Why do molecules like glucose require a carrier protein to get across the cell membrane?
The cell membrane is a made of two layer of phospholipids, and each is made of two parts, a hydrophobic tail and a hydrophilic head. The tails meet making the middle part of the membrane, and the heads go outward making the outer and inner surface of the cell membrane.
The glucose molecule is made of carbon atoms connected to many OH groups and H protons. This makes it a polar molecule which is a hydrophilic one.
Outside the cell when the glucose molecule tends to get inside down concentration gradient, the polarity of it is accepted by the head of the cell membrane so it can pass, but the middle part of the cell membrane repels it. So, there must be channels embedded in the cell membrane with a hydrophilic layer inside that allows the glucose molecule to pass.