I love these romps through broken chords (around the CIRCLE of FIFTHS) and my adult students “work out” weekly. (The challenge is to have smooth shifts between octaves, a seamless playing in legato minus thumb CLUNKS, and in staccato to play crisp and evenly.
Arpeggios (root, third, fifth patterns) can be practiced in parallel motion (same direction both hands); contrary motion, starting in the middle of the piano, for example and fanning out Right and Left hand in opposite directions; in tenths (parallel motion, naturally; and in 4-note groupings with a doubled root: example, CEGC, EGCE, GCEG, CEGC or *Four-note arpeggios played through inversions)
This last form can also be arranged in 10ths where the right hand starts ten notes above the left, or is basically a “skip” from the root, transposed into a higher octave. (aka beginning in first inversion, RH) This scheme can be reversed with the LEFT HAND starting in first inversion or any other against ten notes above in the right. (Various choices-inversions are possible.)
Arpeggios can and should be played in many articulations and dynamic levels. (legato–smooth and connected; staccato-crisp at forte and piano levels) Forearm, finger, and “wrist staccato” renderings are recommended.
Blocking techniques, especially, are very helpful in executing arpeggios, where notes between the thumb are clumped or chunked.
Here are some snatches of lessons-in-progress and tutorials from my ARPEGGIO video archive:
A minor Arpeggio in parallel motion (Tutorial)
G# minor in TENTHS
Tutorial, E Major arpeggio :
Piano Lesson in progress by Skype, E Major Arpeggio, snip legato into staccato
C Major arpeggio practice in blocks.. (Skype supplement)
D Major Arpeggio in contrary motion (Skype supplement)
Diminished 7th form, and then Diminished 7th to Dominant 7th arpeggios