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This is the common problem with gaming kidz that they download malware or some type of viruses with games. But The online games specially flash games, run on the browser and no need to download them for playing. Some common and very intersting games sites that i found for my nephew are
Play these games without any hasitation of malware.
Parents have always been concerned about the growth of their Kids. Well letting your kids play outside or inside will help strengthen their confidence and boost their immune system. It is a fact that sitting in front of computers for long time will have negative effects on the body, but 2-3 hours a day does not cause any problem to them. There are many benefits of playing online games, almost all the schools are using computer for teaching, so they can easily grasp the things that are taught to them. You can find educational games, skill enhancement games at online portals like Dr-games.com that will help children learn and enjoy at the same time. If you are concern is about playing games on facebook, then I would say if your kid is too small to follow such social networking site, you can give them an option to play free games at other gaming website portal.
The most unsafe things on the internet as far as malware is concerned are games, porn, and p2p downloads (of music, movies, games, and porn). If you want to allow your kids to play online games, don't allow them to go to just any site -- make sure that you help them connect to a few sites that you have vetted first yourself. You might look into setting up a free account at Hi5.com, which is a large social networking site that's focused on gaming. You're allowed to play games anonymously (providing safety from predators) and since they're a commercial entity based in the US, it's highly unlikely that their games would get infected with malware -- that would be devastating for them if it happened. I'm not a big fan of FaceBook games, but Hi5 has an engine that allows their Facebook games to be played at Hi5.com.