Chapter 4: Analyze Proportional Relationships and Use Them to Solve Real-World Problems (6 Weeks)

Table of Contents

4.0 Anchor Problem: Tasting Lemonade

Section 4.1: Understand and Apply Unit Rates

• 4.1a Class Activity & Homework: Equivalent Ratios, Fractions, and Percents (Review from 6th grade)
• 4.1b Class Activity & Homework: Equivalent Ratios and Proportional Relationships
• 4.1c Class Activity & Homework: Model and Understand Unit Rates
• 4.1d Class Activity & Homework: Finding Unit Rates
• 4.1e Class Activity & Homework: Comparing Unit Rates
• 4.1f Class Activity & Homework: Using Unit Rates to Solve Problems
• 4.1g Self-Assessment: Section 4.1

Section 4.2: Construct and Analyze the Representations of Proportional Relationships

• 4.2a Class Activity & Homework: Graphs of Proportional Relationships
• 4.2b Class Activity & Homework: More Graphs of Proportional Relationships
• 4.2c Class Activity & Homework: Equations of Proportional Relationships
• 4.2d Class Activity & Homework: More Equations of Proportional Relationships
• 4.2e Class Activity & Homework: Equations of Proportional Relationships y = kx
• 4.2f Class Activity & Homework: The Representations of Proportional Relationships
• 4.2g Class Activity & Homework: Proportional and Nonproportional Relationships
• 4.2h Class Activity & Homework: More Proportional and Nonproportional Relationships
• 4.2i Self-Assessment: Section 4.2

Section 4.3: Analyze and Use Proportional Relationships and Models to Solve Real-World Problems

• 4.3a Class Activity & Homework: Writing Proportions
• 4.3b Class Activity & Homework: Solving Proportions
• 4.3c Class Activity & Homework: Odds and Probability — Chance Proportions (part-to-part and part-to-whole) Problems
• 4.3d Class Activity & Homework: Percent Proportions (Write and Solve Three Percent Problems)
• 4.3e Class Activity & Homework: Proportional Constants in Markups and Markdowns
• 4.3f Self-Assessment: Section 4.3
• 4.3g Extra Practice with Part-to-Part and Part-to-Whole Relationships

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Standards Covered in This Chapter

1. Find unit rates using fractions. This includes ratios of lengths, areas, and other measurements — even when the units are different. For example, if someone walks half a mile in a quarter of an hour, you can write this as the complex fraction (1/2) ÷ (1/4) miles per hour, which equals 2 miles per hour. (Standard 7.RP.1)

1. Recognize and represent proportional relationships — meaning two quantities that grow or shrink together at a constant rate. (Standard 7.RP.2)
2. a. Decide whether two quantities are proportional. You can test this by checking for equivalent ratios in a table, or by graphing the values and seeing if they form a straight line that passes through the origin (the point where the x-axis and y-axis meet, at (0,0)). (7.RP.2a)
3. b. Identify the constant of proportionality — also called the unit rate — in tables, graphs, equations, diagrams, and word descriptions. (7.RP.2b)
4. c. Write equations for proportional relationships. For example, if the total cost t depends on the number of items n you buy at a fixed price p, you can write the equation as t = pn. (7.RP.2c)
5. d. Explain what a point (x, y) means on the graph of a proportional relationship, especially the points (0, 0) and (1, r), where r stands for the unit rate. (7.RP.2d)

1. Use proportional relationships to solve multi-step ratio and percent problems, such as simple interest, tax, markups and markdowns, tips and commissions, fees, percent increase and decrease, and percent error. (Standard 7.RP.3)

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Chapter Overview

This chapter builds on what you already know about ratios and helps you understand proportionality — the idea that two quantities increase or decrease together at a steady rate. You'll use this idea to solve both one-step and multi-step problems.

The chapter starts by reviewing ideas from 6th grade, along with earlier chapters in this book, before moving into algebra-based ways of representing these relationships. You'll use what you've already learned to find unit rates, identify proportional constants, compare rates and situations shown in different formats, write expressions and equations, and read tables and graphs. By the end of the chapter, the goal is for you to feel comfortable using many different methods — tables, graphs, equations, and words — to represent and solve ratio and proportion problems.

One important idea to keep in mind: a ratio can be written in several different ways, including as a part-to-part comparison or a part-to-whole comparison. Fractions specifically represent a part-to-whole relationship (for example, the fraction 3/4 compares a part to the whole amount).

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Key Vocabulary

Bar/tape model, comparison model, constant of proportionality (proportional constant), equation, part-to-part ratio, part-to-whole ratio, percent change, proportional relationship, proportion, rate, table, ratio, and unit rate.

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How This Chapter Connects to Other Learning

What You Should Already Know

Before starting this chapter, you should be able to draw models showing part-to-part and part-to-whole relationships. Using these models, you should be able to work confidently with fractions and decimals — especially simplifying fractions, writing division as a fraction, converting between mixed numbers and improper fractions, and solving percent problems. Many of these skills were covered in Chapters 1 and 3, and this chapter will review them briefly. You also studied ratios extensively in 6th grade, where you learned to write ratios as fractions, with a colon (like 3:4), or in words.

At the start of this chapter, it's assumed you can already use models to solve percent, fraction, and whole-number ratio problems. From there, you'll connect ratio and proportional thinking to a variety of multi-step problems.

Why This Matters for the Future

Having a strong understanding of proportions is essential for success throughout middle school and high school math.