In this activity, students work with changing temperatures to build understanding of equations that represent situations with negative coefficients, variables, and solutions. Students choose from a bank of equations to find two equations, one that represents the situation using a variable and the other that represents a path to solve for the variable. Students contextualize and decontextualize between the contexts of changing temperatures and the equations that represent them (MP2). In this activity, students also critique a statement or response that is intentionally unclear, incorrect, or incomplete and improve it by clarifying meaning, correcting errors, and adding details (MP3).

This goal of this discussion is for students to connect the equation that represents a situation to the equation that represents the solution strategy. Begin by inviting students to share examples of how they chose:

• The equation that represents the situation.
• The equation that represents the solution strategy.
• The order in which they chose the two equations.

Then use Critique, Correct, Clarify to give students an opportunity to improve a sample written response for the 3rd situation, in which the temperature is dropping by 3 degrees per hour and is -6 degrees at a certain time, by correcting errors, clarifying meaning, and adding details.

• Display this first draft:

  “The equation represents a situation where the temperature is dropping by 3 degrees per hour and is -6 degrees at a certain time. The variable represents the number of hours that it takes to reach -6 degrees, and . It takes 2 hours to go from 0 degrees to -6 degrees, and since the temperature is decreasing, the answer is -2.”

  Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.

• Give students 2–4 minutes to work with a partner to revise the first draft.

• Select 1–2 students or groups to slowly read aloud their draft. Record for all to see as each draft is shared. Then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.

  Here is a sample second draft:

  “The equation represents a situation where the temperature drops by 3 degrees per hour and is -6 degrees at a certain time. The variable represents the number of hours after midnight that it takes to reach -6 degrees, and . Since it takes 2 hours to go from 0 degrees to -6 degrees, the time will be 2 a.m.”

In this activity, students match number line diagrams to situations involving the changing height and depth of sea animals. They reason abstractly and quantitatively when they write equations to represent each situation and interpret their answers in context (MP2).

There is more than one correct equation that represents each situation. Monitor for students who come up with different equations for each situation.

The purpose of this discussion is for students to share their reasoning when writing an equation to match each situation and number line diagram. Invite previously identified students to share their equations, and record them for all to see. For each situation and set of equations, discuss the following questions:

• “Do all of these equations correctly represent this situation?”
• “What do all of the correct equations have in common? How are they different?”
• “How do you see the animal’s final altitude (or change in altitude) in these equations?”

In this activity, students work in groups to create a visual display to represent a given situation. Students reason abstractly and quantitatively when they write an equation to represent their situation, define the variables they use, explain the meaning of each term, and explain how they can use inverses to find the solution (MP2). 

The purpose of this discussion is for students to share their visual display and view other groups' displays. Invite groups to present their solutions or to view all the solutions on display. Consider discussing the following questions: 

• “How did you decide what your variable would represent?”
• “What negative quantities are there in your equation, and what do they mean in terms of this situation?”
• “How did you come up with your equation?”
• “What strategy did you use to solve the equation?”
• “What does your solution mean in terms of this situation?”