Geometry and Pythagoras problems for a 13-year-old — ordering by difficulty, solutions and ACARA v9 mapping

Step-by-step solutions and difficulty ranking for five geometry problems (age 13). Includes assessment of student answers, a 4-point rubric, and a 300‑word teacher comment in a Nigella Lawson cadence, mapped to ACARA v9 Measurement & Number strands.

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Quick summary (correctness)

  • Q1 (rectangle corners): student answer 4 — Correct.
  • Q2 (isosceles triangle): student answer 13 — Correct.
  • Q3 (slackrope walker): student answer 28 — Correct.
  • Q4 (right triangle with consecutive integers): student answer 90 — Correct.
  • Q5 (walking mixed units): student answer 40 — Correct.

Step-by-step solutions

  1. Q1: Let the rectangle sides meeting at F be x and y. FD=3 so one side is 3. FI is the diagonal: sqrt(3^2+y^2)=5 => 9+y^2=25 => y^2=16 => y=4. So FA=4. (Student: 4 — correct.)
  2. Q2: Area = (1/2)*base*height so height = 2*60/24 = 5. Half the base = 12. Equal side = sqrt(12^2+5^2)=sqrt(144+25)=sqrt(169)=13. (Student: 13 — correct.)
  3. Q3: Model rope as two straight segments from pole tops (15 m) to the walker at (5 m from left pole, height 3 m). Left segment: sqrt(5^2+(15-3)^2)=sqrt(25+144)=13. Right segment: horizontal 9, vertical 12 => sqrt(81+144)=15. Total rope = 13+15 = 28 m. (Student: 28 — correct.)
  4. Q4: Let the other leg be n and hypotenuse n+1. 9^2 + n^2 = (n+1)^2 => 81 = 2n+1 => n=40. Sides 9, 40, 41 => perimeter = 90. (Student: 90 — correct.)
  5. Q5: North 9 m then south (9 m + 32 ft) leaves a net south displacement of 32 ft (metres cancel). East displacement 24 ft. Distance = sqrt(24^2+32^2)=sqrt(1600)=40 ft. (Student: 40 — correct.)

Order of problems by difficulty (easiest → hardest) with reasons

  1. Q2 (isosceles area → side): direct area formula then Pythagoras; short, procedural.
  2. Q5 (walking): neat trick with mixed units cancelling — needs unit awareness but short computation.
  3. Q1 (rectangle corners): spatial visualisation and recognising a 3-4-5 right triangle.
  4. Q4 (consecutive integers): requires setting up and solving a simple algebraic equation from Pythagoras.
  5. Q3 (slackrope): needs to model rope as two segments and compute two distances — multi-step geometry.

Assessment rubric (per question, 4 marks)

  • 4: Correct answer, clear method and units shown.
  • 3: Correct answer, minimal or missing justification.
  • 2: Method shown but arithmetic or small reasoning error; partial progress.
  • 1: Attempt shown but incorrect approach.
  • 0: No attempt or irrelevant work.

Teacher comment (ACARA v9 mapped)

Mapped to ACARA v9 strands: Measurement & Geometry (use of Pythagoras and spatial reasoning), Number & Algebra (algebraic set-up and manipulation), and Problem-solving & Reasoning. Below, a short teaching comment.

Imagine each problem as a small recipe — you measure, you fold in a theorem, you taste the arithmetic. You have plated five tidy dishes: the flavours are familiar, each step confident. Your work shows that you know how to extract a height from area, recognise that metres can cancel when the same distances appear north and south, set up an equation from ‘‘consecutive integers’’ and identify a classic 3–4–5 right triangle. There is a lovely economy to your thinking: the answers are correct and the computations concise. For growth, always write one clear sentence saying which theorem you used (‘‘Pythagoras’’ or ‘‘area = 1/2 bh’’) and include units — that keeps your solution readable and exam‑proof. Try explaining aloud why the slackrope creates two straight segments: that verbal step solidifies the model. Keep treating geometry like a recipe: visualise, choose the right tool, compute, and taste the result. Bravo — neat, warm, and satisfying.

Final notes

All five student answers are correct. Suggested scoring: give 4 marks for each question if workings are shown clearly; otherwise 3 if only final answers appear without justification.

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