Information for course "Unit 3 - Unit Rates And Percentages"

Description: In the previous unit, students began to develop an understanding of ratios and rates. They started to describe situations using terms such as “ratio,” “rate,” “equivalent ratios,” “per,” “constant speed,” and “constant rate” (MP6). They understood specific instances of the idea that a:b is equivalent to every other ratio of the form sa:sb, where s is a positive number. They learned that “at this rate” or “at the same rate” signals a situation that is characterized by equivalent ratios. Although the usefulness of ratios of the form ab:1 and 1:ba was highlighted, the term “unit rate” was not introduced.

In this unit, students find the two values ab and ba that are associated with the ratio a:b, and interpret them as rates per 1. For example, if a person walks 13 meters in 10 seconds at a constant rate, that means they walked at a speed of 1310 meters per 1 second and a pace of 1013 seconds per 1 meter.

Students learn that one of the two values (ab or ba) may be more useful than the other in reasoning about a given situation. They find and use rates per 1 to solve problems set in contexts (MP2), attending to units and specifying units in their answers. For example, given item amounts and their costs, which is the better deal? Or given distances and times, which object is moving faster? Measurement conversions provide other opportunities to use rates.

Students observe that if two ratios a:b and c:d are equivalent, then ab=cd. The values ab and cd are called unit rates because they can be interpreted in the context from which they arose as rates per unit. Students note that in a table of equivalent ratios, the entries in one column are produced by multiplying a unit rate by the corresponding entries in the other column. Students learn that “percent” means “per 100” and indicates a rate. Just as a unit rate can be interpreted in context as a rate per 1, a percentage can be interpreted in the context from which it arose as a rate per 100. For example, suppose a beverage is made by mixing 1 cup of juice with 9 cups of water. The percentage of juice in 20 cups of the beverage is 2 cups and 10 percent of the beverage is juice. Interpreting the 10 as a rate: “there are 10 cups of juice per 100 cups of beverage” or, more generally, “there are 10 units of juice per 100 units of beverage.” The percentage—and the rate—indicate equivalent ratios of juice to beverage, e.g., 2 cups to 20 cups and 10 cups to 100 cups.

In this unit, tables and double number line diagrams are intended to help students connect percentages with equivalent ratios, and reinforce an understanding of percentages as rates per 100. Students should internalize the meaning of important benchmark percentages, for example, they should connect “75% of a number” with “34 times a number” and “0.75 times a number.” Note that 75% (“seventy-five per hundred”) does not represent a fraction or decimal (which are numbers), but that “75% of a number” is calculated as a fraction of or a decimal times the number.

Work done in grades 4 and 5 supports learning about the concept of a percentage. In grade 5, students understand why multiplying a given number by a fraction less than 1 results in a product that is less than the original number, and why multiplying a given number by a fraction greater than 1 results in a product that is greater than the original number. This understanding of multiplication as scaling comes into play as students interpret, for example,

35% of 2 cups of juice as 35100⋅2 cups of juice. 250% of 2 cups of juice as 250100⋅2 cups of juice.