Getting Started 
Misconception/Error The student is unable to solve the system of equations either graphically or algebraically. 
Examples of Student Work at this Level The student
 Does not attempt to solve the system algebraically. The student attempts to graph each equation but is unable to do so correctly.
 Attempts to solve the system using one or both approaches but is unable to correctly find the solution using either approach.

Questions Eliciting Thinking Can you explain how you graphed these equations? What method did you use?
Is there another way to write the equation that would make it easier to graph?
What is a system of equations? What methods have you learned for solving systems of equations?
What does the solution of a system of equations look like? 
Instructional Implications Review standard and slopeintercept forms of linear equations in two variables and how to graph equations in each of these forms. Assist the student in transforming equations from other forms into one or both of these forms.
Review what it means for an ordered pair to be a solution of a linear equation in two variables and a solution of a system of linear equations in two variables. Demonstrate graphically finding solutions of systems of equations. Expose the student to graphs of systems of equations that result in intersecting lines, parallel lines, and the same line. Relate the solution outcome in each case to the nature of the graph.
Review algebraic methods for finding solutions of systems of linear equations in two variables. Encourage the student to use the method that is best suited to the form of the equations given in the system. Expose the student to systems of equations that result in one, none, and infinitely many solutions. Relate the solution outcome in each case to the graph of the system of equations.
Provide the student with additional opportunities to solve systems of equations both graphically and algebraically. Encourage the student to compare the solution found by graphing to the solution found using an algebraic method and to reconcile differences when they occur. Then consider implementing this task again using another set of equations. 
Moving Forward 
Misconception/Error The student solves the system of equations graphically but is unable to correctly solve the system algebraically. 
Examples of Student Work at this Level The student correctly solves the system by graphing, but when attempting to solve algebraically, the student:
 Uses substitution but substitutes back into the same equation and cannot finish solving.
 Uses elimination but does not add the yterms correctly and cannot finish solving.
The student does not attempt to solve the system algebraically.

Questions Eliciting Thinking What method did you use to solve this system of equations? What other methods could you use?
It looks like you solved one equation for y. Which equation should you substitute the resulting expression into? Does it matter?
I see you added the two equations so the x variables canceled out. What is the sum of the yterms?
What should the solution of the system of equations be? 
Instructional Implications Review algebraic methods for finding solutions of systems of linear equations in two variables. Encourage the student to use the method that is best suited to the form of the equations given in the system. Expose the student to systems of equations that result in one, none, and infinitely many solutions. Relate the solution outcome in each case to the graph of the system of equations.
Provide the student with additional opportunities to solve systems of equations both graphically and algebraically. Encourage the student to compare the solution found by graphing to the solution found using an algebraic method and to reconcile differences when they occur. Then consider implementing this task again using another set of equations. 
Almost There 
Misconception/Error The student demonstrates an understanding of how to solve the system of equations both graphically and algebraically but makes an error when using one or both methods. 
Examples of Student Work at this Level The student makes an initial algebraic error (e.g., when solving one or both equations for y) but then continues to solve and graph making no additional errors. The student may have difficulty determining his or her error.

Questions Eliciting Thinking If your solution using algebra is different from the solution you found by graphing, what can you do to determine which solution is correct?
Once you know which solution is correct, what can you do to locate your error? 
Instructional Implications Assist the student in identifying his or her error and provide feedback. Allow the student to make corrections as needed. Then provide the student with examples of completed systems of equations that contain errors and have the student identify and correct the errors. These examples may come from the work of other Almost There students.
Provide the student with additional opportunities to solve systems of equations both graphically and algebraically. Encourage the student to compare the solution found by graphing to the solution found using an algebraic method and to reconcile differences when they occur.
Encourage the student to consider when one algebraic method is more efficient or easier to use than another method. Guide the student to consider the form of the equations given in the system and to transform one or both equations to another form if that would facilitate solving the system.
Consider implementing MFAS tasks Apples and Peaches (AREI.3.6), Solving a System of Equations  1 (AREI.3.6), and Solving a System of Equations  3 (AREI.3.6). 
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 both equations and identifies the solution as (8, 1). The student then correctly uses an algebraic method to find the solution of the system.

Questions Eliciting Thinking Which algebraic method did you use to solve this system? Why did you choose that method?
How can you determine which algebraic method will be the most efficient to use on a system?
When might it not be advisable to use graphing to solve a system of equations?
How can you check your answer to see if it is correct? 
Instructional Implications Challenge the student with word problems thatÂ require first writing a system of equations. Consult websites such as NCTM Illuminations for additional activities and exercises on systems of equations (http://illuminations.nctm.org/Lesson.aspx?id=2783).
Ask the student to describe the algebraic outcomes when solving systems of equations that have no solution or infinitely many solutions.
Expose the student to a simple system of three equations in three variables. Model solving the system using the method of substitution. Provide additional challenge problems for the student to attempt.
Consider implementing MFAS tasks Apples and Peaches (AREI.3.6), Solving a System of Equations  1 (AREI.3.6), and Solving a System of Equations  3 (AREI.3.6). 