Algebra II—Semester A (2014-2015)

Nothing complex here...except complex numbers.

  • Credit Recovery Enabled
  • Course Length: 18 weeks
  • Course Type: Basic
  • Category:
    • Math
    • High School

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We've recently upgraded our Algebra II course in order to give you everything you could ever want from, uh, an Algebra II course. You can still use this version, but we recommend hopping over to the new one here.

You won't be sorry.

And if you are, then we are, too.



Unit Breakdown

1 Algebra II—Semester A (2014-2015) - Seeing Structure in Expressions

We'll start the course off by diving headfirst into expressions! With the help of factoring, radicals, and even sequences and series, we'll be able to pluck out different parts of an expression and figure out if two expressions are two peas in a pod or two peas in…separate pods. Because sometimes, one pod is just too cramped.

2 Algebra II—Semester A (2014-2015) - Arithmetic with Polynomials and Rational Expressions

You may have dealt with polynomials and rational expressions in the past, but you ain't never seen 'em like this before. Not only will we gain some new tips and tricks to help us deal with these pesky expressions, we'll learn and even prove a few theorems along the way. Yeah, we mean business.

3 Algebra II—Semester A (2014-2015) - The Complex Number System

You might think imaginary numbers are about as helpful as Maurice, your imaginary friend from kindergarten. We'll fill you in on a little secret: while Maurice can't deal with negatives under the radical or tell you about the Fundamental Theorem of Algebra, imaginary numbers can—and will. Then again, Maurice can make a mean imaginary apple cobbler. We all have our strengths.

4 Algebra II—Semester A (2014-2015) - Creating Equations

This unit is all about using the magic of the equal sign to create and solve problems. (We'd use the magic of Houdini, but we're still working on that rope escape trick.) By creating and graphing every type of equation you can possibly think of, we'll learn how to tell them apart and understand which type of equation is applicable to which situation.

5 Algebra II—Semester A (2014-2015) - Reasoning with Equations and Inequalities

For this unit, we're going how to put all the equations we created to use. We're going to graph our creations—both separately and together—and see how to use them to solve problems. More often than not, they can do a significant portion of the work for us, and who doesn't love outsourcing their work? So put your feet up, grab a piña colada, and let equations and inequalities do the work for you. (Not really.)


Recommended prerequisites:

  • Algebra I—Semester A
  • Algebra I—Semester B
  • Geometry—Semester A
  • Geometry—Semester B

  • Sample Lesson - Introduction

    Lesson 3.05: Plotting Complex Numbers on the Coordinate Plane

    The spheres are the same size? Seems imaginary to us.

    (Source)

    Optical illusions can break our brains. At first, you think you see one thing, but then you realize (or are told) that things are not what they appear to be.

    Our mind is about to shift again. The optical illusion: an ol' fashioned coordinate plane.

    We might have always looked at an axis one way, thinking they are all the same. The x-axis represents the x-values and the y-axis represents the y-values, right? When it comes to complex numbers, it isn't that simple. 

    The usual Cartesian plane is only useful when we're working with real numbers. We're upgrading our numbers, though, going from real to complex, so we'll have to upgrade our plane, too.

    When we throw complex numbers into the mix, we need to find a place to plot the i's. Imaginary numbers prefer hiding in dark places like under the bed or in the attic. Silly imaginary numbers.

    We're going to let the imaginary numbers hang out along the imaginary-axis. That means that the real numbers will be along the x-axis. When real numbers and imaginary numbers mix it up, we'll be ready to plot them on the complex plane.


    Sample Lesson - Reading

    Reading 3.3.05: Plotting Complex Numbers on the Coordinate Plane

    Graphing complex numbers is a little different than graphing real numbers. First of all, the graphs are different. If there is an imaginary component to a complex number it can't be graphed on a regular old real number axis. That wouldn't be complex enough.

    We're joking, though. Really, graphing on the complex number plane isn't that bad. Like the real number plane, the complex number plane also has an x- and y-axis. However, each axis represents something different in the complex plane.

    Remember that complex numbers can be split into a real and imaginary component: a + bi. The x-axis is going to be pretty much the same as ever. The y-axis, though, is going to represent the imaginary part. In other words, we can label it i, 2i, 3i, etc., going up from the origin, and -i, -2i, ­-3i, etc. going down.

    Adding i's to the y-axis has made it possible to take any complex number in a + bi form and to locate it on the plane. The x-coordinate is the "a" and the y-coordinate covers the "bi" part. Please, hold their applause.

    So, to plot 3 + 7i, we start at the origin. We go over 3 units on the x-axis, and up 7i units on the y-axis. Our point would be in the Quadrant I, right here.

    If our point was -4 – 11i, we would go 4 to the left and 11i down, and plot the point in the Quadrant III. See—even though we've made things more complex, they aren't really any more complex. Y'know, in a manner of speaking.

    Recap

    Since complex numbers have both a real and imaginary component, we graph each of these components separately. The real component is plotted along the x-axis while the imaginary coordinate is plotted along the y-axis.


    Sample Lesson - Activity

    1. What quadrant is the point 5 – 6i in?
    2. Which quadrant is 7 – 2i in?

    3. Where is -6 on the complex plane?

    4. What quadrant is -10 + 10i in?
    5. What quadrant is 4i in?

    6. How is the complex coordinate plane different than the real number coordinate plane?
    7. How are a complex number and its conjugate related on the complex plane?

    8. How is a complex number and its conjugate related?

    9. Why can't 7 + 4i be plotted on the real x and y coordinate grid?
    10. Can -6 + 4i2 be plotted on the complex coordinate plane? Why or why not?
    11. Conjugates are reflected across the imaginary axis. Is this true or false?

    12. A complex number can never be on the left side of the imaginary axis. Is this true or false?