K12MATH012: Precalculus I

Course Syllabus for "K12MATH012: Precalculus I"

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Mathematics comes together in this course. You enter precalculus with an abundant array of experience in mathematics, and this course offers an opportunity to make connections among the big ideas you encountered earlier. It also assists you in developing fluency with the tools used in learning calculus. The focus of this course is the concept of function - it’s with functions that mathematicians and scientists can model the world and make leaps of invention, sending rockets to far planets, determining the future size of populations, and finding the amount of earth to be moved when creating new roads. The unit begins by defining and exploring certain attributes of functions and continues with specific kinds of functions - linear, polynomial, rational, logarithmic, and exponential. In addition to precalculus being important for many fields, the subject is obviously designed to prepare you for calculus, which is the mathematics of things that are changing. Calculus allows us to find areas of strangely curved shapes and to determine how fast something is changing at any particular instant. Throughout the course, you will compare and contrast ideas and apply those concepts to real-world situations. It’s a great course to help you braid ideas together and build an internal framework of understanding. As you travel through precalculus, you will look back over earlier topics and use them to look forward to new concepts. Here we go!

Learning Outcomes

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

  • Summarize, contrast, and interpret characteristics and properties of functions and inequalities - linear, piecewise, quadratic, rational, exponential, and logarithmic.
  • Solve equations involving linear, piecewise, quadratic, rational, exponential, and logarithmic functions.
  • Graph linear, piecewise, quadratic, rational, exponential, and logarithmic functions.
  • Combine functions and determine inverse functions.
  • Interpret roots and maxima and minima of functions.
  • Solve and interpret solutions to real-world problems, using concepts of functions, including linear, piecewise, quadratic, rational, exponential, and logarithmic functions.
  • Interpret and re-represent functions in other forms, specifically numeric, graphic, algebraic, and verbal.
  • Use functions to model real-world phenomena.

Course Requirements

In order to take this course, you must:

√    Have access to a computer.

√    Have continuous broadband Internet access.

√    Have the ability/permission to install plug-ins or software (e.g., Adobe Reader or Flash).

√    Have the ability to download and save files and documents to a computer.

√    Have the ability to open Microsoft files and documents (.doc, .ppt, .xls, etc.).

√    Have competency in the English language.

√    Have read the Saylor Student Handbook.

√    Have access to a graphing calculator.

√    Have taken Algebra II or a comparable course.

Course Information

Welcome to Precalculus I. Below, please find general information on this course and its requirements.
Course Designer: Dr. Virginia D. Gray, PhD                                 
Primary Resources: This course is composed of a range of different free, online materials. However, the course makes primary use of the following:

  • Lippman and Rasmussen’s Precalculus: An Investigation of Functions
  • Khan Academy videos

Requirements for Completion: There are two trails of success in mathematics, and both are important. First of all you want to see the big picture, understanding the concepts, using them in the given problem situations, and eventually being able to apply them in atypical situations. The second success trail has to do with the focused, detailed picture in mathematics, which means developing accuracy and precision. Thus you want to be disciplined in the details when completing the assigned problems. There is a big difference between 50 and -50, although it can seem insignificant on the written page. (After all, it's just a little minus sign!) In order to grasp not only the big ideas but also these accuracy details offered in this course, do the assigned problems with careful attention. They not only assist you in understanding and applying the mathematical concepts in each unit, but also prepare you to grasp subsequent material in future units.

In order to pass this course, you will need to earn a 70% or higher on the final exam. Your score on the exam will be tabulated as soon as you complete it. If you do not pass the exam, you may take it again.
Time Commitment: This course should take a total of 69 hours to complete, depending upon the background, skill, and effort you bring into it. Each unit includes a “time advisory” that lists the amount of time you are expected to spend on each subunit. These should guide you along the way.
Tips/Suggestions: Focus on understanding what you are doing. The early material creates the tools that you will use in the later material, so make sure you don’t skip over it. The answers in the back of the text are a learning tool; occasionally you will get the wrong answer and a tough problem can be cracked by working backward to understand the process. Be careful of how you use answers. Sometimes students look at the answer in the back of a text and see that they made a small mistake - maybe they were missing a minus sign - and it makes the rest of the problem incorrect. Remember, this means that although you may have been close, you were NOT correct, and you need more practice. You need to keep working more problems until you have correct answers consistently.
If you are struggling with a topic and the given materials are not making sense to you, remember that the Internet offers incredible resources. You can use a search engine to explore resources on whatever topics you want help with.
Mathematics involves a lot of writing--not only in the symbolic notation, but also in taking notes, making charts, and writing paragraphs of explanation. Please read Lab Space’s Bridge to Success: “Writing Mathematics” and “Tips for Writing Mathematics”. These two web pages have helpful explanations and tips regarding how to write in mathematics.

You will need a scientific graphing calculator. Many smartphones have scientific calculators embedded (turn the iPhone calculator on the side, and you will find one). There are also scientific, graphing calculator apps that you can download.  
Here are some options for graphing calculators:

Texas Instruments, TI Graphing Calculators
Texas Instrument calculators can range from $100 - $300, depending on the model you purchase. Graphing calculators not only allow you to perform basic actions such as addition and subtraction, but they also allow higher order functions such as solving equations. The best feature of a graphing calculator is its ability to graph functions and produce a line plotted on a coordinate plane. The best reason for having a graphing calculator is to use during class or during standardized exams when computers and mobile devices are not permitted.

Desmos offers the same capabilities of a graphing calculator, but it is available on your computer for free! The browser page allows for basic functions, and it can also complete equations and graph lines on the same page. What makes Desmos better than a graphing calculator is its ability to show the equation next to the graph of the line and your ability to change the equation for experimentation. The only con is that it cannot be used during standardized tests, as it requires a computer.

Microsoft Math
Microsoft Math is the same as Desmos, although only available on Windows operating systems. Microsoft Math can also create 3-D graphs for higher level functions such as those found in calculus. The program also allows you to save your work so that it can be preserved or printed out according to your needs. The only con is that it cannot be used during standardized tests, as it requires a computer.
Mobile Apps: Quick Graph (iOS) and Algeo (Android)
Both apps are capable of computing lower order operations and higher level functions such as graphing equations. They also allow for the ability to graph in 2-D and 3-D, which means they can be used for calculus level work. The apps cannot, however, transfer your work to other devices, which prevents them from leaving your mobile device. Since the apps are only for mobile devices, they cannot be used on standardized tests.

HRW Graphing Calculator

Table of Contents: You can find the course's units at the links below.