First of all, let me echo all the comments -- the key point here is that these surfaces are homeomorphic, but this homeomorphism cannot be realized by an isotopy. This is an important distinction! It is probably worth taking some time to explain the issue. In particular, students should understand that topological spaces don't naturally come embedded in $\mathbb{R}^n$ and that homeomorphism is a property of the abstract spaces.
I think I would draw this picture on the black board and ask the students to complete the bottom diagram.

I haven't done this particular example in class, but I do use this sort of pictures for cylinders, Mobius strips, tori, Klein bottles and projective planes all the time, so it would tie into that familiarity.
I can imagine a fun activity here. Pass out the three types of strips to the students and have them triangulate them by drawing with pen. Have them carefully record which vertices are joined by edges and which three cycles are filled with triangles. Then pass that list of combinatorial data to another team and see if that team can determine which simplicial complex is which strip. If this works for you, let me know how it goes!