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CAn I say,

(a) We entertained the children by sinigng lovely songs. Our teacher performed the piano.

(b) They liked the songs. Some orphans joined in the singing / song (with us).

(c) They clapped hands to the song. They thanked us for visiting them.
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It is correct to say "she performed on the piano" or "she played the piano".

"She performed on the piano"gives an impression that she is"showing off" when playing the piano.
Comments  
Vincent TeoCAn I say,

(a) We entertained the children by sinigng lovely songs. Our teacher performed the piano.[Y]

(b) They liked the songs. Some orphans joined in the singing / song (with us). I would prefer singing with us.

(c) They clapped hands to the song. They thanked us for visiting them. [Y]

cheers[H]
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(a) Is there correct to say, "performed the piano"? not "played the piano"? \

(b) Do we say, " ... joined in the song with us."?
Vincent Teo(a) Is there correct to say, "performed the piano"? not "played the piano"? \

(b) Do we say, " ... joined in the song with us."?

Both are correct, but play is much more common.

Why not simply say:

the children joined in (with us)

as the song has already been mentioned?
 Feebs11's reply was promoted to an answer.
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CAn I say,

(a) Some of the children joined in our singing .

(b) They joined in our singing group.