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

Utah Core Standards

Number Sense:

1. Change a rational number (a number that can be written as a fraction) into a decimal by doing long division. Know that the decimal form either stops (ends in zeros) or repeats a pattern forever.

1. Solve real-world and math problems using addition, subtraction, multiplication, and division with rational numbers.

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. A bigger number means the event is more likely. A probability close to 0 means the event probably won't happen. A probability around ½ means the event is just as likely to happen as not. A probability close to 1 means the event will probably happen.

1. Estimate the probability of a chance event by gathering data about it and looking at how often it actually happens over many tries (called the relative frequency). Also, use a known probability to predict how often something should happen. 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.

Equations and Expressions:

1. Solve multi-step real-life and math problems using positive and negative rational numbers in any form — whole numbers, fractions, or decimals — choosing the best tools and methods. Use the properties of operations to calculate, and switch between fraction, decimal, and percent forms when it's helpful. Check whether your answer makes sense using mental math and estimation.

Example 1: If a woman earning $25 an hour gets a 10% raise, she earns an extra 1/10 of her pay per hour — that's $2.50 — making her new hourly wage $27.50.

Example 2: If you want to center a towel bar that is 9¾ inches long on a door that is 27½ inches wide, you'll need to leave about 9 inches of space on each side. You can use this estimate to check your exact answer.

Chapter 1 Summary

This chapter starts with a short introduction to probability — the study of how likely something is to happen. This topic is a great way to review and practice working with whole numbers and fractions. Along the way, students will learn basic counting methods and how to list all the possible outcomes of an event (called a sample space). Students will also learn the difference between theoretical probability (what you'd expect to happen based on math) and experimental probability (what actually happens when you test it out).

There are two big reasons this chapter starts the year with probability. First, it helps students see math as a tool for exploring real-world situations. Second, doing hands-on probability activities early in the year helps build a classroom culture where students talk, share ideas, and work together.

Throughout the chapter, students will also review and strengthen their skills with fractions, percents, and decimals — skills they began learning in earlier grades. It's important to understand that fractions, percents, and decimals are all ways of describing a part of a whole. Students will also practice comparing and ordering fractions, including both positive and negative ones. The chapter ends with a section on solving percent and fraction problems, including those involving discounts, interest, taxes, tips, and percent increase or decrease.

Vocabulary

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

Connections to Other Content

What Students Already Know

In earlier grades, students learned about ratios — ways of comparing numbers. They should already understand the difference between a part-to-whole relationship (like "3 out of 5 students") and a part-to-part relationship (like "3 boys for every 2 girls"), though this understanding may not be fully solid yet. In this chapter, we focus only on part-to-whole relationships, since probability, fractions, decimals, and percents are all part-to-whole ideas. Later, in Chapter 4, students will practice telling part-to-part and part-to-whole relationships apart, and they'll learn about odds, which are a part-to-part relationship.

Students have already used addition, subtraction, multiplication, and division with fractions and decimals in earlier grades. They should also be familiar with using number lines and bar (or "tape") models to picture fractions, percents, and decimals.

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

What's Coming Next

As students move through this chapter, they'll begin by studying probability. (This chapter only introduces the topic — students will explore probability in much more depth in Chapter 7.) The ideas they learn here about chance events, along with theoretical and experimental probability, will be built upon later.