So many students have a disconnect between learning basic technical exercises and integrating them into their playing and musicality. I have found that using transposition for the intermediate and advanced student has multiple benefits.
I have also used this technique in teaching beginning theory with instruments only to a group of mixed instruments. In that instance we work with "Happy Birthday" in all keys and use no music.
After the student is reasonably comfortable with all the major scales and arpeggios and understands that scales have set patterns, I choose a simple melody that has no additional accidentals to be transposed into all the keys. We look at how it is put together, and whether it starts on the tonic or dominant or on another scale degree.
The first transposition is just changing the key signature to 1/2 step above or below the original note. The letter names of the notes remain the same. So we might go from C Major to C# major.
The next transposition goes to a letter name above or below the original key. If the original key is F, then we go to G. I use a visual approach with the transposition going either up or down to the very next line or space. A note on a line will then go to the space above or below and the note on the space will go to the line above or below. (ex. F-G)
The appropriate key signature needs to be applied. We review the scale of the key and the tonic and dominant chords in the new key.
After that we transpose further away from the original key using the visual context of two lines or two spaces up for transposition of a fifth above or one line and a space up for transposition to a fourth above.
The visual reference gives the students a clear tool to use while developing their sense of pitch using cerebral knowledge as well as auditory.
Transposition works with tone development and intonation because the student tries to make the melody sound the same in all keys.
After the initial transposition we go to more complicated melodies with additional accidentals and then analyze those patterns.
I use the "Tone Development Through Interpretation" by Moyse as the source for the transposed melodies. Many of them are in the "belle canto" style which encourages the student to play with an expressive tone. Then I work with the advanced student using the tuner to compare pitch tendencies in the various keys.
Transposition is an additional tool to bring variety and added interest to your teaching as well as help your students to sharpen their skills.