IN THE CLASSROOM COURT: ACARA v9 — Pythagorean Path
Case: A 7×7 grid with 4 pre-marked dots. Objective: connect the four given points so the consecutive segment lengths, in order, are √10, 5, √10, √10 to form one continuous path.
Statement of method (brief, formal — with a little Ally McBeal cadence)
Counsel for the pupil states: "We observe, we label, we calculate, we connect — and then we celebrate."
- Label the points. Call the pre-marked grid points A, B, C and D in whatever order makes reading easiest.
- Remember useful Pythagorean displacements. On a square grid the squared distance between two points (x1,y1) and (x2,y2) is (Δx)^2 + (Δy)^2. For our required lengths:
- √10 arises when (Δx, Δy) is a permutation of (1, 3) because 1^2 + 3^2 = 10.
- 5 arises when (Δx, Δy) is a permutation of (3, 4) because 3^2 + 4^2 = 25 and √25 = 5.
- Compute and match. For each candidate ordered pair of marked points (for example A→B, B→C, C→D, D→A if making a loop, or the given order), calculate Δx = x2 − x1 and Δy = y2 − y1. Then compute Δx^2 + Δy^2. Compare to the required squared lengths (10, 25, 10, 10). If it matches, the segment is valid.
Ally aside: "Measure twice, connect once."
- Search systematically. If the order of the four points is not fixed, try permutations or use graph reasoning:
- Create a list of which point pairs are at distance √10 and which are at distance 5 by computing squared distances for all 6 pairs.
- Find a path that visits all 4 points once with the distances in the required order. Because there are only 4 points, this is a short search.
- Verify continuity and grid boundaries. Make sure the line segments sit entirely on the grid (they connect grid points directly) and the path is continuous — the end of one segment is the start of the next.
- Show your work clearly. For each segment show the coordinates, Δx, Δy, calculation of Δx^2 + Δy^2 and the square root. Example calculation format:
A→B: (xA,yA) = (2,3), (xB,yB) = (3,6) → Δx=1, Δy=3 → Δx^2+Δy^2=1+9=10 → distance=√10 (valid).
Exemplary teacher comments (ACARA v9 report style, delivered as a legal brief)
To the learner (13 years old):
Findings: Student has demonstrated exemplary achievement. The pupil accurately identified the necessary Pythagorean displacements, used coordinate differences to verify segment lengths, and produced a single continuous path connecting the four pre-marked points in the required order. Mathematical notation was correct and calculations were fully justified. Work was neatly presented and the grid diagram clearly annotated.
Evidence of achievement:
- Correct recognition that √10 corresponds to moves of 1 and 3 units (or 3 and 1) and that 5 corresponds to moves of 3 and 4 units (or 4 and 3).
- Systematic computation of Δx, Δy and Δx^2 + Δy^2 for each segment, showing that the squared distances matched 10 or 25 as required.
- Logical reasoning in searching permutations when point order was not predetermined; efficient elimination of impossible sequences.
- Clear diagram and labelling enabling any reviewer to follow and re-check each computed distance.
Teacher judgement (summary): The student meets and in places exceeds year-level expectations for geometric reasoning under ACARA v9. They used Pythagoras and coordinate geometry effectively to solve a non-routine problem, demonstrated accurate calculation and clear communication of method.
Next steps / Extension:
- Challenge: Try larger grids with more points and longer lists of required lengths; consider writing a short algorithm to search possible paths (introduce basic programming).
- Reasoning extension: Prove why √10 segments must be (±1, ±3) permutations on integer grids and discuss parity constraints on path closure.
Verdict (with Ally's flourish): Sustained. Objection overruled. Excellence recorded. The pupil has earned an exemplary outcome.
Signed, The Teacher — ACARA v9 Maths Department