Here's an easy way to keep your child's LEGO under control, at least somewhat.
Grab a fitted sheet and lay it on the floor as shown, with the elastic upwards, so that any stray pieces are contained. The sheet is big enough so that it will accommodate a lot of LEGO and give your child space to play within.
When you want to pack it away, it's a simple mater of gathering up the sheet by the elastic, and moving it aside. Pro tip: ensure your child hasn't left any important creations within the maelstrom of loose LEGO before doing this!