In this activity, students revisit two contexts seen previously and ultimately find equations for the proportional relationships. As students find missing values in the table, they should see that they can always multiply the number of food items by the constant of proportionality to find the number of people served. When students see this pattern and represent the number of people served by cups of rice as (or by spring rolls as ), they are expressing regularity in repeated reasoning (MP8).

Monitor for students who use these strategies to complete the tables:

• Find the unit rate, interpret it, and use it to calculate other values.
• Use a scale factor to find the values for the first row and then scale those values to find other rows.
• Find the constant of proportionality that relates the left column to the right column.

Regardless of whether students reason based on the meaning of the unit rate in context or based on the structure of the table, the key takeaway is the constant multiplicative relationship.

The goal of this discussion is to show how an equation can be used to represent the proportional relationship shown in each table. Display 2–3 approaches to the rice problem from previously selected students for all to see. Use Compare and Connect to help students compare, contrast, and connect the different representations. Here are some questions for discussion:

• “What do the approaches have in common? How are they different?”
• “How does the constant of proportionality, 3, show up in each method?”
• “Are there any benefits or drawbacks to one method compared to another?”

The key takeaway is that any value in the right column can be found by multiplying the corresponding value in the left column by 3. For the last row, we can represent times 3 as .

Next, suggest to students that we let represent the number of people who can be served by cups of rice. Ask students to write an equation that gives the relationship between and . Display the equation and help students interpret its meaning in the context: “To find , the number of people served, we can multiply the number of cups of rice, , by 3.”

Lastly, ask students to write an equation that represents the relationship for the spring rolls. Record and display students’ equations. Ask them to interpret what the equations tell us about the situation. (To find the number of people served, , we can divide the number of spring rolls, , by 2, or multiply it by or 0.5.)

In this activity, students write an equation to represent a proportional relationship between distance and time. The context of airplane flight is similar to  that of a previous activity, but not exactly the same. This activity prompts students to move back and forth between the abstract representation and the context (MP2) as they create an equation and then use it to find other values that aren’t in the table. This task increases the level of difficulty by having so much missing information and by using decimals in the table.

There are various ways students may approach the last question. Monitor for students who:

• Multiply the elapsed times by the rate 610 miles per hour, without referencing the table or equation.
• Use the structure of the table, adding rows for 3 and 3.5 hours.
• Use the equation to find the distances for 3 and 3.5 hours.

Plan to have students present in this order to support moving them from arithmetic methods towards algebraic methods.

The purpose of this discussion is to show the value of finding an equation that represents a proportional relationship.

Ask previously selected students to share their solutions to the last question (the distances the plane would travel in 3 and 3.5 hours). Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display their work for all to see.

Connect the different responses to the learning goals by asking questions such as:

• “How are these methods the same? How are they different?”
• “How does the constant of proportionality show up in each representation?”
• “Are there any benefits or drawbacks to one strategy compared to another?”

If not mentioned by students, display the equation for all to see, and ask students to interpret its meaning in the context of the situation. (To find , the distance traveled by the plane in miles, multiply the hours of travel, , by the plane’s speed in miles per hour, 610.)

The key takeaways are:

• An equation is an efficient way to represent a proportional relationship.
• The equation shows the constant of proportionality, and the variables indicate how the quantities are related.
• You can substitute a value in for one of the variables and then multiply (or divide) to find the value for the other variable.
   

This activity gives students more practice writing an equation that represents a proportional relationship. It revisits a context examined in a previous lesson—the amounts of coconut milk and flour in a bread recipe. Students can then use their equation to answer additional questions about the situation.

Ask students to compare answers with their partner and discuss their reasoning until they reach an agreement. Then invite students to share with the class how they used their equation from the second question to answer the third question.

Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response about how to get the equation for this relationship by correcting errors, clarifying meaning, and adding details.

• Display this first draft:
  “First I wrote the equation . Then I wanted to solve it for one of the variables, so I divided each side by 200 to get the equivalent equation .”
  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.
• Here is an example of a second draft:
  “For each row of the table, the grams of flour is 1.8 times the milliliters of coconut milk. For example . An equation that represents this relationship is .
• 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.