7WB4 - 1

Table of Contents

Chapter 4: Study Proportional Relationships and Use Them to Solve Real-World Problems (6 Weeks) ......... 2

4.0 Warm-Up Problem: Tasting Lemonade ......... 6

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

• 4.1a Class Activity: Equal Ratios, Fractions, and Percents (Review from 6th grade) ......... 14
• 4.1a Homework: Equal Ratios, Fractions, and Percents (Review from 6th grade) ......... 22
• 4.1b Class Activity: Equal Ratios and Proportional Relationships ......... 26
• 4.1b Homework: Equal Ratios and Proportional Relationships ......... 31
• 4.1c Class Activity: Show and Understand Unit Rates ......... 33
• 4.1c Homework: Show 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-Check: Section 4.1 ......... 57

Section 4.2: Build and Study Ways to Show 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: Ways to Show Proportional Relationships ......... 108
• 4.2f Homework: Ways to Show Proportional Relationships ......... 116
• 4.2g Class Activity: Proportional and Non-Proportional Relationships ......... 122
• 4.2g Homework: Proportional and Non-Proportional Relationships ......... 131
• 4.2h Class Activity: More Proportional and Non-Proportional Relationships ......... 137
• 4.2h Homework: More Proportional and Non-Proportional Relationships ......... 141
• 4.2i Self-Check: Section 4.2 ......... 143

Section 4.3: Study 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 Chance — Part-to-Part and Part-to-Whole Problems ......... 168
• 4.3c Homework: Odds and Chance — Proportions ......... 172
• 4.3d Class Activity: Percent Problems Using Proportions ......... 173
• 4.3d Homework: Write and Solve Three Percent Problems ......... 176
• 4.3e Class Activity: Proportional Numbers in Markups and Markdowns ......... 177
• 4.3e Homework: Proportional Numbers in Markups and Markdowns ......... 181
• 4.3f Self-Check: Section 4.3 ......... 183
• 4.3g Extra Practice with Part-to-Part and Part-to-Whole Relationships ......... 186

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7WB4 - 2

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

What Students Will Learn

1. Find unit rates (how much of one thing there is for just one of another thing) when working with ratios of fractions, including lengths, areas, and other measurements. For example, if someone walks 1/2 mile in 1/4 hour, you can figure out they walk 2 miles in one hour. This is called the unit rate.

1. Notice and describe proportional relationships — this means two amounts that grow or shrink together in a matching pattern.
2. Figure out whether two amounts are proportional. You can check this by testing if their ratios stay equal in a table, or by graphing them on a coordinate grid and seeing if you get a straight line that passes through the starting point (0, 0).
3. Find the constant of proportionality (also called the unit rate) in tables, graphs, equations, pictures, and word descriptions of proportional relationships.
4. Write proportional relationships as equations. For example, if the total cost (t) depends on the number of items (n) you buy at a steady price (p), you can write this as: t = pn.
5. Explain what a point (x, y) means on a graph of a proportional relationship — especially the points (0, 0) and (1, r), where r stands for the unit rate.

1. Use proportional relationships to solve problems with several steps, involving ratios and percents. Examples include simple interest, tax, markups and markdowns, tips, fees, percent increase and decrease, and percent error.

Chapter Overview

This chapter builds on what students already know about ratios and helps them understand proportionality — a special kind of relationship between two amounts. Students will use this understanding to solve problems with one step or several steps.

The chapter starts by reviewing ideas from 6th grade and from Chapters 1–3 of 7th grade. Then it helps students move toward using algebra to describe these ideas. Students will use what they've already learned to find unit rates, find proportional constants, compare rates and situations shown in different ways, write expressions and equations, and study tables and graphs. The goal is for students to feel comfortable using many different ways to show ratios and proportions so they can solve all kinds of problems.

One important idea for students to understand: a ratio can be written in different ways, including part-to-part ratios (comparing one part to another part) and part-to-whole ratios (comparing one part to the whole amount). Fractions always show a part-to-whole relationship.

Vocabulary

Bar/tape model, comparison model, constant of proportionality (also called 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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Connections to Other Learning

What Students Should Already Know

Students should already be able to draw models showing part-to-part and part-to-whole relationships. Using these models, students should be able to work confidently with fractions and decimals — especially simplifying fractions, writing division as a fraction, changing mixed numbers into improper fractions, and solving percent problems. Many of these skills were covered in Chapters 1 and 3 and will get a quick review in this chapter. Students also studied ratios a lot in 6th grade, learning to write ratios as fractions, with a colon (like 3:4), or in words.

At the start of this chapter, students are expected to already be able to use models to solve percent, fraction, and whole-number ratio problems. From there, students will connect their ratio and proportion thinking to many different multi-step problems.

What Students Will Need This For Later

Having a strong understanding of proportions is very important for success in middle school and high school math classes that come after this one.