Lesson 3

This warm-up prompts students to review work in grade 5 of converting across different-sized standard units within a given measurement system.

Arrange students in groups of 2. Ask students to use the following numbers and units to record as many equivalent expressions as they can. Tell them they are allowed to reuse numbers and units. Give students 2 minutes of quiet think time followed by 1 minute to share their equations with their partner.

Invite a few students to share equations that they had in common with their partner and ones that were different. Record these answers for all to see. After each equation is shared, ask students to give a signal if they had the same one recorded. Display the following questions for all to see and discuss:

• “What number(s) did you use the most? Why?”
• “If you could include two more cards to this selection, what would they be? Why?”

This activity has two purposes. First, it involves an important context that can be represented by proportional relationships, namely measurement conversion. Second, it introduces the idea that there are two constants of proportionality, and that they are reciprocals (also known as multiplicative inverses). Students understand why this is the case later when they use equations to represent proportional relationships. Students start to use “is proportional to” language to distinguish between the two constants of proportionality. During the discussion, students should be reminded that dividing by a number is equivalent to multiplying by its reciprocal. In principle, this is something students learn in grade 6, but they often need to be reminded. This will be needed in future lessons, so it is important to discuss it here.

Conversion of centimeters to millimeters has a constant of proportionality of 10 while conversion of millimeters to centimeters has a constant of proportionality of \(\frac{1}{10}\). One way to explain why these two constants of proportionality are multiplicative inverses is to imagine starting with a measurement in centimeters, 15 cm for example. When we convert to millimeters we multiply by 10: 15 cm = 150 mm. If we convert the measurement in millimeters back to centimeters, we know it will be 15, so the constant of proportionality we need to multiply by is \(\frac{1}{10}\).

Tell students, “Let’s look at how centimeters and millimeters are related and how it is related to what we have been doing recently.”

Some students may say that the constants of proportionality are both 10 since you can divide by 10 in the second table. Tell students, “The constant of proportionality is what you multiply by. Can you find a way to multiply the numbers in the first column to get the numbers in second column?”

The goal of this discussion is to help students see the connection between this situation and the earlier tasks, so they can use the structure of the table (MP7) to find the constants of proportionality. Ask questions like, “Can you use any of the strategies we have been discussing in earlier problems to help you solve this problem?” as students work to help them make this connection.

If there is disagreement about the constants of proportionality, ask students to explain their thinking.

Look for students who have seen the connection between dividing by 10 and multiplying by \(\frac{1}{10}\). Ask the students if there is a generalization to be made here. Help them articulate the idea that when you divide by a number, that is the same as multiplying by the reciprocal.

This activity focuses on making connections between constant speed and proportional relationships, with special attention to the constant of proportionality. In grade 6, students solved problems involving constant speed, but they need opportunities to make the connection to proportional relationships; students who successfully make this connection are reasoning abstractly about contexts with constant speed (MP2). Students who need support in understanding the context can trace the segments in the map, labeling the distances they know and putting question marks for unknown distances. An empty double number line could also be a useful tool in helping students reason about the context.

The numbers are chosen so that students are more likely to use unit rate rather than scale factors. The goal of the discussion is for students to understand that when speed is constant, time elapsed and distance traveled are proportional. The constant of proportionality indicates the magnitude of the speed. For example, if distance is given in miles and time in hours, then the constant of proportionality indicates the speed in miles per hour.

Students might wonder about the route. If they ask about it, teachers can share some reasons why airplane routes are complex, e.g., the need to avoid congested areas and the fact that the shortest distances on the curved surface of Earth do not always correspond to lines on a map.

It may have been some time since students reasoned about constant speed, so before they start on this activity, consider posing some simple problems about constant speed. For example:

• Someone walks at a constant speed of 4 miles per hour. How much time does it take them to walk 4 miles? 8 miles? 20 miles? 2 miles? \(\frac12\) mile? How far do they get in 3 hours? In 10 minutes?
• Someone rides a bike at a constant speed. They go 30 miles in 2 hours. What was their speed? If students did some work with double number lines when learning about constant speed in grade 6, they may be inclined to create one to help them think through these launch questions and when they get into the activity. If no one suggests it, it might be worth showing a double number line when talking through this launch, so that while working on the activity, students recall that these are useful tools. For example:

The goal of the discussion is for students to understand that when speed is constant, then distance traveled is proportional to elapsed time. The constant of proportionality is the speed. For example, if distance is given in miles and time in hours, then the constant of proportionality indicates the speed in miles per hour.

Begin by having a student share a table that is all in hours, miles, and miles per hour. Use this example to point out that every number in the first column can be multiplied by the speed to get the number in the second column. Ask:

• “Which quantities are in a proportional relationship? How do you know?”
• “What is the constant of proportionality in this case?”

If a student wrote their times in minutes and speed in miles per minute, you can have the same discussion if the class seems ready for that leap, but it is not required. In any case, summarize by making it explicit that when time and distance are in a proportional relationship, the constant of proportionality is the speed.