I would start every lesson with an explanation as to which objectives you are trying to meet and why. I would also make any practice problems more relevant to their lives b changing the names in the problems to the names of your students and by creating realistic problems that they might encounter. As for the first student who asked "why it was relevant", they are obviously acting out to avoid doing the work, probably because it is too hard for them. I would take the student aside after instruction and explain it to them privately, as a tutor would, until they grasp whatever the concept you are trying to teach them is.

Telling students real world scenarios is a great idea, but I thnk that often times allowing students the opportunity to work out real world problems themselves is even more effective. Since this case has a student that is refusing to do work, I think that it would be necessary to include these in the guided practice portion so they are forced to participate and pay attention. It may even be beneficial to bring in realia and hands-on materials (i.e. if you are talking about how fractions will be needed when students are cooking, bring in cooking supples) to "hook" the students into the idea that you are trying to portray. If these ideas are incorporated into the lesson, then more students will become engaged in the topic and less students will refuse to do the work.

I don't have any solutions for the student at hand. But I do have some advice for you. I know as a student struggling in math my whole life I hated math and never felt like the teachers understood me. After watching this video my mind was blown, this is a must see for math teachers. http://www.ted.com/talks/dan_meyer_math_curriculum_makeover

It seems like this behavior is not due to students not having a purpose. The students are using this "lack of purpose" as an excuse because they either do not want to do the work or because they don't understand it. Try reteaching the lesson or grouping students to work together so they have a better understanding of the content.

This is a super difficult question for teachers to answer, and aside from giving them some real-world examples, there are some other things you can tell them. For starters, you can tell them "This won't make sense to you right now, but you'll understand it later in life." Ask them to trust in you that what they're learning is important. You can also follow up real-world scenarios with the fact that these problems help them develop critical thinking and problem-solving skills, which can be applied in all sorts of places outside of school or a job. Go the extra mile by giving example of how these extraneous skills are applied outside of the academics or the workplace. You can also point out how certain skills develop inside math classes. Point out from time to time (in passing) that your math class and the work they do helps them with other skills. Time management, problem-solving skills, critical thinking, etc. Finally, while you can try to integrate the real world into math problems, you can instead talk about how various jobs use math when you wouldn't expect them to. Waiters and cashiers need to be able to count money quickly, firefighters need to be able to calculate PSI to effectively spray water with a hose, and they use "fire dynamics" math to calculate the stability of a building. Even pottery requires you to know algebra. Showing your students that math up to the pre-calculus level is used in all sorts of jobs will motivate them more to learn.

I would tell the class that if you do not learn this stuff now and remember it, you will literally have to pay for it later on in life if you go to college. I will explain that if you go to college, they will have remedial courses called Pre-Algebra, Elementary Algebra, and Intermediate Algebra and that textbooks and all cost about $600.00 So seriously, unless you wanna quit school, you need to learn this stuff or else you're going to have to pay for it. Learn it now for free so you can pass many tests on it now and in the future, or learn it later when it'll literally cost you.