WASHINGTON, March 16 — More than half of American families with teenagers use filters to limit access to potentially harmful online content – a 65% increase from the number of those who used filters in 2000. But big majorities of both teens and parents believe that teens do things on the internet that their parents would not approve of. A new survey of 1,100 youth – those ages 12 to 17 – and 1,100 of their parents shows that 54% of internet-connected families now use some sort of internet filter or monitoring software, up from 41% of internet-connected families who used filters in 2000, the most recent time the Pew Internet & American Life Project surveyed on the issue. In all, about 19 million youth live in homes with internet connections and the number of children living in homes with filters has grown from 7 million in 2000 to 12 million today. The filters tend to be used by parents who themselves are frequent users of the internet and who have middle-school-age children. Parents who have older children and who are less tech-savvy are less likely to use filters. At the same time, parents are showing higher levels of vigilance about protecting their children online, there is striking consensus among parents and their teens that the teenage population is not as careful as it should be online and that teens do things online their parents don’t know about.