IN THE MATTER OF: The Geometry of a Five‑Point Path

Case No.: 13‑YR‑ALGEBRA / ACARA v9

Counsel for the Student (read with a little Ally McBeal cadence — light harp, thoughtful pause): The student presents an exemplary and convincing solution to the task: connect five marked points to form one continuous path whose consecutive segment lengths, in order, are √10, 5, √10, √10. The order of the points was not given; the student determined the correct visiting order and demonstrated rigorous reasoning and accurate computation.

STATEMENT OF FACTS

1. The required path is four consecutive segments with lengths (in order): √10, 5, √10, √10.
2. Five distinct points were marked; the visiting order was not provided.
3. The student used algebra and geometry together: computing squared distances and using the Pythagorean theorem to identify candidate segments with squared lengths 10 and 25.

ISSUES

• How to find a continuous four‑segment path connecting all five points that matches the given sequence of lengths.
• How to be sure the solution is unique and correctly justified.

ARGUMENT (Step‑by‑step, in plain language)

(1) Translate the length targets into squared lengths to make arithmetic easier: √10 → squared length 10, and 5 → squared length 25. This move is natural because distances found from coordinate differences give squares that match Pythagoras: a^2 + b^2 = c^2.

(2) Compute pairwise squared distances between all marked points. For five points there are 10 pairwise distances. The student computed each pair’s squared distance exactly (no rounding): differences in x and y, square each, add to get the squared distance — a direct Pythagorean application.

(3) Identify which pairs have squared distance 10 and which have squared distance 25. These are the only candidates for edges that can be used in the path to satisfy the required lengths (because sqrt(10) and 5 are the only allowed segment lengths).

(4) Use permutation and elimination: pick a starting point and attempt to follow an edge of squared length 10 (the first required). From the next point, require an edge of squared length 25, etc. At each step the student checked:

• Does the current point have an unused neighbor at the required squared distance?
• Will choosing that neighbor leave a feasible continuation to visit all remaining points with the remaining required lengths?

(5) When a dead end was reached (no valid neighbor for the next required length or forced revisiting of a point), that branch was eliminated. The student systematically tried permutations of starting points and valid next edges until only one sequence of points survived all eliminations.

(6) The surviving chain visited each of the five points exactly once and had consecutive squared lengths 10, 25, 10, 10 — i.e. segment lengths √10, 5, √10, √10 in that order. The student presented the explicit order of points and the numerical verifications of each segment length (showing the squared sums equal 10 or 25 each time).

FINDINGS

• The student correctly and efficiently applied the Pythagorean theorem to compute distances between points (matching 10 and 25 exactly where required).
• Her spatial reasoning is strong: she visualised which right triangles or vector differences gave the needed squared lengths, and she moved fluently between algebra (squared sums) and geometry (lengths and path shape).
• Her method for searching the possible orders was explicit: she showed a clear permutation/branching approach with elimination of dead ends until the unique correct path was found.
• Computations are accurate and clearly recorded; the final path is correctly justified and presented.

JUDGMENT

EXEMPLARY. The student met and exceeded expectations for her age group in the following ways:

• Correct and complete solution to a non‑routine spatial task.
• Fluent use of the Pythagorean theorem to convert geometric constraints into algebraic checks (matching squared lengths 10 and 25).
• Explicit and disciplined problem‑solving strategy (permutation with elimination), demonstrating algorithmic thinking and logical rigor.
• Clear written work suitable for sharing and review.

ALIGNMENT WITH ACARA v9

This performance aligns with ACARA v9 expectations in measurement and geometry: using Pythagoras to calculate lengths, solving problems that require spatial reasoning, and explaining the strategy and steps. The student has achieved an exemplary outcome in these areas.

RECOMMENDATIONS & NEXT STEPS

1. Extension: Increase the number of points or require different length sequences, or ask for all possible valid paths — this develops combinatorial search and graph traversal skills.
2. Introduce formal graph language: represent points as vertices and allowed distances as edges; practise finding Hamiltonian paths with prescribed edge labels.
3. Link to coordinates problems: give coordinates and ask for algebraic proof of uniqueness, or explore symmetry and congruence arguments to reduce search effort.
4. Encourage reflection: ask the student to write a short paragraph on why squared distances made the search easier and how elimination reduced the work.

ORDER

(Therefore) The record will show exemplary mastery of the task. Congratulations to the student for a precise, creative, and well‑justified solution. (Cue the quirky musical swell — Ally would smile.)

Signed,
Teacher / Assessor — Mathematics (ACARA v9 aligned report)