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

Table of Contents

Chapter 4 Analyze Proportional Relationships and Use Them to Solve Real-World Problems (6 Weeks) ................. 2
4.0 Anchor Problem: Tasting Lemonade ................................................................................................................. 6

Section 4.1: Understand and Apply Unit Rates ................................................................................................... 13

• 4.1a Class Activity: Equivalent Ratios, Fractions, and Percents (Review from 6th grade) ............................... 14
• 4.1a Homework: Equivalent Ratios, Fractions, and Percents (Review from 6th grade) ..................................... 22
• 4.1b Class Activity: Equivalent Ratios and Proportional Relationships ............................................................. 26
• 4.1b Homework: Equivalent Ratios and Proportional Relationships .................................................................. 31
• 4.1c Class Activity: Model and Understand Unit Rates ......................................................................................... 33
• 4.1c Homework: Model and Understand Unit Rates .............................................................................................. 39
• 4.1d Class Activity: Finding Unit Rates .................................................................................................................... 40
• 4.1d Homework: Finding Unit Rates .......................................................................................................................... 44
• 4.1e Class Activity: Comparing Unit Rates .............................................................................................................. 47
• 4.1e Homework: Comparing Unit Rates .................................................................................................................... 50
• 4.1f Class Activity: Using Unit Rates to Solve Problems ...................................................................................... 52
• 4.1f Homework: Using Unit Rates to Solve Problems ........................................................................................... 55
• 4.1g Self-Assessment: Section 4.1 ............................................................................................................................... 57

Section 4.2: Construct and Analyze the Representations of Proportional Relationships ................................ 60

• 4.2a Class Activity: Graphs of Proportional Relationships .................................................................................. 61
• 4.2a Homework: Graphs of Proportional Relationships ....................................................................................... 68
• 4.2b Class Activity: More Graphs of Proportional Relationships ...................................................................... 71
• 4.2b Homework: More Graphs of Proportional Relationships .......................................................................... 77
• 4.2c Class Activity: Equations of Proportional Relationships ............................................................................ 81
• 4.2c Homework: Equations of Proportional Relationships ................................................................................. 88
• 4.2d Class Activity: More Equations of Proportional Relationships ............................................................... 92
• 4.2d Homework: More Equations of Proportional Relationships .................................................................... 96
• 4.2e Class Activity: Equations of Proportional Relationships y = kx ............................................................... 99
• 4.2e Homework: Equations of Proportional Relationships y = kx ................................................................... 104
• 4.2f Class Activity: The Representations of Proportional Relationships ..................................................... 108
• 4.2f Homework: The Representations of Proportional Relationships .......................................................... 116
• 4.2g Class Activity: Proportional and Nonproportional Relationships ........................................................ 122
• 4.2g Homework: Proportional and Nonproportional Relationships .............................................................. 131
• 4.2h Class Activity: More Proportional and Nonproportional Relationships ............................................. 137
• 4.2h Homework: More Proportional and Nonproportional Relationships .................................................. 141
• 4.2i Self-Assessment: Section 4.2 ............................................................................................................................. 143

Section 4.3: Analyze and Use Proportional Relationships and Models to Solve Real-World Problems ....... 155

• 4.3a Class Activity: Writing Proportions ................................................................................................................ 156
• 4.3a Homework: Writing Proportions ...................................................................................................................... 161
• 4.3b Class Activity: Solving Proportions ................................................................................................................ 162
• 4.3b Homework: Solving Proportions ...................................................................................................................... 167
• 4.3c Class Activity: Odds and Probability — Chance Proportions (Part-to-Part and Part-to-Whole) Problems .... 168
• 4.3c Homework: Odds and Probability — Chance Proportions ........................................................................ 172
• 4.3d Class Activity: Percent Proportions ................................................................................................................. 173
• 4.3d Homework: Write and Solve Three Percent Problems .............................................................................. 176
• 4.3e Class Activity: Proportional Constants in Markups and Markdowns .................................................. 177
• 4.3e Homework: Proportional Constants in Markups and Markdowns ....................................................... 181
• 4.3f Self-Assessment: Section 4.3 .............................................................................................................................. 183
• 4.3g Extra Practice with Part-to-Part and Part-to-Whole Relationships ...................................................... 186

---

Common Core Standards

1. Compute unit rates using ratios of fractions, including ratios that involve lengths, areas, and other measured quantities—whether the units are the same or different. For example, if someone walks 1/2 mile every 1/4 hour, you can express the unit rate as the complex fraction (1/2)/(1/4) miles per hour, which simplifies to 2 miles per hour. (Standard 7.RP.1)

1. Recognize and represent proportional relationships between quantities. (7.RP.2)
2. a. Determine whether two quantities have a proportional relationship. You can test this by checking for equivalent ratios in a table, or by graphing the values on a coordinate plane and checking whether the result is a straight line that passes through the origin (the point (0,0)). (7.RP.2a)
3. b. Identify the constant of proportionality (also called the unit rate) when it's shown in a table, graph, equation, diagram, or written description. (7.RP.2b)
4. c. Write equations to represent proportional relationships. For example, if the total cost t depends proportionally on the number of items n purchased at a fixed price p, you can write this relationship as t = pn. (7.RP.2c)
5. d. Explain what a specific point (x, y) means on the graph of a proportional relationship, paying special attention to 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 problems involving ratios and percents. Examples include simple interest, tax, markups and markdowns, tips and commissions, fees, percent increase and decrease, and percent error. (7.RP.3)

---

Chapter Overview

This chapter builds on students' existing understanding of ratio to develop a deeper understanding of proportionality—the idea that two quantities grow or shrink together at a constant rate—so students can solve both one-step and multi-step problems. The chapter opens with a review of ideas from 6th grade, along with material from Chapters 1–3 of 7th grade, before moving students toward algebraic representations (using equations and variables instead of just numbers).

Students will draw on what they've already learned to find unit rates, identify proportional constants, compare rates and situations presented in different formats, write expressions and equations, and analyze tables and graphs. The overall goal is for students to build a flexible understanding of the different ways ratios and proportions can be represented, so they can apply that understanding to a wide range of problems.

One key idea for teachers to emphasize is that a ratio can be written in several different forms, including part-to-part and part-to-whole relationships. Fractions specifically represent a part-to-whole relationship.

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.

---

Connections to Content

Prior Knowledge

Students should already be able to draw models showing part-to-part and part-to-whole relationships. Using these models, they should be able to work confidently with fractions and decimals—especially simplifying (reducing) fractions, expressing division as a fraction, converting between mixed numbers and improper fractions, and solving percent problems. Much of this material was covered in Chapters 1 and 3 and will be reviewed briefly here. Students also studied ratio extensively in 6th grade, where they learned to write ratios as fractions, with a colon (such as 3:4), or in words.

At the start of this chapter, students are expected to be able to use models to solve problems involving percents, fractions, and whole-number ratios. From there, they will connect their understanding of ratio and proportional reasoning to a variety of multi-step problems.

Future Knowledge

A solid foundation in proportional reasoning is essential for success throughout middle school and high school mathematics. [The passage ends here in the source material.]