Is showing acts of worship with the intention that children emulate them, or to bring joy to one's mother, or to encourage friends and relatives to do good, considered a form of showing off (riyaa')?

The fundamental principle in Islamic law is that concealing good deeds is preferable to disclosing them, as it keeps one further from showing off (riya') and closer to sincerity (ikhlas). However, disclosing a good deed is praiseworthy if there is a legitimate Sharia interest, and this should be assessed according to the need.

What is mentioned in the question does not fall under dispraised showing off (riya'), but it is recommended to dispense with disclosure if possible, and to conceal the deed for fear of the infiltration of showing off and the love of people's praise.

As for what is necessarily a means to achieve a legitimate Sharia interest, it should be disclosed, while striving to purify the heart and perfect sincerity.