1.3 Writing Proofs & Constructed-Response Strategy

Constructed Response: Where Points Are Won and Lost

Parts II-IV are worth 32 of the 80 credits, and every one of those 11 questions asks you to show something — work, a reason, a graph, a construction, or a proof. Because they are scored holistically against a NYSED rating guide, communication matters as much as the final answer. Aim to make your reasoning explicit enough that a rater can follow every step without guessing your intent; unexplained leaps are where credit quietly disappears.

Two-Column and Paragraph Proofs

A two-column proof lists statements on the left and a reason for each on the right. Every statement needs a justification — a given, a definition, a postulate, or a theorem. You begin from the Given and end at the Prove.

A paragraph proof presents the same chain of reasoning in complete sentences, and a flowchart proof is also accepted. Whatever the format, the graders look for the same thing: correct statements, each tied to a valid reason, leading logically to the conclusion.

A Coordinate-Proof Example

To prove quadrilateral ABCD is a parallelogram on the grid, you can compute the slope of each side and show both pairs of opposite sides are parallel (equal slopes), or find the midpoint of each diagonal and show they coincide (the diagonals bisect each other). Either method earns credit — but only if you finish with a stated conclusion, for example: "Because both diagonals share the midpoint (3, 2), the diagonals bisect each other, so ABCD is a parallelogram." A page of correct calculations with no concluding sentence frequently loses the final credit.

Common Constructed-Response Formats

• Transformations: describe or perform a translation, reflection, rotation, or dilation. State that rigid motions preserve distance and angle, so the image is congruent, while a dilation produces a similar figure.
• Coordinate proofs: use distance, slope, and midpoint to prove a figure is a parallelogram, rectangle, right triangle, or isosceles triangle. Always finish with a sentence stating the conclusion and the reason.
• Similarity / congruence proofs: name the exact criterion (SSS, SAS, ASA, AAS, HL for congruence; AA, SAS~, SSS~ for similarity) and invoke CPCTC when you need corresponding parts after proving triangles congruent.
• Modeling: density, unit cost, and capacity problems set in real contexts. Track units carefully, and answer the exact quantity asked ("how many bags," "least cost," "will it fit"), usually with a written justification and sometimes a rounding rule ("round to the nearest dollar").
• Constructions: use only a compass and straightedge, and leave all arc marks visible — the arcs are the evidence that you used a valid method rather than eyeballing the figure.

Show Work to Earn Partial Credit

The rating guide is holistic, but a few patterns recur across administrations:

• Correct answer + complete, correct work → full credit.
• Correct method with a computational error → usually lose one credit (you keep the method credit).
• Correct final answer with no work → usually only 1 credit, no matter the item's value.
• A conceptual error (wrong theorem, wrong setup) costs more than a small arithmetic slip.

How Responses Are Rated

Open-ended answers are read against a published rating guide that includes model responses and a model response set, and schools may not rescore an open-ended answer once it has been rated. The guide tells raters exactly how many credits a specific error costs, which is why a clean, well-organized response that mirrors the expected steps is far easier to award full credit than a cramped, out-of-order one. Label your work, keep each part of a multi-part question separate, and box or underline the final answer.

Point-Losing Mistakes to Avoid

1. Writing only a final answer with no work shown.
2. Filling the statements column but leaving reasons blank or vague ("obvious," "it looks like").
3. Rounding too early or dropping units in a modeling problem.
4. Reversing a transformation (up vs. down, clockwise vs. counterclockwise) or the center of a rotation/dilation.
5. Confusing similar (proportional) with congruent (equal), or naming the wrong triangle criterion.
6. Freehand constructions or erasing the compass arcs — the arcs are the evidence of your method.
7. Answering a different question than asked (area instead of perimeter, x instead of the length).
8. Stopping before the conclusion — a coordinate proof needs the closing line, e.g., "therefore the quadrilateral is a parallelogram because both pairs of opposite sides have equal slopes."

Treat every constructed response as a short argument: state what is true, say why, and end with the answer the question actually requested.