The purpose of this task is to introduce students to the idea of a proportional relationship. Students examine two different tables of equivalent ratios and use them to solve problems. In the Activity Synthesis they learn that there is a proportional relationship between two quantities when the quantities are characterized by a set of equivalent ratios.

The contexts and numbers used in this activity are intended to be accessible to all students. This way they can focus on its mathematical structure (MP7) and the new terms introduced in the lesson without being distracted.

Monitor for groups who use these different strategies to complete the first table:

• Create a drawing or diagram that depicts 15 cups of rice and 45 people, organized into 3 people per cup
• Use a scale factor to scale from one row of the table to another (For example, since , then we can do to get 30.)
• Calculate and use a unit rate (In other words, one cup of rice serves 3 people, so 10 cups of rice must serve 30 people.)
• Apply the constant relationship between the columns of the table (In other words, multiply the number in the left column by 3 to get the number in the right column.)

Plan for students to present in this order, from concrete to more abstract.

The purpose of this discussion is to introduce the concept of a proportional relationship. Ask previously selected groups to share their reasoning for completing the first table. 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:

• “Why do the different approaches lead to the same outcome?”
• “What do you notice about the columns in the completed table?”
• “How does the unit rate (people per cup of rice) show up in each method?”

After students have shared their reasoning, introduce the term proportional relationship. For example, say, “Whenever we have a situation like this where two quantities are always in the same ratio, we say there is a proportional relationship between the quantities.” Display these statements for all to see:

• “The relationship between the number of cups of rice and the number of people is a proportional relationship.”
• “The number of people is proportional to the number of cups of rice.”
• “There are 3 people for every 1 cup of rice.”

Next, ask students to work with their partner to write a sentence that describes the proportional relationship in the second table. Then, invite students to share their sentences. Record and display their sentences for all to see.
Sample responses:

• The relationship between the number of spring rolls and the number of people is a proportional relationship.
• The number of people is proportional to the number of spring rolls.
• There are 2 spring rolls for each person.
• There are 0.5 people per spring roll. (The unit rate 0.5 people per spring roll may sound strange, but it means that 1 spring roll only halfway satisfies a person.)
   

The purpose of this activity is to introduce the term constant of proportionality. Students continue to find missing values in a table representing a proportional relationship. However, the numbers are purposefully chosen so that it is difficult to identify scale factors that scale one row to another. This encourages students to look for a unit rate relationship between the columns.

This is the first time Math Language Routine 6: Three Reads is suggested in this course. In this routine, students are supported in reading a mathematical text, situation, or word problem three times, each with a particular focus. During the first read, students focus on comprehending the situation. During the second read, students identify quantities. During the third read, the final prompt is revealed and students brainstorm possible starting points for answering the question. The intended question is withheld until the third read so students can make sense of the whole context before rushing down a solution path. The purpose of this routine is to support students’ reading comprehension as they make sense of mathematical situations and information through conversation with a partner.

The key takeaway from this discussion is that the constant of proportionality for this proportional relationship is 1.8.

Invite students to share their strategies for completing the table, especially the row with 375 milliliters of coconut milk. Highlight strategies that make use of the relationship between the columns of the table, that is, any value in the first column can be multiplied by 1.8 to get the corresponding value in the second column. Tell students that 1.8 is the constant of proportionality for this proportional relationship. Note that, like unit rate, the constant of proportionality can always be determined by finding how much of the second quantity there is per one of the first quantity.

Ask students to interpret the constant of proportionality in the context: “What does the 1.8 tell us about the situation?” (There are 1.8 grams of flour per milliliter of coconut milk.)

If time permits, consider inviting students to share their strategies for the question about the amount of coconut milk to use with 450 grams of flour. Possible strategies include:

• Multiplying the values in the second row of the table by the scale factor
• Dividing the given amount of flour by the constant of proportionality, 1.8

Highlight the structure that dividing any value in the second column by the constant of proportionality gives the corresponding value in the first column.

The purpose of this activity is to practice finding and using the constant of proportionality in another context.

Ask students for the value that belongs next to the 1 in the table. Invite several students to explain the significance of this number.