In this activity, students recall how to write solutions to inequalities in one variable, then they begin to move those solutions onto a plane to expand into thinking about inequalities in two variables. As students connect graphs of inequalities on a number line and on a coordinate plane they look for and express regularity in repeated reasoning (MP8).

The purpose of the discussion is to begin to move students' thinking from drawing solutions to one-variable inequalities on a number line toward drawing solutions to two-variable inequalities on a coordinate plane.

Select students to share their solutions. Poll the class about their coordinate pairs for the last question. Plot the points on a coordinate plane for all to see. Then ask, "What do you notice about the points we found that are solutions to the inequality? How is it related to the number line you drew for the same inequality?" (It includes all the points to the left of -2 on the plane. It represents the same region as the number line, but shaded above and below the line.)

Here are some other questions for discussion:

• "Why do some of the number lines have open dots and others have closed dots?" (An open dot means the point is not included in the solution. A closed dot means the point is included in the solution.)
• "How do you think we could indicate on the plane whether points like should be included in your shaded region?" (Students do not need to learn the usual convention for this at this point. Students will learn the convention in the associated Algebra 1 lessons.)

In this activity, students begin to see a line that represents an equation as a boundary condition that breaks the plane into two halves. Students graph a line, determine whether a given point is on or off the line using both the equation and its graph, and then choose a coordinate so that a point will be on the line, above the line, or below the line. As students explore what values would place a point on, above, or below the line they reason abstractly and quantitatively (MP2).

The purpose of the discussion is for students to recognize that a line divides the plane into three different parts: points that are on the line and points that are to either side of the line. Select students to share their solutions and reasoning.

Here are some questions for discussion:

• "Is it easier to decide whether a point is on the line based on the equation or the graph?" (It is faster to use the graph to see if it is reasonable that the point is on the line, but the equation seems like a more accurate way to know.)
• "How many choices did you have for so that it was on the line? How many choices did you have for so that it is above the line?" (There is only 1 choice for so that the point is on the line. There are infinitely many choices for so that it is above the line.)
• "What rule could you write to make sure the point is below the line?" (All of the possible choices for make the inequality true.)