1

I know that some kanji are educational and others are general use, with kyouiku (1006 characters) learned in grades 1-6 included in jouyou (2136 characters). What is the other set of kanji (1136 characters) learned in grades 7-12 called?

2

As far as I know there is no official name for the characters outside the prescribed kyouiku kanji list but inside the jouyou kanji list.

In addition, there are only guidelines for teaching the rest of the characters, rather than strict amounts per year. These guidelines are called 中学校学習指導要領 (Junior High School educational curriculum guidelines), as below:

Actually the 2136 jouyou kanji are all taught by the end of junior high school:

  • JHS 1st year (Year 7) - approx 1400 kanji learned (including kyouiku)
  • JHS 2nd year (Year 8) - approx 1800 kanji learned (including kyouiku)

  • JHS 3rd year (Year 9) - 2136 kanji learned (all jouyou kanji)

Depending on the textbook used (which varies), the order of the kanji learned will also be different. Jinmeiyou kanji are generally nor taught officially in JHS or SHS, but they are common enough that students are exposed to them regularly and would be able to at least read many of them by the end of year 12.  

In 2020, the kyouiku kanji list will be updated to the 学年別漢字配当表 , which will include an additional 20 characters, bringing the total of characters learned in elementary school to 1026.
See here for more details (Japanese)

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