to join in / get involved
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to join in
To actively take part in something, participate, or get involved, often without having been explicitly asked to do so. The expression is usually used colloquially.
Bei der Diskussion über das neue Projekt wollte jeder mitmischen.
In the discussion about the new project, everyone wanted to join in.
'Mitmischen' (to join in / get involved) is a separable verb. The prefix 'mit-' comes at the end in a main clause. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Bei dieser Entscheidung mischt der Chef immer mit.
The boss always gets involved in this decision.
The word is made up of the prefix 'mit-' and the verb 'mischen' (to mix). Here, the prefix 'mit-' means 'together with others' or 'being part of it', which makes the meaning 'to participate' logical.
In der Politik wollen viele neue Leute mitmischen.
In politics, many new people want to get involved.
In the meaning of 'sich beteiligen' (to take part), 'mitmischen' (to join in / get involved) is clearly colloquial. It sounds active and sometimes also a little cheeky or self-confident.
Unser kleines Team mischt jetzt bei den Großen mit.
Our small team is now joining in with the big players.
If you want to say what someone is taking part in, you usually use the preposition 'bei' followed by the dative.
Er will bei der Planung des Festes unbedingt mitmischen.
He absolutely wants to get involved in planning the party.
to participate / sympathize
'teilnehmen' and 'mitmischen' can both mean that someone is part of something and does something. 'teilnehmen' sounds neutral and fits well with courses, events, competitions, or official activities. 'mitmischen' is more colloquial and emphasizes that someone actively has a say or gets involved, sometimes even without a clear invitation. If you only want to say that someone is present at a course or appointment, 'teilnehmen' is more natural. If someone gets heavily involved or actively tries to influence things, 'mitmischen' fits better.
Bei der Planung des Straßenfests will auch der neue Nachbar mitmischen.
The new neighbor also wants to get involved in planning the street festival.
Trotz des schlechten Wetters nehmen viele Kinder am Training teil.
Despite the bad weather, many children are taking part in the training.