Getting Started 
Misconception/Error The student is unable to determine the number of solutions using any strategy. 
Examples of Student Work at this Level The student:
 Does not understand what it means for an ordered pair of numbers to be a solution of a system of equations.
 Does not understand what slopes and yintercepts indicate about the relationship between equations and their graphs.
 Associates parallel lines with infinitely many solutions and coinciding lines with no solutions.
 Says there are no solutions when â€śthe slopes are different and the yintercepts are different.â€ťÂ

Questions Eliciting Thinking How did you determine the number of solutions for the first (second, third, fourth) system?
What does it mean to be a solution of a linear equation in two variables?
What does it mean to be a solution of a system of equations?
How many solutions will systems represented by parallel (coinciding or intersecting) lines have? Why? 
Instructional Implications Review what it means for an ordered pair to be a solution of a system of linear equations in two variables. Demonstrate finding the solution of a system of two independent equations graphically. Ask the student to use the equations to show that the solution satisfies each equation in the system and is, consequently, a solution of the system. Emphasize the onetoone relationship between solutions of equations and points on their graphs. Make it clear that the point of intersection of the two graphs represents a solution of each equation in the system so is, consequently, a solution of the system. Next, expose the student to graphs of systems of equations that result in parallel lines and coinciding lines. Relate the graphs to the equations they represent and the nature of the solutions. Consider using MFAS task Identify the SolutionÂ and MFAS task Solving System of Linear Equations by Graphing.
Review slope, yintercept, and the slopeintercept form of a linear equation. Be sure the student understands how to identify the slope and the yintercept of the graph of a line from its equation written in slopeintercept form. Model how to graph linear equations using the yintercept and slope. Next, provide the student with systems of linear equations in slopeintercept form (no solution, infinitely many solutions, and one solution). Have the student identify the slopes and yintercepts from the equations and use them to graph the lines. Guide the student to realize parallel lines will have the same slope but different intercepts; coinciding lines will have both the same slope and same yintercept; and intersecting lines will have different slopes. Finally, help the student identify parallel, coinciding, and intersecting lines from their equations. Provide the student with additional opportunities to determine the number of solutions of given systems of linear equations and to justify the answers. Eventually, introduce systems in which the equations are written in standard form. 
Moving Forward 
Misconception/Error The student is unable to determine the number of solutions of systems of linear equations without solving. 
Examples of Student Work at this Level The student attempts to solve the system:
 By graphing.
 Algebraically.
Note: Errors may occur when the student attempts to solve.

Questions Eliciting Thinking What is a solution? How would you determine the solutions without solving the system?
How many solutions do systems represented by parallel lines (intersecting lines or coinciding lines) have? How do you know?
How do you know if the lines are going to be parallel (intersecting or coinciding)? Can you tell by looking at the equations if the slopes are going to be the same?
What does an equation in slopeintercept form (standard form) look like? Can you convert an equation in standard form to slopeintercept form? 
Instructional Implications Make explicit the difference between the phrases â€śdetermine the solutionâ€ť and â€śdetermine the number of solutions.â€ť Discuss how many solutions systems represented by parallel lines, intersecting lines, and coinciding lines will have and why. Confirm the studentâ€™s understanding by providing systems of linear equations and their graphs. Ask the student to identify the solutions of each system, if they exist, and justify his or her responses.
Provide the student with systems of linear equations in slopeintercept form (no solution, one solution, and infinitely many solutions). Emphasize the relationship between the equations and their graphs, and guide the student to interpret the graphical outcomes to determine the number of solutions of each system. Provide guided and independent practice as needed.
Finally, address systems of linear equations written in standard form. Model how to convert equations written in standard form to slopeintercept form for easier comparison. Ask the student to describe the relationship between the slopes and the yintercepts that result in each case (no solution, one solution, and infinitely many solutions). Provide both guided and independent practice. Encourage the student to justify his or her answers. 
Almost There 
Misconception/Error The student makes a minor error when determining the number of solutions or justifying an answer. 
Examples of Student Work at this Level The student:
 Identifies the last system as perpendicular lines when comparing the slopes.
 Reads as in the last system. Upon questioning, the student acknowledges the oversight.
 Writes â€śmultiple solutionsâ€ť instead of â€śinfinitely manyâ€ť solutions.
 Determines the second system has no solution because the â€śslopes are the same and the lines will be parallel.â€ť Upon questioning, the student acknowledges that he or she made an error by not converting to slopeintercept form (or made an error in his or her mental conversion).

Questions Eliciting Thinking What do you mean by â€śmultiple solutionsâ€ť?
How do you know if the lines will be perpendicular?
How did you determine that the second system has no solution? 
Instructional Implications Guide the student to find and fix any errors made. Have the student explain the error and how it can be corrected. Encourage the student to check over his or her work carefully. Provide the student with additional practice opportunities, and then consider the Instructional Implications for a Got It student. 
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 identifies the number of solutions of each system of linear equations, without solving the systems, as: no solution, one solution, or infinitely many solutions.
Upon questioning, the student explains how to convert the second and third systems from standard form to slopeintercept form for easier comparison.

Questions Eliciting Thinking What is meant by a solution of a system of equations?
Why do systems represented by parallel lines have no solution?
How did you determine the slopes were different in the second system? Why might a student think the slopes were the same in the second system?
For the third system, what do you mean by â€śthe lines are the sameâ€ť? What does â€śinfinitely manyâ€ť solutions mean?
For the fourth system, do you think the lines will be perpendicular? Why or why not? 
Instructional Implications Challenge the student to write three systems of linear equations in standard form (one having no solution, one having infinitely many solutions, one having only one solution). Next, have the student verify the solutions both by graphing the systems and by solving them algebraically. Consider using MFAS task System Solutions.
Provide instruction and guided practice on how to write a system of linear equations in order to solve a word problem. Consider implementing CPALMS Lesson Plan Exploring Systems with Piggies, Pizzas and Phones (ID 26881). Then, consider using MFAS task Writing System Equations. 