Finite Math—Semester A
You can't handle the truth tables.
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-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 1. 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.
Unit 2. 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.
Unit 3. 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.
Unit 4. 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.
Unit 5. 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.
Unit 6. 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.)
Unit 7. 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.
Sample Lesson - Introduction
Lesson 5: Standard Minimization Problem
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 6.5: 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.
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
Activity 6.5c: Problem Set
Sample Lesson - Activity
Quiz 6.5d: Quiz
- Credit Recovery Enabled
- Course Length: 18 weeks
- Grade Levels: 10, 11, 12
- Course Type: Basic
Algebra II—Semester A (2014-2015)
Algebra II—Semester B (2014-2015)
Algebra I—Semester A
Algebra I—Semester B
Just what the heck is a Shmoop Online Course?
Common Core Standards
The following Common Core Standards are covered in this course:CCSS.Mat.Content.HSA-REI.B.3.1