RWM101: Foundations of Real World Math

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 ): $\mathbb{R}$

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.

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.

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.

Fractions are amazingly helpful; we can use them to tell so many different kinds of stories, but this can also make them seem overwhelming or confusing. How likely will a coin flip result in heads? We use a fraction to express the answer to this probability $\frac{1}{2}$: our outcome (a heads up) should occur 1 out of 2 times or one-half of the time.

If you serve 10 pizzas to 40 students, how much pizza should each student receive? We use fractions to determine that each student should receive $\frac{1}{4}$ or one-fourth of a single pizza. We also use fractions to discuss ratios and rates, such as this car gets 40 miles per gallon.

Combining fractions using addition, subtraction, multiplication, or division is a bit more complicated than combining integers, but these combinations are useful in many situations and problems.

For example, if your friend Andy has one-half of a chocolate bar and Tai has one-third of a bar, how much do they have together? To answer this question, we need to add the rational numbers $\frac{1}{2}+ \frac{1}{3}$ (see Section 4.8). In this unit, we explore these kinds of topics, beginning with a basic review of how to notate a fraction and ending with a discussion on how to use them in multi-step problems.

Completing this unit should take you approximately 8 hours.

In this unit, we study decimals, which are simply another way to express fractions. We use this way of notating fractions (including improper ones and mixed numbers), most notably when dealing with money or currencies.

For example, if the price of your aunt's amazing vanilla latte is $3.50, we understand this to mean it costs three whole dollar bills and fifty cents. The period or decimal point, as mathematicians call it, separates whole numbers from the fractions in this expression – it serves this purpose in this price and in all other decimal expressions. Our whole numbers (in this case, 3) sit to the left of the decimal point, and fractional quantities sit to the right of the decimal point.

Since the digit 5 appears immediately to the right of the decimal, it really means five-tenths. The zero means zero hundredths. Of course, we can reduce the fraction five tenths as

$\frac{5}{10} = \frac{1}{2}$.

Altogether this means we can translate $3.50 literally or mathematically to three whole dollars and one-half of a dollar. That is

$3.5 = 3 + \frac{1}{2} = 3\,\frac{1}{2}$.

Because decimals are just a different way to express factions, they are just as useful and omnipresent. In this unit, we explore how to convert decimals into explicit fractions and how to add, subtract, multiply, and divide them.

Completing this unit should take you approximately 6 hours.