A Ferris wheel is m across, its centre m above the ground, and it turns once every four minutes. You board at the bottom. The question sounds innocent: what does your height above the ground look like as a graph, from the moment you board until the ride ends two turns later?
The task
Sketch the graph at the board — axes labelled, key heights and times marked — and be ready to defend its shape against the room’s favourite wrong answers: is it made of straight lines? Half-circles? Something else? Then push on it: mark every moment you are exactly m up. Mark where your height is changing fastest, and where it is barely changing at all. Finally, redesign the ride: what happens to your graph if the wheel turns twice as fast? If the boarding platform is raised by a metre? If the wheel grows?
Facilitation notes — for the teacher
The two seductive wrong shapes — zigzag lines and stacked semicircles — should both appear on boards; engineer it by asking “could it be straight lines?” of a group that jumped to the right answer. The defeater is the speed question: near the top your height barely changes for many seconds, so the graph must flatten there, which kills both the zigzag and the semicircle. Groups that finish the redesigns can chase the ghost question: where is your horizontal distance from the centre, as a graph, and why does it look like the height graph shifted sideways? That is sine and cosine’s relationship, one day early.
What mathematics tends to surface
Periodicity, first: the graph must repeat exactly, turn after turn, which gives cycle and period their meanings. The flattening at top and bottom is the deep observation — circular motion projected onto height is not linear, and the smooth wave that satisfies every constraint is the sinusoid. The redesign questions are transformations in disguise: faster wheel, shorter period; higher platform, raised axis; bigger wheel, larger amplitude. Sinusoidal Functions gives the wave its name and its letters.
Where it leads
Tides, daylight hours, heartbeats on a monitor — every repeating phenomenon in this unit graphs like your wheel, and the redesign moves become the recipe . The unit’s task puts real tide data behind the same four letters.
The answer is not on this page
No finished graph appears here. The shape — and the argument that forces it — belongs to your group at the boards.
Curriculum connection
D2.1
describe key properties (e.g., cycle, amplitude, period) of periodic functions arising from real-world applications (e.g., natural gas consumption in Ontario, tides in the Bay of Fundy), given a numeric or graphical representation
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D3.4
predict the effects on a mathematical model (i.e., graph, equation) of an application involving periodic phenomena when the conditions in the application are varied (e.g., varying the conditions, such as speed and direction, when walking in a circle in front of a motion sensor) Sample problem: The relationship between the height above the ground of a person riding a Ferris wheel and time can be modelled using a sinusoidal function. Describe the effect on this function if the platform from which the person enters the ride is raised by 1 m and if the Ferris wheel turns twice as fast.
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