Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 59692
How Many Solutions?
Students are asked to determine the number of solutions of each of four systems of linear equations without solving the systems of equations.
Subject(s): Mathematics
Grade Level(s): 8
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, system of linear equations, solution, intersecting lines, parallel lines, coinciding lines
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_HowManySolutions_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 How Many Solutions? worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
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.
page 1 of 5 Does not understand what slopes and y-intercepts 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 “the slopes are different and the y­intercepts are different.” 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 one-to-one 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 Solution (8.EE.3.8) and
MFAS task Solving System of Linear Equations by Graphing (8.EE.3.8).
Review slope, y-intercept, and the slope-intercept form of a linear equation. Be sure the student understands how to identify the slope and the y-intercept of the graph of
a line from its equation written in slope-intercept form. Model how to graph linear equations using the y-intercept and slope. Next, provide the student with systems of
linear equations in slope-intercept form (no solution, infinitely many solutions, and one solution). Have the student identify the slopes and y-intercepts 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 y-intercept; 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.
page 2 of 5 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 slope-intercept form (standard form) look like? Can you convert an equation in standard form to slope-intercept form?
Instructional Implications
Make explicit the difference between the phrases “determine the solution” and “determine the number of solutions.” Discuss how many solutions systems represented by
parallel lines, intersecting lines, and coinciding lines will have and why. Confirm the student’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 slope-intercept 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 slope-intercept form for easier comparison.
Ask the student to describe the relationship between the slopes and the y-intercepts 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.
page 3 of 5 Reads
as
in the last system. Upon questioning, the student acknowledges the oversight.
Writes “multiple solutions” instead of “infinitely many” solutions.
Determines the second system has no solution because the “slopes are the same and the lines will be parallel.” Upon questioning, the student acknowledges that he or
she made an error by not converting to slope-intercept form (or made an error in his or her mental conversion).
Questions Eliciting Thinking
What do you mean by “multiple solutions”?
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.
page 4 of 5 Upon questioning, the student explains how to convert the second and third systems from standard form to slope-intercept 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 “the lines are the same”? What does “infinitely many” 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 (8.EE.3.8).
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 (8.EE.3.8).
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
How Many Solutions? 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 5 of 5