Getting Started |
| Misconception/Error The student is unable to correctly graph linear equations and to use the graphs to identify a solution. |
| Examples of Student Work at this Level The student makes errors when using the slope or y-intercept (or both) to graph the equations. The student is unable to identify a solution even given graphing errors.
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| Questions Eliciting Thinking How did you graph these lines?
Can you identify the slope and y-intercept of the graphs from the equations?
Once you have the lines graphed, how can you find the solution of the system? |
| Instructional Implications 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. Then model how to graph linear equations using the y-intercept and slope. Make explicit the difference between the x-intercept and the y-intercept, and the difference between positive and negative slopes. Provide opportunities for the student to identify the slope and y-intercept from equations and use them to graph lines.
Consider implementing CPALMS Lesson Plan Graphing Equations on the Cartesian Plane: Slope (ID 31376) or Graphing Lines (ID 53119) to address basic slope and intercept issues. Consider implementing CPALMS Lesson Plan Whose Line Is It Anyway? (ID 41660) or Exploring Slope Intercept Form with Graphs and Physical Activity (ID 37212) to address graphing the slope-intercept form of a linear equation.
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. |
Moving Forward |
| Misconception/Error The student is unable to identify the solution of a system of equations from its graph. |
| Examples of Student Work at this Level The student correctly graphs each equation but:
- Does not extend the lines until they intersect.
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- Identifies a point other than the point of intersection as the solution of the system.
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- Graph lines that are not straight so is unable to correctly identify the solution of the system.
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| Questions Eliciting Thinking How did you identify the solution of the system?
Can you identify a solution of the first equation from its 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 = 12 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.
Consider implementing CPALMS Lesson Plan Exploring Systems of Equations Using Graphing Calculators (ID 39089), Where does my string cross? (ID 41691), or A Scheme for Solving Systems (ID 54390). |
Almost There |
| Misconception/Error The student makes a minor error in identifying the solution or cannot adequately explain or confirm the solution. |
| Examples of Student Work at this Level The student correctly graphs each equation and locates the point of intersection. However, the student:
- Incorrectly identifies the solution as (7, -3). Upon questioning, the student says the point is up seven and back three, confusing the convention for writing ordered pairs with the convention for calculating slope (rise over run).
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- Correctly identifies the solution as (-3, 7) but is unable to clearly explain that the coordinates of this point satisfy each equation.
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| Questions Eliciting Thinking How do you write an ordered pair, (x, y) or (y, x)? Which variable goes first?
How do you write slope? Which variable goes first?
How did you know that (-3, 7) is the solution? What makes it a solution?
How can you check if your ordered pair is the solution? How will you know you made a graphing error? |
| 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 graphed correctly.
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 demonstrate how (-3, 7) 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 The student correctly graphs both lines and identifies (-3, 7) as the solution of the system. The student explains that it is the solution of the system because it is the point of intersection of the two graphs and will satisfy each equation in the system.
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| Questions Eliciting Thinking Why is the solution the point at which the two lines meet? What is the significance of this point?
Is (-3, 7) the only solution? How do you know?
Could you predict how many solutions there would be before you graphed the equations?
How many solutions will systems that graph as parallel lines have? Coinciding lines? |
| 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 (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.
Consider implementing MFAS task Solving Systems of Linear Equations or MFAS task How Many Solutions? |
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