This activity introduces students to ratio language and notation through examples based on a collection of everyday objects. Students learn that a ratio is an association between quantities, and that this association can be expressed in multiple ways.

After discussing examples of ratio language and notation for one way of categorizing the objects in the collection, students write ratios to describe the quantities for another way of categorizing objects in the collection.

As students work, circulate and monitor for those who:

• Create different categories from the given collection.
• Create categories whose quantities can be rearranged into smaller groups (for instance, 6 A’s and 4 B’s can be expressed as “for every 3 A’s there are 2 B’s”).
• Express the same ratio in opposite order or by using different words (for instance, “the ratio of A to B is 7 to 3,” and “for every 7 A’s there are 3 B’s”).

Have a collection of objects ready to display for the Launch. Make sure that there are different ways in which the collection can be sorted.

Invite several students to share their categories and sentences. Display them for all to see, attending to correct ratio language. Be sure to include students who express the same categories in reverse order, in different words, or with a different set of numbers (which students will later call an equivalent ratio). Leave several sentences displayed for students to see and use as a reference while working on the next task.

In this activity, students continue to draw connections between a diagram and the ratios it represents. Students discuss with a partner different ways to use ratio language to describe discrete diagrams. They first identify statements that would correctly describe a given diagram. Then they create both a diagram and corresponding statements to represent a new situation involving ratio. 

Monitor for the different kinds of diagrams that students create and the different statements that they write to describe the play clay recipe. Here are some ways in which students may create diagrams, from more elaborate to more simple:

• Draw literal drawings of cups and pints.
• Draw rectangles, circles, or other shapes.
• Use letters or other symbols.

They may write statements that:

• Use the given values, such as: “the ratio of cups of corn flour to pints of glue is 8 to 2.”
• Regroup the quantities, such as: “for every 4 cups of corn flour, Jada used 1 cup of glue.”
• Use equivalent ratios, even though these have not been explicitly taught, such as: “the ratio of cups of corn flour to pints of glue is .” While this is correct and we have regrouped quantities and used language, such as “for every 4 of something, there is 1 of another thing,” we have not yet asserted that the ratio can be written as . The idea of equivalent ratios is sophisticated and will be developed over the next several lessons.
• Include fractions, such as: “for every 1 cup of corn flour, Jada used pint of glue.” Although students are not expected to work with fractions in this lesson, responses involving fractions are fine.

Writing and using ratio language requires attention to detail. This activity further develops students’ ability to describe ratio situations precisely by attending carefully to the quantities, their units, and their order in the ratio.

Students are given a set of cards showing discrete diagrams and statements that represent ratios of objects in pencil cases. They work with a partner to find diagrams and statements that represent the same ratio. Students take turns making a match and explaining their reasoning. In doing so, they practice constructing viable arguments and critiquing the reasoning of others (MP3).

This is the first Card Sort activity of the course. An important aspect of this routine is to allow students time at the start to sort the cards into categories of their choosing. This step gives students the opportunity to familiarize themselves with the content of the cards without the additional pressure of organizing them in a specific fashion. It also provides insight into the aspects of each card students attend to and the language they have to describe their observations.