Chapter 1: Probability, Percent, and Rational Number Equivalence

Usually, seventh grade math starts by reviewing everything students learned about numbers before. The goal is to help students get better at math operations while also understanding how the number system works underneath. But this kind of review can feel boring. Students who already understand the material don't want to repeat it, and students who don't understand it yet often don't learn much better the second time around either.

That's why this book uses a different plan. Instead of a plain review, we mix the review into a brand-new topic: probability. Probability is the study of how likely something is to happen. Learning about probability is interesting on its own, and it also gives students a real reason to practice arithmetic. Starting the year this way also helps build a classroom where students think of math as a tool for understanding the real world, and where they talk and work together to solve problems.

Throughout this chapter, students will review and strengthen their skills with fractions, percents, and decimals, and learn how these are all different ways of showing the same rational numbers. Students should understand that fractions, percents, and decimals all depend on agreeing what counts as "the whole." Students will also learn to compare and order fractions, including negative fractions. The chapter ends with a section about solving real-world percent and fraction problems — things like discounts, interest, taxes, tips, and increases or decreases in percent. As students work through these problems, they will start using the basic rules of arithmetic (called "field axioms") without formally naming them yet. Those rules will be explained more fully in Chapter 3.

This chapter is students' first real introduction to probability. In earlier grades, students worked with sets of data and looked at different ways to organize and understand that data. Now, in seventh grade, students begin learning statistics — the set of tools used to analyze data. Probability is the foundation that statistics is built on, which is another reason to begin the year with this topic. Statistics help us describe, in general terms, what data tells us about a group of people or things. For example, a medical study testing a new medicine might conclude: "The data suggest that Medicine A cures 80% of patients with Disease B, but causes side effects in 4% of patients." Those percentages come from the basic ideas of probability — assigning a number to how likely a certain outcome is (in this case, being cured).

In short, probability is a branch of math that supports statistics. Statistics, in turn, is at the heart of the scientific method: making a guess (a hypothesis), collecting data, studying the data, and figuring out how likely it is that the guess was correct.

Section 1.1: Investigate Chance Processes and Build Probability Models

In the first section, students will study chance processes — experiments or situations where we know all the possible outcomes but can't be sure which one will happen each time. Students will look at probability as a ratio (a comparison) between a part and a whole, written as a fraction, decimal, or percent. Later, in Chapter 4, students will learn about part-to-part ratios, and telling the difference between the two types of ratios will become important. Eventually, students will see that a part-to-whole ratio can always be turned into a part-to-part ratio, and the other way around. For example, if 3/5 of a class are girls, then we know 2/5 must be boys, and the ratio of girls to boys is 3:2. Later, in Chapter 7, students will learn about "odds," which are also part-to-part ratios.

Probabilities are often found by running experiments. Students will learn that the sample space is the list of every possible outcome for an experiment. They'll learn that the probability of any single event (one possible result from the sample space) can be described using words like impossible, unlikely, equally likely, likely, or certain — or as a number between 0 and 1. Students will focus on two types of probability: experimental probability (based on results from actually doing the experiment) and theoretical probability (based on reasoning about what should happen). They will learn how these two types are alike and different. This wraps up the first section.

While learning probability, students will also keep studying rational numbers. They will practice changing rational numbers into decimals and percents, and will place them on a number line. This sets up important ideas for 8th grade, when students will discover that some numbers (called irrational numbers) are needed to completely fill in the number line.

With that in mind, the next section of the chapter helps students build strong number sense by reviewing fractions, decimals, and percents. This section has two main goals: first, students should be able to confidently explain how fractions, decimals, and percents relate to each other, using words, pictures, and symbols. Second, students should be able to use models to find a portion of different-sized wholes.

Understanding equivalent fractions naturally leads to comparing and estimating numbers. Students will learn to order positive and negative fractions using the number line. It's important for students to build strong estimation skills as they order and work with both positive and negative rational numbers. Finally, students will learn that percent means "per hundred" — in other words, a fraction with 100 as its denominator. In this section, percent and fraction problems will be explored using hands-on models.

The chapter finishes with a section where students keep solving real-world fraction, decimal, and percent problems — but now they begin moving from just using models to writing out number expressions. In later chapters, students will build on this by writing equations using letters to stand for unknown numbers.

Why Start With Probability?

Probability shows up everywhere in daily life, which is why it's such an important topic to study. One example is weather forecasting. Imagine you have outdoor plans for a certain day, and the weather report says there's a 70% chance of rain. Should you go ahead with your plans, or cancel them?

Another everyday example comes from sports. A baseball player's batting average is actually a probability — it tells us how likely the player is to get a hit. A batting average is calculated using real past data (hits divided by at-bats), but then it's used to predict the future: a batter with a .300 average is 50% more likely to get on base than a batter with a .200 average. So if your favorite player has a .300 batting average, that means every time they step up to bat, there's only a 30% chance they'll get a hit!

Playing the lottery is another example of probability in real life. Millions of people around the world buy lottery tickets, hoping to win a huge prize. But do they realize just how small their chances of winning really are?

Learning about probability gives students a new topic to explore while also practicing arithmetic with whole numbers and rational numbers. It also prepares students for statistics, which they'll study in Chapter 7. Probability gives us a way to describe randomness mathematically — like the differences we see in results when we run experiments or take random samples. Students will build this understanding by talking with classmates about all kinds of probability situations.

The History of Probability

In the 1400s, as mathematicians began using math to understand how things work, many started looking for ways to calculate how likely different events were. Most historians believe this area of math began with an unfinished dice game.

A French mathematician named Blaise Pascal got a letter from his friend, Chevalier de Méré, a professional gambler. Chevalier de Méré had a problem involving two different dice games.

In the first game, he made money by betting that he could roll a six at least once in four rolls of a die. From experience, he believed he would win more often than he'd lose — and he was right! He correctly figured out that the chance of rolling a six on a single roll is 1/6. But then he made a mistake: he thought that in four rolls, the chance of getting at least one six would be 4 × 1/6 = 2/3. Even though his math was wrong, he still won money over the years with this bet. Today, we know the real probability of winning this bet is about 51.8%.

Once people stopped taking that bet, de Méré changed the game. This time, he bet that if he rolled a pair of dice 24 times, he'd get "double sixes" (a six on both dice at once) at least once. It seemed like a good bet — but he started losing money. He correctly figured that the chance of rolling double sixes on one roll of two dice is 1/36. But again, he made a mistake, thinking that over 24 rolls, his chance of getting a double six would be 24/36, or 2/3.

Since he kept losing money, de Méré knew something was wrong with his reasoning, even though he couldn't figure out what. So he asked his brilliant friend Blaise Pascal for help. Pascal worked on the problem with another mathematician, Pierre Fermat, and together they solved it. Many people consider this the beginning of the mathematical study of probability.

A Closer Look: Where the Chevalier Went Wrong

Studying probability also helps us understand how to properly add and multiply fractions — which is exactly where the Chevalier made his mistake.

In the first game, he assumed that since the chance of rolling a six once is 1/6, the chance of rolling a six in two tries would be double that, or 1/3, and in four tries, 2/3. He seemed to think that every time you repeat an experiment, the probability of a good outcome just keeps adding up. But if he'd kept going with that same idea, he would have found the flaw: for three rolls, adding 1/3 + 1/3 + 1/3 equals 1 — meaning it would be certain you'd roll a six. But that's not true at all! It's completely possible to roll a die three times and never get a six. In fact, you could roll a die 500 times and still never get one (even though it's very unlikely).

When de Méré asked Pascal what was wrong, Pascal realized that probabilities don't simply add together the way de Méré thought. Instead, Pascal figured out that when you repeat an experiment, the probability of getting the same result every single time is found by multiplying, not adding. For example, to find the probability of rolling a six twice in a row, you multiply the probability of each roll together.