Lesson 12

Negative Rates

12.1: Grapes per Minute (5 minutes)

Warm-up

The purpose of this warm-up is to review "per" language.

Launch

 Arrange students in groups of 2. Give students 1 minute of quiet work time followed by 1 minute of partner discussion, then follow with whole-class discussion.

Student Facing

  1. If you eat 5 grapes per minute for 8 minutes, how many grapes will you eat?
  2. If you hear 9 new songs per day for 3 days, how many new songs will you hear?
  3. If you run 15 laps per practice, how many practices will it take you to run 30 laps?

Student Response

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

Activity Synthesis

For each question, ask a student to share their response and reasoning. Resolve any disagreements that come up. Remind students that whenever we see the word "per", that means "for every 1".

12.2: Water Level in the Aquarium (10 minutes)

Activity

This activity builds students understanding of how negative rates can be used to model directed change. Students use their knowledge of dividing and multiplying negative numbers to answer questions involving rates. They are not expected to express these as relationships of the form $y = kx$ in this activity, though some students might. In the second question students will also need to convert between different rates. Pay close attention to students conversion between units in the last question as the rates are over different times. Identify those who convert to the same units (either minutes or hours) (MP6).

Launch

For each question, ask students to read the whole prompt individually before they start working on it. Remind students that they need to justify their answers. At the end of the activity put the students into groups of two. Use MLR 8 (Discussion Supports) to help students create context for this activity by providing or guiding students in creating visual diagrams that illustrate what is happening with the aquariums.

Representation: Access for Perception. Read the statements aloud. Students who both listen to and read the information will benefit from extra processing time. If students are unsure where to begin, suggest that they draw a diagram to help organize the information provided.
Supports accessibility for: Language
Reading: MLR6 Three Reads. Use this routine with the first problem to support students’ reading comprehension. In the first read, students read the situation with the goal of understanding what the situation is about (e.g., an aquarium has a system to maintain the correct water level; too much water and it will overflow; too little water and the fish will get sick). If needed, discuss the meaning of unfamiliar terms at this time, or display a diagram to clarify the context. Use the second read to identify the important quantities by asking students what can be counted or measured, without focusing on specific values. Listen for, and amplify, the important quantities that vary in relation to each other: amount of water in the aquarium, in liters; amount of time spent filling or draining water, in minutes. After the third read, ask students to brainstorm possible strategies to answer the questions. This helps students connect the language in the word problems with the reasoning needed to solve the problems.
Design Principle(s): Support sense-making

Student Facing

  1. A large aquarium should contain 10,000 liters of water when it is filled correctly. It will overflow if it gets up to 12,000 liters. The fish will get sick if it gets down to 4,000 liters. The aquarium has an automatic system to help keep the correct water level. If the water level is too low, a faucet fills it. If the water level is too high, a drain opens.

    One day, the system stops working correctly. The faucet starts to fill the aquarium at a rate of 30 liters per minute, and the drain opens at the same time, draining the water at a rate of 20 liters per minute.

    1. Is the water level rising or falling? How do you know?
    2. How long will it take until the tank starts overflowing or the fish get sick?
  2. A different aquarium should contain 15,000 liters of water when filled correctly. It will overflow if it gets to 17,600 liters.

    One day there is an accident and the tank cracks in 4 places. Water flows out of each crack at a rate of \(\frac12\) liter per hour. An emergency pump can re-fill the tank at a rate of 2 liters per minute. How many minutes must the pump run to replace the water lost each hour?

Student Response

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

Activity Synthesis

Ask students to compare their simplifying assumptions and solution methods. In a whole group discussion bring out the assumptions students have made. Compare the different rates in the answers to the last question.

12.3: Up and Down with the Piccards (15 minutes)

Activity

This activity builds on students' previous work with proportional relationships, as well as their understanding of multiplying and dividing signed numbers, to model different historical scenarios involving ascent and descent, and students must explain their reasoning (MP3). While equations of the form \(y = kx\) are not technically proportional relationships if \(k\) is negative, students can still work with these equations. Identify students who convert between seconds and hours.

Launch

Arrange students in groups of 2. Introduce the activity by asking students where they think the deepest part of the ocean is. It may be helpful to display a map showing the location of the Challenger Deep, but this is not required. Explain that Jacques Piccard had to design a specific type of submarine to make such a deep descent. Use MLR 8 (Discussion Supports) to help students create context for this activity. Provide pictures of Jacques Piccard with a submersible and Auguste Piccard with a hot air balloon. Guide students in creating visual diagrams that illustrate what is happening with the trench, ocean surface, and balloon. Remind students that they need to explain their reasoning. Provide for a quiet work time followed by partner and whole-class discussion.

Representation: Internalize Comprehension. Provide appropriate reading accommodations and supports to ensure students have access to written directions, word problems and other text-based content.
Supports accessibility for: Language; Conceptual processing

Student Facing

  1. Challenger Deep is the deepest known point in the ocean, at 35,814 feet below sea level. In 1960, Jacques Piccard and Don Walsh rode down in the Trieste and became the first people to visit the Challenger Deep.

    A photo of the Trieste. 
    1. If sea level is represented by 0 feet, explain how you can represent the depth of a submarine descending from sea level to the bottom of Challenger Deep.
    2. Trieste’s descent was a change in depth of -3 feet per second. We can use the relationship \(y=\text-3x\) to model this, where \(y\) is the depth (in feet) and \(x\) is the time (in seconds). Using this model, how much time would the Trieste take to reach the bottom?
    3. It took the Trieste 3 hours to ascend back to sea level. This can be modeled by a different relationship \(y=kx\). What is the value of \(k\) in this situation?
  2. The design of the Trieste was based on the design of a hot air balloon built by Auguste Piccard, Jacques's father. In 1932, Auguste rode in his hot-air balloon up to a record-breaking height.

    1. Auguste's ascent took 7 hours and went up 51,683 feet. Write a relationship \(y=kx\) to represent his ascent from his starting location.
    2. Auguste's descent took 3 hours and went down 52,940 feet. Write another relationship to represent his descent.
    3. Did Auguste Piccard end up at a greater or lesser altitude than his starting point? How much higher or lower?

Student Response

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

Student Facing

Are you ready for more?

During which part of either trip was a Piccard changing vertical position the fastest? Explain your reasoning.

  • Jacques's descent
  • Jacques's ascent
  • Auguste's ascent
  • Auguste's descent

Student Response

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

Anticipated Misconceptions

Some students may be confused by the correct answer to the last question, thinking that \(51,\!683 - 52,\!940 = -1,\!257\) means that Auguste landed his balloon below sea level. Explain that Auguste launched his balloon from a mountain, to help him reach as high of an altitude as possible. We have chosen to use zero to represent this starting point, instead of sea level, for this part of the activity. Therefore, a vertical position of -1,257 feet means that Auguste landed below his starting point, but not below sea level.

Activity Synthesis

First, have students compare their solutions with a partner and describe what is the same and what is different. This will help students be prepared to explain their reasoning to the whole class.

Next, select students to share with the class. Highlight solutions that correctly operature with negatives, those convert between seconds and hours, and those that state their assumptions clearly. Help students make sense of each equation by asking questions such as:

  • After 1 second, by how much have they changed vertical position? After 10 seconds? After 100 seconds?
  • How can you tell from the equation whether they are going up or down?
  • How can you tell from the equation the total distance or total time of their ascent or descent?
  • What does 0 represent in this situation?

The most important thing for students to get out of this activity is how different operations with signed numbers were helpful for representing the situation and solving the problems.

Writing, Representing: MLR3 Clarify, Critique, Correct. To begin the whole-class discussion, display the following incorrect statements for all to see: “Since they are under water, submarines have negative ascent and descent, so the \(k\) will always be negative." and "A hot air balloon will always have a positive ascent and descent because it is in the air, so the \(k\) will always be negative." Give students 1–2 minutes of quiet think time to identify the errors, and to critique the reasoning. Invite students to share their thinking with a partner, and to work together to correct one of the statements. Listen for the ways students interpret each statement, paying attention to the connections they make between vertical distances, and how changes in vertical position are represented. For each situation, select 1–2 groups to share their revised statements with the class. This will help clarify language and understanding about how ascent and descent can be represented by an equation.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness

Lesson Synthesis

Lesson Synthesis

Key takeaways:

  • Recall we can represent speed with direction (velocity) using signed numbers. We can do this with vertical movement (in fact with any rate).
  • The convention is that up is the positive direction and down is the negative direction.

Discussion questions:

  • What other rates have you encountered where it makes sense to have positive and negative values?

12.4: Cool-down - Submarines (5 minutes)

Cool-Down

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

Student Lesson Summary

Student Facing

We saw earlier that we can represent speed with direction using signed numbers. Speed with direction is called velocity. Positive velocities always represent movement in the opposite direction from negative velocities.

We can do this with vertical movement: moving up can be represented with positive numbers, and moving down with negative numbers. The magnitude tells you how fast, and the sign tells you which direction. (We could actually do it the other way around if we wanted to, but usually we make up positive and down negative.)