Lesson 24

Reasoning with Graphs of Inequalities

These materials, when encountered before Algebra 1, Unit 2, Lesson 24 support success in that lesson.

24.1: Notice and Wonder: Shady Graphs (5 minutes)

Warm-up

The purpose of this warm-up is to discuss what students attend to when they look at graphs with multiple shaded regions on the same coordinate plane, which will be useful when students interpret and create these graphs in the associated Algebra 1 lesson. While students may notice and wonder many things about these images, what is shaded and how the regions overlap are the important discussion points.

Through articulating things they notice and things they wonder about the graph, students have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or precise language, and then restate their observation with more precise language in order to communicate more clearly, such as starting to use the word “solutions” to describe the shaded regions.

Launch

Display the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion

Student Facing

What do you notice? What do you wonder?

A graph of two intersecting inequalities on a coordinate plane.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

If students are struggling to interpret the shading done with lines, help them see that as shading.

24.2: Some (Admittedly Silly) Riddles (15 minutes)

Activity

In the associated Algebra 1 lesson, students use the riddle context, with other numbers, to help them recall systems of equations. In this activity, the riddle context helps students reason about situations that can have multiple solutions, and begin thinking about how solutions to a system of linear inequalities are a subset of the solutions to each inequality. The goal of this activity is for students to reason, using graphs or mental math, about solutions that satisfy multiple inequalities at once.

Monitor for strategies that students use to find equations or inequalities to represent each statement, including:

  • guessing values and checking them for the conditions
  • rewriting verbal descriptions into equations or inequalities

Launch

If desired, introduce one or more of the riddles by using a Three Reads routine.

Student Facing

Each riddle consists of two conditions. For each riddle,

  • find a pair of numbers that satisfy the first condition.
  • find a pair of numbers that satisfy the second condition.
  • find a pair of numbers that satisfy both conditions.
  • determine whether there could be more than one solution.
  • write a system of equations or inequalities that represent the riddle.
  1. I’m thinking of two numbers. Their sum is 15. Their difference is 1.
  2. I’m thinking of two numbers. Their sum is more than 15. Their difference is more than 1.
  3. I’m thinking of two numbers. One is more than twice the other. Their sum is less than 30.
  4. Think of your own riddles about a pair of numbers.
    1. Make a riddle with multiple pairs of numbers that will work.
    2. Make a riddle with one pair of numbers that will work.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

The purpose of this discussion is for students to generalize their strategies for solving two-number riddles and connect them to what they know about inequalities. The discussion focuses on preparing students to use a graph to interpret solutions to systems of inequalities

Ask previously identified students to share their strategies in this order:

  • guessing values and checking them for the conditions
  • rewriting verbal descriptions into equations or inequalities

Ask students:

  • “Did both strategies arrive at the same answer?” (Yes)
  • “Does anyone feel one strategy takes more time than another?” (In some cases, it may be easy to guess and check quickly, but sometimes it's hard to think of numbers that work, then writing equations or inequalities can end up being quicker.)

The rest of this discussion will focus on the riddle about finding two numbers whose sum is more than 15 and whose difference is more than 1.

Display the image of a blank graph for all to see.

Blank coordinate plane.

Ask students to share the pairs of values they found that had a sum greater than 15. Record those in one color. Call on a student to graph the inequality they found for numbers with a sum greater than 15. Ask students what they notice. Ask students to share the pairs of values they found that had a difference greater than 1. Record those in another color. Call on a student to graph the inequality they found for numbers with a difference greater than 1. Ask students what they notice. Ask students to share pairs of values they found that had a sum greater than 15 and a difference greater than 1. Record those in a third color. Ask students what they notice. If no student mentions it, point out that the solutions to the riddle are in the area of the graph where the two linear inequalities overlap.

Graph of system of inequalities on grid.
 

24.3: Which Side Are You On? (20 minutes)

Activity

In this activity, students get a chance to practice graphing lines, then they are directed to recognize that lines divide the plane into sections. They practice referring to sections by describing them to a partner. In the associated Algebra 1 lesson, students will need to recognize that regions of the plane represent solutions to systems of inequalities and that any point in that region is a solution to the system.

Monitor for students who describe a region based on:

  1. Non-mathematical language such as "over there" or "kind of to the left."
  2. Verbal relationships to the lines such as "above the first line, but below the second."
  3. Using a coordinate pair for a point in the region such as "the region containing the point \((1,2)\)."

Some students may use the axes to further divide the plane into regions, but this should be discouraged. The regions of the plane should be bound only by the graphed lines, not the axes.

Launch

For additional support, consider demonstrating how to play the game by graphing two intersecting lines and using a student volunteer to select a point in one of the 4 regions. Then ask the student questions to determine which region they are thinking of.

Student Facing

For each question:

  • Draw a graph that represents the equation or equations.
  • Determine how many regions the lines split the plane into.
  • Play a game with your partner
    • Partner A: Think of one of the regions.
    • Partner B: Without pointing to the coordinate plane, use the lines to ask questions of your partner about the region they have chosen. When you think you know which region they are thinking of, point to the region and ask your partner if you are correct. If you are correct, shade the entire region.
    • Change roles for the next question.
  1. \(x = 3\)

    Coordinate plane 
  2. \(y = \text{-}1\) and \(x = 2\)

    Coordinate plane 
  3. \(y = x\) and \(y = \text{-}x\)

    Coordinate plane 
  4. \(y = 2x + 2\) and \(y = 2x - 3\)

    Coordinate plane 
  5. \(x + y = 4\) and \(x - y = 6\)

    Coordinate plane 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

The purpose of the discussion is to recognize how lines can determine regions of the coordinate plane and methods for describing the regions.

Select previously identified students to share their methods for describing a particular region in the order listed.

  1. Non-mathematical language such as "over there" or "kind of to the left."
  2. Verbal relationships to the lines such as "above the first line, but below the second."
  3. Using a coordinate pair for a point in the region such as "the region containing the point \((1,2)\)."

Ask students:

  • "What do the graphs of equations that have 4 regions have in common?" (There are 2 lines that intersect at one point.)
  • "Will the plane always be split into 3 regions when there are 2 parallel lines graphed?" (Yes, unless the two lines are actually the same line and there are only 2 regions.)
  • "Into how many regions could 3 lines divide the plane?" (2 regions if all three lines are the same. 3 regions if two of the lines are the same and one is parallel, but different. 4 regions if two of the lines are the same and the third intersects it at one point or if all 3 lines are distinct, parallel lines. 6 regions if all three lines intersect at the same point or there are two parallel lines and one that is not parallel. 7 regions if each line intersects the other two in distinct points.)