Lesson 9

Dealing with Negative Numbers

9.1: Which One Doesn’t Belong: Rational Number Arithmetic (10 minutes)

Warm-up

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.

Launch

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.

Student Facing

Which equation doesn’t belong?

\(15 = \text- 5 \boldcdot \text- 3\) 

\(2 + \text- 5 = \text- 3\) 

\(4 - \text- 2 = 6\) 

\(\text- 3 \boldcdot \text- 4 = \text- 12\) 

Student Response

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Activity Synthesis

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.

9.2: Old and New Ways to Solve (15 minutes)

Activity

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.

Launch

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

Representation: Internalize Comprehension. Activate or supply background knowledge about strategies for performing operations on signed numbers, as well as different representations of equations. This will help students make connections between new problems and prior work.
Supports accessibility for: Memory; Conceptual processing

Student Facing

Solve each equation. Be prepared to explain your reasoning.

  1. \(x + 6 = 4\)
  2. \(x - \text- 4 = \text- 6\)
  3. \(2(x - 1) = \text- 200\)
  4. \(2x + \text- 3 = \text- 23\)

Student Response

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Anticipated Misconceptions

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.

Activity Synthesis

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.

Representing: MLR7 Compare and Connect. Use this routine when students explain how they solved their equations. Ask students, “What is the same and what is different?” about the strategies. Draw students’ attention to the connection between the approaches of ‘finding the value that makes the equation true’ and ‘doing the same to each side.’ These exchanges strengthen students’ mathematical language use and reasoning based on ways to solve equations that involve negative numbers.
Design Principle(s): Support sense-making; Maximize meta-awareness

9.3: Keeping It True (15 minutes)

Activity

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.

Launch

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.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. Consider pausing after the first question for a brief class discussion before moving on.
Supports accessibility for: Organization; Attention
Speaking: MLR8 Discussion Supports. To support students to produce statements about how their partner transformed the equation \(\text-5=x\), ask students to revisit the moves that were used earlier in the lesson to transform \(x = \text-6\). Students should use different combinations of the moves on the list in conjunction with sentence frames such as “I noticed ___ so I . . .” or “I know ___ because . . . .” This will help students explain their thinking. Design Principle(s): Optimize output (for explanation)

Student Facing

Here are some equations that all have the same solution.

\(\begin {align} x &= \text-6\\ x - 3 &= \text-9\\ \text-9 &= x - 3\\ 900 &= \text-100(x - 3)\\ 900 &= (x-3) \boldcdot (\text-100)\\ 900 &= \text-100x + 300\\ \end {align}\)

  1. Explain how you know that each equation has the same solution as the previous equation. Pause for discussion before moving to the next question.
  2. Keep your work secret from your partner. Start with the equation \(\text-5 = x\). Do the same thing to each side at least three times to create an equation that has the same solution as the starting equation. Write the equation you ended up with on a slip of paper, and trade equations with your partner.
  3. See if you can figure out what steps they used to transform \(\text-5=x\) into their equation. When you think you know, check with them to see if you are right.

Student Response

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Anticipated Misconceptions

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.

Activity Synthesis

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?”

Lesson Synthesis

Lesson Synthesis

Ask students to think of one or two important things they learned in this lesson, and share them with a partner. Points to highlight include:

  • Doing the same thing to each side of an equation still keeps the equation balanced, even when there are negative numbers.
  • Doing the same thing to each side of an equation still keeps the equation balanced, even when the moves don’t get you closer to a solution.

9.4: Cool-down - Solve Two More Equations (5 minutes)

Cool-Down

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Student Lesson Summary

Student Facing

When we want to find a solution to an equation, sometimes we just think about what value in place of the variable would make the equation true. Sometimes we perform the same operation on each side (for example, subtract the same amount from each side). The balanced hangers helped us to understand that doing the same to each side of an equation keeps the equation true. 

Since negative numbers are just numbers, then doing the same thing to each side of an equation works for negative numbers as well. Here are some examples of equations that have negative numbers and steps you could take to solve them.

Example:

\(\begin{align} 2(x-5) &= \text-6 \\ \tfrac12 \boldcdot 2(x-5) &= \tfrac12 \boldcdot (\text-6) & \text{multiply each side by }\tfrac12 \\ x-5 &= \text-3 \\ x-5+5 &= \text- 3 + 5 & \text{ add 5 to each side} \\ x &= 2 \\ \end{align}\)

Example:

\(\begin{align} \text-2x + \text-5 &= 6 \\ \text-2x + \text-5 - \text-5 &= 6 - \text-5 & \text{subtract -5 from each side} \\ \text-2x &= 11 \\ \text-2x \div \text-2 &= 11 \div \text-2 & \text{divide each side by -2} \\ x&=\text- \tfrac{11}{2}\\ \end{align}\)

Doing the same thing to each side maintains equality even if it is not helpful to solving for the unknown amount. For example, we could take the equation \(\text-3x +7=\text-8\) and add \(\text-2\) to each side: 

\(\begin{align} \text-3x+7 &= \text-8 \\ \text-3x + 7 + \text-2 &= \text-8 + \text-2 & \text{add -2 to each side}\\ \text-3x+5 &= \text-10 \\ \end{align}\)

If \(\text-3x+7=\text-8\) is true then \(\text-3x+5=\text-10\) is also true, but we are no closer to a solution than we were before adding -2. We can use moves that maintain equality to make new equations that all have the same solution. Helpful combinations of moves will eventually lead to an equation like \(x=5\), which gives the solution to the original equation (and every equation we wrote in the process of solving).