Chapter 1: Probability, Percent, and Rational Number Equivalence

Traditionally, seventh grade begins by reviewing everything students have learned about numbers and arithmetic. The goal is to help students become more flexible with math operations while showing them the deeper structure of the number system. But experience shows this approach often doesn't work well. Both strong and struggling students tend to find this kind of review boring—it either repeats what they already know or repeats material they aren't likely to learn better just by seeing it again. Since review is still important, it makes more sense to build that review into a new topic. We chose probability as that new topic. Probability is naturally interesting, so it grabs students' attention while showing them why arithmetic operations matter in real situations. Starting the year with probability activities also helps build a classroom culture where students think about math as a tool for investigating the real world, and where discussion and teamwork are valued from day one.

Throughout this chapter, students will review and strengthen their skills with fractions, percents, and decimals, building on what they learned in earlier grades. They'll practice recognizing when these different forms represent the same rational number. Students should understand that fractions, percents, and decimals all depend on agreeing what counts as the "whole" or the "unit." Students will also compare and order fractions, both positive and negative. The chapter ends with a section on solving percent and fraction problems—including discounts, interest, taxes, tips, and percent increase or decrease. As students work through these real-world math problems, they will begin using the basic properties of operations (sometimes called the "field axioms") in an informal way. These properties will be explained more formally in Chapter 3.

This chapter is students' first formal introduction to probability. They've already worked with data sets before, learning different ways to display data to get useful information. In seventh grade, students get their first real taste of statistics—the set of tools used to analyze data. Probability lays the theoretical groundwork for statistics, which is another reason to start seventh grade with this topic. We use statistics to make qualitative statements about what data tells us about a group of people or things. For example, a medical study testing how well Medicine A treats Disease B might conclude: "The data suggest that Medicine A is 80% effective in curing Disease B, while causing side effects in 4% of patients." Those percentages come from the basic principles of probability, which involve assigning numerical values to how likely certain outcomes are (in this case, related to Disease B).

In short, probability is a branch of math that provides the foundation for analyzing data statistically. That kind of analysis is at the heart of the scientific method: forming a hypothesis, gathering data, analyzing the data, and then estimating how likely it is that the hypothesis is true.

In the first section, students will study chance processes—experiments or situations where we know all the possible outcomes but can't predict which one will happen on any single try. Students will look at probabilities as ratios, written as fractions, decimals, or percents, comparing a "part" to a "whole." (In Chapter 4, we'll look at part-to-part relationships, where it becomes important to tell the difference between the two types.) Eventually, students will see that a part:whole relationship and a part:part relationship can be converted into each other, depending on what we want to emphasize. For example, if 3/5 of a class are girls, then we know 2/5 are boys, and the ratio of girls to boys is 3:2. Later, in Chapter 7, students will discuss odds, which are part:part relationships.

Probabilities are often figured out by running experiments and looking at the results. Students will learn that the sample space is the set of all possible outcomes for an experiment. They'll learn that the probability of any single event (a subset of 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 (empirical) probability, based on actual results from trials, and theoretical probability, based on reasoning about equally likely outcomes. They'll explore what these two types have in common and how they differ. This wraps up the first section.

While studying probability, students will keep working with rational numbers. They'll convert rational numbers into decimals and percents and locate them on the number line. This sets the stage for eighth grade, where placing numbers on a number line leads students to discover irrational numbers—numbers needed to fill in all the remaining points on the line. With this in mind, the next section of this chapter helps students strengthen their understanding of rational numbers by carefully reviewing fractions, decimals, and percents. This section has two main goals: (a) students should be able to confidently explain, using words, models, and symbols, how fractions, decimals, and percents relate to each other; and (b) students should understand and use models to find portions of different wholes.

Understanding equivalent fractions naturally leads students to think about ordering numbers and estimating. Ordering positive and negative fractions will be connected to the number line. It's important for students to build estimation skills alongside their skills in ordering and calculating with positive and negative rational numbers. Finally, students will look at percent as "per hundred"—a fraction with a denominator of 100. Percent and fraction problems in this section will be explored intuitively, using models.

The chapter ends with a section where students continue solving real-world problems involving fractions, decimals, and percents. Here, they begin moving away from relying only on models and start writing numeric expressions instead. In later chapters, they'll build on this by writing equations and proportional equations using variables.

Section 1.1: Investigate Chance Processes; Develop and Use Probability Models

The math in this chapter shows how important probability is in today's world. We run into probability constantly—for example, in weather forecasts. 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 and try another day?

Sports offer another everyday example of probability. A baseball player's batting average is really a calculation of the probability that the player will hit the ball. It's a statistic—hits divided by at-bats—based on the player's past performance. But it's also 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 baseball player has a .300 batting average, that means every time he or she steps up to the plate, there's only a 30% chance of getting a hit!

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

Probability gives students a way to explore new math ideas while reviewing and practicing arithmetic with whole numbers and rational numbers. It also prepares students for the study of statistical inference later on (Chapter 7), since probability provides the mathematical language for describing randomness—like the natural variation we see in the results of randomized experiments and random samples. Students will build this understanding by working through and discussing many different probability situations with their classmates.

The origins of probability. In the 15th century, as mathematicians began using math to understand how the world works, they turned their attention to figuring out how to calculate the likelihood of events. Most historians believe this field of study began with an unfinished dice game.

The French mathematician Blaise Pascal received a letter from his friend Chevalier de Méré, a professional gambler who was trying to make money betting on dice. Chevalier de Méré's puzzle involved two different dice games.

In the first game, he made money by betting that he could roll at least one 6 in four rolls of a single die. From experience, he believed he would win more often than he would lose. He correctly figured out that the chance of rolling a 6 on any one roll is 1/6. But then he mistakenly assumed that, over four rolls, the chance of getting at least one 6 would simply be four times that amount: 4 × 1/6 = 2/3. Even though his math was wrong, he still made a good amount of money over the years with this bet. Today, we know the actual probability of winning this bet is 1 − (5/6)⁴, which equals about 51.8%.

Once people stopped taking that bet, de Méré changed the game. This time, he bet even money (meaning he would either double his money or lose it) that he could roll double sixes at least once in 24 rolls of a pair of dice. This seemed like a smart bet, but he started losing money instead. He had correctly calculated that the chance of rolling double sixes with a pair of dice is 1/36. However, he made the same kind of mistake as before, assuming that over 24 rolls, the chance of getting at least one double-six would be 24 × 1/36 = 2/3.

Based on the real results—he was losing money—de Méré knew something was wrong with his reasoning in the second game. So he asked his famous friend Blaise Pascal for help figuring out why. Pascal shared the problem with another mathematician, Pierre Fermat, and together they solved it. This moment is often considered the beginning of the mathematical study of probability.

Extension. There's one more good reason to begin the school year reviewing fractions through probability: it gives students a real context for understanding addition and multiplication of fractions. Chevalier de Méré's mistake, in fact, came from misunderstanding how probabilities add together—and that misunderstanding cost him money.

In the first game, he reasoned that since the probability of rolling a six on one roll is 1/6, the probability of rolling a six in two rolls must be twice that, or 1/3, and in four rolls, four times that, or 2/3. He seems to have assumed that repeating an experiment increases the chance of at least one success by simply adding the probabilities together each time. But if he had pushed this reasoning just one step further, he would have seen the problem: for three rolls, his method would give 1/3 + 1/3 + 1/3 = 1, meaning he'd be claiming it was certain to roll a six within three tries. But we know that's not true—you could roll a die three times, or even 500 times, and never roll a six.

When de Méré brought his puzzle to Pascal, hoping to understand what had gone wrong (and how to fix it), Pascal likely worked through this same kind of thought experiment and quickly realized that probabilities don't simply add together in situations like this. So what was the right approach? Pascal's thinking probably went something like this: when we repeat an experiment, the probabilities of getting a favorable outcome every single time should be multiplied together, not added. For example, consider this question: what is the probability of rolling a six twice in a row? Well, the probability of rolling a s—