a square
- Lesson 12 - Area of circle and a triangle
- Lesson 13 - Right triangle problem without coordinates

**Functions (General) and Graphing**

- Lesson 1 - Relations
- Lesson 2 - Graphing Relations using Set Notation
- Lesson 2b - Set notation in a plane
- Lesson 3 - Function notation
- Lesson 4 - Domains and Ranges of Functions
- Lesson 5 - Linear Functions: Slope
- Lesson 6 - Linear Functions Graphing
- Lesson 7 - Linear Equations: Solving
- Lesson 8 - Linear inequalities: Graphing
- Lesson 9 - Linear Functions Worksheet
- Lesson 10 - Step-wise Functions (Piece-wise)
- Lesson 11 - Absolute Value Functions
- Lesson 12 - Absolute Value Inequalities
- Lesson 13 - Absolute Value Inequalities Graphing

** **

**Quadratic Functions**

- Lesson 1 - Graphing Quadratic Functions
- Lesson 2 - Graphing Quadratic Functions with x intercepts
- Lesson 3 - Imaginary Numbers
- Lesson 4 - Complex Numbers
- Lesson 5 - Graphing Quadratic Equations with no x intercepts
- Lesson 6 - Solving Quadratic Equations by Completing the Square
- Lesson 7 - Solving Quadratic Equations using the Quadratic Formula
- Lesson 8 - Graphing Quadratic Inequalities
- Lesson 9 - Minimums and maximums of quadratic functions
- Lesson 10 - Optimization of quadratic functions word problems

**Polynomial Functions **

- Lesson 1 - Fundamental Theorem of Algebra
- Lesson 2 - Polynomial Function Definition
- Lesson 3 - What is a Polynomial Function?
- Lesson 4 - Function Arithmetic
- Lesson 5 - Finding a “Zero” of a Function
- Lesson 6 - Multiplicities of a Zero
- Lesson 7 - The Factor and The Remainder Theorems
- Lesson 8 - Polynomial Division
- Lesson 9 - Synthetic Polynomial Division
- Lesson 9b - Synthetic Polynomial Division another example
- Lesson 10 - Difference Quotient Function
- Lesson 11 - Average Rate of Change Function
- Lesson 12 - Function Composition
- Lesson 14 - One-to-one and Onto Functions
- Lesson 15 - Inverse Functions

**Exponents and Logarithms**

- Lesson 1 - Rules of Exponents
- Lesson 2 - Rational Exponents
- Lesson 3 - Exponential Equations
- Lesson 4 - Properties of Exponential Functions
- Lesson 5 - Graphing an Exponential Function in Excel
- Lesson 6 - Exponential Formulas (pdf)
- Lesson 7 - Logarithms: General
- Lesson 8 - Properties of Logarithmic Functions
- Lesson 9 - Algebraic Properties of Logarithms
- Lesson 10 - Solving Logarithmic Equations
- Lesson 11 - Euler's Number
- Lesson 12 - Logarithms - Solving Natural Logarithmic Equations
- Lesson 13 - Graphing Logarithmic Functions in Excel
- Lesson 14 - Applications (INTEREST problems only)
- Lesson 15 - Logarithms: Change of Base Rule
- Lesson 16 - Some Exponential and Logarithmic Problems Worked Out

** Systems of Equations**

- Lesson 1 - Systems of Linear Equations
- Lesson 2 - Graphing Systems of Linear Equations
- Lesson 3 - Solving by Substitution
- Lesson 4 - Gaussian Elimination
- Lesson 5 - Consistent and Independent Systems (3 variables)
- Lesson 6 - Inconsistent Systems (3 variables)
- Lesson 7 - Dependent and Consistent Systems (3 variables)
- Lesson 8 - Coffee mixture problem
- Lesson 9 - Changing a System into a Matrix
- Lesson 10 - The Matrix - Aij Identifying Rows and Column
- Lesson 11 - Determinants of a Matrix
- Lesson 12 - The Augmented Matrix
- Lesson 13 - Row Echelon Form
- Lesson 14 - Reduced Row Echelon Form
- Lesson 15 - Solving a 3 x 4 matrix
- Lesson 16 - Matrix - Inconsistent and/or Dependent
- Lesson 17 - How to Use a Matrix to Find a Quadratic Equation
- Lesson 18 - Scalars
- Lesson 19 - Properties of Matrices
- Lesson 20 - Matrix Addition and Subtraction
- Lesson 21 - Matrix Multiplication (non-square matrices)
- Lesson 22 - Multiplying Matrices (square matrices)
- Lesson 23 - The Identity Matrix
- Lesson 24 - Matrix Inverses
- Lesson 25 - Eigenvalues and Eigenvectors

**Sequences**

- Lesson 1 - Sequences
- Lesson 2 - The nth Term
- Lesson 3 - Arithmetic Sequences: Finding the nth Term
- Lesson 4 - Geometric Sequences: Finding the nth Term
- Lesson 5 - Summation Notation
- Lesson 6 - Summation of Arithmetic Sequences
- Lesson 7 - Summation of Geometric Sequences When |r| ≥ 1
- Lesson 8 - Summation of Geometric Sequences When |r| < 1
- Lesson 9 - Principal of Mathematical Induction (PMI)
- Lesson 10 - The Binomial Theorem

- Teacher: Richard
- Areas of expertise: Algebra, Statistics and Probability, Trigonometry, Calculus, Discrete Math, Linear Algebra, How to Teach Math (for future Math instructors), Business Math, and others.
- Education: BA in Mathematics (SDSU), MA in Curriculum and Instruction Emphasis in Math Instruction (SDSU), BCLAD Teaching Credential (SDSU)
- Interests: Watching students succeed in mathematics, Emerging Learning Technologies, Puzzles, Soduko, Space Exploration, Science Fiction, Zombie movies (good ones, not those cheap ones!), Music, Video Production, Website Construction, World Peace, Family, others
- Skills: I have an easy non-threatening approach to the teaching/learning of mathematics. I break down the harder concepts into easier to understand steps. I am one of those weirdos who actually loves math. I see math like art and/or music - it gives me goose bumps!
- Associations: TODOS: Mathematics For All Greater San Diego Math Teachers Association National Council of Math Teachers (NCTM) Hispanic Association of Colleges and Universities (HACU) White House Initiative on Educational Excellence for Hispanic Americans’ Hispanic Family Learning Initiative
- Issues I care about: I am very concerned with how our country has watered down math education in our public schools. There is no doubt that the courses that keeps students from succeeding in high school and/or college are all mathematics courses. I don't blame the students - I blame the educators; not the teachers, but the Educational Leaders who make the big decisions. Public school teachers have zero choice in what and how to teach math.

I have been a Math Learning Specialist and Director of our University's Tutoring Center for the last 15 years. I have personally worked with thousands of struggling math students. I have also taught all of the math courses that our school offers, at both undergraduate and graduate level. A very sad fact is that around 90% of our incoming students are not prepared for college level mathematics. For some strange reason, students who are not successful at math in high school believe that they will understand it when they get to college - just the opposite is true. Students who master the concepts covered in this 11th grade course will succeed in their math courses in college.

What you should already know before beginning this course

**Section 1: Set Theory Quiz ANSWERS**

**Section 2: Cartesian Plane and Coordinate Geometry Quiz ANSWERS**

**Section 3: Functions (General) and Graphing Quiz ANSWERS**

**Section 4: Quadratic Functions Quiz ANSWERS**

**Section 5: Polynomial Functions Quiz ANSWERS**

**Section 6: Exponents and Logarithms Quiz ANSWERS**

**Section 7: Systems of Linear Equations Test ANSWERS**

**Section 8: Sequences Test ANSWERS**

Midterm 1 ANSWERS

Midterm 2 ANSWERS

Midterm 3 ANSWERS

Final Exam ANSWERS

Graphing Linear Equations (pdf)

Graphing blanks for practice

Basic Geometry Formulas used in this course.

List of Powers 2 - 20 - KNOW THEM!

Systems of Linear Equations: Example