In this lesson, students use two more familiar representations, tape diagrams and tables, to reason about percentages.

In prior grades, students used tape diagrams to reason about fractions and multiplicative comparisons. Here, tape diagrams can serve similar purposes, enabling students to see connections between percentages and fractions.

For example, this tape diagram shows that 25% of a whole is the same as of that whole since 25% of the whole is one part when 100% of the whole is divided into four equal parts.

Students observe that when reasoning about percentages, it is important to indicate the 100%, just as it is important to indicate the whole when working with fractions. This is an opportunity to practice attending to precision (MP6).

If we are finding 25% of 60, then we can assign 100% to 60 and represent this on the tape diagram, which can then help us reason about 25% of 60.

A table can also be used to find 25% of 60. The structure of both a double number line diagram and a table encourages students to reason about equivalent ratios. Students may find the latter to be more flexible and efficient than the former, however, especially when dealing with percentages greater than 100%, or when finding the value for 100% when a percentage is known.

For instance, if 125% of a fundraising goal is \$50, what is the fundraising goal? Here is one way to find 100% of the goal: