Ooo. I see lots of things that need fixin iluv!
A 5g is WAY too small!! Talk to your mom; you're going to need a 40g tank to keep those cute lil guys happy and healthy.
You also MUST have a filter. Its obvious you have a local Wallymart where the folks there don't know the first thing about fish.
Your mom deserves a hug for making you change that water; if you didn't the fish would have been dead from ammonia poisoning already. Fish produce ammonia as waste (they pee it, breathe it out, the food rots and turns into it) and ammonia will kill them.
So to explain and clarify:
Goldfish need 10g EACH. They are really dirty fish (they give off a LOT of ammonia waste) and they also grow BIG. The bigger the tank the easier it is to keep the water clean enough for them to live in.
A filter is necessary; it sucks up the bigger bits of waste out of the tank. It also gives good bacteria, bacteria which eats ammonia, a place to live.
You really need test kits to test for ammonia, nitrite and nitrates.
My guess is even tho you are changing the water frequently, your guy is suffering from ammonia poisoning. Take a look at this article; it explains about ammonia in fishtanks and all: http://faq.thekrib.com/begin-cycling.html