See Christmas 2013 on Mathematics learning and Web 2.0 if you are looking for Christmas activities generally. Here I will concentrate on those activities suitable as December starters.
Perhaps use an advent calendar such Alex Pett’s very beautiful version complete with history and problems for each day (and turn the sound on!). It is an ActivInspire resource but Alex has also provided a pdf version or use as a Google document.
Also worth checking for starters are the various Nrich advent calendars. Look at problem 1 on the secondary calendar for example; Special Numbers would make an ideal starter. The calendars link to posters of Nrich problems but you can easily search on Nrich for the problem if you want to look at the teachers’ notes and solution. Take problem 1 for example – Special Numbers. Go to the Nrich website and enter special numbers in the search box, the first entry is the required problem.
If you want number properties for each day in December then you could of course use WolframAlpha, eg 19 or Tanya Khovanova’s wonderful Number Gossip site. A good starter can be to ask students to come up with a question which has an answer of 19 and then go on to look at properties of that number.
From TES Resources these Christmas grid puzzles or Operation Christmas Tree would make rather nice starters. Back to the subject of calendars. this very well presented PowerPoint advent calendar could easily be adapted for your students. You could use it to plan your December starters!