Deriving the equations of a cycloid, epicycloid, and hypocycloid

First, students will learn two ways of deriving the equations of a cycloid.

Watch these videos:

Tim Hodges

Deriving the Equations of a Cycloid by Xander Gouws

Next, students will learn how to derive the equations of an epicycloid.

Deriving the Equations of an Epicycloid by Xander Gouws

After that, students will learn how to derive the equations of a hypocycloid.

Hypocycloid derivation of parametric equations and examples of hypocycloid animations. by Zak's Lab

Conclude by giving your students these challenges:

Spot the Card by NRICH
Factorising with Multilink by NRICH
Concrete Calculation by NRICH
2005 AMC 8, Problem 12
Chords in a Circle

Draw four circles, such that each circle contains an odd number of roses, each circle contains a different number of roses, and no two circles touch or intersect:

Here's the solution:

Source: Test Your Math IQ by Steve Ryan