1) Order the problems by difficulty (for a 13-year-old)
- Harder: Rectangle corners problem (F, I, D, A) — requires correct interpretation of which distances are sides vs diagonal and using Pythagoras plus reasoning about labels and 'minimum possible'.
- Easier: Slackrope walker problem — straightforward application of Pythagoras to two right triangles and adding the two segment lengths.
2) Solutions and evaluation of student answers
Problem A (rectangle corners)
Question summary: Four people F, I, D, A occupy the four corners of a rectangle. We are told F is 3 m from D and 5 m from I. What is the minimum possible distance from F to A?
Key idea and interpretation: If two opposite corners are (F and I) and the other two are (D and A), then from F the three other corner distances are: one side length, the other side length, and the diagonal. The phrase 'F is 3 m from D and 5 m from I' fits the case where one of those is a side and the other is the diagonal. Specifically, if FD is a side of length 3 and FI is the diagonal of length 5, then the other side FA must satisfy Pythagoras: FA^2 + FD^2 = FI^2, so FA^2 + 3^2 = 5^2.
Calculation: FA^2 = 25 - 9 = 16, so FA = 4 m. The distances from F to the other corners could be {3,4,5}; the minimum possible of these is 3 if A were the same corner as D, but with the labelling that makes FI the diagonal and FD a side, FA = 4. The question asks the minimum possible distance that F could be from A, given the labels; with the usual reading (F and I are opposite corners, D and A are the other opposite corners), the computed value is 4 m.
Student answer: 4 — correct. The student has given the correct numeric result.
Problem B (slackrope walker)
Question summary: Two poles, each 15 m high, are 14 m apart. A rope is tied between their tops. The walker stands 5 m from one pole (so 9 m from the other). At that point the walker is 3 m above the ground, so the rope at that point is 3 m high. Find the total rope length.
Key idea: Each straight segment of rope from a pole top (height 15 m) down to the walker (height 3 m) forms a right triangle. The vertical drop from pole-top to the walker is 15 - 3 = 12 m. The horizontal distances are 5 m and 9 m.
Calculations: Left segment length = sqrt(5^2 + 12^2) = sqrt(25 + 144) = sqrt(169) = 13. Right segment length = sqrt(9^2 + 12^2) = sqrt(81 + 144) = sqrt(225) = 15. Total rope length = 13 + 15 = 28 m.
Student answer: 28 — correct.
3) Comparison of difficulty (concise)
The slackrope question is a straightforward Pythagoras application twice and addition; it is procedural. The rectangle-corners question requires careful reading to decide which distances are sides and which is the diagonal, and uses reasoning about labels and the phrase 'minimum possible'. That extra interpretation step makes it more difficult for many students.
4) Rubric (how I mark these answers)
- Accuracy (4 marks): Correct numeric answer (2 marks), correct units (1 mark), correct exact form or explanation if required (1 mark).
- Method and Reasoning (4 marks): Correct diagram or clear description of which distances are sides/diagonal (2 marks), correct use of Pythagoras or algebra showing steps (2 marks).
- Communication (2 marks): Clear labelling, clear final statement (2 marks).
- Total: 10 marks per problem.
Performance band: 9–10 excellent, 6–8 competent, 4–5 partial, 0–3 needs major improvement.
5) ACARA v9 curriculum mapping
These problems map to the Measurement and Geometry strand for middle secondary years (around Years 7–9 / age 12–15):
- Use of Pythagoras' theorem to solve problems involving right triangles (apply and interpret lengths) — essential content for Year 8–9.
- Geometric reasoning about rectangles, sides and diagonals, and representing problems with diagrams — links to developing spatial reasoning.
- Problem-solving and communication of mathematical reasoning (showing steps, labelling, justifying choices).
6) Teacher comments (in a strict, Amy Chua–style cadence)
I am pleased that you wrote the correct numbers for both questions — 4 and 28. But correct numbers without clear, labelled working are like catching a fish and not knowing what bait you used: lucky and not repeatable. You have the right instincts: recognising the 3-4-5 relationship in the rectangle problem and using Pythagoras in the slackrope problem. That shows you are beginning to see structure. Now refine the habit.
For the rectangle question, too many students panic at the labelling. Always draw the rectangle. Label the corners F, D, I, A and place the known distances on the drawing. Decide if each given length is a side or a diagonal. If you find a 3 and a 5, test whether 5 could be the diagonal: if so, compute the other side by Pythagoras. Write these steps down. You wrote 4 — excellent — but you must show the equation FA^2 + 3^2 = 5^2 next time. No marks given for memory; all marks are given for reasoning.
For the slackrope question, you used Pythagoras well. But again: always draw both right triangles, label heights (15 at tops, 3 at the walker), calculate the vertical drop (12) explicitly, and show each square root calculation (sqrt(169)=13, sqrt(225)=15). Then add the segments. That level of clarity moves you from ‘I guessed correctly’ to ‘I can teach this to someone else.’
Next steps and deliberate practice: do 10 more problems where one length is a diagonal and one is a side — practice deciding which is which. Do another 10 problems where a point is interior to a rectangle or square (distances to corners given) so you are forced to organise information and not panic. For slackrope-style problems, practise variants where the walker is not between the poles symmetrically — change distances so you use two different horizontal lengths each time.
Specific feedback: bring a neat diagram, label heights and horizontal distances, write the Pythagoras equation before you calculate, box your final answer with units, and write one sentence explaining why that value is minimal if the question asks for 'minimum possible'.
Finally, remember: accuracy is necessary but not sufficient. Mathematical maturity is shown by clear diagrams, correct labelling, explicit equations, and tidy final answers. You got the right numbers; now make your working as reliable and fierce as your answers. I expect that next time you will present complete working for full marks.