Scaffolding a Rectangle Diagonal Proof

The responses show three different obstacles, not one.

1. Circular reasoning (17 students). These students use the statement to be proved as a reason. This is a logical obstacle: they may know the rectangle properties, but they have not internalized that a proof must begin from givens, definitions, and previously established theorems, and cannot assume the conclusion. The textbook sequence-presenting rectangle properties before the diagonal theorem-may encourage this by making "rectangles have congruent diagonals" feel like a known fact rather than a claim under investigation.
2. Empirical justification (6 students). These students treat measurement or appearance as proof. This is an epistemological obstacle about the nature of proof: a diagram can suggest a conjecture, but congruence requires a deductive warrant. This is different from circular reasoning because the students are not misusing a theorem; they are using observation as evidence.
3. Procedural congruence error (4 students). These students know a proof should use triangle congruence but misapply the criteria, citing SSA or failing to identify the included angle. This is a technical obstacle, not a failure to understand what proof is for.

The two most important obstacles are circularity and empirical justification, because both prevent students from understanding the purpose of proof. The SSA group is closer to a valid proof and can be addressed with targeted feedback. The teacher should not treat all 27 incorrect responses as the same error.

Scaffold (25 minutes):

1. Separate given from goal (3 min). Have students write only the given and the prove statement: Given rectangle ABCD; prove AC ≅ BD. Ask: "Which statement are we not allowed to use as a reason?" Students should identify "diagonals of a rectangle are congruent" as the goal, not a tool.
2. Inventory allowed tools (5 min). In pairs, students list what a rectangle gives them: opposite sides are parallel and congruent; all four angles are right angles; a side is congruent to itself. Post the list. This makes the givens visible.
3. Locate the diagonals in triangles (7 min). Draw AC and BD. Ask: "Which triangles contain AC? Which contain BD? Can we find a pair of triangles that has one diagonal as a side and enough known parts to prove congruent?" Guide students to triangle ABC and triangle DCB, which share BC and contain AC and BD respectively.
4. Mark and justify (7 min). Have students mark AB ≅ DC (opposite sides of a rectangle), BC ≅ CB (shared side), and ∠ABC ≅ ∠DCB (both right angles). Ask the critical question: "Is the right angle between the two congruent sides in each triangle?" This directly addresses the SSA error. Students then write the SAS statement and use CPCTC to conclude AC ≅ BD.
5. Exit ticket (3 min). "In one sentence, explain why the proof cannot begin with 'a rectangle has congruent diagonals.'"

Representation: a two-column table with a sketch that uses tick marks for congruent sides and right-angle symbols, plus a separate "allowed tools" list. The visual marking supports the logical structure without supplying the proof.

The teacher should assess these students as having a sound overall strategy with a specific congruence-criterion error. They should receive credit for identifying triangle congruence as the route and for using the rectangle's properties, but not full credit for the proof, because SSA is not a valid congruence criterion in general and the reason column is incorrect. The school rubric requires a valid reason for every statement, so the proof fails at that line.

Feedback: Ask the students to label the corresponding parts in the two triangles and circle the angle in each triangle. Then ask: "Is the known angle included between the two known sides?" In the valid proof, AB ≅ DC, BC ≅ CB, and the included angles ∠ABC and ∠DCB are right angles, so SAS applies. If students cited SSA, they likely either mislabeled the correspondence or treated a non-included angle as sufficient. Have them compare SAS and SSA using a counterexample: given two sides and a non-included angle, two different triangles can sometimes be drawn. A dynamic geometry sketch that drags one vertex to produce two noncongruent triangles is useful here.

Follow-up item: "In triangles ABC and DEF, AB = DE, BC = EF, and ∠A = ∠D. Are the triangles necessarily congruent? If not, sketch or describe two noncongruent triangles that satisfy the conditions." This item separates students who understand that the angle must be included from those who are pattern-matching letters. The teacher should also require students to write the congruence statement with vertices in corresponding order, since that habit prevents many SSA errors.