Solving More Systems

5 min

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

Narrative

This Math Talk focuses on systems of equations where one variable is already solved. It encourages students to think about what the equations mean and to rely on the structure of the equations to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students solve additional systems.

To solve the systems, students need to look for and make use of structure (MP7).

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 system mentally.

  • {x=8y=-11\displaystyle \begin{cases} x=8\\ y=\text-11 \end{cases}
  • {x=5y=x−7\displaystyle \begin{cases} x=5\\ y=x - 7 \end{cases}
  • {y=3x−2y=4\displaystyle \begin{cases} y = 3x - 2\\ y=4 \end{cases}
  • {y=2x+3y=12(4x+3)\displaystyle \begin{cases} y = 2x + 3\\ y=\frac{1}{2}(4x + 3) \end{cases}

Sample Response

  • (8,-11)(8,\text{-}11) Sample reasoning: The equations tell us the answers.
  • (5,-2)(5,\text{-}2) Sample reasoning: The first equation tells us the correct value for xx. Then I substituted it into the second equation to find yy.
  • (2,4)(2,4). Sample reasoning: The second equation tells us the correct value for yy. Then I substituted it in for yy in the first equation and solved for xx by adding 2, then dividing by 3.
  • No solution. Sample reasoning: I distributed the 12\frac{1}{2} in the second equation to get y=2x+32y = 2x + \frac{3}{2}. Because the two lines have the same slope and different yy-intercepts, they are parallel lines with no solution.
Activity Synthesis (Teacher Notes)

To involve more students in the conversation, consider asking:

  • “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?”
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
Building On
  • 6.EE.5·Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
  • 6.EE.B.5·Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

15 min

15 min