Cornell Notes for a 13‑year‑old: Grid Path with Segment Lengths 1, √10, 5, √5 — ACARA v9 Aligned

Cues / Questions

• Vocabulary: lattice point, Δx, Δy, Pythagoras, integer triple
• What coordinates will I use?
• How to check lengths quickly?
• One-line justification for each segment
• ACARA v9 alignment & teaching tips

Notes / Worked solution

Given idea: build a continuous path between lattice points so the 4 successive segment lengths are 1, √10, 5 and √5 (in that order). Use integer Δx and Δy so Pythagoras gives those values.

Chosen path (coordinates): Start at A = (0,0) → B = (1,0) → C = (4,1) → D = (7,5) → E = (8,7).

Segment checks (tiny Pythagorean computations):

1. AB: A(0,0) to B(1,0). Δx = 1, Δy = 0.
   Length = √(1^2 + 0^2) = √1 = 1.
   One-line reason: horizontal step of 1 so length = 1.
2. BC: B(1,0) to C(4,1). Δx = 3, Δy = 1.
   Length = √(3^2 + 1^2) = √(9 + 1) = √10.
   One-line reason: vector (3,1) gives √(9+1)=√10.
3. CD: C(4,1) to D(7,5). Δx = 3, Δy = 4.
   Length = √(3^2 + 4^2) = √(9 + 16) = √25 = 5.
   One-line reason: (3,4) is a 3–4–5 triple so length = 5.
4. DE: D(7,5) to E(8,7). Δx = 1, Δy = 2.
   Length = √(1^2 + 2^2) = √(1 + 4) = √5.
   One-line reason: vector (1,2) gives √5.

Plotting instructions for a printable grid: draw a grid covering x = 0..8 and y = 0..7. Mark points A (0,0), B (1,0), C (4,1), D (7,5), E (8,7). Connect A→B→C→D→E with straight segments. Label each segment with its computed length.

Compact summary (what to write on the printed sheet):

Path: (0,0) → (1,0) → (4,1) → (7,5) → (8,7). Segment lengths 1, √10, 5, √5 computed by Δx/Δy and Pythagoras. Each check: AB uses (1,0); BC uses (3,1); CD uses (3,4); DE uses (1,2).

Teaching tip (student version):

Always write Δx and Δy next to each segment, make the right triangle you used for Pythagoras, and add a one-sentence reason like “(3,4) → 5 by Pythagoras.” Keep diagrams tidy—neat labels speed up checks.

ACARA v9 mapping (classroom language):

• Measurement & Geometry: use the Pythagorean theorem to determine lengths between lattice points; construct and reason with right triangles.
• Number & Algebra: perform squares and square roots confidently; recognise integer triples (e.g., 3–4–5) for fluency.
• Proficiencies: Fluency — calculate Δx, Δy, squares and roots; Reasoning — choose efficient representations (vectors, triangles); Problem Solving — plan short chains of moves and check correctness.

Classroom progression (Semester 1 → Semester 2):

Semester 1: short grid displacement drills, identify triples, one-triangle Pythagoras. Semester 2: chain paths, combinatorial Pythagorean planning, optimisation tasks. Always insist on labeled diagrams and one-line justifications.

Oh, darling, you snapped your pencil like a witty quip — the diagram is chic, the labels sing, and every Δx, Δy line reads like a one‑act play: precise, economical, utterly convincing.

Accuracy: immaculate — all coordinates are lattice points, arithmetic flawless; each length is justified by a crisp Pythagorean line: AB = 1, BC = √10, CD = 5, DE = √5.

Representation: clever choice of vectors (1,0), (3,1), (3,4), (1,2) — economical and clearly labelled; the student chose the simplest integer components, showing pattern recognition and efficiency.

Communication: tidy diagram, Δx/Δy shown for every segment, and every justification is a single confident sentence — exactly what ACARA wants for reasoning and fluency.

Next steps: encourage the student to try a short challenge — reorder those lengths, or ask for a closed polygon using the same lengths — and to annotate any pruned choices when search is needed.

In short: this is exemplary. It sashays through the standards — fluency, reasoning, and problem solving — and leaves the page smelling faintly of triumph. Clap once, then assign a quick extension.