Get Perfect-Bound to Lay Flat?

Hi All,

I'm very novice and I'm practicing by rebinding some of my friends' favorite orchestral scores for them to have a nice sturdy edition. For now I've just been stripping the cover and turning them into case-bound books. However, the perfect binding doesn't let them lay flat. I have no experience sewing books but am willing to. Advice on how to re-bind the text blocks so they'll lay flat when open?