The Blues Scale
The C blues scale are these notes: C Eb F F# G Bb C
Obviously it doesn't follow the normal music theory guidelines: namely, that you can't repeat a musical alphabet letter. These collection of notes should be mastered first before anything else, right hand only. A suggested fingering would be: 1234123. However, this fingering is not necessary for every situation. Going up and down with the fingering: 13131313... is just as acceptable.
Instruct the student to learn the blues scale playing it up and down for at least a week until they are very comfortable with the notes.
Playing to a backing track and 12 bar blues
Once they are very comfortable playing up and down the blues scale, the next step is to have them play to a backing track in order to practice soloing. Get their family ipad or the student or parent's phone, and open youtube. Search for: C blues backing track. Anything with a 12 bar blues pattern will work. The pattern should go: 4 measures of C7, 2 measures of F7, 2 measures of C7, 1 measure of G7, 1 measure of F7, 1 measure of C7, 1 measure of G7, and that pattern repeated over and over. That way they can make up a cool little jazz solo. Definitely demonstrate a solo for them. 12 bar blues has a couple variations, but the above pattern is popular.
Encourage them to make a 'story' with their soloing, making it lyrical and understandable. They can combine notes too. The most popular example is the "blues note." The blues note is actually not possible on the piano. It's actually the quarter tone between F and F#. We get close by playing F and F# together. Other combinations can be used effectively as well.
Teach them the 12 bar blues pattern. You could even write it down, showing the C7 is played for 4 measures, F7 for 2, etc. No need to explain 7th chords yet.
How to swing 8th notes
Once they get good enough to do the 8th note pattern, be sure to teach them to swing the 8th notes. If the 8th note falls on a beat (1,2,3, or 4), then the note is played longer. If it is played off the beat (on "and"), then it is played short.
Teach them to either say: 1 ah 2 ah 3 ah 4 ah, or Looong short Loooong short looong, etc
During their solos, they could actually play triplets as well, and triplets would sound quite at home with the swing pattern.
Playing with them
Having them solo while you play an accompaniment is a great way to connect with the student. There are several possibilities. A "walking bass" sounds great and there are many variations. But basically you could "walk" up to the Bb, starting on C, thus simulating the C7 chord a bit. The right hand could "comp" jazzy chords: C6, C9, C7, F7, F9, G7,G9, and all their inversions, etc. The pattern below is an example of the LH.
Another example of a LH pattern with chord examples.
Call and response
As the student progresses being able to play the right hand really well, you can introduce the concept of call and response. There are many variations you could teach, but the simplest would probably be to have you do 2 measures of solo, then the student 2 measures soloing, switching back and forth.
Obviously, you must have a command of soloing and playing a left hand pattern at the same time. You are not ready to teach the student call and response until you have mastered this skill.
Controlling tempo with the student is important, be sure to go slow enough. It's the blues afterall. But it can be taken at more of a rock pace when you and the student become proficient.
Call and response can be more fun when there is a "conversation" involved. Explain to the student that you will listen to what they say and say something similar. This is a very loose concept though, and there is no harm in "changing the subject" from now and then. Thus you have to be listening to what each other plays and play off of your ideas. Give them an example of what could be done to respond to an idea, maybe changing 1 or 2 notes, or the rhythm a bit.
These guys do it pretty good :) If you need more of an example:
Playing hands together
The eventual goal is to have them do their own jazz improv without a backing track or teacher. You can start this process by having them practice the following jazz exercise that utilizes the 12 bar blues pattern: jazz improv exercise.
Eventually, after having very thoroughly mastering the 12 bar blues pattern, they can try making up their own notes. They must know the left hand pattern very well though. If not, then they will likely make mistakes in the 12 bar blues pattern as they strive to make up ideas.
Make sure that they make up ideas very simply at first. Perhaps the 8th note pattern can be modified slightly, little by little, to help them make up