Getting Started

The student’s proof shows no evidence of an overall strategy or logical flow.

The student:

• States the given information, but is unable to go any further.
  
  
  
  
• Makes some observations about rectangles without regard to the statement being proven.
  
  
  
  
• Uses the theorem to be proven in its own proof.

What do you know about this figure? What are you trying to prove?

What have you already proved about rectangles?

Did you think of a plan for your proof before you started?

Did you draw the diagonals as the problem asked?

Provide the student with the statements of a proof of this theorem and ask the student to supply the justifications. Then have the student analyze and describe the overall strategy used in the proof.

Emphasize that a theorem cannot be used as a justification in its own proof. Encourage the student to first question what is available to use in a proof of a particular statement.

Review the ways to prove two triangles congruent (SSS, SAS, ASA, AAS, HL) and what must be established in order to conclude two triangles are congruent when using each method. Remind the student that once two triangles are proven congruent, all remaining pairs of corresponding parts can be shown to be congruent.

Suggest strategies for working with overlapping triangles in diagrams (e.g., draw each triangle separately carefully labeling its vertices). Assist the student in identifying shared sides and marking congruent sides and angles. Then ask the student to identify a congruence theorem (e.g., SSS, SAS, ASA, AAS, HL) that can be used to justify the congruence of the triangles.

If necessary, review notation for naming sides (e.g., ) and describing lengths of sides (e.g., XY) and guide the student to use the notation appropriately.

Moving Forward

The student’s proof shows evidence of an overall strategy but fails to establish major conditions leading to the prove statement.

The student constructs the two diagonals and attempts to prove that  in order to prove that the diagonals on the parallelogram are congruent. However, the student:

• Outlines the proof without providing details.
  
  
  
  
• Understands the need to show is congruent to without adequately showing it.
  
  

What is your general strategy for this proof?

How can you show the two triangles are congruent?

What do you need to show in order to use the SAS congruence theorem? Have you done this in your proof?

Review the ways to prove two triangles congruent (SSS, SAS, ASA, AAS, HL) and what must be established in order to conclude two triangles are congruent when using each method. Remind the student that once two triangles are proven congruent, all remaining pairs of corresponding parts can be shown to be congruent.

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 collaborate on completing the proof.

Suggest strategies for working with overlapping triangles in diagrams (e.g., draw each triangle separately carefully labeling its vertices). Assist the student in identifying shared sides and marking congruent sides and angles. Then ask the student to identify a congruence theorem (e.g., SSS, SAS, ASA, AAS, HL) that can be used to justify the congruence of the triangles.

If necessary, review notation for naming sides (e.g., ) and describing lengths of sides (e.g., XY) and guide the student to use the notation appropriately.

Almost There

The student fails to establish a condition that is necessary for a later statement.

The student fails to establish the congruence of one pair of angles or sides necessary to use the congruence theorem cited.

I see that you stated that these triangles are congruent. Can you show me all of the steps needed to use the theorem you used? Did you include all of them in your proof?

Using a colored pencil or highlighter, encourage the student to mark the statements which support the congruence theorem chosen. Remind the student that each letter of the theorem name represents a pair of parts that must be shown to be congruent (e.g., if using ASA to prove the triangles congruent, the proof must include showing the congruence of two pairs of corresponding angles and their included sides and a reason or justification must be provided for each).

If necessary, review notation for naming sides (e.g., ) and describing lengths of sides (e.g., WY) and guide the student to use the notation appropriately.

Consider using MFAS tasks Proving Parallelogram Side Congruence (G-CO.3.11), Proving Parallelogram Angle Congruence (G-CO.3.11), Proving Parallelogram Diagonals Bisect (G-CO.3.11) or Proving a Rectangle Is a Parallelogram (G-CO.3.11) if not previously used.

Got It

The student provides complete and correct responses to all components of the task.

The student provides a complete proof with appropriate justification such as:

Since quadrilateral WXYZ is a rectangle, it is also a parallelogram. Since opposite sides of a parallelogram are congruent, . Since, by definition, the angles of a rectangle are right angles,  and  are right angles. Since all right angles are congruent to each other, . By the Reflexive Property, .  because of the SAS Congruence Theorem. Therefore  because corresponding parts of congruent triangles are congruent (or by definition of congruent triangles).

Is there another method you could have used to prove that the diagonals of the parallelogram are congruent?

Challenge the student to prove other statements about parallelograms, squares, rectangles, and rhombi. Provide the student opportunities to write proofs using a variety of formats some of which include narrative paragraphs, flow diagrams, and two-column format.

Consider using MFAS tasks Proving Parallelogram Side Congruence (G-CO.3.11), Proving Parallelogram Angle Congruence (G-CO.3.11), Proving Parallelogram Diagonals Bisect (G-CO.3.11) or Proving a Rectangle Is a Parallelogram (G-CO.3.11) if not previously used.