For the segment where Nana and notKomari are having their battle of wits, they reference a few "advanced" topics from the Japanese school curriculum.

The first is a poem from this collection https://en.wikipedia.org/wiki/Shin_Kokin_Wakash%C5%AB, which has an official translation that can be read here:
https://books.google.co.jp/books?id=szAABAAAQBAJ&pg=PA217&lpg=PA217&dq=omokage+no/+kasumeru+tsuki+zo&source=bl&ots=d2KZQfFfTn&sig=JOpv81_SS-dd1ZlvR-XiTM6KEuI&hl=en&sa=X&ved=2ahUKEwj306ft2-LcAhXMFIgKHT_VAacQ6AEwAXoECAkQAQ#v=onepage&q=omokage%20no%2F%20kasumeru%20tsuki%20zo&f=false

For a more detailed explanation in moon of what the poem means (as translating poetry from Japanese to English can only preserve so much meaning), this is a nice and succinct interpretation that seems to be along the lines of most explanations of it online:
https://detail.chiebukuro.yahoo.co.jp/qa/question_detail/q1250831767

Next up were really old Japanese conjugations of verbs meaning "to be" that you won't ever really see in modern literature. There isn't really any way to explain it in English, but if you're really interested in learning old verb conjugations that you'll probably never read in Japanese, have a look here:
https://ja.wikipedia.org/wiki/%E3%83%A9%E8%A1%8C%E5%A4%89%E6%A0%BC%E6%B4%BB%E7%94%A8

Next is a mnemonic device for remembering the start of the Kamukura shogunate. The old way used to use いい国 (ii kuni - 1192CE) for the start date, but the new date they teach is 1185CE, so it's now いい箱 (ii hako). The end of the era is still 1334CE, and as a result, the last part of the mnemonic device is still いざ見よ (iza miyo).

Finally, they start listing off the elements in order, so we used the periodic table song that everyone who got through at least high school chemistry should have heard once or twice.