As a math teacher, I've found that my students are much more interested in learning new math concepts and skills when they have a sense of when and where it was developed, and why. (In addition to current applications, like careers that employ it.)
There's a great BBC documentary called The Story of One that addresses this fairly well. There's also a book called The Math Book, which has separate entries for different events in the history of math.
Mainly, I'd like to have some rich information on the various cultures that developed different math concepts as solutions to problems in their societies or improvements.
Hello, I'm happy to write an answer to your question, but since there is so much literature on the history of mathematics, and more specifically, there are only few overview books I would really recommend (as opposed to very many, very good monographies on special topics) I would need some additional information:
Do you want your students to read the texts themselves or do you want to read the texts and tell the students about it (I believe it is safe to assume you have much more knowledge about mathematics and history than your students, so you will be able to read much more sophisticated texts).
When you say "cultures", are you interested in different approaches, e.g. China, India, Japan and Europe in comparison? Or do you mean mathematical culture in the middle ages as opposed to the 19th century (as an example)?
If you say you want to motivate your students, do you mean highschool students or university student in math/a discipline that requires math like physics (because lots of different math here)?
How deep do you want to go? Are you the kind of person that loves to get deep and dirty into technical details or do you prefer to have a rather broad overview instead?
Next to English: can you read any other languages (French, Italian and German in particular)?
Sorry for all the additional questions here, but as I said, the really good literature is usually specialized as far as a subject/period/single mathematician is concerned and it feels senseless to list e.g. my favourite books on projective geometry and perspective in drawing, if you are looking for the history of calculus