Describe the jacket that the Fours, Fives, and Sixes wore, and the reason it was designed the way it was. Also describe the jacket the Sevens wore, and what it symbolized

Fours, Fives, and Sixes wore jackets with buttons on the back, so they could learn interdependence. The Sevens wore jackets with buttons on the front, so they can learn to become independent. It symbolized them growing up.

From the text:

The little girl nodded and looked down at herself, at the jacket with its row of large buttons that designated her as a Seven. Fours, Fives, and Sixes all wore jackets that fastened down the back so that they would have to help each other dress and would learn interdependence.

The front-buttoned jacket was the first sign of independence, the first very visible symbol of growing up.