When I tried to heat a high-carbon steel fish hook so that I could bend it to a different shape, it broke. How do I heat and bend this steel without it breaking? Should I cool it slow or fast or cool and reheat it again to keep its strength?

When you heat and slow cool steel, you soften it (the process is called annealing). When you heat and rapidly cool (quench) steel, you harden it (the process is called hardening). If you want to bend a steel part that is either hard or brittle, you need to anneal it first so that it is soft (has a lot of ductility) then after it is bent go back and harden it. Hardening keeps the steel from weakening in the final product.

If you heated the fish hooks up, bent them and then found that some broke when you cooled them down, it was more than likely due to the fact that you put too much (internal) stress in the material (from the bending process). That's why you should bend only annealed steel.

To anneal, you want to slow cool the fish hook. If you are heating it in a furnace, just turn the furnace off. If you are heating it in air, put the fish hook into sand or into an insulated container so that it won't cool down too quickly. Then try bending them. After they are bent, you can reheat and fast cool (quench in oil or water) to harden it.

Follow-up Question
I heated the hooks with a torch then bent the eye closed tighter (so it wouldn’t catch in the weeds). This made them brittle, and I noticed that they all broke in the same place - at the base of the hook. I then heated the other hooks up to a red glow then cooled (in air), and after they had been bent, heated them back up and quenched them as you suggested. They did not break. I took some pliers to the eye and tried to break it with fairly hard pressure, and they still didn’t break. I am wondering if I should put them in the oven on low for a while or just go with heating and bending then heating again and cooling? I am wondering just how strong they will become. Do they get harder in the oven?

Follow-up Answer
You want to put them into an oven at or around 300ºF in order "to take the curse off." That is, temper them to improve their ductility while only very slightly reducing strength.