What I did was write up a proposal and presented it to them. I found out the pro's for them (and yes, time off for kids out of school is one) as well as a few cons. I suggested that we gradually adjusted to working from home starting with 1 day a week and moving up to full time. I outlined recommended goals and suggested that if I was unable to meet the goals, I understood that the work from home would change back to work from the office.
In the proposal I included my home equipment and suggested I bring it in so the IT guys could check it out and prep. I agreed to shoulder the cost of all of my equipment including repairs. This is important since your computer will be used for other than work.
Also, make sure you state how you will handle meetings. How much notice do you need to get into the office? What will you do with your children?
Working with the kids up is NOT easy. It is very difficult to meet your quota's/goals with them wanting your attention. I am not sure what you do, but think ahead of time how you will respond if you are on a business call and your child wants your attention NOW and starts yelling or grabbing at your clothes. I had to get up early to get work done, and stay up late when I was on a project that required reading and focus. Will you be able to do that? Make sure you have a dedicated office, not a dining table that you spread out on in the morning and clean up at night. Remember, it is still a job.
By having a written proposal with timelines and guidelines (that you need to be flexible with), they will understand that this is not a momentary thing and that you have thought of how the company can and will be impacted. You have done what you can to min. the "pain" felt by the company. They may say no, but the may say yes.
Sorry this is so long. If you have any q's feel free to email me.