Does the slope of an indifference curve change?

Yes, usually, because both goods are desirable, so you would be equally happy (for example) with 5 puppies and 2 cats as you would be 2 puppies and 5 cats. If you connect these points, you will create a negatively sloped line.

There are instances where the slope is positive because one of the options involved is undesirable. For example, if I have puppies and garbage. Now, I want more puppies, but less garbage. So, I may be equally happy with 2 puppies and 2 piles of garbage as I would be with 5 puppies and 5 piles of garbage. This is because the benefit of having more puppies is offset by the negative of having more garbage. This creates a positively sloped line. There are also instances where there is 0 slope or infinite slope if the quantity of a good is irrelevant to your enjoyment of it, like air. You would be happier with more puppies, but you would not be happier with more air. You would also not be less happy with more air. The amount of air is irrelevant to your happiness. This would be a zero slope line.