arcgis desktop - Splitting large number of polygons at same time into smaller polygons?

I would like to split the large polygons in the picture below into smaller polygons using the border. I have about 4000 lots that I would like to divide and find the area of each of the smaller sections of the polygons but every process I have used so far will not allow me to work on more then one polygon at a time.

I am a student that does not have much experience with Python but can try it out.