Chapter 1: Probability, Percent, Rational Number Equivalence (3–4 Weeks)

Utah Core Standards

Number Sense:

1. Convert a rational number to a decimal using long division. Understand that the decimal form of a rational number either ends in repeating zeros or eventually repeats a pattern of digits. (7.NS.2d)

1. Solve real-world and mathematical problems using addition, subtraction, multiplication, and division with rational numbers. (7.NS.3)

Probability and Statistics:

1. Understand that the probability of a chance event is a number between 0 and 1 that shows how likely the event is to happen. Larger numbers mean greater likelihood: a probability near 0 means the event is unlikely, a probability near ½ means the event is just as likely to happen as not, and a probability near 1 means the event is likely. (7.SP.5)

1. Estimate the probability of a chance event by collecting data on the process that produces it and observing how often it happens over many trials (its long-run relative frequency). Use a known probability to predict roughly how often an event will occur. For example, if you roll a number cube 600 times, you'd expect a 3 or a 6 to come up about 200 times — though probably not exactly 200. (7.SP.6)

Equations and Expressions:

1. Solve multi-step real-life and mathematical problems involving positive and negative rational numbers in any form (whole numbers, fractions, decimals), choosing appropriate tools. Apply the properties of operations to calculate with numbers in any form; switch between forms when it's useful; and check whether an answer makes sense using mental math and estimation. For example: if a woman earning $25 an hour gets a 10% raise, she gains an extra 1/10 of her hourly wage — $2.50 — making her new wage $27.50. Or: if you want to center a towel bar that's 9¾ inches long on a door that's 27½ inches wide, you'll need to leave about 9 inches of space on each side; this estimate lets you check your exact calculation. (7.EE.3)

Chapter 1 Summary

This chapter opens with a brief introduction to probability, using it as a way to review and practice working with whole numbers and fractions. Along with learning basic counting methods and how to list all possible outcomes (called a sample space), students will learn to tell the difference between theoretical probability (what should happen based on math) and experimental probability (what actually happens when you test something out repeatedly). Starting the year with probability activities serves two purposes: it builds a habit of using math to explore real-world situations, and it helps create a classroom culture built on discussion and teamwork.

Throughout the chapter, students will also review and strengthen their skills with fractions, percents, and decimals from earlier grades. A key idea to understand is that fractions, percents, and decimals are all ways of describing a part in relation to a whole. Students will also practice comparing and ordering fractions, including negative ones. The chapter wraps up with a section focused on solving percent and fraction problems — including discounts, interest, taxes, tips, and percent increase or decrease.

Vocabulary: chance, decimal, experimental probability, fraction, frequency, outcome, percent, probability, ratio, theoretical probability

Connections to Content

Prior Knowledge

In earlier grades, students learned about ratios — relationships between quantities. While students should already be familiar with two types of ratio relationships — part-to-whole and part-to-part — they likely haven't fully mastered the difference yet. This chapter focuses only on part-to-whole relationships, since probability, fractions, decimals, and percents are all part-to-whole comparisons. Later, in Chapter 4, students will practice telling part-to-part and part-to-whole relationships apart, and they'll explore "odds," which are an example of a part-to-part relationship.

Students have already used all four operations — addition, subtraction, multiplication, and division — with fractions and decimals in previous grades. They should also be familiar with representing fractions, percents, and decimals using number lines and bar (or "tape") diagrams.

In 6th grade, students learned to place both positive and negative numbers on a number line, but they did not yet perform operations with negative numbers. That skill begins in 7th grade, starting in Chapter 2.

Future Knowledge

This chapter offers only an introduction to probability — students will explore it in greater depth in Chapter 7. The ideas they learn here about chance processes, along with theoretical and experimental probability, will serve as a foundation for that later, more advanced work.