Vrftsjtryk

My wife takes a new job, she won't be back home every day, and I find myself
in need of taking care of my kids on Wednesdays (day with no school). I work
on an important project at my company, and I thought that instead of reducing
my work hours (going to 80% for example), I would rather do some remote work
on Wednesdays, not to impede the development.

Indeed, my kids are not so small they just need an adult to be present (just in
case !), so I can do my software programming job from home one day a week.

I talked with my supervisor, he's okay but needs a formal note from me ;
how would you explain the situation properly ? (since the note will go
to upper level management and human resources).

Thanks for the help !

Since you mention in a comment that you would be the first to work remotely in your organization this is going to be extra tough I think. It's not impossible by any means but I think you need to be prepared to respond to likely objections:

Won't the kids be a distraction?

Claiming that they simply never will be is likely to come across as incredibly naive. Most kids who aren't old enough to be left unsupervised aren't up to going all day, every week without requiring some of your attention. Little Johnny won't share with Little Billy or Little Billy hits Little Johnny etc etc.

If you are somehow able to convince your employer that they will never distract you when working from home then I'd expect the to wonder why you need to be there at all.

The best approach IMO is to acknowledge the realities of the situation and provide a resolution. Say that while your kids are old enough to understand that you are there to work not spend time with them you know that there will inevitably be the occasional disruption and that you're prepared to work extra time when that happens to make up for the lost time.

Why can't your wife do it?

Sadly this will be an all-too common objection, either through old fashioned gender-role attitudes or a sense of a "prior claim" in that they had you working Wednesdays first. Explain that the old system/schedule was no longer sustainable for your family and that while it wasn't the hoped for outcome your wife was unable to secure a working pattern that allowed her to remain at home on the Wednesdays.

How will we know you're actually doing any work?

For companies that have never done WFH before the fear that employees will just be treating it as a psuedo-day off is very common. So have some solutions ready to offer them for how they can track your progress and "presence". Whether that's source control commits, or IM or e-mail reports etc, give them some options but remind them that you're happy to work with whatever suits them - the whole point is to make them feel comfortable.

What if we need you for a really important meeting on a Wednesday that can't be rescheduled?

If you can, try and have some back up plans in place - even if they are things that you can only call on occasionally (neighbor, childminder, family, wife uses a day off, etc).

Overall I think the better prepared you can be to demonstrate how you are committed to ensuring this creates the absolute minimal level of disruption to them the more likely you are to get the "yes".

how would you explain the situation properly ?

Just explain it as you have in your question.

Emphasize that you would be working from home and not spending your time with your children. Emphasize that your presence is needed, but not your attention. Emphasize that you won't miss any projects or meetings because of this change.

If your company usually has remote workers, it probably won't be a big deal.

If you would be the first remote worker, then put more effort into describing the benefits to the company resulting from allowing remote work. Talk about how you'll deal with your remoteness - what special processes you will use, etc. Ask your supervisor if you can include the fact that it is okay with him.