• ### Course Introduction

• Time: 40 hours
• Free Certificate
A hungry professor walks into your aunt's coffee shop where you sometimes work as a cashier and manager. After some indecision, your math-professor customer notices she can buy a 6-pack or an 11-pack of delicious, coffee-infused chocolate cannolis. She would like to buy 49 pastries. How many 6-packs and how many 11-packs should you put together for her? Is it even possible to fulfill her order? (Take out a few sheets of paper and explore; your customer will wait patiently).

Our cannoli story has a fascinating answer that speaks to some deep mathematics first explored in the late 1800s and re-explored in the 1980s (thanks to some chicken nuggets from Mcdonald's). We use math in our daily activities, and it plays an important role in nearly every career you can imagine: from business, to cooking, to farming, to medicine and beyond. It is no surprise that many call math a universal language: people across the globe use the same numbers, formulas, and equations to help them navigate the world.

In this course, we study essential math concepts that will enrich your understanding of the world and illuminate a larger, mathematically-rich universe. The three courses in the Real World Math series discuss basic algebra and geometry topics and show you how to apply these concepts to everyday life.

The material focuses on how math relates to common real-world situations, transactions, and phenomena, such as personal finance, business, and the sciences. This real-world focus will help you grasp the importance of the mathematical concepts you encounter in these courses and understand why you need quantitative and algebraic skills to succeed in college and your day-to-day life.

For example, fractions allow us to tell interesting and useful stories that involve measurement, ratios, and proportions. Decimals and percentages are merely fractions in disguise. They help us make financial decisions and measure or compare various types of data. This course will help clarify the different ways we represent data visually, such as with a bar or line graph.

We also examine how to interpret data – no matter how it is presented. This skill will help you read a chart that outlines the current mortgage interest rate or make sense of the latest statistics for your fantasy football league. Let's not forget our coffee-infused chocolate cannolis. You will use addition, subtraction, and multiplication to answer her question. However, we will not help you deliver the bad news to your professor-customer.

A note on numbers. Before we begin, let's clarify what we mean by the word numbers. We usually refer to a quantity, such as five, seven, or 10. But mathematicians have created four different categories or types of numbers.

Here is a summary of four types of numbers (with the fancy symbols mathematicians use to refer to them):

• $\mathbb{N}$ Natural Numbers: all positive whole numbers: $\mathbb{N}=\left \{ 1,2,3,4,5,... \right \}$
• $\mathbb{Z}$ Integers: all positive and negative whole numbers (and zero, too!):$\mathbb{Z}=\left \{..., -3,-2,-1,0,1,2,3,... \right \}$
• $\mathbb{Q}$ Rational Numbers: all positive and negative fractions (including integers): $\mathbb{Q}=\left \{a/b\:where\:a\:and\:b\:are\:integers \right \}$
• $\mathbb{R}$ Real Numbers: all possible positive and negative numbers (including ): $\mathbb{R}$

Since presenting the entire set of real numbers as a collection of quantities is difficult, we usually visualize this set as a solid line of points that stretches on forever in both directions. This set includes all of the four number types in our list above, in addition to some more interesting expressions we won't discuss here. In Unit 1, we discuss the mathematical operations we can perform using all of these numbers: our number properties apply to all real numbers. However, most of our examples will focus on positive whole numbers (integers). In later units, we will explore negative whole numbers (integers) and fractions (also called rationals). You do not need to memorize these symbols, but know that integers can refer to positive and negative whole numbers. Fractions can also be positive or negative.

First, read the course syllabus. Then, enroll in the course by clicking "Enroll me in this course". Click Unit 1 to read its introduction and learning outcomes. You will then see the learning materials and instructions on how to use them.

• ### Unit 1: Number Properties

In this unit, we discuss properties and patterns for adding, subtracting, multiplying, or dividing numbers. You are probably so familiar with these facts that it may seem strange to point them out at all. But there are benefits to naming and using these properties. Some are purely mathematical, but plenty are common, everyday ones too.

First, let's talk about the commutative properties of addition and multiplication. These tell us that it does not matter if you rearrange your numbers when you add or multiply them. So $3+2$ and $2+3$ both equal five and $10 \times 6$ and $6 \times 10$ both equal 60.

But we cannot swap all of our actions around like this. For example, what about when we bathe our dog, Harry, first and then dry him? Subtracting provides a mathematical example. Subtracting is not commutative: $3-2$ is not the same as $2-3$. Division is not commutative, either. You get a different result if you divide a 60-dollar dinner bill among 10 diners than if you divide a 10-dollar dinner bill among 60 diners! The order is important.

In other words, some of our basic operations enjoy certain properties (like commutativity), while others do not. You need to keep track of these rules to avoid confusion, incorrect dinner bills, and wet dogs. These properties will not only help you calculate more complicated equations later on, but they will also help you as you consider less intuitive or obvious properties, such as What happens if I divide 60 by 0? Paying careful attention to all of these rules will help you on your mathematical journey and in the real world.

Completing this unit should take you approximately 2 hours.

• ### Unit 2: GCF and LCM

Photo from Wikipedia

Your aunt has an excellent recipe for a breakfast sandwich that her customers love, but the cafe is all out of eggs and buns. She needs you to run to the grocery store for supplies. Exactly one egg and one bun are required to make each sandwich. Unfortunately, you can only purchase eggs in packages of 12 and buns in packages of eight. If you return with one dozen eggs and one eight-package of buns, four eggs will go unused. If you return with one dozen eggs and two packages of buns, four buns will be left over. What is the smallest number of eggs and buns you can purchase so there are no leftovers of eggs or buns? (Feel free to take out a sheet of paper and work the numbers! It will not take long to see that the answer is 24).

In this brief unit, we explore the concepts of least common multiple (LCM) and greatest common factor (GCF). Our run-of-the-mill questions, such as our egg-and-bun problem, indicate how these concepts are relevant to our everyday lives, but they also show up in lots of other mathematical questions. Here we are talking about sets of whole numbers: in Unit 1, our operations applied to whole numbers, fractions, and every other type of real number. Consequently, notions of least common multiple and greatest common factor belong to the part of mathematics that studies patterns among whole numbers, an area called number theory.

Completing this unit should take you approximately 1 hour.

• ### Unit 3: Order of Operations

We can use our four basic operations to solve so many types of mathematical and real-world problems, but we need to be careful when we combine or repeat them. The commutative and associative laws apply to certain operations but not others. We use the distributive law to guide calculations when we combine different operations. But we need to follow an order of operations, which we apply to any combination of operations we need.

In this unit, we learn how to use exponents and exponential notation to represent repeated multiplication. For example, repeatedly multiplying two against itself five times results in $2\times 2\times 2\times 2\times 2=2^{5}=32$. We will pay careful attention to how negative numbers interact with our operations and clarify how to group different symbols (subjected to different operations) in the correct and clear order.

Completing this unit should take you approximately 4 hours.

• ### Unit 4: Fractions

Fractions allow us to perform calculations with numbers that are not whole. For example, imagine you baked a pie. If your family eats half the pie for dessert one day, you could use fractions to determine that half a pie remains. If you eat another slice, you could use fractions yet again to see how much is left. We use fractions every day to calculate sale prices, measurements, money, and many other situations. In this unit, we explore how to add, subtract, multiply, divide, and reduce fractions.

Completing this unit should take you approximately 11 hours.

In this unit, we study decimals, which are simply another way to write fractions. For example, think about American currency. One dollar is 100 cents and a quarter is 25 cents, or $0.25 when written as a decimal. We can explain a quarter as being 25/100 of a dollar. This reduces to 1/4, which we read as one-quarter. Decimals are everywhere, just like fractions. We use them in money, in measuring lengths, and in amounts. In this unit, we study how to add, subtract, multiply, and divide decimals, and how to convert between fractions and decimals. Completing this unit should take you approximately 8 hours. • ### Unit 6: Ratios and Proportions In this unit, we study ratios and proportions. These are mathematical concepts that we use all the time, probably without even realizing it. For example: • At the grocery store, have you ever compared unit prices for different packages of the same type of food? That is a ratio. • When driving 65 mph (miles per hour) on the highway, have you ever determined how long it will take you to get to your destination? That is a proportion. • In sports, statisticians use proportions to predict an athlete's performance based on what they have accomplished in the past. In this unit, we explore how to write ratios, set up and solve proportions, and apply proportions to real-world scenarios. Completing this unit should take you approximately 3 hours. • ### Unit 7: Percentages Now that we have studied fractions and decimals, we are ready to explore percents. Percents are just fractions and decimals written in a different way. For example, we can describe one half in many ways: 0.5 as a decimal, 1/2 as a fraction, and 50% as a percent. These all mean the same thing! We see percents all the time in the real world, especially in sales at stores. For example, a store might advertise that it is selling clothing at 50% off. So, a$10 shirt would be reduced to \$5. A week later, the store may post a sign saying that there is an additional 20% off the sale price of the shirts. How do we determine the new discounted price of the shirt?

We will learn how to answer that question in this unit. We will also convert between percents and fractions or decimals, and learn about percentage increases and decreases. We will explore how to calculate percents in scenarios that you will see in the real world, such as calculating tips at a restaurant or sale prices at a store.

Completing this unit should take you approximately 2 hours.

• ### Unit 8: Graphs and Charts

Once we have gathered data or performed calculations, we have to visualize the information to make sense of it. It is much easier to read a graph or chart than to interpret meaning from a long list of numbers. We use graphs and charts in almost every field. Businesses use graphs and charts to show trends in growth. Politicians use graphs and charts to explain demographics and voting trends in campaigns and elections. Since we use graphs and charts so often, it is important to know how to read and interpret them.

In this unit, we will discuss the different types of graphs and charts that we use in mathematics. We will interpret the results for each type of graph or chart, learn to create charts and graphs, read charts, and work with the measures of central tendency for a data set.

Completing this unit should take you approximately 5 hours.

• ### Course Feedback Survey

Please take a few minutes to give us feedback about this course. We appreciate your feedback, whether you completed the whole course or even just a few resources. Your feedback will help us make our courses better, and we use your feedback each time we make updates to our courses.

If you come across any urgent problems, email contact@saylor.org or post in our discussion forum.

• ### Certificate Final Exam

Take this exam if you want to earn a free Course Completion Certificate.

To receive a free Course Completion Certificate, you will need to earn a grade of 70% or higher on this final exam. Your grade for the exam will be calculated as soon as you complete it. If you do not pass the exam on your first try, you can take it again as many times as you want, with a 7-day waiting period between each attempt.

Once you pass this final exam, you will be awarded a free Course Completion Certificate.