How do I make this Mexican girl in my class understand that Iberian Spanish is the root of Mexican Spanish?

There is a Mexican girl in my class and we got into an argument about variants of the Spanish language. She says that Mexican Spanish isn't Spanish and that if a Mexican goes to Spain that the Spaniard and the Mexican won't understand each other. I understand that pronunciation is a little bit different, and I also know that Mexican Spanish has indigenous pre-Spanish Mexican words added to it, but I know it's not a completely different language.

Why do Americans think that they can speak Spanish by saying Mexican foods?

Spaniard: Hola. ¿Qué hora es?
American: Burrito taco salsa con queso nacho

Mexicans don’t speak two languages only as most people think?

If they reside in the United States then they most likely speak three languages. English, Spanish and Nahuatl. Most Mexicans don't know that when they speak Spanish they are actually using a lot of vocabulary from the so called Aztec language. As many of you may know. Mexicans are basically a combination of the l Aztec Warrior and the Imperialist Spaniard from Europe. They use a lot of words from the Nahuatl language and dont even realize it. Mexicans speak a hybrid of the Nahuatl language and Spanish. Now three with English haha.