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Defining Regions Using Inequalities
Before the
  • Assessment Task: Combining Inequalities (15 minutes)
    • Set this task, in class or for homework, a few days before the formative assessment lesson. This will give you an opportunity to assess the work, to find out the kinds of difficulties students have with it. You will then be able to target your help more effectively in the follow-up lesson.

      • Give each student a copy of Combining Inequalities, a pencil and a ruler.
      • Briefly introduce the task:
        • Spend 15 minutes individually answering these questions.
        • Show all your work so that I can understand your reasoning.

      It is important that students answer the questions without your assistance, as far as possible.

      Students should not worry too much if they cannot understand or do everything because in the next lesson they will engage in a similar task, which should help them. Explain to students that by the end of the next lesson, they should expect to answer questions such as these confidently. This is their goal.

  • Assessing Students' Responses
    • Collect students' written work for formative assessment. Read through their papers and make informal notes on what their work reveals about their current levels of understanding.

      We strongly suggest that you do not write scores on students' work. Research shows that this is counterproductive, as it encourages students to compare scores, and distracts their attention from what they could do to improve their mathematics.

      Instead, help students to make further progress by asking questions that focus their attention on aspects of their work. Some suggestions for these are given on the table below. These have been drawn from common difficulties observed in trials of this unit.

      We suggest that you write your own lists of questions, based on your own students' work, using the ideas that follow. You may choose to write questions on each student's work. If you do not have time to do this, select a few questions that will help the majority of students. These can then be written on the board at the end of the lesson.

      Assessing Student Responses Suggested Questions and Prompts:

      Student has difficulty distinguishing between > and ≥, or < and ≤.

      • For example: The student includes (2,3) and (5,3) as possible locations for the target (Q1.)
      • Or: The student states the treasure is located at (2,1), (4,3), (5,3), or (3,2) (Q3.)
      • Or: The student does not use a dashed line for < or > inequalities (Q2 or Q3.)
      • Write the inequalities into words.
      • What is the difference between > and ≥?
      • What is the difference between < and ≤?
      • The point (2,5) is outside the region where the treasure is located. Which clue tells you this?
      • Are points on the line x = 2 possible locations for the treasure? Are points on the line 2y − x = 0 possible locations for the treasure? How can you distinguish graphically between the two?
      • Which points are not allowed?

      Q2. Student uses guess and check to figure out the possible location for the treasure.

      • The student does not draw the inequality boundaries as lines on the grid but instead guesses possible locations for the treasure and checks to see if they fit the clues.
      • Can you think of a quicker way to figure out the possible locations?
      • How can you convince me there are no other possible points?
      • How can you use the graph to show the region where the treasure is located?

      Q3. Student provides insufficient reasoning.

      • For example: The student does not explain the reason why Clue 4 is unhelpful.
      • How does the clue affect the region where the treasure is located?
      • Does this clue help you find the position of the treasure?

      Q3. Student assumes the treasure is located at one of the points chosen in Q1.

      • Check to see if your point fits your new clue.

      Student correctly answers all the questions.

      • The student needs an extension task.
      • Another treasure is at (6,5). Write just two clues that will locate the exact position of the treasure. Your clues should use the inequality symbols >, <, ≥, or ≤.
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