It sounds like you're really thinking about what you have learned. That's great. It's true that some arthropods are able to expand. As you probably know, arthropods get their names from having movable joints. These are places where the hard parts of the exoskeleton are joined together by soft tissue.

Think about wearing a suit of armor. If the whole suit were metal, you couldn't move your arms and legs. At the point where your elbow, knee or other joint is, you have to allow for movement.

Arthropods have soft tissue there. It's hard to see this on insects and spiders, but see if your teacher can help you find a lobster or crab exoskeleton. You can also see this sort of design in the hard leg protectors that baseball catchers wear. There are hard plates attached to a softer cloth pad.

When a honey pot ant expands, the soft parts get a lot bigger, but the hard parts stay the same size. If you have Internet access, check out the photographs of honey pot ants at this site:
ant

You will see that there are one or more plates stuck onto what the "blown up" abdomen of the ants. That is the hard exoskeleton plate. It didn't get bigger at all. Just the soft parts expanded. It is now useless at protecting the ant, but the full ants just stay inside the nest where they are fairly safe.

By the way, their scientific name is Camponotus inflatus. Why do you think they were named that? If exoskeletons don't stretch, how do arthropods grow?

If arthropods had none of the thin cuticle, then they would not be able to move. Yes, it is thin and flexible at least for joints, and in those animals, it is thin in other places as well.

The exoskeletons of arthropods are made of a polymer sugar called chitin. It's the same stuff that makes up the cuticles of annelids (segmented worms), as well as the cell walls of fungi. Particularly thick layers of chitin are stiffer than thinner layers, and arthropods have made their chitin cuticles thick enough that it functions as a skeleton. Marine arthropods further fortify their cuticles with the mineral calcite (the principal component of cement), to make their cuticles even harder.

Along the arthropod body, there are locations where the stiff, thick cuticle becomes in-pocketed and very thin, and thus much more flexible- these are the joints. The arthropods with flexible exoskeletons that you just named have these thin,flexible, portions of their cuticles able to puff out,connecting their stiffer portions with a thinner,balloon-like portion.