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.”

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes, such as interpreting, explaining, and comparing. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Interpret

• Statements and notations describing ratios (Lesson 2).
• Different representations of ratios (Lessons 2 and 6).
• Situations involving equivalent ratios (Lesson 8).
• Situations with different rates (Lesson 9).
• Tables of equivalent ratios (Lessons 11 and 12).
• Questions about situations involving ratios (Lesson 17).

Explain

• Reasoning about equivalence (Lesson 4).
• Reasoning about equivalent rates (Lesson 10).
• Reasoning with reference to tables (Lesson 14).
• Reasoning with reference to tape diagrams (Lesson 15).

Compare

• Situations with and without equivalent ratios (Lesson 3).
• Representations of ratios (Lessons 6 and 13).
• Situations with different rates (Lessons 9 and 12).
• Situations with same rates and different rates (Lesson 10).
• Representations of ratio and rate situations (Lesson 16).

In addition, students are expected to describe and represent ratio associations, represent doubling and tripling of quantities in a ratio, represent equivalent ratios, justify whether ratios are or aren't equivalent and why information is needed to solve a ratio problem, generalize about equivalent ratios and about the usefulness of ratio representations, and critique representations of ratios.

The table shows lessons where new terminology is first introduced in this course, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.

lesson	new terminology
	receptive	productive
6.2.1	ratio ___ to ___ ___ for every ___
6.2.2	diagram
6.2.3	recipe batch same taste equivalent	ratio ___ to ___ ___ for every ___
6.2.4	mixture same color check (an answer)	batch
6.2.5	equivalent ratios
6.2.6	double number line diagram tick marks representation	diagram
6.2.7	per
6.2.8	unit price how much for 1 at this rate	double number line diagram
6.2.9	constant speed meters per second
6.2.10	same rate	equivalent ratios
6.2.11	table row column
6.2.14	calculation	per table
6.2.15	tape diagram parts suppose
6.2.16		tape diagram