Lesson 2

This warm-up encourages students to look for regularity in how the tiles in the image are growing and describe this pattern (MP8) using ratios as a review of their work in grade 6. Students may use each color to reason about missing figures while others may reason about the way the tiles are arranged.

Arrange students in groups of 2. Display the image for all to see and tell students that the collection of tiles is growing, but we can only see the second and fourth images. Ask students to look for a pattern in the sequence of figures and to give a signal when they have thought of one. Give students 1 minute of quiet think time, and then time to discuss their patterns and arrangements for Figures 1 and 3 with a partner. Tell students to use what they discussed in figuring out their answers to the next questions.

Invite students to share their responses and reasoning. Record and display the different ways of thinking for all to see. If possible, record the relevant reasoning on or near the images themselves. After each explanation, ask the class if they agree or disagree and to explain alternative ways of thinking, referring back to what is happening in the images each time.

Students may use each color to reason about missing figures while others may reason about the way the tiles are arranged. Emphasize both insights in addition to the usefulness of Figure 1 as students share their strategies.

This activity is intended to build confidence and facility in writing an equation for a proportional relationship. Students build on their understanding that measurement conversions can be represented by proportional relationships, which they studied in an earlier lesson and will revisit in future lessons. Students are expected to use methods developed earlier: organize data in a table, write and solve an equation to determine the constant of proportionality, and use repeated reasoning (MP8) to arrive at an equation. In addition, they are expected to identify the relationship between the constant of proportionality when going from centimeters to meters and vice versa (reciprocals). Since students have already explored a similar relationship (centimeters and millimeters) in a previous lesson, this activity may go very quickly.

As students work, look for students writing an equation like \(100k = 1\) (for table 2) as a step to finding the constant of proportionality. Encourage them to say how they would solve the equation. Ask students to say why using this equation makes sense in the scenario.

Introduce the task: “In a previous lesson, you examined the relationship between millimeters and centimeters. Today we will examine the relationship between centimeters and meters.”

Invite students to share their answers. These questions can guide the discussion:

• "How can we find an equation for each table?"
• "Where does the constant of proportionality occur in the table and equation?"
• "What is the relationship between the two constants of proportionality? How can you use the equations to see why this should be true?"

The equations can help students see why the constants of proportionality are reciprocals:

\(\displaystyle y = 100x\)

\(\displaystyle \left(\frac{1}{100}\right)y = \frac{1}{100} \boldcdot (100x)\)

\(\displaystyle \left(\frac{1}{100}\right)y = x\)

\(\displaystyle x = \left(\frac{1}{100}\right)y\)

This line of reasoning illustrated above 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, multiply the number of centimeters by 100.
• Given the length in centimeters, to find the the length in meters, multiply the number of meters by \(\frac{1}{100}\).

The theme continues by asking students to make sense of the two rates associated with a given proportional relationship. Here, students are asked to reason from an equation rather than a table, although they may find it helpful to create a table or graph (MP5). In this particular example, students work with both number of gallons per minute and number of minutes per gallon.

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

Give students 5 minutes quiet work time followed by sharing with a partner and a whole-class discussion.

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.

Select students to share their answers. Elicit both responses from the class, and be sure to identify connections between them.

Select students who used different explanations to share their answers to the last question.

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 they 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 first few questions ask about 1 shrimp. The question about feeding 10 shrimp helps prepare students for work they will do in the next lesson with multiple quantities that are in proportional relationships to each other.

Arrange students in groups of 2. Give students 6 minutes of partner work time followed by whole-class discussion.

Invite students to share their answers. Ask students which equation was most useful to answer each of the last two questions.