This section introduces students to ways to represent ratios and the concept of equivalent ratios.

Students begin by using ratios and ratio language to describe collections of physical objects. Next, students draw diagrams to represent situations involving one or more ratios. They learn that simple diagrams can be useful and efficient for reasoning about ratios.

Students then make sense of equivalent ratios through concrete experiences involving recipes. They learn that scaling a recipe up or down—to create multiple batches or a fraction of a batch—produces a result that is the same as the original recipe in some important way. For example, tripling the amount of each ingredient in a drink recipe makes three times as much drink that tastes the same as the original recipe. Doubling the amount of blue water and red water in a color mixture makes twice as much water that is the same shade of purple. The taste of the drink or the shade of the color mixture are determined by the ratio of the ingredients.

In each case, students see that scaling a recipe involves multiplying each quantity by the same factor, producing a ratio of ingredients that is equivalent to that in the original recipe. They then generalize that multiplying and by the same name number produces a ratio that is equivalent to .

In this section, students reason about situations in which the quantities in the ratio have the same units and questions can be asked about the individual quantities (the parts) and their sum (the total). Students learn to use tape diagrams as a way to represent such situations. They also interpret ratios expressed in “parts” rather than standard units, such as cups, ounces, meters, and so on.

For instance, a recipe calls for 5 parts of blue paint for every 3 parts of yellow paint. If 80 cups of green paint are made, how much blue paint was used? A tape diagram, such as shown here, could be used to reason about this problem.

As they solve application problems, students have greater flexibility to choose reasoning strategies and representations (such as double number lines, tables, or tape diagrams) that make sense to them and to the problem at hand.

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.”

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.

This section introduces new ways to represent and describe equivalent ratios. Students see that double number line diagrams are useful for reasoning about equivalent ratios. For example, this diagram shows that the ratios , , , , and are equivalent. Mixing cranberry juice and soda water in these amounts will create drinks that taste the same.

Students reason about equivalent ratios where one quantity has a value of 1 and express them using language such as:

• “2.5 cups of cranberry juice for 1 cup of soda water”
• “2.5 cups of cranberry juice per cup of soda water”

They relate the phrases “at this rate” and “at the same rate” to equivalent ratios in contexts, such as constant speed, uniform pricing, and recipes.

Later in the section, students use tables to reason about situations involving equivalent ratios. They see that the values in a table don’t need to be listed in order, so they can choose the multipliers strategically. This makes a table a more flexible reasoning tool than a double number line.

For instance, to find the pay for 8 hours given a pay of $90 for 5 hours, students may use division to calculate the pay for 1 hour or find a multiplier that relates the values in two rows.

While certain ways of working with tables are presented, students can use other representations for support. They should be encouraged to explain their choice and compare the efficiency of different methods.

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.