Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 59690
Solving Systems of Linear Equations
Students are asked to solve three systems of linear equations algebraically.
Subject(s): Mathematics
Grade Level(s): 8
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, system of linear equations, solution
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_SolvingSystemsOfLinearEquations_Worksheet.docx
FORMATIVE ASSESSMENT TASK
Instructions for Implementing the Task
This task can be implemented individually, with small groups, or with the whole class.
1. The teacher asks the student to complete the problems on the Solving Systems of Linear Equations worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
Getting Started
Misconception/Error
The student does not have an effective strategy for solving systems of linear equations.
Examples of Student Work at this Level
The student:
Is unable to start the problem.
Tries to add the equations as they are.
Tries to solve each equation individually.
Uses “guess and test.”
Questions Eliciting Thinking
page 1 of 4 What is a system of linear equations? What does a solution of a system of equations look like?
What is meant by “solve” the system?
What are you trying to accomplish by adding the equations together?
Instructional Implications
If needed, review what it means for an ordered pair of numbers to be a solution of a linear equation. Emphasize the one-to-one relationship between solutions of linear
equations and points on the lines that represent them. Give the student a linear equation such as y = 2x + 3 along with its graph. Ask the student to use the graph to
identify several points on the line and then demonstrate that each point satisfies the equation. Next ask the student to identify a point not on the line and use the
equation to show that it is not a solution.
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.
Next, provide instruction on solving systems of linear equations using algebraic methods such as Substitution and Elimination. Provide additional opportunities to use both
methods to solve systems of equations. Be sure to include equations with no and infinitely many solutions. Assist the student in recognizing algebraic outcomes associated
with these two cases. Eventually, guide the student to consider which method might be better suited to the equations in a particular system.
Moving Forward
Misconception/Error
The student has a strategy for solving systems, but makes significant algebraic errors.
Examples of Student Work at this Level
The student:
Combines terms on opposite sides of the equal sign when adding the equations together.
Uses the wrong operation when attempting to eliminate a variable term from an equation.
Combines like terms incorrectly.
Questions Eliciting Thinking
When using the elimination method, can you eliminate (or combine like terms) if the like terms are on opposite sides of the equal sign? Can you find the error in this step?
What happens when you try to eliminate the same term from each side of an equation? What happens to the 4x on the right side of this equation when you try to
eliminate it from the left side of the equation: 4x + 28 = 4x + 10?
Did you check your solutions in the original equations?
Instructional Implications
Review the process for using the elimination and substitution methods. Model the use of the elimination and substitution method for each system and explain the rationale
behind each method. Encourage the student to identify which method is better suited to each system. Emphasize the importance of checking the solution in each of the
original equations.
Provide the student with guided practice in areas of need. For example, combining like terms, operations with integers, working with equations with variables on both sides
of the equation.
page 2 of 4 Provide additional opportunities to solve systems of equations algebraically and provide feedback.
Almost There
Misconception/Error
The student has a strategy for solving systems, but makes an easily correctable error.
Examples of Student Work at this Level
The student:
Makes a sign error.
Does not understand the significance of the algebraic outcome in the second equation.
Rewrites part of an equation incorrectly which leads to subsequent errors.
Checks the solution in only one of the equations.
Questions Eliciting Thinking
I think you made an error in this problem? Can you find and fix your error?
For the second system, your work is completely correct but what do you think this means in terms of the solution?
Instructional Implications
Provide feedback to the student concerning the error made and allow the student to revise his or her work. Offer strategies to help the student avoid minor errors. For
example, suggest that the student check solutions in both of the original equations and check that each equation was rewritten correctly.
Be sure the student understands the significance of algebraic outcomes when solving systems that result in no or infinitely many solutions. Ask the student to summarize the
various possibilities when solving systems of equations in terms of the algebraic outcomes, the nature of the graphs, and the number of solutions.
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 solves each system correctly getting (4, 9) for the first system, “no solution” for the second system, and (­8, 24) for the third system. For the first and third
page 3 of 4 system, the student checks the solution to see that it satisfies both equations. The student can explain why the second system has no solution.
Questions Eliciting Thinking
Is (4, 9) the only solution for the first system? Why or why not?
What will the graph of each system look like?
Can you predict how many solutions a system will have by looking at the equations (not graphing the equations)?
Is there another way you could have solved each system?
How do you decide which method to use when solving a system? Do you always use the same method?
Instructional Implications
Discuss how many solutions systems represented by intersecting lines, parallel lines, and coinciding lines will have. Present three different systems of linear equations (e.g.,
one solution, no solution, infinitely many solutions) with each equation written in slope-intercept form. Ask the student to analyze the slopes and y-intercepts of the
equations in each system and to predict how many solutions each system will have.
Next, present the student with a system of linear equations where the two equations are not written in the same form (e.g., y = 3x – 4 and ­3x + y = 2). Show the
student how to rewrite the equations for easy comparison. Consider implementing MFAS task How Many Solutions? (8.EE.3.8).
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Solving Systems of Linear Equations worksheet
SOURCE AND ACCESS INFORMATION
Contributed by: MFAS FCRSTEM
Name of Author/Source: MFAS FCRSTEM
District/Organization of Contributor(s): Okaloosa
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.8.EE.3.8:
Description
Analyze and solve pairs of simultaneous linear equations.
a. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection
of their graphs, because points of intersection satisfy both equations simultaneously.
b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the
equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because
3x + 2y cannot simultaneously be 5 and 6.
c. Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given
coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line
through the second pair.
Remarks/Examples:
Examples of Opportunities for In-Depth Focus
When students work toward meeting this standard, they build on what they know about two-variable linear
equations, and they enlarge the varieties of real-world and mathematical problems they can solve.
page 4 of 4