Getting Started |
| Misconception/Error The student makes significant errors when graphing points in the coordinate plane. |
| Examples of Student Work at this Level The student:
- Graphs some or all of the vertices incorrectly.
- Reverses the x- and y-axes, reverses x- and y-coordinates, or interchanges the positive and negative portions of the axes.
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- Graphs more points than were given.
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| Questions Eliciting Thinking On a coordinate plane, which is the x- and which is the y-axis? In an ordered pair, which is the x- and which is the y-coordinate?
Where did you start counting? Which direction did you move to graph the x-value and which direction did you move to graph the y-value?
Can you show me how you graphed the vertices?
What type of figure did you graph? Does it look like a polygon? |
| Instructional Implications Review graphing points in the coordinate plane. Be sure to include points in all four quadrants and on both axes. Ask the student to both graph points given their coordinates and to give the coordinates of graphed points. Consider implementing the CPALMS Lesson Plan Chameleon Graphing (ID 5728). Provide the student with additional opportunities to graph specified figures given the coordinates of their vertices.
Define a polygon as a closed figure with three or more sides. Show the student examples and non-examples of polygons. Provide the student with additional opportunities to graph polygons in all four quadrants with integer and rational number coordinates. Clarify for the student the correct way to name points, line segments, and polygons. Remind the student that the order the points are listed indicates the order of the vertices in the polygon. |
Moving Forward |
| Misconception/Error The student is unable to determine the lengths of sides and if the sides are parallel. |
| Examples of Student Work at this Level The student correctly graphs the polygon but is unable to determine the lengths of the sides. The student attempts to determine lengths by counting unit lengths on the graph, but the student:
- Does not count the half units.
- Includes the halves as whole units.
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| Questions Eliciting Thinking What is the length of each side of the polygon? Can you show me how you counted? Did you count the grid lines or the unit lengths?
What do you notice about the coordinates for points B and C (A and D)? How could you use the coordinates to find the length of each side? |
| Instructional Implications Review the concept of length and give the student opportunities to find lengths on a number line by counting unit lengths. Relate finding length on a number line to using a ruler. Explain how vertical and horizontal lengths are measured in the coordinate plane. Directly address the misconception that length is calculated by counting notches or grid lines or that length can be negative. Remind the student that partial units must be included in the length. Emphasize the meaning of the unit length by providing a unit of measure such as centimeters. Eventually, transition the student to calculating distances (rather than counting unit lengths) between points with the same first coordinate or the same second coordinate on a coordinate plane. Guide the student to use absolute value symbols to represent lengths [e.g., represent the distance from A(6, 3) to B(6, -4) as AB = |3 – (-4)| = 7 or as AB = |-4 – 3| = 7].
Define parallel lines as lines in the same plane that never intersect or have no points in common. Model an explanation using appropriate math vocabulary. For example, the endpoints of each segment, AB and DC, have the same y-coordinate which indicates they are both horizontal, and all horizontal segments in the coordinate plane are parallel. Likewise, the endpoints of each segment, AD and BC, have the same x-coordinate which indicates they are both vertical, and all vertical segments in the coordinate plane are parallel. |
Almost There |
| Misconception/Error The student is unable to adequately explain why the opposite sides are parallel. |
| Examples of Student Work at this Level The student correctly graphs the polygon and determines the lengths of the sides. In addition, the student identifies the parallel sides but is unable to explain what makes them parallel. The student says the sides are parallel because the lines are:
- Running next to each other.
- The same length.
- Straight lines.
The student identifies the parallel sides but gives no explanation.
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| Questions Eliciting Thinking What are parallel lines? How do you know if lines are parallel?
What do you mean by “running next to each other”?
Do line segments have to be the same length in order to be parallel? |
| Instructional Implications Define parallel lines as lines in the same plane that never intersect or have no points in common. Model an explanation using appropriate math vocabulary. For example,  and   are horizontal segments because the endpoints of each segment have the same y-coordinate which indicates that they are both horizontal and all horizontal segments in the coordinate plane are parallel. Likewise,  and  are vertical segments because the endpoints of each segment have the same x-coordinate which indicates that they are both vertical and all vertical segments in the coordinate plane are parallel. Avoid describing parallel lines as lines that are “the same distance apart” since the distance between two lines had not yet been well defined. |
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:
- Graphs the polygon correctly,
- Determines the length of each side correctly (AB = 3 units, BC = 5 units, CD = 3 units, DA = 5 units), andÂ
- Explains that the opposite sides are parallel because they are vertical and horizontal segments.Â
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| Questions Eliciting Thinking Can you determine the perimeter (or area) of the polygon? How would you determine the perimeter (or area) of the polygon using only the coordinates (and not the graph)?
What type of quadrilateral would be drawn if point A were located at (1, 3.5) instead of (2, 3.5)? Explain. |
| Instructional Implications Challenge the student to determine the perimeter (or area) of a polygon without graphing the coordinates when given vertices with the same first coordinate or the same second coordinate. Guide the student to use the corresponding coordinates from adjacent vertices to calculate lengths. Guide the student to use absolute value symbols to represent lengths [e.g., represent the distance from A(6, 3) to B(6, -4) as AB = |3 – (-4)| or as AB = |-4 – 3|].
Consider implementing the MFAS task Patio Area (6.G.1.3) which involves determining the area of a rectangle graphed on the coordinate plane. |
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