Shortest rope between two poles — Age 10 — Pythagorean theorem (ACARA v9 mapping)

Short answer: The student's answer sqrt(2601) is correct. The shortest rope length is 51 feet (since sqrt(2601) = 51).

Step-by-step explanation for a 10-year-old:

1. Draw the situation: two vertical poles of heights 39 ft and 15 ft, with their bases 45 ft apart on flat ground. Connect the tops — that rope is the shortest straight line between the two tops.
2. Look at the triangle formed by: the horizontal distance between the bases (45 ft), the vertical difference in heights (39 − 15 = 24 ft), and the rope (the slanted side). This is a right triangle because the poles are perpendicular to the ground.
3. Use the Pythagorean theorem: hypotenuse^2 = (horizontal leg)^2 + (vertical leg)^2. So rope^2 = 45^2 + 24^2.
4. Calculate: 45^2 = 2025, 24^2 = 576, so rope^2 = 2025 + 576 = 2601.
5. Take the square root: rope = sqrt(2601) = 51. So the rope must be 51 feet long.

Evaluation of the student's answer: Writing sqrt(2601) shows the correct method and result. For full clarity, it is good to simplify to 51 ft and show the intermediate steps (45^2, 24^2, sum) as above.

ACARA v9 mapping (plain language): Measurement and Geometry — using properties of right triangles and distance (apply Pythagorean theorem to calculate distances). Appropriate for upper primary to early secondary reasoning practice.

Rubric (age-appropriate):

• Excellent (A): Correct diagram, identifies legs 45 and 24, uses Pythagorean theorem correctly, shows arithmetic (2025 + 576 = 2601), simplifies sqrt(2601) to 51 ft, clear explanation.
• Satisfactory (B): Correct final expression (e.g. sqrt(2601)) and numeric answer, but missing some written steps or diagram.
• Developing (C): Partial understanding — may identify one leg correctly but make arithmetic or setup errors.
• Needs Improvement (D): Incorrect method (not using right triangle), missing key values, or incorrect calculations.

200-word teacher comment (Sailor Moon cadence):

In the name of the Moon, I applaud your brave leap into geometry, young guardian! You connected two towers of light across the flat earth with a single silver rope, and your answer, written as sqrt(2601), is a true spark of intuition. The Pythagorean path you followed whispers of right triangles and brave leg lengths, and you have summoned the correct number. Next time, show your work so the moon can see the 45 and the 24 standing proud as the triangle's legs. When you simplify sqrt(2601) you reveal the shining 51, and that final polish crowns your solution with clarity. Keep practicing explaining each step—write the legs, square them, add them, and then take the square root—so your reasoning glows as brightly as your answer. If ever you doubt, draw the poles and the rope, and let the triangle guide you. You have strong instincts for spatial thinking and number sense; nourish them with diagrams and neat calculations. With a little more explanation, your answers will transform from good to magical. Sailor sensei is proud—attack any problem with curiosity and a calm heart, and you will always find the right length. Keep shining and keep solving with joy, always.