# Introduction to periodic behavior

## See a basic real-world model of periodic behavior and make sense of what data or graph it might generate.

Students have had a lot of exposure to graphing linear, quadratic, and exponential functions. They have yet to see a function that behaves periodically and understand how it might connect to a specific context. The ferris wheel provides a familiar scenario for students to see how the height of the cart will go up and down continuously, and to connect this information to a possible graph of the height. Students can make a rough sketch after watching the demo, or can use the more specific tools available in the Desmos activity to attempt to get a more accurate graph.

## External Resources

#### Description

WHAT: Students watch three different animations of motion along a curve, and then attempt to graph distance as a function of time. They can then watch an animation of the motion described by their graph, compare it with the actual motion, and refine their graph. The last motion is a ferris wheel cart, leading to a trigonometric function.

WHY: This activity reminds students of the relationship between motion and features of a distance-time graph, and prepares them to think about trigonometric functions describing periodic motion (F-IF.B.4$^\star$). A discussion of periodic behavior would be appropriate after students complete the ferris wheel scenario, as students think about what the graph would look like as the ride continues through multiple cycles.

#### Description

WHAT: In this task students compute the speed of motion at the earthâ€™s surface caused by the earthâ€™s rotation at various different latitudes, given the latitude and longitude of various different cities and the distances between them. They use right triangle trigonometry and the relationship between angle and arc length.

WHY: Although this activity does not directly relate to the use of trigonometric functions to describe periodic behavior, it activates student thinking about circular motion and reminds them of previous work with the relationship between angle and arc length (G-C.B.5) and with trigonometric ratios (G-SRT.C.8$^\star$), all of which are useful as they work on extending trigonometric functions using the unit circle and on understanding radian measure. The activity also provides an opportunity for students to apply several distinct topics and see the connections between them in order to solve a larger problem (MP.1).

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