Learning About Ratios and Proportions

This is Chapter 4. It takes 6 weeks to learn. In this chapter, you will learn about proportional relationships. That's a fancy way of saying "things that grow together in the same pattern."

What You Will Learn

Here are the big ideas in this chapter:

1. Unit rates. You will learn to figure out a rate for "one" of something. For example, if you walk half a mile in a quarter of an hour, you can figure out how many miles you walk in one whole hour. (The answer is 2 miles per hour!)

1. Spotting proportional relationships. You will learn how to tell if two things grow together in a matching pattern. You can check this with a table of numbers or by drawing a graph. If the graph makes a straight line that passes through the very start (called the origin), then it's proportional!

1. Finding the constant number. In a proportional relationship, there's a special number called the constant of proportionality. This number is like the "rule" connecting the two things. You will find this number in tables, graphs, equations, and word problems.

1. Writing equations. You will learn to write a math sentence (equation) for a proportional relationship. For example, if buying more items costs more money at the same price each time, you can write an equation for that.

1. Understanding graph points. You will learn what each point on a graph means. Two special points are (0, 0) — the very start — and (1, r) — where r is the unit rate.

1. Solving real problems. You will use everything you learn to solve real-life problems, like figuring out tax, discounts (markdowns), extra charges (markups), tips, and how much something has increased or decreased.

Chapter Overview

This chapter builds on what you learned before about ratios (comparing two amounts). It helps you move from just comparing numbers to using math sentences (equations) to solve problems. You will use what you already know to:

• find unit rates
• find the special constant number
• compare rates written in different ways
• write expressions and equations
• read tables and graphs

The goal is for you to get really comfortable using ratios and proportions to solve all kinds of problems.

Important tip: A ratio can be written in different ways. Some ratios compare a part to another part (part:part). Some ratios compare a part to the whole thing (part:whole). Fractions always show a part:whole relationship.

New Words You Will Learn

Here are some important words in this chapter:

• Bar/tape model – a drawing using bars to show amounts
• Comparison model – a drawing that compares amounts
• Constant of proportionality – the special number that connects two amounts that grow together (also called the proportional constant)
• Equation – a math sentence with an equal sign
• Part-to-part ratio – comparing one part to another part
• Part-to-whole ratio – comparing one part to the whole group
• Percent change – how much something goes up or down, shown as a percent
• Proportional relationship – when two things grow together in a matching pattern
• Proportion – a math sentence showing two ratios are equal
• Rate – a comparison of two different kinds of things, like miles and hours
• Table – a chart of numbers
• Ratio – a comparison of two amounts
• Unit rate – a rate for just "one" of something

What You Should Already Know

Before starting this chapter, you should be able to:

• draw pictures (models) that show part-to-part and part-to-whole relationships
• work with fractions and decimals, including making fractions simpler
• show division as a fraction
• change mixed numbers into fractions (and back)
• solve percent problems

You learned a lot about ratios last year. You learned to write ratios as fractions, with a colon (like 3:4), or using words.

At the start of this chapter, we expect you can already use drawings to solve simple percent, fraction, and whole number ratio problems. Now, you will connect these ideas to bigger, multi-step problems.

Why This Matters Later

Understanding proportions well now will help you do well in math for many years to come, all through middle school and high school.