In this activity students use equations to create tables and decide whether the relationships are proportional. Students have previously looked at measurement conversions that can be represented by proportional relationships. This task introduces a measurement conversion that is not associated with a proportional relationship. 

Monitor for students who use these different strategies to justify whether each relationship is proportional:

• Look for patterns in the table, such as scale factors between rows or constant of proportionality between columns
• Use the structure of the equation, whether it represents multiplying one quantity by a constant of proportionality to get the other quantity

Some students may struggle with the fraction in the temperature conversion. Teachers can prompt them to convert the fraction to its decimal form, 1.8, before trying to evaluate the equation for the values in the table.

Some students may think of the two scales on a thermometer like a double number line diagram, leading them to believe that the relationship between degrees Celsius and degrees Fahrenheit is proportional. Point out that when a double number line is used to represent a set of equivalent ratios, the tick marks for 0 on each line need to be aligned.
 

The goal of this discussion is to start to move students from determining whether a relationship is proportional by examining a table, to making the determination from the equation. Display 2–3 approaches from previously selected students for all to see. Use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:

• “What do the strategies have in common? How are they different?”
• “How does the constant of proportionality show up in each method?”
• “Are there any benefits or drawbacks to one approach compared to another?”

The key takeaway is that the equation for the proportional relationship is of the form , while the equation for the nonproportional relationship is not.

• When we use an equation of the form to create a table, the values in the columns will be related by the constant of proportionality .
• When we use an equation that is not of the form to create a table, the values in the columns will likely not be related by a constant of proportionality. (There’s a small possibility that a given equation could be equivalent to an equation of the form , but showing this would require algebraic skills that students have not yet developed. Examples:  or )

This activity uses a geometric context to give students more practice with deciding whether relationships are proportional. The context of surface area and volume of a cube should be familiar from grade 6. 

The units for the quantities are purposely not given in the task statement to avoid giving away which relationships are not proportional. However, discussion should raise the possible units of measurement for edge length, surface area, and volume.

Some students may struggle to complete the tables. Teachers can use nets of cubes (flat or assembled) partitioned into square units to reinforce the process for finding total edge length, surface area, and volume of the cubes in the task. Snap cubes would also be appropriate supports.

If difficulties with the fractional side length keep students from being able to find the surface area and volume or write the equations, the teacher can tell those students to replace with 10 and retry their calculations. Their answers for surface area and volume will be different for that row in the table, but their equations and proportionality decisions will be the same. That way they can still learn the connection between the form of the equations and the nature of the relationships.

The goal of the discussion is for students to recognize that the proportional relationship has an equation of the form while the nonproportional relationships do not.

Direct students' attention to the reference created using Collect and Display. Ask students to share what they notice about the equations for the relationships. Invite students to borrow language from the display as needed and update the reference to include additional phrases as they respond. Display words and phrases such as “factor,” “number next to the variable,” “coefficient,” “exponent,” “operation,” “not proportional.” Make sure students see that the equation for the proportional relationship is of the form , and the others are not.

If time permits, consider asking:

• "What could be possible units for the side lengths?" (linear measurements: centimeters, inches)
• "Then what would be the units for the surface area?" (square units: square centimeters, square inches) 
• "What would be the units for the volume?" (cubic units: cubic centimeters, cubic inches)

Connect the units of measurements with the structure of the equation for each quantity: the variable that represents the side length and the units are raised to the same power.

This activity involves checking for a constant of proportionality in tables generated from simple equations. Students should be able to evaluate these equations from their work with expressions and equations in grade 6. The purpose of this activity is to generalize about the forms of equations that do and do not represent proportional relationships. The relationships in this activity are presented without a context so that students can focus on the structure of the equations (MP7) without being distracted by what the variables represent.

Students might struggle to see that the two proportional relationships have equations of the form and to characterize the others as not having equations of that form. Students do not need to completely articulate this insight for themselves. This synthesis should emerge in the whole-class discussion.

Invite students to share what the equations for proportional relationships have in common, and by contrast, what is different about the other equations.

Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response about which of these equations represent a proportional relationship, by correcting errors, clarifying meaning, and adding details.

• Display this first draft:
  “Just the equation that uses multiplication is proportional, and all the other equations are not proportional.”
  Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.
• Give students 2–4 minutes to work with a partner to revise the first draft.
• Select 1–2 individuals or groups to read their revised draft aloud slowly enough to record for all to see. Scribe as each student shares, then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.

The key takeaway is that any equation that can be written in the form represents a proportional relationship. At first glance, the equation does not look like our standard equation for a proportional relationship, . Suggest to students that they rewrite the equation using the constant of proportionality they found after completing the table: which can also be expressed . If students do not express this idea themselves, remind them that they can think of dividing by 4 as multiplying by .