Lesson 12
Negative Rates
12.1: Grapes per Minute (5 minutes)
Warmup
The purpose of this warmup 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 wholeclass discussion.
Student Facing
 If you eat 5 grapes per minute for 8 minutes, how many grapes will you eat?
 If you hear 9 new songs per day for 3 days, how many new songs will you hear?
 If you run 15 laps per practice, how many practices will it take you to run 30 laps?
Student Response
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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.
Supports accessibility for: Language
Design Principle(s): Support sensemaking
Student Facing

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.
 Is the water level rising or falling? How do you know?
 How long will it take until the tank starts overflowing or the fish get sick?

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 refill 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
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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 wholeclass discussion.
Supports accessibility for: Language; Conceptual processing
Student Facing

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.
 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.
 Trieste’s descent was a change in depth of 3 feet per second. We can use the relationship \(y=\text3x\) 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?
 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?

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 hotair balloon up to a recordbreaking height.
 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.
 Auguste's descent took 3 hours and went down 52,940 feet. Write another relationship to represent his descent.
 Did Auguste Piccard end up at a greater or lesser altitude than his starting point? How much higher or lower?
Student Response
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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
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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.
Design Principle(s): Optimize output (for explanation); Maximize metaawareness
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: Cooldown  Submarines (5 minutes)
CoolDown
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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.)