Lesson 5

Two Equations for Each Relationship

5.1: Missing Figures (5 minutes)

Warm-up

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.

Launch

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.

Student Facing

Here are the second and fourth figures in a pattern.

An increasing rectangular tile pattern with 2 rows of blue and yellow tiles. 
  1. What do you think the first and third figures in the pattern look like?
  2. Describe the 10th figure in the pattern.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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.

5.2: Meters and Centimeters (10 minutes)

Activity

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.

Launch

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.”

Student Facing

There are 100 centimeters (cm) in every meter (m).

length (m) length (cm)
1 100
0.94
1.67
57.24
\(x\)
length (cm) length (m)
100 1
250
78.2
123.9
\(y\)
  1. Complete each of the tables.
  2. For each table, find the constant of proportionality.
  3. What is the relationship between these constants of proportionality?
  4. For each table, write an equation for the proportional relationship. Let \(x\) represent a length measured in meters and \(y\) represent the same length measured in centimeters.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

  1. How many cubic centimeters are there in a cubic meter?
  2. How do you convert cubic centimeters to cubic meters?
  3. How do you convert the other way?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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 \(\frac{1}{100}\).
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation such as: “First, I _____ because….,” “I noticed _____ so I….,” “Why did you . . .?,” and “I agree/disagree because….”
Supports accessibility for: Language; Social-emotional skills
Speaking: MLR8 Discussion supports. Provide students with sentence frames such as “I found the constant of proportionality by _____” to help explain their reasoning. Revoice student ideas to model mathematical language use by restating a response as a question in order to clarify, and apply appropriate language.
Design Principle(s): Maximize linguistic and cognitive awareness

5.3: Filling a Water Cooler (15 minutes)

Activity

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.

Launch

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

Action and Expression: Internalize Executive Functions. To support development of organizational skills, check in with students within the first 2–3 minutes of work time. Check to make sure students have identified the correct equations that represent the relationship between \(w\) and \(t\) and can justify their reasoning.
Supports accessibility for: Memory; Organization
Writing, conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their written mathematical argument for whether the cooler is filling faster before or after Priya changed the rate of water flow. Give students time to meet with 2–3 partners to share and get feedback on their initial drafts. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “Can you explain how . . .”, “Is there another way to say . . . ?”, and “How do you know . . . . ?  Invite students to go back and revise or refine their written argument based on the feedback from peers. This will help students understand situations in which two different rates are associated with the same proportional relationship through communicating their reasoning with a partner.
Design Principle(s): Optimize output (for justification); Cultivate conversation

Student Facing

It took Priya 5 minutes to fill a cooler with 8 gallons of water from a faucet that was flowing at a steady rate. Let \(w\) be the number of gallons of water in the cooler after \(t\) minutes.

  1. Which of the following equations represent the relationship between \(w\) and \(t\)? Select all that apply.

    1. \(w = 1.6t\)
    2. \(w = 0.625t\)
    3. \(t = 1.6w\)
    4. \(t = 0.625w\)
  2. What does 1.6 tell you about the situation?

  3. What does 0.625 tell you about the situation?

  4. Priya changed the rate at which water flowed through the faucet. Write an equation that represents the relationship of \(w\) and \(t\) when it takes 3 minutes to fill the cooler with 1 gallon of water.

  5. Was the cooler filling faster before or after Priya changed the rate of water flow? Explain how you know.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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.

Activity Synthesis

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.

5.4: Feeding Shrimp (10 minutes)

Optional activity

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.

Launch

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

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, present one question at a time and monitor students to ensure they are making progress throughout the activity.
Supports accessibility for: Organization; Attention
Speaking, Reading: MLR5 Co-Craft Questions. Use this routine to help students interpret the language of proportional relationships, and to increase awareness of language used to talk about proportional relationships. Display only the first sentence of this problem (“At an aquarium, a shrimp is fed \(\frac{1}{5}\) gram of food each feeding and is fed 3 times each day.”), and ask students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the remainder of the question. Listen for and amplify any questions involving relationships between the two quantities in this task (amount of food and number of feedings).
Design Principle(s): Maximize meta-awareness; Support sense-making

Student Facing

At an aquarium, a shrimp is fed \(\frac{1}{5}\) gram of food each feeding and is fed 3 times each day. 

  1. How much food does a shrimp get fed in one day?

  2. Complete the table to show how many grams of food the shrimp is fed over different numbers of days.

    number of days food in grams
    1
    7
    30
    A picture of a shrimp.
  3. What is the constant of proportionality? What does it tell us about the situation?
  4. If we switched the columns in the table, what would be the constant of proportionality? Explain your reasoning.
  5. Use \(d\) for number of days and \(f\) for amount of food in grams that a shrimp eats to write two equations that represent the relationship between \(d\) and \(f\).
  6. If a tank has 10 shrimp in it, how much food is added to the tank each day?
  7. If the aquarium manager has 300 grams of shrimp food for this tank of 10 shrimp, how many days will it last? Explain or show your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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

Lesson Synthesis

Lesson Synthesis

Briefly revisit the first two activities to summarize the important points of the lesson. For example:

  • We examined the proportional relationship between meters and centimeters. Why were we able to write two equations for this situation? What were they? What were the constants of proportionality? 
  • We examined a proportional relationship where we knew how long it took to fill a water cooler with a certain amount of water. What were the constants of proportionality for this relationship? What equations did we determine would represent this situation?
  • In each case, what was the relationship between the two constants of proportionality and between the two equations?

5.5: Cool-down - Flight of the Albatross (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

If Kiran rode his bike at a constant 10 miles per hour, his distance in miles, \(d\), is proportional to the number of hours, \(t\), that he rode. We can write the equation \(\displaystyle d = 10 t\) With this equation, it is easy to find the distance Kiran rode when we know how long it took because we can just multiply the time by 10.

We can rewrite the equation:

\(\begin{align} d &= 10 t\\ \left( \frac{1}{10} \right) d &= t \\ t &= \left( \frac{1}{10} \right) d \end{align}\)

This version of the equation tells us that the amount of time he rode is proportional to the distance he traveled, and the constant of proportionality is \(\frac{1}{10}\). That form is easier to use when we know his distance and want to find how long it took because we can just multiply the distance by \(\frac{1}{10}\).

When two quantities \(x\) and \(y\) are in a proportional relationship, we can write the equation \(\displaystyle y = k  x\) and say, “\(y\) is proportional to \(x\).” In this case, the number \(k\) is the corresponding constant of proportionality. We can also write the equation \(\displaystyle x = \frac{1}{k} y\) and say, “\(x\) is proportional to \(y\).” In this case, the number \(\frac{1}{k}\) is the corresponding constant of proportionality. Each one can be useful depending on the information we have and the quantity we are trying to figure out.