Fiddle geekery, and new question for the string players

(Doing this as a blog post instead of a post to Facebook, since this is really too long for a status update.)
Okay, so my monthly-or-so lessons with Lisa Ornstein have been going swimmingly. Yesterday I had another lesson with her, and we started talking about how to do string transitions so that I could start to do simple arpeggios and if I’m feeling really ambitious, really simple tunes.
The arpeggio drill has been good, letting me practice walking from the tonic, to the third, to the fifth, and then up to the octave, and then back down again. So yay!
We’ve also been talking about four types of string transitions:

  • Open string to open string
  • Finger on a string to open string
  • Open string to finger on a string
  • Finger to finger

And, Lisa’s advised me that when I’m doing scales in particular, and I’m coming down from an open string (the fifth) back down to the fourth on the string below, that’s an open-to-finger transition. And she’s got me doing a “stop, drop, and roll” thing that’s seeming to click well with my brain to try to make the scale as smooth as possible. I’ve just tried it today, and it’s gotten me the smoothest scales I’ve managed to play yet.
Then I tried to get a bit ambitious, and this is where the question for string players who follow me comes in.
I’ve been toying with “Frere Jacque” since it’s a real simple little children’s tune, and I figured playing with something like that to start with would be within my capabilities. So we did a bit of that in yesterday’s lesson, applying to it the same techniques I’ve done in workshops learning session tunes: i.e., breaking it down into pieces and thinking about how to play each piece.
I also asked Lisa when I should be changing bow directions, and she told me, I should change direction when I change notes. (IMPORTANT NOTE: I already know just from observing fiddle players in session that there are plenty of times when this is not in fact the case, and just from screwing around on my own instrument, I discovered that oh okay playing a bunch of notes on a single stroke is apparently how you do slurs? But for purposes of this question, I’m assuming I need to keep it simple for my newbie self and stick to “change directions when I change notes.”)
Given this, and given breaking “Frere Jacque” down into its constituent pieces, it seems to me like the bowing pattern gets a little weird and I’m not entirely sure how to handle it. The pieces look like this:
1st piece: Fre-re Jac-que, Fre-re Jac-que (Down-up down-up, down-up down-up)
2nd piece: Dor-mez vous? Dor-mez vous? (Down-up-down, down-up-down)
3rd piece: Son-nez les ma-ti-nes, Son-nez les ma-ti-nes (Down-up-down-up-down-up, down-up-down-up-down-up)
4th piece: Ding-dong-ding, ding-dong-ding (Down-up-down, down-up-down)
So it’s the 2nd and the 4th pieces that are confusing me a bit, because those are tuples, not doubles. And I can’t do two down strokes in a row, right? So should I go down-up-down, up-down-up? That would seem like the right thing to do, but I am not a hundred percent sure.
Any string players want to advise me?