2.1, Due on January 10

Difficult:
I understood the congruence classes, but I got confused about the second part of Corollary 2.5. It says that there are exactly n distinct congruence classes. I am confused about the meaning of distinct. Does this just mean they do not overlap with each other? I also found it difficult to understand that the set of all congruence classes has exactly n elements. I was confused because at first it points out that the congruence class of 2, 5, -1, and 14 are the same modulo 3, but that only the congruence class of 2 is in the set of all congruence classes modulo 3. I just didn't understand why all of the other ones wouldn't be considered to be in the set of congruence classes modulo 3 because they are the same as the congruence class of 2 modulo 3. I guess overall, the most difficult part of this section for me was understanding congruence classes and properties of them.

Reflection:
I really like that the reflexive, symmetric, and transitive properties also apply to congruence modulo n. I know that as a secondary mathematics teacher, I will encounter these three properties quite often when teaching algebra and other subjects. I like that these are properties that I am familiar with and understand so that I can apply the information that I already understand, and will one day teach, to something that I am less familiar with, congruence and congruence classes.  I am excited to be able to teach my students about these properties and how they apply in algebra classes.