1. Huiguanʼs explanation divides the teaching into gradual 漸 and sudden 頓, further dividing gradual into the following five:
   1. the distinct teaching of the three vehicles (Āgama Sutras, etc.) 三乘別教;
   2. the shared teaching of the three vehicles (Prajñāpāramitā Sutras, etc.) 三乘通教;
   3. the restraining and praising teaching (Vimalakīrti-sūtra 維摩經, etc.) 抑揚教;
   4. the teaching of the shared intent (Avataṃsaka-sūtra 華嚴經, etc.) 同歸教 and
   5. the eternal teaching (Nirvāṇa-sūtra 涅槃經, etc.) 常住教. This is the taxonomy of the 'Nirvāṇa school.' 〔三論玄義 T 1852.45.5b3〕
2. In the Huayan teaching taxonomy of Liuqiu 劉虬 the Buddhâvataṃsaka-sūtra 華嚴經 is called the 'sudden teaching' and the 'gradual teaching' is divided into the following five:
   1. the teaching of men and gods 人天教;
   2. the teaching of the existence of form (the Āgamas 阿含經 and so forth, that accept the existence of discriminated forms) 有相教;
   3. the formless teaching (prajñāpāramitā 般若經 teaching, etc. that denies the real existence of form) 無相教;
   4. the teaching of shared intent (Lotus Sutra, etc.) 同歸教 and
   5. the eternal teaching (Nirvana Sutra 涅槃經 etc.) 常住教
3. Zhiyi altered the above #2 somewhat. According to Tiantai 天台 teaching, the five periods are:
   1. The Huayan period 華嚴時. After the Buddhaʼs enlightenment, he expounded the Avataṃsaka-sūtra for 21 days for bodhisattvas. This was a teaching for those of very sharp faculties, and according to this teaching such bodhisattvas may directly awaken to the reality-principle.
   2. The Deer Park period 鹿苑時. After delivering the Avataṃsaka-sūtra, ordinary people of undeveloped faculties are not able to understand, so he discards this teaching in favor of guiding people through expedient teachings. Thus he delivered the lesser vehicle teachings at Deer Park near Benares. This period was twelve years, and the scriptures that developed from this are the Āgamas, so it is also called the 'Āgama period'  阿含時.
   3. The Vaipulya period 方等時 (Elementary Mahāyāna). For the people that had grasped the Hīnayāna teachings, the Buddha now teaches Mahāyāna Sutras such as the Vimalakīrti-sūtra 維摩經, the Sutra of the Golden Light 金光明經, and the Śrīmālā Sūtra 勝鬘經, broadly explaining the Tripiṭaka, Shared, Distinct and Perfect Teachings. This is intended to convert people from Hīnayāna to Mahāyāna. This period lasts for eight years.
   4. Perfection of Wisdom 般若時. For 22 years the Buddha teaches the Prajñāpāramitā sutras in order to awaken people to the principle of emptiness.
   5. The Lotus and Nirvāṇa period 法華涅槃時. The Buddha makes clear that the final goal of Hīnayāna and Mahāyāna is the same. Thus it is also called the Teaching of the One Vehicle, wherein the presence of Buddha-nature in each person is affirmed.

References:

Mun, Chanju. 2005. “Wonhyo (617–686): A Critic of Sectarian Doctrinal Classifications.” Hsi Lai Journal of Humanistic Buddhism 6 : 290–306.

----. 2005. The History of Doctrinal Classification in Chinese Buddhism: A Study of the Panjiao System. Lanham, MD:  University Press of America.