We rencently bought a home in Buffalo, NY, and there is an addition to it where the first floor used to be the garage and a new upstairs that is not finished at all. There is just plywood for the walls and floor, no siding, no heating. We installed insulation but noticed that frost would form between the insulation and plywood. It has been in the negative numbers the past week. Is that going to be a problem? And if so how do we fix it?

The exterior of the plywood sheathing needs to be covered with a vapor barrier. And siding is a definite must.

Sounds like you did not put a vapour barrier on with the insulation.

This is easy to do, but I am a bit confused about what you have done. Can you just confirm a few things, and someone will help with more directions:
a) You insulated the former garage ceiling, and there is frost in the uninsulated & unfinished room above it?
b) Are you ever going to finish the upstairs room
c) do you have drywall on the ceiling of the former garage, or just bare insulation?

a)The former garage ceiling is not insulated just a plywood ceiling, ans insulation on the outside walls, we just insualted the room above it and noticed frost behind the insulation we just put up
b) yes we are trying to turn that into the master bed room
c)there is no drywall

There is Tytex paper on the outside of the house. Also when we put up the insulation we unrolled it, cut it, and stapled it up. We know how to put the insulation on the inside we are just concerned about the frost and where it would be coming from and what damage may occur. The room is not heated.

Here's a link on vapor barrier.
Rona.ca - Installing a vapour barrier It really is important to do it right - with a correctly installed vapour barrier, you won't have moisture problems when the house is finished.

You may have moisture problems in your unheated rooms until you finish and heat them. This is because moisture is coming from the warmer rooms and then condensing/ freezing in the colder rooms. (this is the same reason that causesyour basement to have a moldy smell in the summer). If you'll be finishing these rooms within the next year, then I would not worry about: just make sure everything is dry before you start the renovation.

Yes, you should also put your siding on in the spring/ summer just to protect the house from weather. In our jurisdiction, we also need to have a rain screen under the siding for ventilation. Check with your local building department for your requirements

The biggest problem is if your insul has gotten frost and it melts in the exposed insul, and 5% or more moisture in it. Its now lost about 95% of its properties.
101 on the vapor It must be installed on the warm side towards room. The ceiling insul also must have a vapor in it and IT must be towards the inside room. Here where most mistakes are made in a garage. The vapor barrier in the floor must also be towards the heated side of the floor right against the sub floor ply. Not the bottom of the floor joist of that upper floor nor another vapor barrier in the bottom of that floor joist and most want to do that because its easier and seems to make sense to them because of stapling onto that bottom floor joist like it's a stud wall but forget its actually a big crawl space big enough to be a garage.

Check your codes on this below but this in Pa is my building code for us builders here

The exterior walls should be form out side in:
Siding
Tyvex house wrap
Ply sheeting.
Insulation
Vapor barrier on inside of room/wall studs
Drywall

Starting from the Inside room of floor of sub floor
Ply sub floor
Vapor barrior
Insulatuon in floor joist bay
5/8" fire rated drywall on bottom of joist. (the ceiling for the garage)

Here's a site tha shows some problems insulation above a garage.
YouTube - Holmes on Homes Cold Bonus Room above Garage

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I have a sub floor consisting of 1x4's lsif on diagonol to the floor joist. This is covered with felt paper.
Now I have another floor laid on top of this at 90 deg to the floor joist. Also 1x4's.
Now I have a foam pad for a floating floor and the floor.
The problem is that the 1x4's that are sandwiched between the felt and the foam is either damp or wet. What's happening?