Getting Started 
Misconception/Error The student does not understand how to graph linear inequalities (or lines) in the coordinate plane. 
Examples of Student Work at this Level The student incorrectly graphs boundary lines and may also shade solution regions incorrectly.

Questions Eliciting Thinking Can you explain how you graphed the boundary lines?
Can you explain how you determined the solution regions? 
Instructional Implications Review graphing lines written in slope intercept form and the role of the boundary line in graphing the solution region of an inequality. Demonstrate for the student how to use the yintercept and the slope to graph a line. Emphasize that solutions of a linear equation are points on the line, but all possible solutions of an inequality lie in a halfplane separated by a boundary line. Guide the student to understand the relationship between the inequality symbol and whether or not points on the boundary line are solutions of the inequality. Provide instruction on conventions for graphing boundary lines (i.e., drawing solid versus dashed lines). Provide additional examples of strict (< or >) and nonstrict ( or ) inequalities for the student to graph. Emphasize the relationship between the shaded part of the graph and the solutions of the inequality.
Introduce graphing systems of inequalities. Emphasize that the solution region of a system of inequalities is the intersection of the solution regions of the individual inequalities. Explain that any point in the intersection satisfies each inequality in the system so, by definition, is a solution of the system. Review the logic used when testing a point to determine the solution region.
Provide additional opportunities to graph systems of inequalities. After each graph is completed, ask the student to identify a point on the graph that is a solution of the system and a point that is not. Then ask the student to justify his or her choices by determining whether or not the points satisfy the system of inequalities. 
Moving Forward 
Misconception/Error The student does not shade or incorrectly shades the solution region. 
Examples of Student Work at this Level The student correctly draws boundary lines for each linear inequality but does not know how to determine the solution region or incorrectly identifies it. For example, the student correctly uses a testpoint to determine which halfplane contains the solutions, but makes a mistake when interpreting the results and shades the wrong region.

Questions Eliciting Thinking Can you explain how you determined the solution regions?
What does it mean if both test statements are true? What does it mean if one is true but the other is false? 
Instructional Implications Explain that the solution region of a system of inequalities is the intersection of the solution regions of the individual inequalities. Demonstrate that any point in the intersection satisfies each inequality in the system so, by definition, is a solution of the system. Review the logic used when testing a point to determine the solution region.
Provide additional opportunities to graph systems of inequalities. After each graph is completed, ask the student to identify a point on the graph that is a solution of the system and a point that is not. Then ask the student to justify his or her choices by determining whether or not the points satisfy the system of inequalities. 
Almost There 
Misconception/Error The student makes a minor graphing error. 
Examples of Student Work at this Level The student demonstrates an understanding of how to graph the solution region of a system of inequalities but makes a minor graphing error. For example, the student:
 Uses a solid line instead of a dashed line when graphing a strict (< or >) inequality.
 Interchanges the yintercepts of the boundary lines but all other work is correct.
 Graphs the boundary of the second inequality as y = x + 2 but all other work is correct.

Questions Eliciting Thinking Why is the boundary line for an inequality sometimes solid and sometimes dashed?
Can you explain how you graphed the boundary lines? You made an error when graphing one of them. Can you find your error? 
Instructional Implications Provide feedback to the student concerning any errors made. If needed, review the conventions for graphing boundary lines (i.e., drawing solid versus dashed lines). Be sure the student understands points on the boundary line of nonstrict ( or ) inequalities are contained in the solution region, but points on the boundary line of strict (< or >) inequalities are excluded from the solution region.
Provide opportunities to graph systems of inequalities. After each graph is completed, ask the student to identify a point on the graph that is a solution of the system and a point that is not. Then ask the student to justify his or her choices by determining whether or not the points satisfy the system of inequalities. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level The student correctly graphs each boundary line and identifies the solution region. When asked, the student can identify whether or not particular points represent solutions of the system.

Questions Eliciting Thinking What does a solid line signify? What does a dashed line signify?
What does the shading signify? Where in your graph are solutions of the system found?
Is (2, 4) a solution of the system? What about (0, 2)? (0, 0)? (3, 9)? 
Instructional Implications Challenge the student to identify points in the solution region that are outside of the area shown on the graph such as (100, 2).
Provide a graphed system of inequalities and ask the student to identify examples and nonexamples of solutions. Give the student examples of points that are in shaded regions of some, but not all, of the inequalities, on dashed boundary lines, on solid boundary lines that are part of the solution region, and on solid boundary lines that are not part of the solution region. Ask the student to decide if each point is a solution. 