In this activity, students practice writing an equation to represent a proportional relationship given in a table. This activity revisits the idea that there are two reciprocal constants of proportionality between two related quantities. Students use repeated reasoning (MP8) to arrive at the equations and to identify the constants of proportionality as reciprocals.

We return to the context of measurement conversion, again examining the same distances measured in two different units. Previously students compared centimeters with millimeters and saw that the constants of proportionality were 10 and . Here, students compare meters with centimeters and see that the constants of proportionality are 100 and . The similarities between this activity and the earlier one may cause this activity to go very quickly.

The purpose of this discussion is to highlight how the two equations illustrate the reciprocal relationship. Invite students to share how they found the equation for each table. Consider asking:

• “Where does the constant of proportionality occur in each table and equation?” (In the table, the constant of proportionality is in the right column on the row that has 1 in the left column. In the equation, the constant of proportionality is right next to the variable that represents the quantity that was in the left column of the table.)
• “What is the relationship between the two constants of proportionality? How can you use the equations to see why this should be true?” (They are reciprocals. This makes sense because the variables are in opposite places.)

Display and discuss this sequence of equivalent equations to help students see why the constants of proportionality are reciprocals:

This line of reasoning should be accessible to students, because it builds on grade 6 work with expressions and equations.

Ask students to interpret the meaning of the equations in the context: “What do the equations tell us about the conversion from meters to centimeters and back?”

• Given the length in meters, to find the length in centimeters we can multiply the number of meters by 100.
• Given the length in centimeters, to find the length in meters we can multiply the number of centimeters by .

In this activity, students make sense of the two rates associated with a given proportional relationship (MP2). Students are asked to identify the two equations that represent a situation, working with both the number of gallons per minute and the number of minutes per gallon. This activity is the first time that no table is given to help students make sense of the proportional relationship, though students may find it helpful to create a table.

Monitor for students who use different ways to decide if the cooler was filling faster before or after the flow rate was changed.

For the first question, if students struggle to identify the correct equations, encourage them to create two tables of values for the situation. Encourage them to create rows for both unit rates, in order to foster connections to prior learning.

The goal of this discussion is to connect the meaning of each constant of proportionality with the structure of each equation that represents the relationship. Invite students to share how they decided which equations represent the situation. Ask students to interpret what the equations tell us about the situation.

• To find the amount of water, , we can multiply the elapsed time in minutes, , by 1.6.
• To find the elapsed time, , we can multiply the number of gallons of water, , by 0.625.

If not mentioned by students, highlight the fact that 1.6 and 0.625 are reciprocals. Since these constants of proportionality are given as decimals in the equations, it may be harder for students to recognize this relationship. Consider asking half the class to calculate while the other half calculates .

Next, invite students to share their responses to the last two questions about Priya changing the rate of water flow. Two possible approaches for the last question are:

• Comparing the new equation to the previous equation . In these equations, the constant of proportionality represents the numbers of minutes per gallon. Since 3 is greater than 0.625, that means it takes Priya more time to get the same amount of water after changing the rate of water flow.
• Comparing the new equation to the previous equation . In these equations, the constant of proportionality represents the number of gallons per minute. Since is less than 1.6, that means Priya gets less water in the same amount of time changing the rate of water flow.

It is not necessary to demonstrate every possible approach. The goal is for students to see how keeping in mind the meaning of the numbers and variables is helpful for making sense of the situation.

This activity provides an additional opportunity for students to represent a proportional relationship with two related equations in a new context. This situation builds on the earlier work students did with feeding a crowd, but includes more complicated calculations. Students interpret the meaning of the constants of proportionality in the context of the situation and use the equations to answer questions.

The goal of this discussion is to highlight the structure of the two equations that represent the proportional relationship, including the meaning of the two constants of proportionality. Invite students to share their answers. Ask students which equation was most useful to answer each of the last two questions and to explain their reasoning.