By what two methods can amendments to the US Constitution be proposed and ratified?

There are actually four methods.

1. An Amendment may be proposed by two-thirds vote in each house of congress, and ratified by three-fourths of the state legislatures.
2. An Amendment may be proposed by Congress and then ratified by conventions, called for that purpose in three-fourths of the state3.
3. An Amendment may be proposed by a national convention, called by Congress at the request of two-thirds of state legislatures, and then ratified by three-fourths of the state legislatures.
4. An amendment may be proposed by a national convention and ratified by conventions in three-fourths of the states.

Answered directly from the "American Government" Textbook.