Short step-by-step solution (for a 10-year-old):
Imagine a straight line from the top of the taller pole to the top of the shorter pole. That line is the rope and it forms the hypotenuse of a right triangle. The horizontal distance between the pole bases is 45 ft. The difference in pole heights is 39 - 15 = 24 ft. So the two legs of the right triangle are 45 and 24.
Use the Pythagorean theorem: hypotenuse = sqrt(45^2 + 24^2) = sqrt(2025 + 576) = sqrt(2601) = 51. Therefore the shortest rope is 51 feet. Always write the unit, ft.
Evaluation of the student's answer:
The student wrote: sqrt 2601. That is mathematically correct as a value, but it should be simplified and presented with units and a brief explanation. The simplified answer is 51 ft. The student needed to show the reasoning (45, 24, Pythagoras) and include units.
ACARA v9 mapping: Relevant to Measurement and Geometry (Years 5–6): apply Pythagoras to solve problems involving right-angled triangles and use appropriate units when reporting answers.
Rubric (total 10 marks):
- Correctness (4 marks): full marks for 51 ft; 3 marks for correct unsimplified radical; fewer if incorrect.
- Method (3 marks): clear use of Pythagoras and identification of legs = 3; partial method = 1–2.
- Working & Communication (2 marks): shows steps, arithmetic and units = 2; missing steps or units = 0–1.
- Presentation (1 mark): final answer simplified and boxed/clearly stated with units = 1.
Sample score for the student: sqrt 2601 = correct radical but no simplification, no units shown, no working presented → about 4/10. To improve: simplify to 51 ft, show 45^2 + 24^2 = 2025 + 576 = 2601, and write a one-sentence reason why Pythagoras applies.
200-word teacher comment (Amy Chua Tiger Mother cadence):
Good attempt, but we will not let sloppiness slide. You identified the correct relationship — the tops form the hypotenuse of a right triangle — and you used the squares of the horizontal distance and the difference in heights. That is precisely the right idea. But writing sqrt 2601 and stopping there is like doing 90% of the job and refusing to finish: mathematics is about neat answers and clear communication. Simplify: sqrt(2601)=51, so the rope must be 51 feet long. Always include units and a brief sentence explaining why the Pythagorean theorem applies: the rope, the ground, and the vertical pole difference create a right triangle. Show the arithmetic steps: 45^2+24^2=2025+576=2601, sqrt(2601)=51. For full marks, label your diagram, state the triangle's legs (45 and 24), show the calculation, and box the final answer with units. Next time avoid leaving an unsimplified radical and write in sentences so anyone can follow your reasoning. You have the right instincts; now learn to finish cleanly and present like a mathematician. Work neatly, check arithmetic, and don't be proud of shortcuts that hide steps. I expect an improved response next time — show clear labeling, units, and the final simplified number. No excuses, please.