# Unit 4 Proportional Relationships and Percentages

• Students began their work with ratios, rates, and unit rates in grade 6, representing them with expressions, tape diagrams, double number line diagrams, and tables. They used these to reason about situations involving color mixtures, recipes, unit price, discounts, constant speed, and measurement conversions. They extended their understanding of rates to include percentages as rates per 100, reasoning about situations involving whole-number percentages. They did not use the terms “proportion” and “proportional relationship” in grade 6.

A proportional relationship is a collection of equivalent ratios, and such collections are objects of study in grade 7. In previous grade 7 units, students worked with scale factors and scale drawings, and with proportional relationships and constants of proportionality. Although students have learned how to compute quotients of fractions in grade 6, these first units on scaling and proportional relationships do not require such calculations, allowing the new concept (scaling or proportional relationship) to be the main focus.

In this unit, students deepen their understanding of ratios, scale factors, unit rates (also called constants of proportionality), and proportional relationships, using them to solve multi-step problems that are set in a wide variety of contexts that involve fractions and percentages.

In the first section of the unit, students extend their use of ratios and rates to problems that involve computing quotients of fractions, and interpreting these quotients in contexts such as scaling a picture or running at constant speed (MP2). They use long division to write fractions presented in the form abas decimals, e.g., 1130=0.36⎯⎯⎯.

The section begins by revisiting scale factors and proportional relationships, each of which has been the focus of a previous unit. Both of these concepts can be used to solve problems that involve equivalent ratios. However, it is often more efficient to view equivalent ratios as pairs that are in the same proportional relationship rather than seeing one pair as obtained by multiplying both entries of the other by a scale factor. From the scaling perspective, to see that one ratio is equivalent to another or to generate a ratio equivalent to a given ratio, a scale factor is needed—which may be different for each pair of ratios in the proportional relationship. From the proportional relationship perspective, all that is needed is the constant of proportionality—which is the same for every ratio in the proportional relationship.

The second section of the unit is about percent increase and decrease. Students consider situations for which percentages can be used to describe a change relative to an initial amount, e.g., prices before and after a 25% increase. They begin by considering situations with unspecified amounts, e.g., matching tape diagrams with statements such as “Compared with last year’s strawberry harvest, this year’s strawberry harvest increased by 25%”. They next consider situations with a specified amount and percent change, or with initial and final amounts, using double number line diagrams to find the unknown amount or percent change. Next, they use equations to represent such situations, using the distributive property to show that different expressions for the same amount are equivalent, e.g., x−0.25x=0.75x. So far, percent change in this section has focused on whole-number rates per 100, e.g., 75%. The last lesson asks students to compute fractional percentages of given amounts.

In the third section of the unit, students begin by using their abilities to find percentages and percent rates to solve problems that involve sales tax, tip, discount, markup, markdown, and commission (MP2). The remaining lessons of the section continue the focus on situations that can be described in terms of percentages, but the situations involve error rather than change—describing an incorrect value as a percentage of the correct value rather than describing an initial amount as a percentage of a final amount (or vice versa).

The last section of the unit consists of a lesson in which students analyze news items that involve percent increase or decrease. In small groups, students identify important quantities in a situation described in a news item, use diagrams to map the relationship of of the quantities, and reason mathematically to draw conclusions (MP4). This is an opportunity to choose an appropriate type of diagram (MP5), to state the meanings of symbols used in the diagram, to specify units of measurement, and to label the diagram accurately (MP6). Each group creates a display to communicate its reasoning and critiques the reasoning shown in displays from other groups (MP3).

These materials follow specific conventions for the use of language around ratios, rates, and proportional relationships. Please see the unit narrative for the second unit to read about those conventions.