The most common method used to add amendments to the Constitution is the first method. An Amendment may be proposed by two- thirds vote in each House of Congress and ratified by three- fourths of State Legislators.
Article 5 gives the process of amending the US Constitution. Both the Senate and the House of Representatives, in the US Congress, approve by a 2/3 majority voting. This is a joint resolution that amends the Constitution. By this the President's signature is not required.