How to ask for expanding scholarship duration from grad school? I am a second year masters student who studies mathematics and this is my third semester. I found my advisor at the middle of second semester. My college offers scholarship for grad students which includes tuition, housing and some monthly stipend. I am supposed to finish my masters in 2 years, after that they cut out the scholarship benefits. Now, I would like to ask for one more semester before my graduation. The reason is, I want to write a qualified thesis and during that time, I need a house to stay, and some stipend to make a living. 

I think that these reasons are enough, however, I heard that the college science does not support students after 2 years in general. So, what can I do about it? What else can I say? Thank you.
My department offers full fellowships (tuition + stipend, healthcare is publicly funded in the country) to most academic Master's students. They are all limited to two years. The department does not have the ability to extend the scholarship, nor does the university.

That being said, it happens that some students take longer than 2 years to defend (it's an academic Master's, so it's generally expected that the students will have published/submitted at least one or two papers before they graduate). In those cases, the remaining months are fully funded by the advisor, usually in the form a research assistantship from a grant that is directly related to the student's Master's project.

However, if you were at my institution, your request for an extension would sound naive and borderline preposterous and arrogant. An extension is something that occurs organically as you show that you are doing good work and need the extra time to finish and polish your thesis. This is not something that is requested as soon as you start doing research. If your advisor/university is going to invest more money on you, you must demonstrate that you are worth the money and have a good motive not to have finished your thesis in time. Writing a "qualified" thesis is, I'm sorry, a terrible excuse.

The least awkward way to approach anyone about this is to ask about the possibility of (and not demand!) an extension once you are more acquainted with your advisor. A good way to approach them about this issue would be by leading a conversation like this: 


  As you know, I have been studying the effect of wheel geometry in the
  performance of hamster-powered supercomputers. I demonstrated that
  heart-shaped meshes are very efficient, but as you know, that's the
  expected result, given that hamsters tend to work harder in the
  presence of love-displaying figures. I think that it would be very
  promising to investigate whether plastic or steel wheels are more
  efficient. If we demonstrate that plastic wheels are just as good as
  steel, we could see a significant cost reduction in hamster-powered
  supercomputers, and would certainly yield a very impactful paper. But
  as you know, that would require 3D printing the wheels, which would be
  very time-consuming, and would likely require more time than the four
  months I have left. I will start working on the plastic wheels as soon
  as possible, but if more time is necessary to finish the experiments,
  do you think it would be possible to grant me a fully funded extension
  of a few more months so that I can finish the last experiments and,
  hopefully, write the paper?


All of that is, of course, assuming that the possibility of getting a funded extension even exists.