Getting Started 
Misconception/Error The student does not understand how to graph a linear inequality (or line) in the coordinate plane. 
Examples of Student Work at this Level The student incorrectly graphs the boundary line and does not show the graph of the halfplane that represents the solution region. 
Questions Eliciting Thinking What is the slope of the boundary line of the inequality you are trying to graph? Is the slope positive or negative?
What is the yintercept of the boundary line of the inequality you are trying to graph?
Did you graph the boundary line using the correct slope and yintercept?
Is there more to the graph of an inequality than just the boundary line? 
Instructional Implications Review graphing lines written in slope intercept form (y = mx + b) and the role of the boundary line in graphing the solution set of an inequality. Demonstrate for the student how to use the yintercept and the slope to graph the line. Have the student test the coordinates of the yintercept of the boundary line in the inequality to determine whether or not it is a solution. 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 basic instruction on graphing inequalities. Emphasize that solutions ofÂ a linear equation are points on the line, but all possible solutions ofÂ an inequality lie in a halfplane separated by a boundary line. 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. 
Moving Forward 
Misconception/Error The student does not shade or incorrectly shades the halfplane that represents the solution region. 
Examples of Student Work at this Level The student:
 Correctly graphs the boundary line but does not shade the halfplane that represents the solution region.
 Incorrectly shades the wrong halfplane to indicate the solution region.

Questions Eliciting Thinking Did you graph all solutions to the inequality or just the boundary line?
How is graphing the solutions to an inequality different from graphing a line?
How did you determine which halfplane contains the solutions of the inequality?
Identify the coordinates of a point in the plane you shaded. How can you use these coordinates to determine if this point represents a solution of the inequality? 
Instructional Implications Emphasize that solutions ofÂ a linear equation are points on the line, but all possible solutions ofÂ an inequality lie in a halfplane separated by a boundary line. Review the method forÂ testing a point to determine the region of the plane in which solutions are found. Emphasize the relationship between the shaded part of the graph and the solutions of the inequality.
Provide additional examples of strict (< or >) and nonstrict ( or ) inequalities for the student to graph written in both slopeintercept and standard forms. Caution the student not to rely on the direction of the inequality symbol to determine the region to shade (e.g., concluding the less than symbol indicates to shade below the line) as this is only true in special cases. 
Almost There 
Misconception/Error The student makes a minor mistake when graphing the boundary line. 
Examples of Student Work at this Level The student:
 Does not draw arrows at each end of the boundary line.
 Uses a dashed line instead of a solid line when graphing the boundary line for the nonstrict ( or ) inequality.
 Graphs the line as if the slope were Â instead of , but all other boundary line details are correct.

Questions Eliciting Thinking Did you graph a line or a segment? What notation is used to indicate the boundary is a line rather than a segment?
What is the relationship between the inequality and the type of line you draw? Why is the boundary line for an inequality sometimes solid and sometimes dashed?
You made a mistake when graphing your boundary line. Can you find your mistake? 
Instructional Implications Provide feedback to the student concerning the specific error. If needed, review the conventions for graphing boundary lines (i.e., drawing solid versus dashed lines). Be sure the student understands how the graph, both the shaded region and the boundary line, shows the solutions of the inequality.
Provide additional examples of strict (< or >) and nonstrict ( or ) inequalities for the student to graph, written in both slopeintercept and standard forms. 
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 the inequality using a solid boundary line and shading the halfplane aboveÂ it.

Questions Eliciting Thinking Why did you show the boundary line as a solid line? What does the solid line signify? What would a dashed line signify?
How did you determine which halfplane contains the solutions of the inequality?
What does the shading indicate about the solutions of the inequality? 
Instructional Implications Challenge the student to graph a system of linear inequalities and to describe the portion of the plane that represents solutions to the system. 