In this activity, students connect their understanding of a solution to an equation as a value that makes the equation true with their understanding of operations involving negative numbers. Monitor for students who:

• Take an arithmetic approach by using substitution to check whether a given value makes the equation true.
• Take an algebraic approach by using inverse operations to solve for .

In the digital version of the activity, students use an applet to match equations with their solution. The applet allows students to quickly check their answers. The digital version may be helpful for providing students with immediate feedback.

The goal of this discussion is for students to see different ways to match an equation with a given solution. Select an equation for which there is a student who took an arithmetic approach and a student who took an algebraic approach. An example of an arithmetic approach: “I know that -3.5 is the solution to  because I know that . An example of an algebraic approach: “If , then I know that .)

Invite students to share their reasoning for why a solution is correct. Record their work, side by side, for all to see. Tell students that either approach is valid, but in the future there may be more complicated equations for which one approach might be more efficient than the other. If no students used "variable" as they described their approaches, invite them to identify the variables in some of the equations or share the definition of "variable" with the class.

In this activity, students reason quantitatively and abstractly as they interpret equations that represent situations (MP2). Students see that the structure of equations can be used to reason about a path to a solution even when negative values are included or when a variable represents a negative number (MP7). For example, students see that equations of the form can be solved by adding the opposite of to the equation, regardless of whether is positive or negative. Students also see that equations of the form can be solved by multiplying the equation by the reciprocal of .

The purpose of this discussion is for students to share strategies for determining which operation to use when solving problems. Tell students, “We learned four things about the hiking trip in this activity: The students were climbing, the temperature was falling, there were more students this year than last, and the cost of the trip was less this year than last.” Then discuss the following questions:

• “How did you know which operation described the rise in elevation, fall in temperature, rise in number of students, and fall in the cost?" (This conversation can highlight the problem with relying on “key words.” For example, when students see “times as many,” they might simply multiply the numbers they see in the problem. Encourage students to make sense of the situations by acting them out or drawing diagrams.)
• “How did you decide which strategy to use to solve for the unknown quantity?”
• “What are some ways to know that a situation involves negative values?”

In this partner activity, students take turns matching numbers and their additive or multiplicative inverse. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).

The purpose of this discussion is both to make connections and to contrast between a number's additive and multiplicative inverse. A key idea to emphasize is that multiplicative inverses require that the numbers have the same sign in order for the product to be positive, while additive inverses must have opposite signs in order for their sum to be 0.

Select 2–3 groups to share one of their sets of cards and how they matched a number with its additive inverse. Then repeat for 2–3 groups to share how they matched a number with its multiplicative inverse. Discuss as many different sets of cards as the time allows. 

Here are some additional questions for discussion:

• “Which matches were tricky? Explain why.”
• “Did you need to make adjustments in your matches? What might have caused an error? What adjustments were made?”
• "How did matching for additive inverses differ from matching for multiplicative inverses?"