6.2 GeoGebra Illustrative Mathematics - Introducing Ratios

This unit introduces students to ratios and equivalent ratios. It builds on previous experiences students had with relating two quantities, such as converting measurements starting in grade 3, multiplicative comparison in grade 4, and interpreting multiplication as scaling in grade 5. The work prepares students to reason about unit rates and percentages in the next unit, proportional relationships in grade 7, and linear relationships in grade 8.

First, students learn that a ratio is an association between two quantities, for instance, “There are 3 pencils for every 2 erasers.” Students use sentences, drawings, or discrete diagrams to represent ratios that describe collections of objects and recipes.

Next, students encounter equivalent ratios in terms of multiple batches of a recipe. “Equivalent” is first used to describe a perceivable sameness of two ratios, such as two mixtures of drink mix and water that taste the same, or two mixtures of yellow and blue paint that make the same shade of green. Later, “equivalent” acquires a more precise meaning: All ratios that are equivalent to  can be made by multiplying both  and  by the same non-zero number (non-negative, for now).

Students then learn to use double number line diagrams and tables to represent and reason about equivalent ratios. These representations are more abstract than are discrete diagrams and offer greater flexibility. Use of tables here is a stepping stone toward use of tables to represent functional relationships in future courses. Students explore equivalent ratios in contexts such as constant speed and uniform pricing.

A note on using the terms "quantity," "ratio," "rate," and "proportion":

In these materials, a "quantity" is a measurement that can be specified by a number and a unit, for instance, 4 oranges, 4 centimeters, “my height in feet,” or “my height” (with the understanding that a unit of measurement will need to be chosen). 

The term "ratio" is used to mean an association between two or more quantities. In this unit, the fractions  and  are never called ratios, but the meanings of these fractions in contexts are very carefully developed. The word “per” is used with students in interpreting a unit rate in context, as in “$3 per ounce,” and the phrase “at the same rate” is used to signify a situation characterized by equivalent ratios. In the next unit, the fractions  and  will be identified as "unit rates" for the ratio . Students will learn then that if two ratios  and  are equivalent, then the unit rates  and  are equal.

• The terms "proportion" and "proportional" are not used in grade 6. A "proportional relationship" is a collection of equivalent ratios, which will be studied in grade 7. In high school—after their study of ratios, rates, and proportional relationships—students can discard the term “unit rate” and refer to  to , , and  all as “ratios.”