I'm teaching my little daughter the concept of counter - tonic - dominant with tetrads. As a visual aid it seems logic and comprehensible that the dominant with the motrix above his frame resolves down to the tonic (triad) and the counter with the motrix below his frame resolves up to the tonic (triad). What is the 4th tone of the tonic? The octave of the tonic's root note seems not quite logic or enough in this visual concept. Where is the flaw in my thinking?

     If you place the dominant with the motrix above the frame, it is a third away from the frame (ex:GBDF). We usually place the motrix below the frame, a second away from the frame (ex:FGBD). Either disposition is ok when the time comes to play it. The same (or rather the opposite) applies to the counter.
     The fourth note (the motrix) on the tonic is the sixth (A) when the tonic progresses to the dominant. It would be the seventh (B) when the tonic progresses to the counter. However, if you want to take a little breath after the tonic, you make a substitution of the motrix and double the fifth of the tonic (G), and then play your complete dominant. You would double the C on the tonic if you came from the dominant.
     With the 3 chords as tetrads you could have the chord dispositions FACD, GACE, GBDF, with the horizontal lines F-G-G, A-A-B, C-C-D, D-E-F. No matter how the first chord is disposed, the lines will remain the same.
     With the tonic as a triad, you could have the chord dispositions ACDF, GCEG, BDFG, with the horizontal lines, A-G-B, C-C-D, D-E-F, F-G-G. Only the A-G-B is different on account of the substitution.
     You could find the details for all these possibilities on the Diatonic Major Mode page. Follow the Orbits (and Chord Functions) carefully and don't forget the Metamorphosis 1 on the tonic in the middle. We hope this answers your question and that you feel free to write again.