In this partner activity, students take turns matching expressions that have the same value. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3). Students look for and make use of the relationship between inverse operations to make connections (MP7).

If some students struggle to find matches, consider asking:

• "How else can you think of subtraction/addition/multiplication/division?"
• "What do you know about the difference/sum/product/quotient of two signed numbers?"

The purpose of this discussion is to make connections between inverse operations in matching expressions. Select 2–3 groups to share one of their sets of cards and how they matched the two expressions. Discuss as many different sets of cards as the time allows. Highlight strategies that compared the structure of the expressions instead of just the final answers. Consider discussing the following questions:

• "How are a number and its opposite the same or different?" (They have opposite signs but the same magnitude.)
• "What operation is equivalent to subtracting a given number?" (Subtracting a number is equivalent to adding the opposite.) 
• "How are a number and its reciprocal the same or different?" (They have the same sign but different distances from zero. They are found by taking 1 and dividing by the other.)
• "What operation is equivalent to dividing by a given number?" (Dividing by a number is equivalent to multiplying by the reciprocal.)

Tell students that an “inverse operation” is an operation that reverses the effect of another operation, so it results in the original value. For example, an inverse operation for adding 5 could be subtracting 5, or adding -5. 

Explain that the "additive inverse" of a number is its opposite. The sum of a number and its additive inverse is 0. 
Similarly, the "multiplicative inverse" of a number is its reciprocal. The product of a number and its multiplicative inverse is 1.

In this activity, students evaluate expressions using rational numbers. They reason about the relative size of rational numbers and their distance from 0. As they compare the structure of various expressions, they observe that when a variable can be any rational number, it is impossible to compare expressions involving those variables or determine which expression is greater or lesser (MP7). For example, if is positive, then must be greater than , but if can be any rational number, then that is not necessarily true. If is negative, then is less than , and if is 0, is equal to . Students may also use that structure to observe that relative distance from 0 is still possible to tell. For example, is as far away from 0 as , whether that is positive or negative.

Display the completed table for all to see. Invite students to share their values that are largest, smallest, and closest to zero from each set and explain their reasoning.

Ask students if any of these results were surprising and to explain their reasoning. Some possible observations include:

• was both the largest and smallest value at different times. 6 and -6 are both relatively far from 0, so is a large number when is positive and a small number when is negative.
• was often the closest to 0, because it had the same absolute value no matter the sign of and .

In this activity, students use a familiar context to interpret an expression in terms of the position it represents on a vertical number line. Expressions are equal when they represent the same position on a number line. Students use the structure of the number line to reason about the relative values of expressions (MP7).

In a previous course, students have seen fractions with a negative sign in front of the entire fraction, such as . In this activity, students see a fraction with a negative sign in the numerator, such as . Students can apply what they know about dividing signed numbers to make sense of expressions like this.

In the digital version of the activity, students use an applet to place points representing the vertical position of different animals on a vertical number line. The applet allows students to drag points to the number line and quickly check their answers. 

The purpose of this discussion is for students to share their reasoning about the vertical location of each animal and to illustrate the meaning of the equal sign. Begin by displaying the diagram from the Task Statement for all to see. Invite students to share their responses and reasoning, and record them on the diagram for all to see.

As a point for each animal is placed, illustrate the meaning of the equal sign. For example, “We could either label this point with  or with  since we know that . Since these expressions are equal, they represent the same position on the number line.”

If not brought up in students' explanations, consider drawing attention to the equation and discussing how this equation relates to what they have learned about division with signed numbers. A key idea for students to understand is that  is equal to and , but these are not equal to .