Getting Started |
Misconception/Error The student is unable to identify the solution of a graphed system of two independent equations. |
Examples of Student Work at this Level The student writes â€śyes,â€ť â€śone solutionâ€ť or â€śordered pairâ€ť without identifying the specific coordinates of the solution.
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The student lists a different coordinate pair than the point of intersection. For example, the student writes:
- (4, -2), reversing the coordinates.
- The coordinates of another point on either line.
- The x- and/or y-intercepts.
- (1, 5) or (5, 1) by combining the y-intercepts of both lines into one coordinate pair.
The student lists the slopes or writes an equation for each line.
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Questions Eliciting Thinking What does a solution of a linear equation such as x + y = 5 look like?
How is the graph of a linear equation related to the solutions of the equation?
What does it mean to be a solution of a system of equations? What does that look like on a graph? |
Instructional Implications 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 x + y = 10 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. Again, 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.
Provide additional opportunities to identify and justify solutions of graphed systems of equations. Emphasize the relationship between the graphs and the equations they represent and guide the student to interpret the graphical outcomes in terms of the system of equations. |
Moving Forward |
Misconception/Error The student is unable to identify the solution of a graphed system of two inconsistent linear equations. |
Examples of Student Work at this Level The student says there are infinitely many solutions because:
- The lines are parallel.
- They look like the same line.
- The lines go on forever.
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The student gives a definition of parallel lines as never intersecting but does not give an answer indicating there is â€śno solution.â€ť
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The student identifies the y-intercepts of (0, 1) and (0, -2) as solutions or combines the x- and y-intercepts into one point.
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Questions Eliciting Thinking What does it mean for there to be a solution of a system of equations? At how many points do these lines intersect?
If there is an infinite number of solutions, what might some of them be? List a solution and show me how you know it is one.
How are y-intercepts different from points of intersections of two lines? |
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 the same line. Relate the graphs to the equations they represent and the nature of the solutions.
After the student has been exposed to algebraic methods of solving systems of equations, ask the student to summarize the various possibilities in terms of the algebraic outcome, the nature of the graphs, and the number of solutions.
Provide additional opportunities to identify and justify solutions of graphed systems of equations. Emphasize the relationship between the graphs and the equations they represent and guide the student to interpret the graphical outcomes in terms of the system of equations. |
Almost There |
Misconception/Error The student is unable to adequately justify his or her response. |
Examples of Student Work at this Level The student correctly identifies the solution of the first graphed system as (-2, 4). When justifying this response, the student explains that (-2, 4) is a solution to the graphed system because:
- It is on the line.
- It is where they cross the x- and y-axis.
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The student indicates that the second system has no solution because:
- They are different lines.
- They cross the y-axis at different points.
- The equations are not true.
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The studentâ€™s justification refers only to the graph and does not reference the system of equations.
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Questions Eliciting Thinking How did you know that (-2, 4) is the solution? What makes it a solution?
How did you know there was no solution to the second system?
What does it mean for an ordered pair of numbers to be a solution of a system of equations?
Why is the point of intersection a solution of the system? How does it relate to the equations in the system? |
Instructional Implications Review what it means for an ordered pair to be a solution of a system of linear equations in two variables. Guide the student to write an equation for each of the independent lines and then demonstrate how (-2, 4) satisfies each equation. Ask the student to identify a point that is only on one of the graphed lines and demonstrate how it satisfies the equation associated with that line but not the other.
Provide additional opportunities to identify and justify solutions of graphed systems of equations. Emphasize the relationship between the graphs and the equations they represent and guide the student to interpret the graphical outcomes in terms of the system of equations. |
Got It |
Misconception/Error The student provides complete and correct responses to all components of the task. |
Examples of Student Work at this Level For the first system of equations, the student indicates that (-2, 4) is the solution because it is a point on each graph (or line) which means that it is a solution of each equation in the system.
For the second system of equations, the student indicates there is no solution because the lines are parallel and share no points. This means that the equations they represent have no solutions in common, so there is no solution to the system.
The student may initially respond only in terms of the graphs but upon questioning can relate the graphs and the solution outcome to the equations within the system. |
Questions Eliciting Thinking Are there any other possible solution outcomes when solving a system of equations? Is it possible to have two solutions? Is it possible to have an infinite number of solutions? Explain. |
Instructional Implications Have the student write equations in slope-intercept form for each line in each graphed system. Ask the student to analyze the equations for â€ścluesâ€ť to the graphical outcomes, (e.g., intersecting lines versus parallel lines). |