Getting Started |
| Misconception/Error The student does not have an effective strategy for completing the construction. |
| Examples of Student Work at this Level The student:
- Uses a straightedge to draw a triangle with vertices inside (or on) the circle.
- Uses a trial and error strategy.
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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?
Can you describe a strategy for constructing an equilateral triangle in a circle? How can the radius of a circle help you with this construction? |
| 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 the construction. (Several different methods are described below.) 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 the construction and keep them for future reference.
Method 1: Use consecutive 60° arcs to construct an equilateral triangle inscribed a circle.
Step 1: Use the given center and the straightedge to construct a radius of the circle. Use the compass to measure the radius.
Step 2: Using this radius, place the point of the compass at the endpoint of the radius on the circle and make an arc that intersects the circle.
Step 3: Keeping the same radius, place the point of the compass at the point of intersection of the arc (from step 2) and make another arc that intersects the circle.
Step 4: Continue making consecutive arcs around the circle as described in step 3 until the circle has been divided into six 60º arcs.
Step 5: Construct the sides of the equilateral triangle by connecting the endpoints of three consecutive 120° arcs.
Method 2: Use the endpoints of a diameter to construct an equilateral triangle inscribed in a circle.
Step 1: Use the given center and the straightedge to construct a diameter of the circle. Use the compass to measure the radius.
Step 2: Using the radius of the circle, place the point of the compass at an endpoint of the diameter, and make two arcs that intersect the circle at two different points.
Step 3: Construct the sides of the equilateral triangle by connecting these two points to the other endpoint of the diameter.
Allow the student to view the construction process on a website such as www.mathopenref.com. The student is able to watch the construction unfold but can also pause it when necessary. To view a step-by-step process for constructing an equilateral triangle using a compass and straight edge go to: http://www.mathopenref.com/constinhexagon.html and instead of connecting every point of intersection to construct a hexagon, connect every other point of intersection to construct an equilateral triangle.
Have the student construct an equilateral triangle in a circle using paper folding:
Step 1: Given a circle on translucent paper, fold the paper so that two halves of the circle coincide. The crease of the fold is a diameter of the circle. Label the endpoints of the diameter C and D.
Step 2: Fold so that endpoint D coincides with the center of the circle and make a firm crease. This fold produces one side of the equilateral triangle. Label the endpoints of this side points A and B.
Step 3: Use a straightedge and pencil to construct  ,  , and  . Triangle ABC is an equilateral triangle.
Ask the student to compare the paper folding method to one of the construction methods above. |
Moving Forward |
| Misconception/Error The student attempts an effective strategy but makes a significant error. |
| Examples of Student Work at this Level The student constructs six consecutive 60° arcs on the circle but:
- Does not use the intersection of each arc with the circle for the vertices of the equilateral triangle.
- Does not maintain the same radius setting on the compass for the six arcs.
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| Questions Eliciting Thinking Can you explain how you constructed your equilateral triangle?
You constructed six arcs around the circle. What is the next step in the construction process?
What type of triangle are you constructing? What does it mean for a geometric figure to be equilateral?
What must be true of the setting on your compass in order to ensure that the sides of the triangle are congruent and the vertices lie on the circle?
How did you determine the radius setting on your compass?
You drew a number of arcs on your paper. Were all of them necessary for this construction? |
| Instructional Implications Review how to use a compass to measure the radius of a circle and then guide the student through the part(s) of the construction that contain errors. Have the student remove any unnecessary marks or marks made in error. Ask the student to write out the steps of the construction and keep them for future reference.
Explain to the student the need to precisely locate points in constructions. Help the student find a way to hold the compass so as not to inadvertently change the radius setting.
Allow the student to view the construction process on a website such as www.mathopenref.com. The student is able to watch the construction unfold but can also pause it when necessary. To view a step-by-step process for constructing an equilateral triangle using a compass and straight edge go to: http://www.mathopenref.com/constinhexagon.html and instead of connecting every point of intersection to construct a hexagon, connect every other point of intersection to construct an equilateral triangle.
Assist the student in developing a justification for the construction strategy used. Ask the student to use postulates, definitions, or theorems to explain the results of each step of the construction and to justify why the strategy results in an equilateral triangle. Provide feedback as needed. |
Almost There |
| Misconception/Error The student’s construction lacks precision. |
| Examples of Student Work at this Level The student uses an effective strategy to construct an equilateral triangle inscribed in a circle but:
- Draws large dots where each arc intersects the circle, so the intersection is hidden and precision is lost.
- Does not use the exact point where an arc intersects the circle as the vertex of the triangle.
- Leaves unnecessary construction marks on the paper.
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| Questions Eliciting Thinking Can you explain the steps of your construction?
How did you locate the points you used to construct the sides of the equilateral triangle? Did you draw the sides so that they contain these points?
Are there any adjustments you could make to your construction to make it more precise?
What do these arcs (or markings) represent? Did you use them in your construction? |
| Instructional Implications Explain to the student the need to precisely locate and connect points in constructions. Emphasize that drawing large dots at intersections reduces the precision of the construction. Ask the student to explain why large dots (or any at all) can affect the precision of a geometric construction.
Ask the student to remove any unnecessary marks or marks made in error from his or her paper.
Allow the student to view the construction process on a website such as www.mathopenref.com. The student is able to watch the construction unfold but can also pause it when necessary. To view a step-by-step process for constructing an equilateral triangle using a compass and straight edge go to: http://www.mathopenref.com/constinhexagon.html and instead of connecting every point of intersection to construct a hexagon, connect every other point of intersection to construct an equilateral triangle.
Ask the student to use postulates, definitions, or theorems to explain the results of each step of the construction and to justify why the strategy results in an equilateral triangle. |
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 constructs an equilateral triangle inscribed in the circle using an effective strategy. The student’s construction is precise and contains no unnecessary marks or marks made in error.
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| Questions Eliciting Thinking Why does this approach enable one to construct an equilateral triangle in a circle? Why does this method work?
Can you think of another strategy for constructing an equilateral triangle in a circle?
How can you expand your construction of the equilateral triangle to make a regular hexagon in a circle? |
| Instructional Implications Ask the student to use postulates, definitions, or theorems to explain the results of each step of the construction and to justify why the strategy results in an equilateral triangle.
Challenge the student to use other theorems or definitions to find other methods to construct an equilateral triangle inscribed in a circle.
Challenge the student to construct a square and a hexagon inscribed in a circle. Consider implementing the other G-CO.4.13 tasks. |
