In this activity, students use their knowledge of dividing and multiplying negative numbers to answer questions involving rates. In the first situation, students must make sense of the problem by determining whether a faulty aquarium system results in the aquarium filling or draining and persevere in solving the problem (MP1). In the second situation, students encounter another faulty aquarium and will need to convert between different rates to solve the problem. 

The purpose of this discussion is for students to share the strategies they used to solve each aquarium problem and to reflect on how they used integer arithmetic. Begin by arranging students in groups of 2. Ask partners to share their answers and reasoning for each problem. If time allows, instruct each group to create a visual display illustrating what is happening to one of the aquariums. Then consider discussing these questions:

• "What were some assumptions you had to make to solve the first (or second) aquarium problem?" (I assumed that each aquarium was filled correctly when the automatic system stopped working or when the cracks appeared. I assumed the fish swimming in the first aquarium didn't affect the rate of water flowing from the faucet or the drain. I assumed that the cracks in the second pump were all below the level of the water in the aquarium and that no additional cracks formed.)
• "How did you use your knowledge of integer arithmetic to help you solve these problems?"

In this activity, students build on previous work with proportional relationships and their understanding of multiplying and dividing signed numbers to represent two historical scenarios involving ascent and descent. Students reason abstractly and quantitatively when they describe the situation using equations and interpret the meaning of their answer in context (MP2).

Monitor for students who convert between seconds and hours in their equations.

Some students may be confused by the correct answer to the last question, thinking that 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.

The purpose of this discussion is for students to share the equations they wrote to describe each situation. Begin by asking students to compare their solutions with a partner and notice what is the same and what is different. 

Next, invite previously identified students to share their equations with the class. Highlight solutions that compute with negatives correctly, convert between seconds and hours, and state their assumptions clearly. To help students make sense of each equation, consider discussing the following questions:

• "How can you tell from each equation whether they are going up or down?" (When the value of is positive, they are going up, and when the value of is negative, they are going down.)
• "How can you tell from the equation the total time of their ascent or descent?" (The time of the ascent or descent is represented by in each equation.)
• "How can you tell from the equation the total distance traveled?" (The total distance traveled is represented by in each equation.)
• "What does 0 represent in each situation?" (For the submersible, 0 represents sea level. For the hot air balloon, 0 represents the starting elevation of the balloon.)

The purpose of this discussion is to reinforce these key ideas:

• The product of two numbers with different signs is negative.
• The sum of two numbers with different signs will be the same sign as the number with the largest magnitude (or the sum will be 0 if the two numbers have the same magnitude).

Invite students to share their responses and reasoning to the fourth question asking how many parks and clam beds a city would need to absorb the carbon dioxide released by 400,000 cars. As students share their combinations of parks and clam beds, consider discussing the following questions:

• “How did you calculate the amount of carbon dioxide released by the parks?” (I multiplied the average amount of carbon dioxide released by one park and multiplied it by the total number of parks.)
• “Why is the carbon dioxide released by the parks a negative number? How does that make sense in this situation?” (A positive number multiplied by a negative number results in a negative number. This makes sense because trees absorb carbon dioxide.)
• “How do we know when the number of parks and clam beds will be enough to absorb the carbon dioxide released by the cars?” (The sum of all the numbers will be 0.)
• “How can a set of numbers have a sum of 0?” (The total magnitude of the positive numbers is the same as the total magnitude of the negative numbers.)