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Pre-Algebra II—Semester A

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

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If math just sat still, like a bump on a log, it wouldn't be of much use to anyone. Not even mathematicians would get excited about it—and you should see some of the things they get worked up over. Lucky for everyone, then, that math is full of sound and fury, signifying something awesome.

(Levels of awesome may vary.)

In this Common Core-aligned course, we'll peel back the veil and uncover the mysteries of all the algebra that comes before Algebra. With boatloads of problem sets, readings, and quizzes, we'll cover

  • rational and irrational numbers
  • radicals, exponents, and number theory
  • linear equations and inequalities with one and two variables
  • transformations of geometric figures
  • linear functions

P.S. Pre-Algebra II is a two-semester course. You're looking at Semester A, but you can check out Semester B here.

Unit Breakdown

1 Pre-Algebra II—Semester A - Rational and Irrational Numbers

Get ready for the next level of numbers: rational and irrational numbers. Sure, they might be as different as apples and orangutans, but it's good to have some variety. We'll learn how to convert rational numbers into fractions, how to identify those pesky irrationals, and even how to approximate irrational numbers using rationals. We wouldn't suggest approximating apples with orangutans, though; that might get messy.

2 Pre-Algebra II—Semester A - Radicals, Exponents, and Number Theory

If anything proves that good things come in small packages, it's exponents. Well, maybe not always good, but certainly powerful. These tiny little numbers can really pack a punch, so we suggest putting on some headgear. In this unit, we'll learn about the complicated relationship between exponents and radicals, and how one operation always undoes the other. Oh, maybe it isn't that complicated.

3 Pre-Algebra II—Semester A - Equations and Inequalities in One Variable

Equations are the ultimate balancing act. If the two sides don't match up exactly, everything will topple over. With only a little bit of practice, though, we'll be able to juggle, shuffle, and slide numbers around an equation without the slightest wobble. After that, we'll use those same skills to tackle inequalities, the hippie-like cousins to equations. They don't care much about any single value;  as long as the answers fall within a certain range, they go with the flow. Pretty groovy, right?

4 Pre-Algebra II—Semester A - Equations and Inequalities in Two Variables

We'll cover how two-variable equations can describe all kinds of relationships—the good, the bad, the ugly, and the math-y. Mostly that last one, if we're being honest.

5 Pre-Algebra II—Semester A - Geometric Transformations

Sorry, numbers and letters, but we need a break from you. Instead, we'll be movin' and groovin' with figures this unit, except for a few tricky definitions and a quick tango with the coordinate plane. Just be careful; some of these shapes have sharp edges. The last thing we want is for you to get a nasty cut mid-disco.

6 Pre-Algebra II—Semester A - Linear Functions

We're going introduce you to functions nice and slow as we go over the basics: learning the definition of a function, seeing how to identify them by sight in tables and graphs, and visualizing them on the coordinate plane. Then we'll zoom in on linear function and all of their different parts and pieces—no dissection necessary (or wanted). Leave that for your science class.

7 Pre-Algebra II—Semester A - Graphing Linear Equations and Inequalities

Just knowing linear equations inside and out isn't enough: we also have to know how to graph them. Lucky for us, then, that their graphs have a purpose other than looking cool. They can make problems easier to understand and solve—especially when there are real-world applications lurking in the wings. Obviously, looking cool doesn't hurt, either.

Recommended prerequisites:

  • Pre-Algebra I—Semester A
  • Pre-Algebra I—Semester B

    • Sample Lesson - Introduction

    Lesson 3.06: Solving Inequalities Using Multiplication and Division

    Addition and subtraction are great, but sometimes we need another level of solving power. It's just like when you exchange your handsaw for a chainsaw, or add extra thrusters to your rocket ship. Your tool has to be up to the challenge of your problem. For us, the extra boosters are multiplication and division.

    With these added boosters, there's no way the atmosphere is keeping us in this time.

    (Source)

    Thankfully, utilizing this new, heavier-duty solving mechanism doesn't actually make our solution more complex. In fact, multiplication and division in inequalities follow the same rules that addition and subtraction do, which is to say that they follow the same rules regular equations do. It all boils down to our Golden Rule of algebra: Everything you do to one side you have to do to both.

    There is one very important thing to not overlook. The information in this lesson only applies when we are talking about multiplying or dividing by a positive factor. If our factor is negative, the rules are slightly different, and all those rules are covered in the next lesson. Just remember to keep on the positive side of things for now and you'll be just fine.

    Sample Lesson - Reading

    Reading 3.3.06: Solving Inequalities using Multiplication and Division

    Dividing your inequalities by positive numbers won't really look any different than dividing a run-of-the-mill equation by a positive number. Again, we need whatever we do to take place on both sides. (But by now, you know that's basically a given.) We just have to choose the vale to multiply or divide which will allow us to cancel things off and simplify our statement. Then, we can draw conclusions about the values that can solve our inequality.

    This quick video will give us an example of real life equalities and division in action. Try not to get too caught up in the emotional cause at hand. We know blobfish are sympathetic, but keep your eyes on the math.

    Recap

    When we work with inequalities, the same rules for multiplication and division apply that do with normal equations. The exception here is when we are dividing or multiplying by a negative number, but we will get into much greater depth on the topic in the next lesson. For now just remember that when a coefficient or quotient is positive, nothing about the sign has to change. It's beautiful just the way it is.

    Sample Lesson - Activity

    1. What is the solution to the inequality 4b > 24?

    2. What is the solution to the inequality 3x + 8 ≤ 29?

    3. What is the solution to the inequality -3 + 0.5c ≥ 7?

    4. Remember that History exam you forgot to study for? Yeah, well, it's make up time. You need to average a 92 on the next four exams in order to bring your grade back up. Which of the following inequalities represents the total combined score you need on the four exams in order to improve your grade?

    5. What is the solution to the inequality x⁄9 + 15 ≤ 22?

    6. What is the solution to the inequality 12 > p⁄24?

    7. The Shmooze Brothers are coming to town and you are dying to rock out at their concert. Tickets are $75 each and you only have one night waitressing to earn the money. You work an eight-hour shift and earn $35 in tips. Which of the following inequalities shows the minimum you need to make per hour in order to earn enough for the ticket?

    8. Solve the inequality 5a + 4.25 > 9.25.

    9. What is the solution to the inequality ⅛t – 1 ≠ 0?

    10. Solve the inequality -11 ≥ 2b + 11.

    11. Your parents have saved and saved to send you to the finest institution of higher learning: Clown College. After paying for the boring stuff (i.e., tuition, room and board, books), you have $16,260 for entertainment, food, and clothes for the next four years of college. Before you head off to school, your brother decides it is the perfect time to pay you back that $300 he borrowed. After your brother paid you back, how much can you afford to spend per month over the next four years?

    12. Solve 3(x – 2) < 33 for x.