In this activity students get a chance to model a real world sinusoidal phenomenon–the movement of a clock! This task is complex and requires good reasoning skills. Refrain from helping students too early and lowering the cognitive demand of the task!
Students should work on this activity in groups. Have them begin by reading the set-up of the task, which has LOTS of information in it. Students may wish to draw a sketch to keep track of all the measurements. The sequence of questions is designed to get students thinking about how to model the height of a point on the minute hand of a clock as a function of the time in minutes. Early questions have students think about the maximum and minimum heights of Point P, whereas later problems focus on the periodicity of the function and the phase shift. Note that the initial point is neither at the maximum, minimum, nor midline! Students will have to think carefully about what the phase shift is, but the context of the clock should help!
On the back page, students will construct a model by finding the parameters of a cosine function, then compare this to a model that uses a sine function. It is important for students to know that either a sine or cosine function can be used to model any sinusoidal phenomenon, but usually one is more efficient than another. Having students identify which parameters would change and which would stay the same in the alternate model is a key skill.
In the final question, students are challenged to think about the height of a point on the hour hand, which is shorter than the minute hand. Again, students are asked to think about what features of their original function are preserved and which are changed. Decide what level of formality and rigor you want students to use as they compare and contrast the different models and functions. The task does not require students to find the parameters of the alternate model and the new function in questions 8 and 9, but you may wish to have students do this.