Geometry distance problems for a 13-year-old — ordering by difficulty and teacher evaluation (Pythagoras, rectangles, rope sag)

1. Order of the three problems by difficulty (for a 13-year-old) and brief comparison

1. Easiest: The pole tops problem (39 ft & 15 ft poles, 45 ft apart). This is a straightforward application of Pythagoras — treat the difference in heights and horizontal separation as legs of a right triangle.
2. Medium: The rectangle corners problem (F, D, I, A). This requires understanding the geometry of a rectangle, recognising which distances could be side lengths or a diagonal, and reasoning to find the minimum possible distance.
3. Hardest: The slackrope (walker) problem. It requires setting up coordinates, modelling the rope as two straight segments meeting at the walker, using Pythagoras twice and adding lengths. It is more involved algebraically and needs correct set-up of points and distances.

2. Evaluation of each student answer (with corrections and short solution sketch)

• Rectangle corners:

  Student answer: 4 — Correct.

  Reason: Place F at (0,0). The two other corners D and I could be at distances 3 and 5 from F. Two possible arrangements exist: (a) both 3 and 5 are side lengths so the opposite corner A is at distance sqrt(3^2+5^2)=sqrt(34)≈5.83; (b) one of the distances is the diagonal (largest distance from F), so if 5 is the diagonal and 3 is a side, the other side must be 4 (since 3^2+4^2=5^2). Then the remaining corner A is 4 from F. The question asks the minimum possible distance, so 4 is the correct minimum.

• Slackrope walker:

  Student answer: 28 — Correct.

  Reason: Put the left pole at (0,15) and the right pole at (14,15). The walker stands 5 m from the left pole at (5,3). The rope length is the sum of distances from (0,15) to (5,3) and from (5,3) to (14,15): sqrt(5^2+(3-15)^2)=sqrt(25+144)=13, and sqrt(9^2+(15-3)^2)=sqrt(81+144)=15. Total = 13+15 = 28 m.

• Pole tops (39 ft & 15 ft):

  Student answer: sqrt(2601) — Correct (which simplifies to 51).

  Reason: Coordinates (0,39) and (45,15) give horizontal difference 45 and vertical difference 24. Distance = sqrt(45^2 + 24^2) = sqrt(2025 + 576) = sqrt(2601) = 51 ft.

3. Formative teacher comment (about 700 words)

Overall, your answers show a good understanding of right-angled triangle geometry and the Pythagorean theorem. You found the correct numerical answers for all three problems, which is excellent. Each problem needed slightly different reasoning skills: the third problem is plug-and-play Pythagoras; the second required modelling a vertex point and adding two distances; the first required logical consideration of which distances could be sides versus a diagonal and then minimising the remaining distance. You handled those different reasoning demands correctly.

Strengths observed:

• You correctly identified how to place coordinates or imagine right triangles for the pole and rope questions, using horizontal and vertical differences.
• You applied Pythagoras accurately and simplified the arithmetic correctly (for the rope problem you found integer distances 13 and 15, and for the poles you reached 51).
• For the rectangle corners you showed flexibility in reasoning about different geometric arrangements and chose the arrangement that minimised the requested distance.

Suggestions to improve and deepen understanding:

1. Write a brief explanation for each numerical answer. You got the numbers right, but clear written steps — a sentence or two describing the model (coordinate choice, which distances are sides vs diagonal) — help another reader (and your future self) follow your thinking. For example, in the rectangle question explicitly state: "If 5 is the diagonal and 3 is a side, the other side must be 4, so the remaining corner is 4 from F. This gives the minimum possible distance."
2. Practice labelling diagrams carefully. Drawing a small sketch and labelling coordinates (for instance, left pole at x=0, right pole at x=14) will make setting up each equation faster and less error-prone.
3. Reflect on which geometric properties you used: right angles, triangle side relations, and the idea that the diagonal is the longest straight-line distance inside a rectangle. Naming these explicitly helps you recognise where similar ideas apply in other problems.

Common mistakes to watch for:

• Mixing up which distance is a side and which is a diagonal (rectangle question). If you assumed the wrong arrangement you might give a larger value instead of the minimum.
• For the rope problem some students try to treat the rope as a single straight line between pole tops (which would be longer than the V-shape produced by the walker) or forget to add both segments.
• Arithmetic errors—double-check squares and square roots, and simplify radicals where possible.

Next steps / extension:

• Try varying the rope problem: move the walker to different horizontal positions and compute rope length; notice when the total length is minimized (this links to the concept of Fermat points/reflection method).
• For the rectangle question explore all possible labellings and produce a short justification why 4 is the minimum and sqrt(34) is another possible value.
• Practice problems mixing Pythagoras with coordinate geometry (distance formula) and with simple optimisation reasoning.

4. Rubric (short) — criteria and levels

Criteria: (1) Accuracy of numerical answer, (2) Correct setup/model, (3) Explanation/justification, (4) Mathematical communication (labels, units, diagram).

• Excellent (A): Correct answer, correct model, clear steps and diagram, units shown, reasoning explained.
• Proficient (B): Correct answer, mostly correct model, partial explanation, minor omissions in communication.
• Developing (C): Partially correct answer or correct answer with incomplete or unclear reasoning, missing diagrams or labels.
• Beginning (D/E): Incorrect answer with major modelling errors or missing reasoning; needs guided instruction.

5. Mapping to ACARA (v9) — measurement and geometry

These tasks connect with the Australian Curriculum (v9) Measurement and Geometry strand for middle secondary levels (approx. Years 7–9). They require students to:

• Use and apply the Pythagorean theorem to determine distances in right-angled triangles and rectilinear situations.
• Model 2D geometric situations using a coordinate or diagram, calculate distances between points, and reason about shortest paths.

Suggested curriculum links (by description): use Pythagoras to solve problems, represent and reason with 2‑D geometric figures, and solve applied measurement problems involving lengths and distances. These are typical Year 8–9 geometry outcomes in the ACARA v9 framework.