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Defining Regions Using Inequalities
During the
Lesson
  • Whole-Class Interactive Introduction: Hunting the Target (10 minutes)
    • If you have a short lesson, or you find the lesson is progressing at a slower pace than anticipated, we suggest you end the lesson after the paired work, "Preparing to Play Give Us a Clue!", and continue in a second lesson.

      • Give each student either a mini-whiteboard, pen, and eraser, or a sheet of squared paper.
      • Use slide P-1 of the projector resources to project the 6 x 6 coordinate grid on to the board.
      • Write the pair of coordinates (2,2) on a piece of paper, fold it in half (hiding the coordinates), and stick this to the board.
        • I am thinking of a target point on this grid. I have written the coordinates on this paper. Both coordinates are integers.
        • Your task is to guess which point I am thinking of.
        • Here is the first clue: 3y + 2x ≤ 12
        • Does anyone know what this clue means?
      • Student may need careful leading through this idea, so take this stage slowly. Use questions such as the following, asking students to respond using their mini-whiteboards:
        • Show me the coordinates of a point that satisfies the clue.
        • Can you show me another point? ... and another? How do you know?
      • As students suggest possible points, mark these clearly on the grid.
        • Where are all the points that satisfy this clue?
          • [On or below the line 3y + 2x = 12.]
        • Where are all the points that don't satisfy this clue?
          • [Above the line 3y + 2x = 12]
        • Give me a point that just satisfies the clue.
        • Give me a point that easily satisfies the clue.
      • Explain that for this lesson, the region that does not satisfy a clue is to be shaded out.

      To help students keep track of each clue, you may want to use a different color marker for each inequality.

        • Here's the second clue: x > 1.
        • Shade out all the points that are eliminated.
        • Show me the new region.
        • Which points are possible now?
        • Is (1,2) a possible point?

      • Explain that we use a dashed line to show that the point on the line x = 1 are not included as possible points for the target.
        • Here's the third clue: y > x − 1.
        • Shade out all the points that are eliminated.
        • Which points are possible now?
        • Show me the new region.
        • Do you know the point I am thinking of yet?
        • Is (2,2) the only possibility?
        • Why can't (3,2) be a possible point for the target?

      • Although there are many non-integer points that are possible, explain that for this lesson, we will stick with integer coordinates.
  • Preparing to Play Give Us a Clue! (10 minutes)
      • Give each student a copy of the sheet Give Us a Clue!
      • Use slide P-2 of the projector resources to project the 8 x 8 coordinate grid onto the board.
        • You are soon going to play a game called "Give Us a Clue!"
        • You will use the lines on the small graphs on the handout.
        • Before beginning the game, you need to figure out the inequalities for the regions to the left and right of each given line. You will use these inequalities as clues in the game.
        • For example, look at the line 2x − y = 8.
        • Which side of the line are points that fit the inequality 2x − y ≥ 8?
        • Which side of the line are points that fit the inequality 2x − y ≤ 8?

      In order to answer these two questions, it is helpful to test the inequality with specific pairs of coordinates. These are sometimes called test points.

      (0,0) is usually a good choice for a test point, since it makes the arithmetic easy, but if the line itself goes through the origin, then another point should be chosen:

        • Can you put the inequality into words?
        • Let's use the origin (0,0) as a test point. This point is to the left of the line.
        • Which of the two inequalities [2x − y ≥ 8 or 2x - y ≤ 8] does it fit?
        • Now choose your own test point to the right of the line.
        • Use its coordinates to check the inequality for this region.

      Since 2(0) − 0 ≤ 8 is true, the origin is included in the region 2x − y ≤ 8. This region is to the left of the line.

  • Paired Work: Preparing to Play Give Us a Clue! (10 minutes)
      • Organize the class into pairs of students.
      • Explain how students should work collaboratively.
        • Take it in turns to figure out the inequalities for each region of the twelve small graphs.
        • Once you have done this, explain to your partner how you came to your decision.
        • Your partner should either explain that reasoning again in his or her own words or challenge the reasons you gave.
        • You need to agree on and both be able to explain the inequalities for each region of each graph.
        • Make sure you write all the inequalities on your own copy of Give Us a Clue!
        • There is no need to shade the graphs.

      The purpose of this structured paired work is to make each student engage with their partner's explanations and to take responsibility for their partner's understanding.

      You have two tasks during the paired work: to note aspects of the task students find difficult, and support student reasoning.

      Note aspects of the task students find difficult.

      For example, are students having difficulties using a test point? Do they understand the difference between inequality symbols?

      You can use information about particular difficulties to focus whole-class discussion toward the end of the lesson.

      Support student reasoning.

      Try not to make suggestions that move students towards a particular answer. Instead, ask questions to help students to reason together. For students struggling to understand the symbols, it may help if they put the inequalities into words.

          • How did you figure out the inequality for this region?
          • [Select a graph that goes through the origin.] Why is (0,0) not a good test point for this graph?
          • [Select one of the first four graphs.] Why is (4,4) not a good test point to use for this graph?
    • Sharing Work: Preparing to Play Give Us a Clue! (10 minutes)
        • Ask students to check their work with a neighboring pair of students.
          • Check to see which graphs are different.
          • When there is a disagreement, take turns to justify your decision. If you still don't agree, ask for further explanation.
          • Both of you need to agree and understand the math.
    • Students Playing Give Us a Clue! (15 minutes)
        • When students are satisfied with their twelve graphs, use slide P-3 of the projector resource to introduce the game:
          • In your pairs, you are now going to play "Give Us a Clue!"
          • One of you will be the target picker and the other the target hunter.
          • The target picker decides on the position of the target and gives the clues.
          • When giving clues, the target picker can use any inequality sign (≤, <, ≥, >), but not the = sign. Try to give helpful clues! As you give the clues, write them as a list on your mini-whiteboard.
          • The target hunter uses the clues to find the target.
          • The aim of the game for both partners is to find the target in the least number of tries.
          • Both partners should use a blank grid to keep track of the clues that are given.
          • Each time a clue is given, shade out the region where the target cannot be located.

        It is important that students cannot see each other's graphs. They could use a book or folder to hide the graph from their partner.

        • Encourage students to give clues using the correct inequality language, rather than using imprecise language, such as "The point is above the line."
        • When the target picker has used all the useful inequalities on the handout, they could make up their own.
        • At the end of each game, students should check each other's graphs.

        If they are not the same, encourage them to work together to identify mistakes made. The mini-whiteboard listing the clues may help sort out disagreements. This should be seen as a collaborative rather than a competitive activity.

        • Then students can reverse roles.
        • For students who have successfully completed this task, ask them to create their own inequalities and use them to play the game with their partner.
    • Whole-Class Discussion (10 minutes)
      • In the summary discussion, you can explore the best strategy for giving a clue, while revising the main math concepts in the lesson. Students should use their mini-whiteboards to respond to your questions.

        • Use slide P-2 of the projector resource to project the 8 x 8 coordinate grid on to the board.
          • We will now investigate how to give the best clues for targets within an 8 x 8 grid. We are still using the inequalities on the worksheet.
          • Can anyone think of the best first clue for the point (2,5)? [y > 2x.]

        • Ask a few students to justify their answers. Use different color markers to draw their clues on the board.
          • In this case, is the clue y ≥ 2x better than the clue y > 2x? Why?
        • Once students are satisfied that they are using the best first clue, ask:
          • What is the next best clue? [y < x + 4.]
        • Again ask students to explain their reasoning.
          • How do you know y < x + 4 is a better clue than y < x? Show me.
        • If students are struggling with the difference between a clue that uses < and one that uses ≤, ask:
          • How many points could the target be if you use the clue y < x + 4?
          • How many points could the target be if you use the clue y ≤ x + 4?
        • Now ask for a final clue:
          • And what is another good clue? [y > 4]
          • How many places could the target be now?

        • You could extend this further by asking:
          • Can you think of a target point within the 8 x 8 grid that only requires two clues?

        It will be a point on a line. For example, when the target point is (8,4), the clues could be y ≤ 4 and x + 2y ≥ 16.

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