Short answer: P’s method doesn’t work because the 3-4-5 triple must be scaled by the same factor for all three numbers. The correct length of the other leg is √70000 = 100√7 ≈ 264.6 cm.
Why P is wrong (step-by-step):
- P noticed 300 = 3×100 and 400 = 4×100 and tried to use the 3-4-5 triple. But a triple scaled by k is (3k, 4k, 5k) where 5k is the hypotenuse.
- If 400 were 5k, then k = 400/5 = 80, and the legs would be 3k = 240 and 4k = 320 — not 300. So the triangle is not a scaled 3-4-5 triangle with k = 80.
- Or, if P took k = 100 (because 300 = 3×100 and 400 = 4×100), then 5k would be 500, which would make the hypotenuse 500 — but the hypotenuse is actually 400. That’s inconsistent.
- Scaling must use the same k for all three numbers; P mismatched which numbers correspond to which parts of the triple.
Correct calculation (using Pythagoras):
Let the missing leg be x. Then x² + 300² = 400². So x² = 400² − 300² = 160000 − 90000 = 70000. Thus x = √70000 = 100√7 ≈ 264.575...
Evaluation of the student answer:
- The student wrote √70000, which is exactly correct. Award full marks for the correct expression.
- They also correctly noted that the hypotenuse must be the longest side and that there will not be a side of 500 cm. Good reasoning!
- To be complete, the student could give the decimal approximation: about 264.6 cm, and briefly explain why P’s scaling idea was inconsistent.
Simple rubric (mapped to ACARA v9 — Measurement and Geometry, Years 5–6):
- Excellent (3 points): Correct final value (√70000 or 100√7 or ≈264.6 cm), correct Pythagorean working, and clear explanation why scaling claim fails.
- Satisfactory (2 points): Correct final value with calculation but partial or unclear explanation about scaling error.
- Needs improvement (1 point): Incorrect final answer or missing correct working; misunderstanding of hypotenuse vs leg or scaling.
Teacher comment (200 words, in a warm Nigella Lawson cadence):
Darling, imagine numbers like ingredients in a bowl, each with its own flavour and role. P saw 300 and 400 and thought of the lovely 3-4-5 recipe, but then sprinkled the wrong proportions. You can scale a Pythagorean triple only by the same factor for every part — 3k, 4k and 5k together. If 300 were 3k and 400 were 4k then k would be 100, and the hypotenuse would have to be 5k = 500, which it isn’t. So P’s method is like adding salt when the recipe needs sugar. The triangle with sides 300 and 400 is not a scaled 3-4-5 triangle. The correct way is simpler and sweeter: use Pythagoras. Take 400 squared minus 300 squared, that gives 70 000, and the square root of that is the missing leg. That length is 100√7, about 264.6 cm — tidy, warm and sensible. Your student answer, √70000, is spot on, and the note that the hypotenuse must be the longest side is perfectly observed. Encourage them to say the decimal too, and to explain the scaling point next time.
Mapping note: This task develops the ACARA v9 Measurement and Geometry achievement. It asks students to apply Pythagoras’ theorem to find missing side lengths and to reason about numerical scaling of Pythagorean triples.