Lesson 10

This lesson is the first of two that introduce students to percentages as a rate per 100 (MP6) and the ways they are used to describe different types of situations.

Percentages are commonly used in two ways:

1. To describe a part of a whole. For example, “Jada drank 25% of the bottle of water.” In this case, the percentage expressing the amount consumed is not bigger than 100% because it refers to a part of a whole, as shown in the diagram below.

   

   

2. To describe the size of one quantity as a percentage of another quantity. For example, “Jada drank 300% as much water as Diego did.” In this case, there is no restriction on the size of the percentage, because the percentage is describing a multiplicative comparison between two quantities, as shown below.

   

   

In the first usage there is a single quantity and we are describing a part of it; in the second usage we are comparing two quantities. Students may have prior exposure to percentages, but are likely to have only encountered the first usage and might not be able to make sense of percentages above 100% or those used in comparative contexts. This lesson exposes students to both applications of percentages.

Money is the main context for exploring percentages in this lesson and the warm up asks students to convert between dollars and cents providing an opportunity for the teacher to assess students’ current abilities.

For the first several lessons exploring percentages, double number lines are the primary representation presented to students. This choice is intended to strongly communicate that we are working with percent rates, and that students can and should use all of the reasoning they have developed to deal with equivalent ratios and rates when dealing with rates per 100. That said, if students prefer to reason using tables or by multiplying or dividing by unit rates, they should not be discouraged from doing so.