Somewhere on school property stands something you cannot climb, touch the top of, or lower a tape measure from: the flagpole, the gym ceiling, the big tree by the parking lot. Your group’s job is to measure it anyway.
The task
You get a tape measure, a protractor with a straw and a washer on a string — a homemade clinometer — and thirty minutes outdoors. Produce a height, in metres, by two independent methods, and return with a board-ready account: a labelled diagram for each method, the measurements you actually took, and an honest interval — the height is between this and that — with your reasoning for the width. If your two methods disagree, do not average and hide it: say which one you trust more, and why.
Facilitation notes — for the teacher
Send groups out before teaching anything; Grade 10 trigonometry and proportional reasoning are both in the room already. Common second methods: shadows and similar triangles, a metre stick held at arm’s length, pacing plus the clinometer from two different distances. The two-distances approach quietly sets up the oblique-triangle work to come — flag those boards for the consolidation. Expect the measured angle to be the weak joint: a two-degree wobble moves the flagpole by half a metre, which is exactly the conversation you want. Groups who claim centimetre precision should be asked to re-measure and watch their own hands.
What mathematics tends to surface
Every method hides the same skeleton: a right triangle with a known side and a known angle, and a ratio connecting them. The tangent ratio does most of the work whether or not anyone says its name. The interval question surfaces error propagation honestly — small angle wobble, large height wobble. Special Angles then sharpens the tool: some angles give exact answers, no calculator required.
Where it leads
The field method breaks the moment the triangle loses its right angle — measuring across a ravine, or a cliff from the far side of a river. The Sine Law and The Cosine Law are built for exactly that failure, and the two-distances trick your groups improvised is their opening move.
The answer is not on this page
The flagpole’s height is not printed here — the caretaker knows, and we will check the class’s intervals against the truth together.
Curriculum connection
D1.6
pose problems involving right triangles and oblique triangles in two-dimensional settings, and solve these and other such problems using the primary trigonometric ratios, the cosine law, and the sine law (including the ambiguous case)
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D1.7
pose problems involving right triangles and oblique triangles in three-dimensional settings, and solve these and other such problems using the primary trigonometric ratios, the cosine law, and the sine law Sample problem: Explain how a surveyor could find the height of a vertical cliff that is on the other side of a raging river, using a measuring tape, a theodolite, and some trigonometry. Determine what the surveyor might measure, and use hypothetical values for these data to calculate the height of the cliff.
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