Getting Started |
| Misconception/Error The student sketches or draws rather than constructs. |
| Examples of Student Work at this Level The student draws the bisectors rather than constructing them.
The student makes some construction marks on his or her paper that are incomplete or unrelated to the construction.
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| Questions Eliciting Thinking What is the difference between drawing and constructing?
When doing a geometric construction, what tools are typically used?
What is the difference between a straightedge and a ruler?
What is it that you are supposed to construct? |
| Instructional Implications Explain to the student the difference between drawing and constructing. Show the student the tools traditionally used in geometric constructions and explain the purpose of each. Be sure the student understands the difference between a ruler and a straightedge.
Guide the student through the steps of constructing bisectors of segments and angles. Prompt the student to justify each step. Have the student remove any unnecessary marks or marks made in error from his or her paper. Ask the student to write out the steps of each construction and keep them for future reference.
Give the student additional opportunities to construct bisectors. |
Moving Forward |
| Misconception/Error The student attempts the construction, but makes a significant error. |
| Examples of Student Work at this Level The student:
- Did not use the same compass radius for the arcs drawn from each endpoint.
- Drew arcs that did not intersect and therefore did not have two points through which to draw the segment bisector.
- Did not draw the initial arc equidistant from the vertex of the angle, but instead used the last visible point of the ray (the arrow) to construct the angle bisector.
- Only drew the initial arc.
- Only drew a pair of arcs above (or below) the given segment.
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| Questions Eliciting Thinking What is a segment bisector?
What is the midpoint of a segment?
What is an angle bisector? |
| Instructional Implications Explain to the student the need to precisely locate points in constructions and in order to construct lines, rays, and segments, two distinct points must be located. Help the student find a way to hold the compass so as not to inadvertently change the radius setting.
Demonstrate these constructions using an interactive website such as Math Open Reference: Angle bisectors (http://www.mathopenref.com/constbisectangle.html) and Segment bisectors (http://www.mathopenref.com/constbisectline.html) . Provide the student with a variety of angles and segments to bisect while following the step-by-step instructions from the website. |
Making Progress |
| Misconception/Error The student is not able to justify the step for one or both of the constructions. |
| Examples of Student Work at this Level The student writes a statement such as:
- The compass width needs to be slightly longer than half the length so you can get the segment bisector in the exact middle of the line segment.
- The initial arcs help you know where to draw the next set of arcs needed to construct the angle bisector.
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| Questions Eliciting Thinking Can you explain what you meant in your answers to questions 2 and 4?
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| Instructional Implications Have the student draw the initial arcs above and below the given segment to construct the segment bisector, first with a compass width that is less than half the length of the given segment, and second with a compass width that is equal to half of the length of the given segment. Encourage the student to determine and explain from this activity the reason the compass width needs to be slightly longer than half the length of the given segment.
Have the student use points on each side of the angle that are not the same distance from the vertex. Encourage the student to determine from this activity the reason the initial arcs are drawn when bisecting an angle.
Have the students identify the corresponding parts of the two congruent triangles that are formed by the angle bisector, the sides of the angle, and the segment from the initial arcs to point C. From this activity encourage the student to determine the significance in drawing the initial arc for the angle bisector construction and to prove that the angle is bisected.
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Almost There |
| Misconception/Error The student correctly completes and justifies the construction but does not label the construction or leaves unnecessary marks on the paper. |
| Examples of Student Work at this Level The student correctly completes each construction but does not label the bisectors as indicated in the instructions.
The student labels a second point on the bisector as point A.
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| Questions Eliciting Thinking Why is it important to label the construction?
Where would you label the midpoint of the segment in Question 1?
How would you label the ray bisecting  ? |
| Instructional Implications Ask the student to label the constructed bisectors as indicated in the instructions and to remove any unnecessary marks or marks made in error from his or her paper.
Provide the student with completed constructions for segment bisectors and angle bisectors. Have the students label the constructions and then name the segment and the midpoint. Have the student label and then name the angle and the angle bisector. |
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 completes and labels both constructions correctly. The student writes:
- the compass radius needs to be slightly longer than half the length of the segment when drawing the arcs above and below the line so the arcs will intersect at two different points not on the line segment.
- it is necessary to draw the initial arc in order to locate points on each side of the angle that are equidistant from its vertex.
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| Questions Eliciting Thinking If you connected the four points you located in your segment bisector construction, what kind of a figure would be formed?
Why was it important in this construction to locate points on each side of the angle that are equidistant from the vertex?
Does a bisector of a segment have to be perpendicular to the segment? |
| Instructional Implications Review the definition of the angle bisector including the theorem that states that a point lies on the angle bisector if and only if it is equidistant from the sides of the angle. Have the student prove that the point located in the interior of the angle that is used to construct the angle bisector is actually on the angle bisector.
Introduce the student to points of concurrency. Have the student use the perpendicular bisectors construction to locate the midpoints of each side of a triangle and to construct the medians. Ask the student to identify the point of intersection of the medians. Have the student construct angle bisectors for all three angles of a triangle. Ask the student to identify the point of intersection of the three angle bisectors. Ask the student to consider if the medians and the angle bisectors could ever intersect in the exterior of a triangle. |
