Improvements in background checks would be the easiest to pass in Congress if the proposed changes don’t get gummed up with more controversial elements like the much maligned assault weapons ban. Background checks have been identified by gun control advocates as the best way to curb gun violence. Forty percent of gun sales take place at private gun shows where no background checks are required, and 19 states don’t send names to the national database that identifies people who are legally prohibited from owning guns–drug abusers, convicted felons, people with restraining orders on them, etc.

Even so, lawmakers are also focusing on assault weapons, ammunition, video games, and mental health issues. Democrats have yet to tip their hand on how they will proceed: Either to insist on a big package that could be shot down by opponents or offer individual slices of the package, which may doom some ideas but bolster others. Democratic lawmakers and aides have said they want to make sure their response to the Newton, Conn., massacre and other gun crimes is wide-reaching, encompassing mental health and violent imagery in the media as well as gun restrictions. But they also want to be pragmatic.

Rep. John Yarmuth, D-Ky., sits on a House Democratic task force on guns. He told National Journal that he would prefer to deal with background checks first, since it has the broadest support, and tackle other issues after that. “There’s evidence that in certain states where they tightend their background checks, they dramatically reduced the gun crimes. That would be the fastest way to make an impact,” he said. But he also acknowledged that he would support whatever package Democratic leaders adopt, and he doesn’t have the final word on these decisions.