Gower suggests story problems. Many, many story problems.[1]
A much better exercise is to give an absurdly open ended exercise. "I'm at the supermarket, which checkout should I go to?" is one I have used in classes before. You can get a discussion going and generate a lot of interesting ideas, and almost every time I do it in a class someone says something I've not thought of. Once students have given you some good ideas you can massage it into a model and do some more 'proper maths' work. Of course, this takes a good teacher that can engage and steer the class.
It reminded me a lot of Randal Munroe's "What If" blog on the XKCD site [1]. Easy to understand, open ended questions that encourage readers to learn a little about topics _outside_ of math to answer the question. The class gave me an appreciation for math that was lost during all those years of study before that, and it's basically my career now.
I also found it very valuable; it was one of the factors that pushed me over the line into studying STEM at university (I was a better English and economics student in school).