• 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.

• 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?

• 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.

• 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.