In this lesson, students encounter situations in which the sum of two or more quantities is meaningful in context. So far, students have worked with ratios of quantities where the units are the same (such as cups to cups) and ratios of quantities where the units are different (such as miles to hours). In some instances of the former, it makes sense to ask questions about the total amount. For example, mixing 3 cups of yellow paint and 2 cups of blue paint gives 5 cups of green paint. (Note that the sum of quantities in the same unit doesn’t always make sense in context. For instance, 3 cups of water mixed with 2 cups of dry oatmeal will not make 5 cups of soggy oatmeal.)

Students begin by using snap cubes to represent a ratio. For instance, 3 blue cubes and 2 yellow cubes can represent blue paint and yellow paint in a ratio of 3 to 2. Students see that each cube can represent a variety of amounts and use this insight to reason about individual quantities (the parts) in the ratio and the total quantity (the whole). For example, if each cube represents 10 ml, then 30 ml of blue paint and 20 ml of yellow paint would be mixed, producing 50 ml of green paint. 

Students then learn to use tape diagrams to represent and reason about such ratios—sometimes called “part-part-whole” ratios—and solve problems. They also see that ratios can be expressed in “parts” (such as “2 parts of glue and 1 part of water”) rather than in specific units (such as cups or liters).

As they interpret statements about ratios in context and consider ways to represent them, students practice making sense of problems, persevering in solving them (MP1), and reasoning quantitatively and abstractly (MP2). 

This lesson includes an optional activity that allows students to invent problems that involve part-part-whole ratios.