Lesson 2

The purpose of this warm-up is to elicit the idea that you can use a table to see patterns between related quantities, which will be useful when students discuss how to use tables to learn about proportional relationships in a later activity. While students may notice and wonder many things about these images, the relationship between the rows (multiply both entries by the same number to get another row) and the relationship between the columns (multiply the number of cases by 12 to get the number of paper towels) are the important discussion points.

Arrange students in groups of 2. Tell students that they will look at an image, and their job is to think of at least one thing they notice and at least one thing they wonder. Display the image for all to see. Ask students to give a signal when they have noticed or wondered about something. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After each response, ask the class if they agree or disagree and to explain alternative ways of thinking, referring back to the images each time. If the relationship between the number of cases and the number of paper towels does not come up during the conversation, ask students to discuss this idea.

The purpose of this task is to introduce students to the idea of a proportional relationship. From previous work, students should be familiar with the idea of equivalent ratios, and they may very well recognize the table as a set of equivalent ratios. Here, we are starting to expand this concept and the language associated with it to say that there is a proportional relationship between the cups of rice and number of people as well as the number of spring rolls and number of people. More generally, there is a proportional relationship between two quantities when the quantities are characterized by a set of equivalent ratios.

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

While students are working, monitor for groups using each of these approaches:

• Drawing that depicts 15 cups of rice and 45 people, organized into three people per cup. This representation of the problem can support all learners in moving forward. The teacher should highlight the organization of three people per cup, and identify correspondences between parts of the drawing and numbers in the table.
• Moving down the table, multiplying both numbers in a row by the same number. “Since I multiply 2 by 5 to get 10, I will multiply 6 by 5 and get 30. Since 9 times 5 is 45, I will also multiply 3 by 5 to get 15.” If it comes up, the most appropriate name for the multiplier would be scale factor, but it is not necessary for students to call it this.
• Calculation and use of a unit rate. “I reasoned that 1 cup of rice must serve 3 people. So 10 cups of rice must serve 30, and 15 cups must serve 45.”
• Moving across the table. “It looks like we always multiply number of cups of rice by 3 to get the number of people it serves, so I will multiply 10 cups by 3 to get 30, and divide the 45 by 3 to get 15.”

Say, “Rice is a big part of the traditions and cultures of many families. Does your family cook rice, and if so, how?" Invite a student to describe the process (measure rice, measure water, simmer for a while). You use more rice for more people and less rice for fewer people. If students have trouble understanding or representing the context, show them the measuring cup so that they have a sense of its size, or draw a literal diagram that looks something like this:

Similarly, ask students if they have ever eaten a spring roll and invite them to describe what they are. While some spring rolls can be very large, the ones referred to in this activity are smaller.

Select students to share their approaches using the sequence listed in the Activity Narrative.

After students have shared their reasoning and connections have been drawn between the different approaches, introduce the term proportional relationship. For example, “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. So the relationship between the number of cups of rice and the number of people is a proportional relationship. The number of cups of rice is proportional to the number of people.” Write out some of these statements for all to see (in the next activity, students will need to write a statement like these about a different relationship). If the class has a word wall or students keep track of mathematical vocabulary in their notebooks, these new terms can be included there.

Notice that methods 2 and 3 involve identifying relationships of column entries, namely that the entry in the right-hand column is three times the corresponding entry in the left-hand column. This pattern can be explained in terms of the unit rate (people per cup of rice), which tells us how many people we can serve with a given amount of rice. The next activity is intended to build the understanding underlying this explanation.

After discussing the rice context thoroughly, ask students to share their solution approaches to the spring roll context. Ask students to identify and interpret the unit rate in this situation (0.5 people per spring roll may sound strange, but it means that 1 spring roll only halfway satisfies a person.)

In this activity, students grapple with finding missing values for ratios of whole numbers presented in a table where identifying a usable scale factor is not as easy as in the previous activity. This task is designed to encourage students to use a unit rate. Its context is intended to be familiar so that students can focus on mathematical structure (MP7) and the new terms (MP6) constant of proportionality and proportional relationship. If students are having difficulty understanding the scenario, consider drawing discrete diagrams like this:

This can be followed by a double number line diagram. Correspondences among the diagrams can be identified.

Anticipated approaches include use of scale factors, unit rate, and moving across columns. Because 13 is not a multiple of 4 or 8, students are more likely to use and see the value of using the unit rate or the relationship of the columns. Unit rates might be used: \(\frac54\) or 1.25 cups of flour per tablespoon of honey, or \(\frac45\) or 0.8 tablespoon of honey per cup of flour. Both approaches are correct, but the numbers are easier for the first approach, so this is the solution to have students share. While students are working, monitor the approaches used by each student or group.

Tell students that in this activity, they will think about a different proportional relationship. If necessary, show students the measuring cup and the tablespoon side by side to help make the context more concrete. You could even pantomime the first sentence in the activity: “measuring” 8 tablespoons of invisible honey and 10 cups of invisible flour.

Select students to share their solution approaches in this order:

1. Double each entry in the first row to get to row 4.
2. Divide each entry in the first row by 8, yielding the pair \((1, 1.25)\). (1.25 can be called the unit rate, since 1.25 cups of flour are needed for 1 tablespoon of honey.) Multiply by 20 and 13 to get rows 2 and 3. 
3. Notice any entry in the first column can be multiplied by \(\frac54\) or 1.25 to get the corresponding entry in the second column. Name this value the constant of proportionality for the proportional relationship.

Ensure that these are highlighted as part of the discussion and that students describe their methods using mathematical language (MP6). For example,

• Note that even though you multiply by a different scale factor to go from row to row in the table, the unit rate is always the same. Rename this “the constant of proportionality.” Note that it can always be found by finding how much of the second quantity per one of the first quantity.
• Ask students what the proportional relationship is in this situation.
• Ask students to interpret the constant of proportionality in the context: “What does the 1.25 tell us about?” (That there are 1.25 cups of flour per tablespoon of honey.) Because this can help them connect it to their work in grade 6 and prepare them for the next activity.

The purpose of this optional activity is to practice using a proportional relationship in another context. The chosen numbers make the unit rate a useful tool to answer the questions. 

Remind students of the value of a quarter and a dime. 

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