Problem summary (for a 13-year-old)
Two poles 15 m tall stand 14 m apart. A walker on the rope is 5 m from the left pole and 3 m above the ground. Model the rope as two straight segments from each pole top to the walker. Find the total rope length.
Step-by-step method (exact steps you should show)
- Place coordinates: put the left pole top at (0,15) and the right pole top at (14,15). Place the walker at (5,3).
- Draw the diagram and label the horizontal distances (5 and 9) and the vertical drop (12 = 15 − 3). Mark right angles to show the two right triangles.
- Compute the left segment length using Pythagoras: distance between (0,15) and (5,3):
√[(5−0)^2 + (3−15)^2] = √(25 + 144) = √169 = 13 m. - Compute the right segment length using Pythagoras: distance between (14,15) and (5,3):
√[(14−5)^2 + (15−3)^2] = √(81 + 144) = √225 = 15 m. - Add the two lengths: 13 + 15 = 28 m. Finish with a short unit-labelled sentence, e.g. "Total rope length = 28 m."
Worked exemplar (concise)
1. Place top points at (0,15) and (14,15); walker at (5,3).
2. Left: √[(5−0)^2+(3−15)^2]=√(25+144)=√169=13 m.
3. Right: √[(14−5)^2+(15−3)^2]=√(81+144)=√225=15 m.
4. Total = 13+15 = 28 m. Final sentence: "The rope is 28 m long."
Rubric (Slackrope) — total 10 marks
- Diagram & coordinates: 3 marks
- Correct Pythagoras setup for both triangles: 4 marks
- Accurate arithmetic and final sum with units: 2 marks
- Clear justification sentence: 1 mark
ACARA v9 alignment (short)
Measurement & Geometry (Years 7–9): apply Pythagoras' theorem in right triangles; use coordinates to model geometric situations. Mathematical proficiencies: Fluency, Reasoning, Problem Solving.
Teaching tips & common fix
Always draw a clear diagram and label horizontal and vertical distances. A common slip is misreading the horizontal distances (5 vs 9): mark them on the picture to avoid that error. Ask for one brief justification sentence to show modelling understanding (e.g. vertical drop = 15 − 3 = 12).
Teacher comments (Carrie Bradshaw cadence — ~400 words)
There are nights when I think about shoes, and there are nights when I think about geometry — and sometimes, miraculously, the two feel like the same thing. You walked into this problem like someone slipping on a great pair of heels: confident, tidy, and with an eye for the details that make an outfit — or a solution — sing. You placed the pole tops at (0,15) and (14,15), set the walker at (5,3), and in one elegant movement cut the rope into two right triangles. It was like dividing a runway into two perfect panels.
Your arithmetic was the evening gown of the piece: clean lines, no unnecessary frills. √(25+144) becoming 13 and √(81+144) becoming 15 reads like a well-tailored sentence. And when you added them to get 28 m, you delivered the final line as if you’d just stepped onto the red carpet and everyone applauded. The only tiny accessory you might add — and dear, accessories matter — is a brief note about why the vertical distance is 12. Saying 15 − 3 = 12 is the mathematical equivalent of tucking a handkerchief into your pocket; it shows you know you did more than compute — you modelled.
Also, don’t miss the little pattern: those 5–12–13 and 9–12–15 triangles are like classic staples in your wardrobe — they keep appearing and they always work. If you point them out, you start recognising a style, and that recognition makes future problems feel like familiar shoes that you know will fit.
So, sparkle where it counts: keep your diagram labelled, your steps visible, and your final sentence proud. Maths, like fashion, rewards the confident choice. And remember: neatness is never boring. It’s the secret that makes brilliance look effortless.