In this activity, students rewrite multiplication equations as division equations and use their understanding of multiplication and division with integers to solve a problem in context. Students use concrete examples and their prior experiences to more precisely articulate a rule for the sign of a quotient given the signs of the dividend and divisor (MP6).

Monitor for students who identify and describe the rule clearly.

The purpose of this discussion is for students to articulate their understanding about the sign of a quotient given the signs of the dividend and divisor. Note that students do not need to use these terms—they can give examples or use informal language. Begin by inviting students to share a multiplication problem and its corresponding division problem. Then consider discussing the following questions:

• "What patterns do you notice between the multiplication equation and the division equation you wrote?" (The numbers used in both stay the same. The division equations all start with 12 or -12.)
• "What patterns do you notice between the signs of each number in the division equations?" (When the two numbers being divided are both positive or both negative, the answer is positive. When the two numbers being divided have different signs, the answer is negative.)
• "How do these patterns compare to what we saw with multiplication?" (The patterns for the sign of the answer in both multiplication and division problems are the same.)

In this activity, students multiply and divide rational numbers to represent and solve problems in the new context of a drilling rig. They use multiplication and division of negative numbers to work with a relationship that has a negative constant of proportionality. Students use what they know about the structure of proportional relationships to help them represent this situation with a graph (MP7). 

In the digital version of the activity, students use an applet to plot points on a graph. The digital version may reduce barriers for students who need support with fine-motor skills and students who benefit from extra processing time.

The goal of this discussion is for students to share their reasoning. Ask students to share their solutions with each other in groups of 3 and work to come to an agreement.