1) Order the problems by difficulty (easiest → hardest) and compare difficulty:
- Problem 3 (easiest): Two vertical poles, flat ground, distance between bases and heights given → straight use of Pythagoras. Few steps and low chance of misreading.
- Problem 2 (medium): Slackrope problem. Requires recognising that the rope can be treated as two straight segments (piecewise straight lines from pole tops to the walker) and calculating two distances then adding them. A little more arithmetic but still straightforward Pythagoras twice.
- Problem 1 (hardest): Requires a careful reading of which corners are opposite and which are adjacent. Once the geometry is set up it becomes a Pythagoras question, but the interpretation step is the trickiest part for many students.
Why this order? Problem 3 is direct application of Pythagoras to one right triangle (short calculation). Problem 2 needs two Pythagoras calculations and correct placement of the walker between poles. Problem 1 is simple algebraically but can be misinterpreted; settling the correct arrangement of corners is the conceptual hurdle that makes it relatively harder.
Worked solutions and evaluation of the student's answers
Problem 1
Statement (interpreted): F and I are opposite corners of the rectangle, and D and A are the other opposite corners. Given FD = 3 (a side) and FI = 5 (the diagonal), find FA (the other side).
Set the two side lengths as 3 and x. The diagonal length is sqrt(3^2 + x^2) = 5. So 9 + x^2 = 25 → x^2 = 16 → x = 4. Thus the minimum possible distance from F to A is 4 metres.
Student answer: 4 → Correct.
Problem 2
Given: Two poles 15 m high, 14 m apart. Walker stands 5 m from one pole on the rope and is 3 m above the ground at that point. Treat the rope as two straight segments from the top of each pole to the walker.
Place coordinates: left pole top at (0,15), right pole top at (14,15). Walker at (5,3) (5 m from left pole). Compute lengths of each segment:
L1 = distance from (0,15) to (5,3) = sqrt(5^2 + (3-15)^2) = sqrt(25 + 144) = sqrt(169) = 13 m.
L2 = distance from (5,3) to (14,15) = sqrt(9^2 + 12^2) = sqrt(81 + 144) = sqrt(225) = 15 m.
Total rope length = 13 + 15 = 28 m.
Student answer: 28 → Correct.
Problem 3
Given: Two poles of heights 39 ft and 15 ft, bases 45 ft apart on level ground. The shortest rope connecting the tops is the straight line between the two top points. Horizontal separation = 45 ft; vertical difference = 39 - 15 = 24 ft.
Length = sqrt(45^2 + 24^2) = sqrt(2025 + 576) = sqrt(2601) = 51 ft.
Student answer: "sqrt 2601" → mathematically correct but unsimplified; best to simplify to 51. So Correct, but simplify to 51.
Short rubric (out of 10) used to evaluate each response:
- Accuracy (4 points): correct final answer and units.
- Method (3 points): correct geometry setup and use of Pythagoras/distance formula.
- Communication (2 points): clear steps or labels, units stated.
- Reasoning & interpretation (1 point): correct interpretation of diagram/wording (important for Problem 1).
Applying the rubric: All three answers earn top marks for Accuracy and Method. Problem 3 loses a small communication point only because the student left the radical unsimplified. Overall: excellent work.
ACARA v9 mapping (content and proficiencies)
Relevant area: Measurement and Geometry. Level: Stage 4 (approximately Years 7–8) moving into Stage 5 (Year 9) for consolidation of distance ideas.
- Content descriptions matched: applying Pythagoras' theorem to find unknown distances in right-angled contexts; using coordinate-style reasoning or distance formula to determine lengths in plane geometry.
- Proficiencies: Understanding (knowing Pythagoras and the geometry), Fluency (carrying out calculations accurately), Problem Solving (interpreting word problems and choosing strategies), Reasoning (explaining why those strategies work).
Teacher comments (Nigella Lawson cadence — warm, sensory, about 700 words)
My dear, you arrived at each answer with the calm confidence of someone who knows the recipe for a reliable tart — the basic dough is Pythagoras, and once it is mixed the results are simply delicious. I can almost see you in the quiet kitchen of the problem set: sleeves rolled up, the geometry laid out like ingredients. For the first question you handled what often trips up students — the interpretation — with neatness. You recognised the opposite corners, sprinkled a little algebra, and out came a perfect 4. That small moment, where many stumble, is like caramelising sugar just so; subtle, and unmistakable to the practised hand.
In the slackrope problem you were both brave and exact. Visualising the rope as two straight segments is the sort of practical imagination I adore — it is like choosing to deglaze a pan rather than letting the flavours muddle. You computed two right-angled lengths, found 13 and 15, and then combined them into a warm total of 28 metres. The arithmetic was sound; the geometry was confident. Lovely.
The third question was crisp and straightforward and you gave it the simple, elegant treatment it deserved. You wrote sqrt(2601). It is correct; it has the comfort of a dense chocolate cake, but remember: when the oven timer dings, we still slice and serve. Simplify to 51 and present it like a finished slice — clean and ready.
Where could you add tiny refinements? First, label. A small sketch with labels F, I, D, A, and arrows showing which distances are sides and which are diagonals would be like a sprinkling of sea salt over caramel — it enhances everything. Second, in explanations, name the theorem you used: "By Pythagoras..." — this is not only good practice, it lets a reader follow your thought the way guests follow a recipe card. Finally, watch for ambiguity in phrasing: the rectangle question leans on us to interpret corner pairing. A short sentence listing the assumed order of corners is like writing "serves 8" on a pudding recipe — it helps everyone know what to expect.
Overall, your mathematical palate is excellent. You have fluency with right-angled thinking, and you deploy it with calm. On the rubric I would award full marks for accuracy and method across the board; lose a whisper of a point only for leaving the last answer unsimplified. Keep sketching, keep labelling, and keep that tidy numerical handwriting — it reads like good table manners.
These problems are exactly the kind that build confidence: they are honest, they reward careful reading, and they let you see Pythagoras at work in three gentle, distinct ways. Savor them, practise a few more variations (move the walker to different positions, try poles of different heights, swap which corners are opposite), and you will find the same comforting structure every time. Deliciously mathematical.
Final verdict: all three answers are correct. Slight note to simplify sqrt(2601) to 51 next time. Keep going — the clarity you have displayed here will bake into mastery.