The shooting went down before 8 a.m., as the 58-year-old coach supervised a weight-lifting session with about two dozen football players and other students.
Court records show Becker had been arrested before, including for assault and possession of drug paraphernalia. After an arrest in January, he admitted being a methamphetamine user, court records say.
Investigators suggested that law officers lost track of Becker earlier in the week because of a lack of communication with hospital officials after he was arrested for allegedly vandalizing a home and eluding police Saturday night.
Thomas died shortly after arriving at Covenant Medical Center in Waterloo, a hospital spokeswoman said.
No students were wounded. The school is in Parkersburg, about 80 miles northeast of Des Moines.
Thomas had been the coach for nearly three decades at Aplington-Parkersburg (Iowa) High School. He has nearly 300 career victories, and four of his former players have played in the NFL.
Aplington-Parkersburg High School enrollment is just 280 students but is the alma mater of four NFL players Casey Wiegmann of the Kansas City Chiefs, Jared DeVries of the Detroit Lions, Brad Meester of the Jacksonville Jaguars and Aaron Kampman of the Green Bay Packers.
DeVries, a defensive end with the Lions, walked off the practice field in Allen Park, Mich. toward the end of its morning practice, apparently shaken.
Team officials said DeVries was not immediately available for comment.
Thomas made national headlines last year when he insisted that the high school's football field, named in his honor, be rebuilt as a way to help restore community pride in Parkersburg after it was hit by a powerful tornado in May 2008 that killed six people and destroyed the high school.
"A lot of people know coach Thomas for his success as a football coach, but a lot of people here locally know him as a person, as a dad and grandfather, and that's where our thoughts are right now, with coach Thomas," said Superintendent Thompson.
According to Coaching Management Magazine, "Thomas has twice been named Iowa Class 1A Coach of the Year and was a finalist for the 2003 NFL High School Coach of the Year award. In 2004, he was honored as Coach of the Year for Leadership by the Iowa High School Athletic Directors Association. He has been on the Board of Directors of the Iowa Football Coaches Association (IFCA) for 26 years and served as its president in 1991."
In a 2005 interview, Thomas described his philosophy of modern coaching: "First, you'd better make sure your own house is clean. But I think when you see things not being done ethically, you need to report that. On my football team, if we have kids doing what's not right, that's a reflection on our whole program. And if we have coaches doing what's not right, that reflects on the coaching profession. I also think athletic directors don't always take the time to go over ethical guidelines with their coaches especially young coaches coming out of college who might not fully understand the difference between the collegiate and high school levels."
The Associated Press contributed to this report