Quick difficulty ranking (easiest → hardest) and why
- Rectangle-corners distance (student answer: 4) — easier: mostly reasoning about which distances could be side lengths or a diagonal and using a familiar 3-4-5 right triangle.
- Slackrope length (student answer: 28) — harder: needs an optimisation idea (shortest rope given a point on it) and Pythagoras twice; more modelling assumptions to state explicitly.
Question 1 — full reasoning (answer: 4 m)
Four people occupy the four corners of a rectangle. From F, the distances to two other corners are 3 m and 5 m. From one corner point F you can reach the other three corners by either going to the two adjacent corners (these are the side lengths of the rectangle) or to the opposite corner (the diagonal = sqrt(side1^2 + side2^2)).
Two possibilities consistent with distances 3 and 5:
- Both 3 and 5 are the two side lengths. Then the distance to the opposite corner is the diagonal: sqrt(3^2+5^2)=sqrt(34)≈5.83 m.
- One of the given distances is a side (3) and the other is the diagonal (5). If diagonal = 5 and one side = 3, the other side y satisfies sqrt(3^2 + y^2)=5 → y=4 (the 3-4-5 triangle). Then the remaining corner (one adjacent side) is 4 m from F.
We are asked the minimum possible distance F could be from A. Between 4 and ≈5.83, the minimum is 4 m. The student answer 4 is correct.
Question 2 — full reasoning (answer: 28 m)
Two poles of equal height 15 m are 14 m apart. A point on the rope is 5 m from one pole (i.e. horizontal distance 5) and the rope there is 3 m above ground (height 3). We are asked for the rope length. To get the shortest possible rope consistent with the rope passing through that point, we assume the rope is straight on each side of that point (no extra sag). Any extra sag away from that point would only increase length.
So treat the rope as two straight segments from the top of each pole (coordinates (0,15) and (14,15)) to the given point (5,3). The vertical drop from a pole top to that point is 15−3=12 m. Distances:
- Left segment: sqrt(5^2 + 12^2) = sqrt(25+144) = sqrt(169) = 13 m.
- Right segment: horizontal distance from the point to right pole = 14−5 = 9 m, so length = sqrt(9^2 + 12^2) = sqrt(81+144) = sqrt(225) = 15 m.
Total rope length = 13 + 15 = 28 m. The student answer 28 is correct.
Short suggestions for improvement
- Always draw a labelled diagram: mark horizontal distances and heights; label which distances are sides vs diagonals.
- When you make modelling assumptions (e.g. rope approximated by straight segments), say so briefly and justify why that choice gives the shortest possible rope.
200-word teacher comment — Amy Chua (tiger mother) cadence, constructive but strict
You got both answers right. Good. But don’t be smug; getting the number is only the start. For the rectangle question you showed one of the most useful instincts in mathematics: check whether one of the given lengths could be a diagonal and whether a 3–4–5 triangle appears. That is exactly the shortcut a careful thinker uses. For the slackrope: excellent — you recognised that the shortest rope passing through a fixed point is straight on each side of that point, and you used Pythagoras twice to get 13 and 15. Fluent and efficient. Now the next level: always include a quick diagram and a one-line justification of your modelling assumption (why straight segments give the minimum length). Rubric I will use next time: Accuracy (3 marks), Diagram and labelling (2 marks), Justification of modelling choices (2 marks), Clear steps / algebra (3 marks). You scored full marks on accuracy; add the diagram and explicit justification and you will have perfect solutions every time. Keep practising recognition of Pythagorean triples and practice clear labelling — that tiny extra discipline turns a good student into a great one.
ACARA v9 curriculum mapping (age ~13, Years 7–8/9)
- Measurement and Geometry: use Pythagoras’ theorem to solve problems involving right-angled triangles; reason about lengths in rectangles and apply distance calculations.
- Reasoning and problem solving: represent situations with diagrams, justify modelling assumptions, and use efficient calculation strategies (recognise Pythagorean triples).
- Suggested year level: middle secondary (Years 7–9), matching content that develops measurement, geometry and problem-solving skills in ACARA v9.