In this activity, students see how an abbreviated table can be used to solve a problem about a proportional relationship. In grade 6, students learned to use a table with 3 rows, where the middle row had a “1” in the left column. Now they learn to use a table with only 2 rows, making use of structure (MP7).

For teachers accustomed to setting up a proportion and cross-multiplying, this approach is structurally very similar. One benefit to using an abbreviated table is that students have the column headings to help make sure they get the numbers in the right places. In addition, students should have a better idea for why they are multiplying and dividing, which is that they are finding and using either a scale factor or the constant of proportionality. In this case, the numbers are selected such that using the constant of proportionality is easier. It is also a natural way to think about calculating the price of any amount of something.

Some students may struggle to progress with Priya's method because the arrows are not drawn in the image and none of the values given are easily divisible. There are many supporting questions that could be asked.

• What if we knew the price of 1 foot of rope?
• If 6 times something is 7.5, how can we find the something?

The purpose of this discussion is to highlight how solving this problem can be accomplished in two steps—dividing and multiplying—and to interpret what those two operations represent in terms of the situation.

Direct students’ attention to the reference created using Collect and Display. Ask students to share how the table can be used to solve the problem. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases.

If no students come up with one of these methods, display it for all to see.

Scale factor method:

Constant of proportionality method:

Consider asking:

• “How do we find the scale factor between the two rows?” (Divide 50 by 6.)
• “What does the scale factor, mean in this situation?” (The customer is buying “batches” of rope.)
• “How do we find the constant of proportionality?” (Divide 7.50 by 6.)
• “What does the constant of proportionality, 1.25, mean in this situation?” (The rope costs \$1.25 per foot.)
• “Which method do you prefer? Why?”

Although either method will work, there are reasons to prefer using the constant of proportionality to approach problems like these. The constant of proportionality means something important in the situation—it’s the price of 1 foot of rope. Because of that, the 1.25 could be used to easily compute the price of any length of rope. If no students bring it up, point out that the equation could be used to relate any length of rope, , to its price, .

In this activity, students practice using abbreviated tables to solve problems involving proportional relationships. The scaffolding is slowly removed as each problem has less of the table already completed, leaving more of the table for students to fill in.

In this activity, students critique a statement or response that is intentionally unclear, incorrect, or incomplete and improve it by clarifying meaning, correcting errors, and adding details (MP3). 

Some students may struggle to continue working as the scaffolding is decreased. Consider using these questions to prompt students:

• ”What are the two associated quantities in this problem?”
• ”How many quarts of blue paint are needed for 1 quart of white paint?”

The purpose of this discussion is to highlight how the structure of a table can help identify the calculations that are needed to solve a problem involving a proportional relationship. Invite students to share how they solved the third problem about the paint mixture. Highlight strategies that involved setting up a table.

To get students analyzing the structure of the table, consider asking:

• “Does it matter which heading goes in which column?”
• “If you were to switch the columns, would you get a different answer? Why or why not?”
• “If you were to switch the columns, would it change what calculations you had to do? Why or why not?”

Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response to the question “How much white paint would need to be mixed with 4 quarts of blue paint?” by correcting errors, clarifying meaning, and adding details. 

• Display this first draft:

  “There are 0.3 quarts of blue paint to white paint, so 1.2 quarts of white paint would be needed.”

  Ask, “What parts of this response are unclear, incorrect, or incomplete?”

• 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.

If time permits, invite students to share the equations they wrote for each proportional relationship. Consider discussing how there are different, equivalent ways to express each relationship and how the equation can be used to help solve the problem.

In this activity, students solve problems involving proportional relationships and fractions without any tables provided. As students choose to create a table or other representation, they make sense of problems and persevere in solving them (MP1). 

Monitor for students who create a table and use the constant of proportionality to find unknown values.

The purpose of this discussion is for students to share how they solved the problems involving proportional relationships with fractions. Invite students to share their method for finding a solution. If time is limited, pick only one of the problems to discuss.

If not mentioned by students, ask students to describe why they chose to calculate the constant of proportionality for this problem and how that helped them with finding the solution.