Trigonometry 101

This semester I ran a partially flipped open source Trigonometry course using a free textbook and youTube videos. To give back to the OER community, I have recreated my blackboard shell here in a single blog post. I am offering all of the worksheets that I have created below in PDF and source DOCX format under a creative commons license (cc-by-sa).  Please feel free to copy anything here, modify, and/or redistribute.


Open Textbook:  Trigonometry by Michael Corral
Videos: Mathispower4u Trigonometry by James Sousa

End-page Formulas (Trig. cheat sheet): [ Trigonometry Definitions and Formulas | source ]
Free Graphing Utility: Desmos.com
Free CAS: WolframAlfa

Chapter 1: Right Triangle Trigonometry

1.1 Angles: [ Assignment 1a : source ]

   Video notes: Angle Basics
   Video notes: Degrees, Minutes, and Seconds
   Video notes: Angle Relationships and Types of Triangles
   Video notes: The Pythagorean Theorem
   Video notes: Converse of the Pythagorean Theorem

1.2 Trigonometric Functions: [ Assignment 1b : source ]

   Video Notes: Introduction to Trigonometric Functions
   Video Example: Determine Trigonometric Function Values
   Video Example: Trig. Function Values with Hypotenuse Missing

1.3 Applications: [ Assignment 1c : source ] [ Assignment 1d : source ]

   Video Example: Determine the Length of a Side
   Video Example: Perimeter of an Equilateral Triangle

1.4 Trigonometric Functions of any Angle: [ Assignment 1e : source ]

   Video Examples: Determine the Reference Angle
   Video Notes: Introduction to Trig Functions using Angles

1.5 Rotations and Reflections:  [ Assignment 1f : source ]

*** *** *** [ Sample Exam #1 | source ] *** *** ***

 

Chapter 2: General Triangles

2.1 Law of Sines [ Assignment 2a | source ]

   Video: The Law of Sines: The Basics
   Video: The Law of Sines: The Ambiguous Case
   Example: Solve a Triangle Using the Law of Sines

2.2 Law of Cosines [ Assignment 2b | source ]

   Video: The Law of Cosines
   Example: Application of the Law of Cosines
   Example: Determine an Angle given SSS

2.3 Law of Tangents

2.4 Area of a Triangle [ Assignment 2c | source ]

   Example: Determine the Area of a Triangle Using the Sine Function
   Example: Area using Heron's Formula

Chapter 3: Identities

3.1 Basic Identities [ Assignment 3a | source ]
 
   Video: Fundamental Identities
   Video: Verifying Identities (good using a slightly different method.)

3.2 Sum and Difference Formulas [ Assignment 3b | source ]
 
   Video: Sum and Difference Identities for Cosine
   Video: Sum and Difference Identities for Sine
   Video: Sum and Difference Identities for Tangent

3.3 Double and Half-Angle Formulas [ Assignment 3c | source ]
 
   Video: Double Angle Identities
   Example: Double Angle Example Given Information
   Example: Using the Half Angle Formula (use degrees here)
   Example: Using Sine of a Half Angle

*** *** *** [ Sample Exam #2 | source ] *** *** ***

Chapter 4: Radian Measure

4.1 Radian Measure [ Assignment 4a | source ]

   Lecture: Radian Measure (Great video!)

4.2 Arc Length [ Assignment 4b | source ]
 
   Lecture: Arc Length and Area of a Sector

4.3 Area of a Sector [ Assignment 4c | source ]
 
   Lecture: Arc Length and Area of a Sector

4.4 Circular Motion [ Assignment 4d | source ]
 
   Lecture: Linear Velocity and Angular Velocity

Chapter 5: Graphing and Inverse Functions

5.1 Graphing Trigonometric Functions [ Assignment 5a | source ]
 
   Video Lecture: Graphing the Sine and Cosine Function
   Video Lecture: Graphing the Tangent Function
   Video Lecture: Graphing Cosecant and Secant

5.2 Properties of Graphs[ Assignment 5b | source ]

   Video Lecture: Amplitude and period of Sine and Cosine
   Video Lecture: Horizontal and Vertical Translations of Sine and Cosine
   Video Lecture: Graphing Sine and Cosine with Transformations

5.3 Inverse Trigonometric Functions [ Assignment 5c | source ]

   Video Lecture: Inverse Functions
   Video Lecture: Intro. to Inverse Sine, Cosine, and Tangent
   Video Examples: Evaluating Expressions involving the Inverse Trig. Functions

*** *** *** [ Sample Exam #3 | source ] *** *** ***

Chapter 6: Additional Topics

6.1 Solving Trigonometric Equations
[ Assignment 6a | source ] [ Assignment 6b | source ] [ Assignment 6c | source ] 
 
   Video Lecture: Solving Trigonometric Equations I  (Excellent introduction! Linear)
   Video Lecture: Solving Trigonometric Equations II (Quadratic examples)
   Video Lecture: Solving Trigonometric Equations III (Using Identities first, quadratic formula)
   Video Lecture: Solving Trigonometric Equations IV  (Half and multiple angles)

6.2 Numerical Methods in Trigonometry
 
6.3 Complex Numbers [ Assignment 6d | source ]
 
   Video Lecture: Introduction to Complex Numbers
   Video Lecture: Complex Number Operations
   Video Lecture: Trigonometric Form of Complex Numbers
   Video Lecture: De Moivre's Theorem
   Video Lecture: Nth Roots of a Complex Number

6.4 Polar Coordinates [ Assignment 6e | source ]
 
   Video Lecture: Introduction to Polar Coordinates
   Video Lecture: Converting Polar Equations to Rectangular Equations  (sound is bad)
   Video Lecture: Graphing Polar Equations Part I
   Video Lecture: Graphing Polar Equations Part II

*** *** *** [ Sample Exam #4 | source ] *** *** ***

*** Sample Final Exam - coming soon ***

Trigonometry 101 by John Redden is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License. ---

Visually Searchable Algebra Video Compendium

[ OpenAlgebra.com Visually Searchable Video Compendium ]

I have been working with beginning Algebra students for a good many years now. In that time, I have found myself answering many similar questions where students often tell me that when they are alone they get stuck and do not know where to look for help.  This is certainly the case when problems are posed out of sectional context. After some interaction, it becomes clear that folks simply do not have the algebraic vocabulary needed when searching the index or internet for help and quickly become overwhelmed.

"I do not know what to type in when searching for help!"
 

To help, my idea was to create a visually searchable database of commonly asked algebra questions. This compendium of algebra videos is intended to be more of a "how to" type of approach serving students looking to solve a particular problem in a short amount of time.  More instructional or "lecture" type of videos are valuable and quite abundant. For these I would like to recommend James Sousa's MathIsPower4u website.

After authoring two online algebra textbooks published by Flatworld Knowledge, I created a similar visually searchable list of videos for all of the "Thy this!" problems found in the textbook.

[ Elementary Algebra (Algebra 1): Try This Videos ]

[ Intermediate Algebra (Algebra 2): Try this Videos ]

This approach is not the silver bullet for success in Algebra, however, it does review well is appreciated by students who are more inclined toward a visual learning style.  Everything on the openAlgebra.com site is offered free of charge and is openly licensed with a creative commons license.  This means that you are free to and encouraged to link to it, copy-and-paste into your CMS, build upon, improve and share the material.
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Adopt One Open Textbook and Make a Difference

The passage of the Open Textbook bills in California is a significant development in the open educational resources (OER) space. We have been working on getting the word out as to the benefits of Open Textbooks for some time now. However, when the largest higher education system in the US takes action like this – folks will take notice. In fact, I think this news could be the “tipping point” for an Open Textbook revolution.

Consider the $172 cost of the new edition of one of the top selling algebra books for one of our highest enrolled mathematics courses - Algebra. Now consider that I have over 40 students in each of my 3 sections - that is over 120 students:

$172 x 120 students = $20,640

This is a lot of money for one instructor's one semester of Algebra - now scale this up to the over 340,000 Algebra students in California and you start to see the significant amounts of money involved here!  Here is an info-graphic made by the 20 Million Minds Foundation and an open textbook savings calculator. I like to say that adopting an open textbook is like your very own local stimulus plan!

http://www.20mm.org/infographic-open-source-impact.html 

Now some might argue that the top selling textbook is better for any number of reasons... and yes it might be.  But we have to start considering student cost in our decisions to adopt.  

Is it $172 better?

Is it $100 better?

Is it $50 better?

Is it $25 better?

Can I supplement any shortcomings?

Do I have the time to make change?

I happen to think the 21st century college instructor will use a LMS to aggregate OER as a means to present a course that leverages his/her expertise. High quality modular Open Textbooks and resources are the key. Modifying open textbooks, collecting good content and building courses is time consuming. However, the benefits in course relevance and cost far outweigh any upfront effort. New editions are totally up to you and not a product of the bottom line!

We need to take the “all of the above” approach to addressing the soaring costs of education. These bills may not solve the cost issue overnight - but they sure bring the $200 textbook problem into laser focus. I truly hope the use of Open Textbooks tips - where more faculty and students become aware of their viability. 

Instructor Challenge: Start by adopting one open textbook for next semester. They are hard to find, modify, and supplement.  Some links are provided below that might help. Join the OER community:

 http://collegeopentextbooks.ning.com/

Students Challenge: Start by searching out open textbooks and send the links to your instructors.  Sign the petition. This will be your legacy to those who come after you.

My Viable Math Textbook Links

College Open Textbooks: http://www.collegeopentextbooks.org/

Merlot OER repository: http://www.merlot.org/merlot/index.htm

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#Open Textbook Tweet

A quick read by Sharyn Fizpatrick on the what and why of open textbooks.

[ e-version on WikiEducator ]

Make It Your Own

Here is a link to a blog post that I wrote for Flat World Knowledge:

http://blog.flatworldknowledge.com/2012/03/08/3-steps-to-a-perfect-open-textbook/

Unlike most open source materials, the FWK customization tools allow you to easily customize their books.

Related Links

• MIT OpenCourseWare Partners With Flat World Knowledge to Provide Free Courses & Free Textbooks (bostinno.com)
• Flat World Knowledge

Elementary Algebra TOC - Free to Read Online (No Login!)

Here is the table of contents for the open Elementary Algebra textbook found on the FlatWorld Knowledge website.  You will find that the content is fairly modular, I have written it from the ground up with the expectation that instructors will modify it to suit their particular course.  I am confident that everything needed for a Beginning Algebra (Algebra 1) course is covered, but feel free to MIYO (make it your own) by customizing it as you please. This textbook has the advantage over other open source material in that it has been thoroughly reviewed and vetted through the traditional publishing process.  (Note: Be patient with the website, there are many math elements to be rendered for the web.)


Chapter 1 - Real Numbers and Their Operations

1.1 Real Numbers and The Number Line
1.2 Adding and Subtracting Integers
1.3 Multiplying and Dividing Integers
1.4 Fractions
1.5 Review of Decimals and Percents
1.6 Exponents and Square Roots
1.7 Order of Operations
1.8 Review Exercises and Sample Exam

Chapter 2 - Linear Equations and Inequalities 

2.1 Introduction to Algebra
2.2 Simplifying Algebraic Expressions
2.3 Linear Equations: Part I
2.4 Linear Equations: Part II
2.5 Applications of Linear Equations
2.6 Ratio and Proportion Applications
2.7 Introduction to Inequalities and Interval Notation
2.8 Linear Inequalities (one variable)
2.9 Review Exercises and Sample Exam

Chapter 3 - Graphing Lines

3.1 Rectangular Coordinate System
3.2 Graph by Plotting Points
3.3 Graph using Intercepts
3.4 Graph using the y-intercept and Slope
3.5 Finding Linear Equations
3.6 Parallel and Perpendicular Lines
3.7 Introduction to Functions
3.8 Linear Inequalities (Two Variables)
3.9 Review Exercises and Sample Exam

Chapter 4 - Solving Linear Systems 

4.1 Solving Linear Systems by Graphing
4.2 Solving Linear Systems by Substitution
4.3 Solving Linear Systems by Elimination
4.4 Applications of Linear Systems
4.5 Solving Systems of Linear Inequalities (Two Variables)
4.6 Review Exercises and Sample Exam

Chapter 5 - Polynomials and Their Operations 

5.1 Rules of Exponents (Integer Exponents)
5.2 Introduction to Polynomials and Evaluating
5.3 Adding and Subtracting Polynomials
5.4 Multiplying Polynomials and Special Products
5.5 Dividing Polynomials
5.6 Negative Exponents and Scientific Notation
5.7 Review Exercises and Sample Exam

Chapter 6- Factoring and Solving by Factoring

6.1 Introduction to Factoring
6.2 Factoring Trinomials
6.3 Factoring Trinomials ax^2 + bx + c
6.4 Factoring Binomials
6.5 General Guidelines for Factoring Polynomials
6.6 Solving Equations by Factoring
6.7 Applications involving Quadratic Equations
6.8 Review Exercises and Sample Exam

Chapter 7- Rational Expressions and Equations

7.1 Simplifying Rational Expressions
7.2 Multiplying and Dividing Rational Expressions
7.3 Adding and Subtracting Rational Expressions
7.4 Complex Fractions
7.5 Solving Rational Equations
7.6 Applications of Rational Equations
7.7 Variation
7.8 Review Exercises and Sample Exam

Chapter 8- Radical Expressions and Equations 

8.1 Radicals
8.2 Simplifying Radical Expressions
8.3 Adding and Subtracting Radical Expressions
8.4 Multiplying and Dividing Radical Expressions
8.5 Rational Exponents
8.6 Solving Radical Equations
8.7 Review Exercises and Sample Exam

Chapter 9- Solving Quadratic Equations and Graphing Parabolas 

9.1 Extracting Square Roots
9.2 Completing the Square
9.3 Quadratic Formula
9.4 Guidelines for Solving Quadratic Equations and Applications
9.5 Graphing Parabolas
9.6 Introduction to Complex Numbers and Complex Solutions
9.7 Review Exercises and Sample Exam

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Open Precalculus Textbook

I am almost through my first semester using the following open source Precalculus textbook:

Stitz - Zeager Open Source Mathematics

It covered all of the topics that I needed without supplementation and the students liked its readability... and of course the cost. I especially liked the continuing coverage of inequalities and heavy use of sign charts throughout. The pdf is well produced and easy to navigate using the linked TOC. Buying a printed copy was easy and the quality was good -- the Trigonometry portion is printed separately.

I will definitely use it again and I recommend it to anyone teaching this course.

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Isaac Newton's Principia

Isaac Newton's Principia is recognized as the greatest scientific book ever written.  You can read it on Google books for free:  Newton's Principia with notes and illustrations.

Image via Wikipedia

In addition,  you can read about Isaac Newton here and view his childhood home below.

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FWK Technology

An excellent interview with Jon Williams the Chief Technology Officer of flat world knowledge.

Interview: Ed Tech Leader, Jon Williams of Flat World Knowledge

The "make it your own" MIYO platform has great potential. As Jon puts it, "one benefit of thinking of systems as a “platform” is that it forces you to take a longer view, and plan for future business growth and needs."

MIYO

Related articles

• Flat World Knowledge launches platform for customizable digital textbooks (teleread.com)
• Interview with Eric Frank, Flat World Knowledge, by Sue Polanka (teleread.com)
• Random House Invests In Flat World Knowledge (paidcontent.org)

Open Yale Courses

Take a look at the nicely organized open courses at Yale. (  http://oyc.yale.edu/  )
Image via Wikipedia

Open Yale Courses provides free and open access to a selection of introductory courses taught by distinguished teachers and scholars at Yale University. The aim of the project is to expand access to educational materials for all who wish to learn.

Related articles

• Ivy League Education for All - Free Open Courses at Yale (socyberty.com)
• Yale University Website

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Spotlight on Innovation

Webinar today 3:30pm EDT / 12:30 pm PT... Attacking the high cost of textbooks while improving outcomes.

Register now for this webinar to learn about three of the most exciting initiatives aimed at driving costs down while enhancing innovation and outcomes.

Webinar Registration

Moderated by Eric Frank, President and Co-Founder of Flat World Knowledge, this panel will begin with an overview of what open texts are and how they've emerged as a force in past years, and then spotlight the innovative work of the University System of Ohio, Washington Open Course Library Project, and Virginia State University.

While you wait try the cost savings calculator on the flat world knowledge website:

Related articles

• Interview with Eric Frank, Flat World Knowledge, by Sue Polanka (teleread.com)
• Flatword Knowledge Webinars 
• How to Avoid the High Cost of Textbooks (alwaysbcmom.com)

The Benefits of Open Textbooks

Image via WikipediaCost is not the only benefit of open textbooks.  Instructors are encouraged to modify and aggregate content to make their courses more current and relevant.  Add content to your LMS each semester and evolve your course.  Reuse, remix, and share your content.

Before considering an open textbook or any other open educational resource (OER) consider reading:

7 Things you Should Know About Open Textbook Publishing

These concepts are mirrored in Dr. Shu Schiller's video endorsement of Flat World Knowledge textbooks:

Another great resource for open content is the Merlot website. Merlot stands for multimedia educational resource for learning and online teaching.  There is much to sift through out there.  This is why I suggest using an LMS like Blackboard or Moodle, you can add or modify content each semester and evolve your course over time... 21st students will appreciate it.

Related articles

• 1 in 4 College Textbooks Will Be Digital By 2015 (nytimes.com)
• What *is* "Open"? (elearningcentralia.wordpress.com)
• CK-12 Brings Digital Books into the Classroom (ostatic.com)