Lesson 9

This warm-up prompts students to compare four equations. It encourages students to explain their reasoning, hold mathematical conversations, and gives you the opportunity to hear what they recall of arithmetic on signed numbers. To allow all students to access the activity, each equation has one obvious reason it does not belong. Encourage students to move past the obvious reasons and find reasons based on mathematical properties.

During the discussion, listen for strategies for evaluating expressions with rational numbers that will be helpful in the work of this lesson.

Arrange students in groups of 2–4. Display the equations for all to see. Ask students to indicate when they have noticed one that does not belong and can explain why. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, tell each student to share their reasoning why a particular equation does not belong and together find at least one reason each question doesn’t belong.

Ask each group to share one reason why a particular equation does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct. During the discussion, highlight any strategies for adding, subtracting, or multiplying signed numbers.

These are all solvable by thinking “what value would make the equation true.” So, it’s straightforward to figure out what the solution would be, but these equations present an opportunity to demonstrate that “doing the same thing to each side” still works when there are negative numbers. Monitor for students who reason about what value would make the equation true and those who reason by doing the same thing to each side.

Give 5–10 minutes of quiet work time followed by a whole-class discussion.

Some students may need some additional support remembering and applying strategies for performing operations on signed numbers. Draw their attention to any anchor charts or notes that are available from the previous unit.

For each equation, ask one student to explain how they know their solution is correct. If no students mention this approach, demonstrate solving each equation by doing the same thing to each side.

The purpose of this discussion is to make the claim that “do the same thing to each side” also works when subtraction or negative numbers are involved. Tell students that even though it doesn’t make sense to represent negative numbers using the hanger metaphor, we are going to take it as a fact that we can still do the same thing to each side of an equation even when we are working with negative numbers.

The purpose of this activity is to note that doing the same thing to each side of an equation keeps it in balance, even if those moves don’t get us closer to solving the equation. Students first explain how each equation in a sequence follows logically from the previous one. Then, they start with the equation \(\text-5=x\) and repeatedly do the same thing to each side to create a new equation. Their partner tries to guess which moves they made.

Arrange students in groups of 2. Give students 2 minutes of quiet work time on the first question, pause for a discussion, and then time to complete the task with their partner.

After students have a chance to work on the first question, pause for a discussion. Ask students what different types of moves could we do to \(x = \text-6\)? List the different kinds of things in the board, so when students do their own, you can say you have to use different combinations of things on the list. The purpose of this is to prevent students from going wild and generating equations that are far out of the scope of the work in this unit.

Some students may need some additional support remembering and applying strategies for performing operations on signed numbers. Draw their attention to any anchor charts or notes that are available from the previous unit.

Much of the discussion will take place in small groups. Questions for discussion:

• “Did you have any disagreements, and how did you resolve them?”
• “Did anything surprise you? Explain.”
• “What are some important things to keep in mind when working with negative numbers?”