Getting Started 
Misconception/Error The student is unable to use an algebraic method and attempts to solve the system by graphing. 
Examples of Student Work at this Level The student chooses to solve the system by graphing because he or she does not understand how to use an algebraic strategy. Typically, the student is unsuccessful since the solution contains fractions. 
Questions Eliciting Thinking What is a system of equations? What does the solution of a system of equations look like?
Can you explain why you chose to graph these equations? How can you use the graphs to find the solution to the system?
Do you know any algebraic methods for solving the system?
If you know that the solutions consist of fractions, is graphing a good choice of method? 
Instructional Implications 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 finding solutions of systems of equations graphically. 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 consider when an algebraic approach might be a better choice of method. 
Moving Forward 
Misconception/Error The student chooses to solve the system of equations algebraically but is unable to successfully implement the method. 
Examples of Student Work at this Level The student attempts to solve the system using the:
 Elimination method but makes a distribution error and does not consider that a value of a variable in this context cannot be negative.
 Substitution method but errs in solving the second equation for a and does not realize that the solution consists of an ordered pair of numbers.

Questions Eliciting Thinking Can you explain what you did? How did you solve for a?
What should the solution of this system look like? Will it only contain one value?
Your solution contains a negative value; does that make sense in this problem? 
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 chooses to solve the system of equations algebraically but makes a minor error. 
Examples of Student Work at this Level The student:
 Makes a computation error in the final step when solving for a.
 Correctly finds p but substitutes 11 instead of Â to find the value of a.

Questions Eliciting Thinking There is a minor error in your work. Can you find it?
Is your solution reasonable? What do a and p represent?
How can your check to see if your answer is correct? 
Instructional Implications Assist the student in identifying the error and provide feedback. Allow the student to make corrections as needed. Then provide the student with examples of algebraically solved 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 algebraically. Encourage the student to first check the reasonability of the solution and then check the solution by determining if it satisfies both of the original equations.
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 Solving a System of Equations 1 (AREI.3.6), Solving a System of Equations 2 (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 uses an algebraic method to solve the system and determines that the apples cost $1.25 per pound and the peaches cost $2.75 per pound. The student explains that graphing is not practical in this situation since the answers are not integers.

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?
How can you check your answer to see if it is correct? 
Instructional Implications Challenge the student with word problems that require writing a system of equations first. 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.
Consider implementing MFAS tasks Solving a System of Equations 1 (AREI.3.6), Solving a System of Equations 2 (AREI.3.6), and Solving a System of Equations 3 (AREI.3.6). 