In this activity, students practice writing an equation to represent a proportional relationship in a new context. Students reason about quantities and prices, calculating several values before writing an equation. No table is provided for students to organize their thinking in order to encourage them to look for regularity in repeated reasoning (MP8) and to notice the efficiency of using an equation to express the relationship.

This activity also emphasizes the interpretation of the constant of proportionality in the context, as students may choose to express the relationship as 5 cents per bottle, 0.05 dollars per bottle, or 20 bottles per dollar. Students compare these different approaches during the whole-class discussion.

Monitor for students who:

• Write many calculations, without any organization
• Create a table to organize their work
• Write an equation to record their repeated reasoning
• Use 5 as the constant of proportionality
• Use 0.05 as the constant of proportionality
• Use 20 as the constant of proportionality

The two goals of this discussion are:

• To highlight the efficiency of expressing the relationship as an equation
• To contrast different ways the relationship could be expressed as an equation

Invite previously selected students to share how they found the amounts of money and the numbers of bottles. Use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:

• “How could we represent this reasoning with an equation? What does the constant of proportionality in this equation represent?”
• “Why do the different approaches lead to the same outcome?”
• “Are there any benefits or drawbacks to one approach compared to another?”

Here are some different strategies for finding the value of some number of bottles, along with one or more ways to record that reasoning with an equation.

*If students suggest an equivalent equation that uses division, such as , confirm that that equation is correct and also ask if they can think of a way to express that equation in the form .

The key takeaway is that defining variables and writing an equations can be an efficient way to describe the proportional relationship between two quantities. However, there is more than one way to represent a given situation with an equation. It is important to specify what each variable represents so others can interpret the equation.

This activity is intended to further develop students’ ability to write equations to represent proportional relationships. It involves work with decimals and asks for equations that represent proportional relationships of different pairs of quantities, which increases the challenge of the task. As students identify the constants of proportionality between each pair of quantities to represent the relationships with equations, they are reasoning quantitatively and abstractly (MP2).

Students may solve the first two problems in different ways. Monitor for different solution approaches, such as using computations, using tables, finding the constant of proportionality, and writing equations. 

Invite several students to share their methods for solving the first two problems, such as using computations, using tables, finding the constant of proportionality, and writing equations. If students did not use equations to solve the first two problems, ask them how they can use the equations they found later in the activity to answer the first two questions.

If time permits, highlight connections between the equations generated, illustrated by the following sequence of equations.