Graphing Linear Inequalities in Two Variables (Part 1)

10 min

Narrative

This Math Talk focuses on substituting values from an ordered pair into an expression and comparing its value to a given number. It encourages students to think about the meaning of ordered pairs and inequalities and to rely on the structure of ordered pairs to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students test values in a region of the coordinate plane to determine if they are in the solution region of given inequality.

To correctly substitute into the expression, 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.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization

Student Task

Here is an expression: 2x+3y2x+3y.

Decide if the values in each ordered pair, (x,y)(x, y), make the value of the expression less than, greater than, or equal to 12.

  • (0,5)(0, 5)
  • (6,0)(6,0)
  • (-1,-1)(\text-1, \text-1)
  • (-5,10)(\text-5,10)

Sample Response

  • Greater than 12. Sample reasoning: Because x=0x=0 and y=5y = 5, the expression is equivalent to 3⋅5=153 \boldcdot 5 = 15, which is greater than 12.
  • Equal to 12. Sample reasoning: Because x=6x = 6 and y=0y = 0, the expression is equivalent to 2⋅6=122 \boldcdot 6 = 12
  • Less than 12. Sample reasoning: Both values are -1, so the value of the expression is the sum of the opposite of the coefficients. -2+-3=-5\text{-}2 + \text{-}3 = \text{-}5 which is less than 12.
  • Greater than 12. Sample reasoning: Substituting in the values, the expression is equivalent to 2(-5)+3(10)=-10+302 (\text{-}5) + 3(10) = \text{-}10 + 30, and 20 is greater than 12.
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 have 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?”

To help students recall the meaning of a solution to an inequality, ask: "Which pairs, if any, are solutions to the inequality 2x+3y≤122x + 3y \leq 12?" Make sure that students recognize that both (-1,-1)(\text-1,\text-1) and (6,0)(6,0) are solutions because they make the inequality true.

MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because . . . .” or “I noticed _____ so I . . . .” Some students may benefit from the opportunity to rehearse, with a partner, what they will say before they share with the whole class.
Advances: Speaking, Representing
Standards
Building Toward
  • A-REI.12·Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
  • A-REI.12·Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
  • A-REI.12·Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
  • HSA-REI.D.12·Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

15 min

10 min