In this section, students further explore “rates per 1” and solve various problems involving rates.

Students observe that using rates per 1 is a helpful way to make comparisons. They learn that a rate per 1 is a “unit rate,” and that each ratio has two associated unit rates: and . If 8 pounds of apples cost 6 dollars, then 1 pound costs dollar and 1 dollar buys pounds. In interpreting unit rates in context, students see that one unit rate might be more helpful than the other, depending on the question.

Students generalize that when two ratios have the same unit rate, the ratios are equivalent. For instance, 3,000 meters in 20 minutes and 2,550 meters in 17 minutes are equivalent ratios since they both have a unit rate of 150 meters per minute.

A table with two columns. First column, distance in meters, 3000, 1500, 150. Second column, time in minutes, 20, 10, 1. Outside the table, arrows between first and second row say "times one-half." Arrows between second and third row say "times one-tenth." 

A table with two columns. First column, distance in meters, 2550, 150. Second column, time in minutes, 17, 1. Outside the table, arrows between first and second row say "times one-seventeenth."

Likewise, when ratios are equivalent, they have the same unit rates. For instance, , , and are equivalent ratios, so they have the same unit rates, and .

In a table of equivalent ratios, each unit rate is a factor that relates the values in the two columns.

As students progress through the section, they encounter problems with less scaffolding. In the last lesson, students reason about the movement of two objects relative to each other. This lesson is optional because it exceeds the expectations of the standards.

In this section, students make sense of percentages as rates per 100 and solve problems involving percentages.

Students begin by reasoning about percentages of 100 and of 1. Then they work with percentages of other quantities, paying attention to what 100% represents in each situation. To reinforce percentages as rates, double number line diagrams are the primary representation used initially. Later, students also make sense of percentages using tables and tape diagrams.

Tape diagrams help connect benchmark percentages (10%, 25%, 50%, and 75%) to benchmark fractions. They allow students to better see, for example, that “75% of a number” is “ of that number” and can be found by multiplying the number by .

Students go on to solve various problems involving percentages. When they encounter numbers that are difficult to manipulate mentally or by using diagrams (such as finding 67% of 3,200), students generalize the process for calculating any percentage of any number.

In the last lesson, students find a general way to determine what percentage a number is of another number (such as finding what percentage 12 is of 75). Generalizing this calculation is beyond the expectations of the course, so this lesson is optional.

A note about the size of percentages:

Percentages can be used to specify a part of a whole and to describe multiplicative comparisons between two quantities. When used for the former, percentages are often limited to 100% (for example, Jada drank 90% of the water in her bottle). When used for the latter, however, it makes sense that they can exceed 100% (for example, Jada drank 300% as much water as Diego).

A note about percentages and fractions:

Percentages are rates, not numbers. In these materials, statements such as “75% equals ” are avoided as they equate rates and numbers. Instead, connections between percentages and fractions are made by saying, for instance, that “75% of a number” is equal to “ of that number.”

This section extends students’ knowledge of units of measurement. It prompts them to reason about ratios and rates to perform unit conversion, including across different measurement systems.

Students begin by grounding their perception of standard measurement units in benchmark objects. For instance, they relate 1 foot to the length of a ruler and 1 milliliter to the volume of liquid in an eyedropper. This experience builds students’ intuition for the relative sizes of various units, preparing them to better see relationships between units as ratios.

Next, students recall that it takes more of a smaller unit than a larger unit to measure the same quantity. For example, more feet than yards are needed to measure the same length. Although this idea is a foundation for converting units by reasoning about ratios, the lesson is optional because it revisits ideas from earlier grades.

In the last lesson of the section, students learn that two measurements of the same thing given in different units (such as 10 kilograms and 22 pounds) form a ratio. They see that conversion across units can be done by finding equivalent ratios and using familiar representations, and how “rates per 1” can be helpful in solving problems.

In the final section, students have the opportunity to apply their thinking from throughout the unit. As this is a short section followed by an End-of-Unit Assessment, there are no section goals or checkpoint questions.