1. Explain it simply
First state a general situation, then use 甚至 to add an extreme example that goes even further.
Order: Clause A，甚至 + the extreme point. 甚至 heads the second clause, right before the surprising element.
甚至 pairs naturally with 都 or 也 to stress how far things go: 甚至周末都要工作.
Keep 甚至 inside the second clause - it is not a sentence opener and does not sit in the first clause.
2. Show examples
他很忙，甚至周末都要工作。
Tā hěn máng, shènzhì zhōumò dōu yào gōngzuò.
He is so busy that he even has to work on weekends.
甚至 leads the second clause, paired with 都
这个游戏很有意思，甚至大人也喜欢玩。
Zhège yóuxì hěn yǒu yìsi, shènzhì dàrén yě xǐhuan wán.
This game is so interesting that even adults like to play it.
甚至 introduces the surprising case, paired with 也
他很喜欢音乐，甚至上课也听。
Tā hěn xǐhuan yīnyuè, shènzhì shàngkè yě tīng.
He likes music so much that he even listens during class.
the extreme case follows 甚至
3. Share a tip
Say the ordinary situation first, then let 甚至 push it to the extreme - and lean on 都 or 也 right after it.