Lesson 12

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

 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.

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".

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).

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.

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.

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.

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.

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.

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.