mitspielen
to join in
To take part together with others in a game, a sports activity, or a musical or theatrical performance.
Darf mein kleiner Bruder bei eurem Kartenspiel mitspielen?
Can my little brother join in your card game?

to join in / play along
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to join in
To take part together with others in a game, a sports activity, or a musical or theatrical performance.
Darf mein kleiner Bruder bei eurem Kartenspiel mitspielen?
Can my little brother join in your card game?

'Mitspielen' (to join in/play along) is a separable verb. The prefix 'mit-' separates from the verb 'spielen' (to play) in main clauses and goes to the end. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Beim Fußball spielen alle Kinder zusammen mit.
All the children join in together when playing soccer.
To name the activity or event in which someone participates, the preposition 'bei' with the dative is used very often. This applies to both the literal and the figurative meaning.
Sie spielt bei einer Theateraufführung mit.
She is taking part in a theater performance.
In the figurative meaning 'ein Faktor sein' (to be a factor), the subject of the sentence is usually not a person, but an abstract thing. Typical subjects include 'das Wetter' (the weather), 'das Geld' (the money), or 'die Erfahrung' (the experience).
Für den Erfolg des Projekts spielte auch das Glück ein wenig mit.
Luck also played a small role in the success of the project.
In the literal meaning, 'mitspielen' is often used in contexts involving leisure activities. Common areas are sports, music, theater, or board games.
Er spielt seit zwei Jahren in einer Rockband mit.
He has been playing in a rock band for two years.
to play
'spielen' and 'mitspielen' are close in meaning because both can be used with games, sports, music, or theater. 'spielen' generally says that someone is doing an activity: a game, soccer, an instrument, or a role. 'mitspielen' emphasizes that someone is taking part together with others. So you can play guitar alone, but join in a band. In the phrase 'eine Rolle spielen' (to play a role), 'spielen' also means 'to be important'. 'mitspielen' can similarly mean that a factor is involved in something. Learners can easily confuse the words because 'mitspielen' looks like a small variation of 'spielen'. The important part is 'mit': it is often about participation or involvement.

Paul darf heute beim Training der älteren Mannschaft mitspielen.
Paul is allowed to join in with the older team's training today.
Nach der Schule spielen die Kinder im Hof Fußball.
After school, the children play soccer in the courtyard.