Solving Any Linear Equation

5 min

Teacher Prep
Setup
30 seconds quiet think time followed by whole-class discussion for each problem.

Narrative

This Math Talk focuses on solving equations with negative numbers. It encourages students to think about valid moves and to rely on what they know about creating equivalent equations to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students solve equations in this lesson.

In describing their strategies, students need to be precise in their word choice and use of language (MP6). 

Launch

Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:

  • Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
  • Invite students to share their strategies, and record and display their responses for all to see.
  • Use the questions in the activity synthesis to involve more students in the conversation before moving to the next problem. 

Keep all previous problems and work displayed throughout the talk.

Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization

Student Task

Solve each equation mentally.

  • 5−x=85 - x = 8
  • -1=x−2\text-1 = x - 2
  • -3x=9\text-3x = 9
  • -10=-5x\text-10 = \text-5x

Sample Response

  • x=-3x=\text-3. Sample reasoning: I subtracted 5 from each side, then multiplied by -1.
  • x=1x=1. Sample reasoning: I added 2 to each side.
  • x=-3x=\text-3. Sample reasoning: I know that -3 multiplied by -3 is 9, so xx must be -3.
  • x=2x=2. Sample reasoning: I divided each side by -5.
Activity Synthesis (Teacher Notes)

To involve more students in the conversation, consider asking as the students share their ideas:

  • “Who can restate ‾\underline{\hspace{.5in}}’s reasoning in a different way?”
  • “Did anyone use the same strategy but would explain it differently?”
  • “Did anyone solve the problem in a different way?”
  • “Does anyone want to add on to ‾\underline{\hspace{.5in}}’s strategy?”
  • “Do you agree or disagree? Why?”
  • “What connections to previous problems do you see?”

Some students may reason about the value of xx using logic. For example, in -3x=9\text- 3x = 9, the xx must be -3 since -3⋅-3=9\text-3 \boldcdot \text- 3=9. Other students may reason about the value of xx by changing the value of each side of the equation equally by, for example, dividing each side of -3x=9\text- 3x = 9 by -3 to get the result x=-3x=\text- 3. Both of these strategies should be highlighted during the discussion where possible.

MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I ‾\underline{\hspace{.5in}} because . . . .” or “I noticed ‾\underline{\hspace{.5in}} so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking, Representing
Standards
Addressing
  • 8.EE.C·Analyze and solve linear equations and pairs of simultaneous linear equations.
  • 8.EE.C·Analyze and solve linear equations and pairs of simultaneous linear equations.

15 min

15 min