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Finite Math—Semester A

You can't handle the truth tables.

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

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Shmoop's Finite Math course has been granted a-g certification, which means it has met the rigorous iNACOL Standards for Quality Online Courses and will now be honored as part of the requirements for admission into the University of California system.

Whether it was about polynomials or imaginary numbers, we've all thought it at one point or another. We've all asked ourselves the one question that turns math into a bottomless pit of pointlessness: "When am I ever going to use this?"

Well, not anymore.

In this Common Core-aligned course, we'll find that the real world does, in fact, have a place for math—and that place is everywhere. (Shocking, we know.) With loads of problem sets, readings, and quizzes, we'll bring math to life and talk about

  • truth tables, arguments, and logical networks.
  • linear functions and systems of linear equations.
  • matrices and all the nifty things we can do with them.
  • both geometric and algebraic linear programming.
  • the math of finance, including interest and annuity.

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

Technology Requirements

Technology-wise, all you'll need to complete this course is a computer with internet access (a tablet is ok, too), your internet browser of choice, and your calculator of choice (Google works in a pinch). Experience with all these tools will come in handy, too, but if you don't have any, you'll pick it up soon, we promise.

Unit Breakdown

1 Finite Math—Semester A - Logic

Logic is all about making statements. Sometimes they're fashion statements. Other times, they're mission statements. Then there's the occasional thesis statement or mathematical statement. We don't know if all the statements it makes are true, but that's where truth tables and logic circuits come into play.

2 Finite Math—Semester A - Linear Functions

Whether it's review or brand-new information, we'll examine straight lines in a good amount of detail in this unit. We'll learn (or recall) a thing or two about slope, what it means when lines are parallel or perpendicular, and even recognize the various forms linear equations can take—with and without the Groucho glasses.

3 Finite Math—Semester A - Systems of Equations

If you think you know all there is to know about systems of equations, we've got news for you: you don't. In this unit, we'll take you from refreshing your basic algebra skills all the way to solving systems of linear equations using Gauss-Jordan elimination. We'll even solve systems of more than two equations—and interpret our answers in context. Then you'll know all there is to know about these puppies.

4 Finite Math—Semester A - Matrices

In this unit, we'll explore the ins and outs of matrices, taking several twists and turns on our journey in order to become matrix masters. Real-world applications? Challenging math problems? We know you're dying to learn more about matrices already. So strap yourself in, and let's see how deep the rabbit hole goes.

5 Finite Math—Semester A - Linear Programming: Geometric

We'll start off linear programming by learning about linear inequalities. After delving into bounded and unbounded solutions to systems of linear inequalities, we'll get to the meat and potatoes of linear programming: optimization problems, the method of corners, and some really cool shading effects. Get those colored pencils ready.

6 Finite Math—Semester A - Linear Programming: Algebraic

Don't get us wrong: graphing is great. It's cooler than Free Cone Day at Ben & Jerry's—but like ice cream on a hot day, graphing has its limitations. Luckily, we can solve linear programming problems algebraically and get ourselves out of any sticky situation, particularly when several variables are involved. In a nutshell, algebra makes linear programming plain and simplex. (You'll get that joke after you finish this unit.)

7 Finite Math—Semester A - Math of Finance

Money talks—and you'll need to know a thing or two about interest, annuity, and amoritization if you're going to talk back. (Seriously, how do you expect to have a conversation if you don't speak the same language?) By the time we're through learning about loans, compounding interest, and sinking funds, you'll know exactly how to make your first million—and where to invest it.

Recommended prerequisites:

  • Algebra II—Semester A (2014-2015)
  • Algebra II—Semester B (2014-2015)
  • Algebra I—Semester A
  • Algebra I—Semester B

    • Sample Lesson - Introduction

    Lesson 6.05: Standard Minimization Problem

    How appropriate: we've been working on maximization problems, but there's an elephant in the room that we need to address.

    (Source)

    We're fresh off of our triumphant victory over standard maximization problems using the simplex method. We should get a parade in our honor to celebrate our success.

    Solving maximization problems is all well and good, but we should keep in mind that they aren't the only type of problem. There are instances where we don't want to find the greatest amount possible, but the least amount.

    Businesses certainly want to get the highest profit that they can, but that often involves figuring out how to spend the least amount of money. Manufacturers want to make as many products as possible, and that often involves using the least amount of material allowed. Elephants like to eat a lot, but we want to clean up as little of their poop as we can get away with.

    Fortunately, we don't need to deal with elephant poop in this lesson. However, we do need to address what to do when we find ourselves faced with a minimization problem.

    Sample Lesson - Reading

    Reading 6.6.05: The Least of Our Concerns

    We've spent so much time figuring out how to solve a standard maximization problem that you've almost certainly had this thought at some point: "Is there anything else besides standard maximization problems? Surely, there must be more!"

    You're right; there is more. (And don't call us Shirley.)

    In fact, we're ready to move on to what we call a standard minimization problem.

    The criteria for standard minimization problems are pretty similar to those for standard maximization problems:

    • In a standard minimization problem, we are trying to minimize the objective function.
    • The variables of the objective function are defined to be nonnegative; that is, equal to or greater than zero.
    • In addition, the coefficients of the variables in the objective function must also be positive.
    • The constraints must all be inequalities that are greater than or equal to a positive constant.

    The major differences from the maximization problem are 1) we're minimizing the objective function instead (duh), 2) the coefficients of the objective function need to be positive (there were no limits before), and 3) the constraints are now greater than or equal to something positive (where before they were less than or equal to them). Here's an example:

    Minimize w = 3y1 + 9y2 + 2y3, subject to the following constraints:

    Let's look at how this fits the criteria for a standard minimization problem.

    First, we're trying to find the minimum value of the objective function, w. So far, so good.

    The variables y1, y2, and y3 are all defined to be nonnegative. See that last row? Also, the coefficients in the objective function (3, 9, and 2) are all positive, so that's another criteria we've met.

    Finally, each of the constraints is written as an inequality that is greater than or equal to some positive constant. We definitely have a standard minimization problem.

    Now, you might be thinking that we can just forge ahead with building the initial tableau and diving into using the simplex method. Well, don't get cocky; we have a little more to learn before we can go solve this with the simplex method.

    Recap

    Certain characteristics determine whether or not a problem is a standard minimization problem:

    • We are trying to minimize the objective function.
    • The variables from the objective function are all defined to be nonnegative.
    • In addition, the coefficients of the objective function are all positive.
    • The constraints are all written as inequalities greater than or equal to a nonnegative constant.

    Unlike standard maximization problems, we cannot jump straight into using the simplex method. We have to do some work first before we can use it.

    Sample Lesson - Activity

    1. What's the first requirement for a problem to be a standard minimization problem?

    2. What's the second requirement of a standard minimization problem?

    3. How about the third requirement of a standard minimization problem?

    4. You see where we're going with this. What is the fourth and final requirement of a standard minimization problem?

    5. We can solve standard minimization problems directly with the simplex method. Is this true or false?

    6. The following qualifies a standard minimization problem. Is this true or false?

      Minimize z = 18x1 + 19x2 + 20x3, subject to the following constraints:

      18x1 + 2x2 + x3 ≥ 27
      x1 + x2 + x3 ≥ 2
      x1 + x2 + 200x3 ≥ 600
      x1, x2, x3 ≥ 0

    7. Which of the following constraints would disqualify a problem from being a standard minimization problem?

    8. Is this a standard minimization problem? Why or why not?

      Minimize w = y1 + 16y2 – 13y3, subject to the following constraints:

      8y1 + y2 + 8y3 ≥ 59
      y1 + y2 + y3 ≥ 3
      9y1 + 7y2 + 2y3 ≥ 150
      y1, y2, y3 ≥ 0

    9. Why is the following problem not a standard minimization problem (i.e., what's wrong with it)?

      Minimize w = x+ y + z, subject to the following constraints:

      4x + 5y + 6z ≥ 89
      8x + y + 7z ≥ 50
      x + y + z ≤ 300
      x, y, z ≥ 0

    10. In the following problem, which answer will correctly fill in the blank to give us a standard minimization problem (i.e., what's missing)? Minimize w = 65y1 + 47y2 + 38y3, subject to the following constraints: y1 + 16y2 + 18y3 ≥ 91 y1 + 3y2 + 3y3 ≥ 25 [_________]

    11. Is this a standard minimization problem? Why or why not?

      Minimize w = 133y1 + 8y2 + 7y3, subject to the following constraints:

      y1 + y2 + y3 ≤ 5
      8y1 + 99y2 + 10y3 ≤ 480
      y1, y2, y3 ≥ 0

    12. Is this one a standard minimization problem? Why or why not?

      Minimize z = 200x1 + 201x2, subject to the following constraints:

      x1 + 45x2 ≥ 555
      21x1 + x2 ≥ 556
      x1, x2 ≥ 0

    Sample Lesson - Activity

    1. What method is used to solve a linear programming problem with many variables or constraints?

    2. Which of the following could be an objective function?

    3. This is NOT an example of a standard maximization problem. Why?

      Maximize z = 8x1 + 25x2 + 17x3
      x1 + x2 + x3 ≤ 30
      x1 + 15x2 + 39x3 ≤ 75
      5x2 + 5x2 + 5x3 ≤ 123
      x1, x2, x3 ≤ 0

    4. Which of the following could be a constraint of a standard maximization problem?

    5. How do we know we have reached the optimal solution of a standard maximization problem?

    6. If this tableau shows an optimal solution, what is it? Don't keep us in suspense.

    7. Using the tableau below, what number is the pivot point?

    8. What is the optimal solution of the standard maximization problem below?

      Maximize z = 3x1 + 2x2 + x3, subject to the following constraints:
      4x1 + x2 + x3 ≤ 30
      x1 + 2x2 + 3x3 ≤ 40
      2x1 + 3x2 + x3 ≤ 60
      x1, x2, x3 ≥ 0

    9. Which of the following is NOT a requirement for a standard minimization problem?

    10. Does this qualify as a standard minimization problem?

      Minimize w = 2y1 + y2 – 3y3
      y1 + y2 + y3 ≥ 4
      8y1 + 3y2 + 4y3 ≥ 29
      y1 + 10y2 + 18y3 ≥ 68
      y1, y2, y3 ≥ 0