Chapter 1: Probability, Percent, and Rational Number Equivalence

Traditionally, seventh grade math starts by reviewing everything students have learned about numbers and arithmetic, with the goal of making students more flexible with operations while revealing the algebraic structure underneath the number system. But experience shows this approach often falls flat: both strong and struggling students find it a boring rehash of material they either already know or won't learn any better through repetition. Since review is still necessary, a better strategy is to weave that review into the introduction of a new topic. We've chosen probability for this purpose—the idea being that its real-world appeal will grab students' attention while showing why arithmetic operations matter in context. Starting the year with basic probability activities also helps build a classroom culture where math is seen as a tool for investigating the real world, and where discussion and collaboration are the norm.

Throughout this chapter, students will review and strengthen skills from earlier grades—working confidently with fractions, percents, and decimals, and recognizing when different forms of a rational number (a number that can be written as a fraction) are actually equal. Students should understand that fractions, percents, and decimals only make sense relative to an agreed-upon "whole" or unit. They 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 ideas using models and diagrams, they'll begin applying the properties of operations (sometimes called the "field axioms") informally—setting the stage for a more formal treatment in Chapter 3.

This chapter is students' first formal introduction to probability. They've already worked with data sets, exploring different ways to represent data to extract useful information. Now, in seventh grade, they get their first real exposure to statistics—the set of tools used to analyze data. Probability provides the theoretical foundation for statistics, which is yet another reason to begin the year with this topic. We use statistics to draw conclusions about what data tell us about a group or population. For example, a medical study testing a new drug might conclude: "The data suggest that Drug A cures 80% of patients with Disease B, while causing side effects in 4% of them." Those percentages come from basic principles of probability—namely, assigning numerical values to the likelihood of specific outcomes (in this case, outcomes related to contracting Disease B).

In short, probability is the branch of mathematics that underlies statistical analysis of data. That analysis, in turn, is central to the scientific method: form a hypothesis, gather data, analyze it, and estimate how likely it is that the hypothesis is correct.

In the first section, students will study chance processes—experiments or situations where all possible outcomes are known, but which specific outcome will occur on any given try is not. Students will express probabilities as ratios—represented as fractions, decimals, or percents—that compare a part to a whole. (In Chapter 4, we'll look at part-to-part relationships, where the distinction between "part:whole" and "part:part" becomes important.) 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 instance, if 3/5 of a class are girls, then 2/5 must be boys, and the ratio of girls to boys is 3:2. Later, in Chapter 7, students will study "odds," which are part:part relationships.

Probabilities are often calculated from the results of experiments. Students will learn that the sample space is the set of all possible outcomes of an experiment, and 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 (based on actual trial results) and theoretical (based on reasoning about equally likely outcomes). They'll learn how these two types are alike and how they differ. This wraps up the first section.

While studying probability, students continue building their understanding of rational numbers. They will convert rational numbers into decimals and percents and locate them on the number line—groundwork for eighth grade, where representing all numbers on a line leads to the discovery of irrational numbers (numbers that can't be written as simple fractions), which are needed to fill in the remaining gaps on the line. With this in mind, the chapter's second section focuses on solidifying rational number sense through careful review of fractions, decimals, and percents. This section has two main goals: (a) students should be able to confidently explain the relationships among equivalent fractions, decimals, and percents using words, models, and symbols; and (b) students should be able to use models to find a portion of different-sized wholes.

Understanding equivalent fractions naturally leads to questions of ordering and estimation. Ordering positive and negative fractions will be tied to their placement on the number line. It's important for students to build estimation skills alongside their work on ordering and operating with positive and negative rational numbers. Finally, students will explore percent as "per hundred"—that is, a fraction with a denominator of 100. Percent and fraction problems in this section will be introduced intuitively, using models.

The chapter closes with a section where students keep solving real-world problems involving fractions, decimals, and percents, but begin shifting from relying only on models to writing numeric expressions. In later chapters, they'll build on this by writing equations and proportions using variables.

Section 1.1: Investigating Chance Processes and Developing Probability Models

The math in this chapter reflects how important probability is in today's world. We encounter references to it constantly—weather forecasts, for example. Suppose you've made outdoor plans for a certain day, and the forecast says there's a 70% chance of rain. Should you go ahead with your plans, or reschedule? Sports offer another everyday example: a batting average is essentially a probability of a player getting a hit. It's a statistic (calculated as hits divided by at-bats) built from 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 one batting .200. So if your favorite baseball player has a .300 average, that means every time they step up to the plate, there's only a 30% chance they'll get a hit! Playing the lottery is yet another real-life example of probability at work. Millions of people worldwide buy lottery tickets hoping to win a huge jackpot—but do they realize just how small their chances of winning really are?

Probability gives students a way to explore a new mathematical topic while reviewing and practicing arithmetic with whole numbers and rational numbers. It also lays the groundwork for statistical inference, covered in Chapter 7, since probability provides the mathematical language for describing randomness—like the variation we see in the results of randomized experiments and random samples. Students build this understanding by considering and discussing, with their classmates, the outcomes of a variety of probability-based situations.

The origins of probability. In the 15th century, as mathematicians began to see math as a tool for understanding how the world works, attention turned to finding ways to calculate the likelihood of events. Most historians trace this field back to 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. De Méré's problem involved two dice games. In the first, he made money by betting he could roll a six at least once in four rolls of a die. Based on experience, he believed he'd win more often than he'd lose. He correctly reasoned that the chance of rolling a six on a single roll is 1/6. But he incorrectly concluded that across four rolls, the chance of getting at least one six would be 4 × 1/6 = 2/3. Even though his reasoning was flawed, he made a good deal of money on this bet over the years. Today we know the actual probability of winning this bet is 1 − (5/6)⁴, or about 51.8%.

Once people stopped taking that bet, de Méré changed the game: he now bet even money (meaning he'd either double his bet or lose it) that rolling a pair of dice 24 times would produce at least one double-six. This seemed like a solid bet, but he started losing money. He correctly figured that the chance of rolling double sixes on a single roll of two dice is 1/36. But he mistakenly assumed that over 24 rolls, the chance of getting at least one double-six would be 24/36, or 2/3.

Based on the hard evidence of his losses, de Méré suspected something was wrong with his reasoning in the second game. So he asked his famous friend Blaise Pascal to help him figure out the problem. Pascal, in turn, brought the puzzle to Pierre Fermat, and together they solved it—an event often considered the birth of the mathematical theory of probability.

Extension. Another good reason to begin with probability, as part of reviewing fraction arithmetic, is that it gives students a meaningful context for understanding addition and multiplication of fractions. The Chevalier's downfall came from a misunderstanding of addition. In his first bet, he reasoned that since the probability of rolling a six on one roll is 1/6, the probability of rolling a six across two rolls must be twice that, or 1/3, and across four rolls, 2/3. He seems to have assumed that repeating an experiment increases the probability of at least one favorable outcome by simply adding the probabilities together. Had he extended this reasoning one more step, he would have seen the flaw: the probability of rolling a six in three rolls would be 1/3 + 1/3 + 1/3 = 1—meaning it would be a certainty. But we know it's entirely possible to roll a die three times without getting a six; in fact, you could roll it 500 times and never see one.

When de Méré brought this puzzle to Pascal, hoping to understand what had gone wrong and how to fix it, Pascal likely worked through this same thought experiment and quickly realized that probabilities don't simply add together in situations like this. But then what's the right approach? His thinking might have gone like this: it seems clear that when an experiment is repeated, the probabilities of getting the same favorable outcome each time should be multiplied together. For example, consider this question: what is the probability of rolling a six on two rolls in a row? Well, the probability of rolling a s