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...

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...

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...

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...

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....

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...