Different passages in the Bible illuminate that God is the author, originator, sustainer, and causal purpose behind things as "trivial" as feeding the sparrows, causing the grass to grow, and adorning the flowers with their beauty. When we pray and intercede for ourselves and others, we are recognizing that the strength to accomplish all of the activities in a day doesn't originate from us. We are not the authors of change or cause and effect in the world. We are acted upon by God, provided for by God, strengthened and fulfilled by God. God supplies all of the strength, energy, intellectual capacity to do everything throughout the course of our daily lives. Prayers attribute these things to Him. By not praying, and being seemingly "independent" sends a message to God and the rest of the world that we have (and originate) what we need (strength, gifts, intelligence, time) to do all of the normal and extraordinary happenings during our day.
1st Peter 4:11 says: Whoever speaks, is to do so as one who is speaking the utterances of God; whoever serves is to do so as one who is serving by the strength which God supplies; so that in all things God may be glorified through Jesus Christ to whom belongs the glory and dominion forever and ever, amen.
It is quite obvious that whoever supplies the strength or capacity to "do" things gets the credit for their being accomplished. Which is why prayer is so important. It is a declaration that God is the one who is giving the strength to accomplish daily tasks, so He rightly deserves the credit. Prayer is humbling for all of us, its a way of recognizing the truth about abilities, talents, and resources. God supplied them all, so when we do things all day (or week) without requesting strength to "do" them, we are rejecting the Truth that God is the supplier of the ability to go to work, take care of the kids, pay the bills, and so on and so forth.
We need to make prayer a habit, a daily hourly habit of asking God for help and assistance. He provides it so often that we begin to feel entitled to it and spurn the gift that His assistance is. How much easier, smoother, and truer are all of our activities that are bathed in prayer for the strength to see them done. I think you will find yourself with much more freedom, both in humility of actions and attitudes, and the ease with which all activities come when we don't have to be the one who provides all of the energy, finances, and strength to accomplish them.