Getting Started |
| Misconception/Error The student’s proof shows no evidence of an overall strategy or logical flow. |
| Examples of Student Work at this Level The student:
- Makes a series of unsupported statements which do not logically lead to the conclusion that the medians are concurrent.
- Writes one or two statements that may or may not be relevant but are presented out of context and show no indication of a strategy or plan for the proof.
|
| Questions Eliciting Thinking What are you trying to prove?
Did you think through a plan for your proof before you started? Did you consider what you already know that might help you to prove that the medians are concurrent?
What do you know as a consequence of the fact that points D, E, and F are midpoints?
What happens when you connect the midpoints of two sides of a triangle? What do you know about a midsegment of a triangle? |
| Instructional Implications Review the meaning of the following: median, midpoint, midsegment, concurrent, parallel lines or segments, and alternate interior angles. Review the methods of proving triangles are similar. Guide the student to observe and justify that the pair of triangles formed by drawing two medians and the midsegment that connects their endpoints are similar.
Provide the student with the statements of the proof of the Concurrent Medians theorem and ask the student to supply the justifications. Then have the student analyze and describe the overall strategy used in the proof.
Provide the student with frequent opportunities to make deductions using a variety of previously encountered definitions and established theorems. For example, provide diagrams as appropriate and ask the student what can be concluded as a consequence of:
- Point M is the midpoint of
 .
- < A and < B are supplementary and m < A = d.
 and PQ = m units. is the bisector of < DE.
Provide opportunities for the student to determine the flow of a proof. Give the student each step of a proof written on a separate strip of paper and ask the student to determine the order of the steps so that there is a logical flow that makes sense. Ask the student to then provide the justification for each step.
Provide the student with additional examples of proofs of statements about triangles. Ask the student to prove simple statements and provide feedback. |
Moving Forward |
| Misconception/Error The student’s proof reveals some evidence of an overall strategy, but the student fails to establish major conditions leading to the prove statement. |
| Examples of Student Work at this Level The student:
- States that
 with no justification but attempts to use the similarity to continue the proof.
- Proves and states that the ratio of corresponding lengths of the sides of
 and  is 2:1 but does not complete the proof.
|
| Questions Eliciting Thinking How do you know ?
Can you explain how knowing that corresponding sides of these triangles have lengths in the ratio 2:1 leads to the concurrency of the medians?
How do you know that the third median contains point G? |
| Instructional Implications Provide additional feedback to the student regarding any other missing statements or justifications. Guide the student through the details of the proof that are missing. Encourage the student to critically read proofs that he or she has written and to question statements given without justification. Explain that such statements should be justified before proceeding since the inability to do so may call into question the rest of the proof.
Encourage the student to begin the proof process by developing an overall strategy. Provide another statement to be proven and have the student compare strategies with another student and to collaborate on completing the proof.
Consider using the NCTM lesson Pieces of Proof (http://illuminations.nctm.org/Lesson.aspx?id=2561) in which the statements and reasons of a proof are given separately and the student must arrange the steps in a logical order. Encourage the student to use multiple proof formats including flow diagrams, two-column, and paragraph proofs. Allow the student to work with a partner to complete these exercises. |
Almost There |
| Misconception/Error The student’s proof shows evidence of an overall strategy, but the student fails to establish a minor condition that is necessary to prove the theorem. |
| Examples of Student Work at this Level The student fails to:
- Explicitly state that the ratio of the length of the midsegment to the length of the parallel side of the triangle is 2:1, but the student uses this result in the proof.
- Names the point of intersection of medians
 and  and the point of intersection of medians and  in the same way, namely as point G, before proving that these points are the same.
- Clearly establish that the point of intersection of medians and and medians and are the same point.
|
| Questions Eliciting Thinking How do you know that the ratio of the lengths of the sides of and is 2:1?
You are trying to prove that the point of intersection of medians and is the same as the point of intersection of medians and . Does it make sense to name these points the same before you have shown they are the same?
Can you explain how you showed that the point of intersection of medians and is the same as the point of intersection of medians and ? |
| Instructional Implications Provide the student with direct feedback on his or her proof. Prompt the student to supply justifications or statements that are missing. For example, assist the student in clearly establishing the point of intersection of medians and and medians and are the same point.
Correct any misuse of notation. If necessary, review notation for naming angles (e.g.,  ) and describing angle measures (e.g.,  ) and guide the student to write equations and congruence statements using the appropriate notation.
Ask the student to use one of the descriptions of a proof that the medians are concurrent given in the Instructional Implications for “Got It” and supply the missing details.
Consider implementing MFAS tasks Isosceles Triangle Proof (G-CO.3.10), Triangle Sum Proof (G-CO.3.10), and Triangle Midsegment Proof (G-CO.3.10). |
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 devises a complete and correct proof using, for example, one of the approaches described below.
The Similar Triangles Approach
Let G be the point of intersection of medians and . Construct midsegment  and use the consequences of  to show that . Since the length of the midsegment is half the length of  , the ratio of corresponding lengths of the sides of  and  is 2:1 so that AG:GE = 2:1. Likewise, let H be the point of intersection of medians and . By constructing midsegment  , it can be shown that  so that BH:HF = AH:HE = 2:1. So, AG:GE = AH:HE = 2:1. There can be only one point on that divides into two parts in the ratio 2:1, so, points G and H must coincide. Consequently, each median contains point G so that the three medians are concurrent, intersecting in point G.
The Coordinate Geometry Approach
Situate  in the coordinate plane so that its vertices are A(0, 0), B(2a, 2b), and C(2c, 0). Use the coordinates of the vertices to write the equation of each median. Show that both the point of intersection of medians and and the point of intersection of medians and are the point given by  . Therefore, medians , , and are concurrent.
The Areas Approach
Given , let G be the point of intersection of medians and . Construct  , and let E be the point of intersection of and  . The objective of this proof is to show that is a median (e.g., that BE = EC or p = q). , , and divide into six triangles. The areas of these triangles are indicated by  , and  in the figure. Since D and F are midpoints it can be shown that  and  . Next  =  since each sum represents one-half the area of  ; therefore  so that  . Since  and  =  , it follows that p = q. So, E is the midpoint of  , and is a median. Thus, all three medians contain point G.
|
| Questions Eliciting Thinking In addition to being concurrent, what else do you know about the medians as a consequence of your proof? |
| Instructional Implications Ask the student to use one of the descriptions of another proof given in the Instructional Implications for “Got It” and supply the missing details.
Ask students to extend the Areas Proof to show that all six areas formed by constructing the three medians are equal. Then introduce the centroid as the center of mass of a triangle of uniform density.
Consider implementing MFAS tasks Isosceles Triangle Proof (G-CO.3.10), Triangle Sum Proof (G-CO.3.10), and Triangle Midsegment Proof (G-CO.3.10). |
