Welcome to my world...
I’m also a private maths tutor and I’ve had this situation quite a few times, so I can sympathise.
Basically, there are quite a few things you can try. It’s your job as a maths tutor to try to see what works i.e. what motivates and engages them- what makes them “tick”.
Try: visual learning/including shapes when explaining things (e.g. shapes when teaching fractions really helps).
Also asking the student what their interests are and then trying to relate the maths to their interests often works.
But that’s just 2 things. There’s lots of things you can try. Search google and ME SE for more decent advice/articles on what else you can do to get them engaged in lessons.
Then, just keep doing those things- the things that actually work for that individual student, even if you find them unusual. It’s really important they don’t find the lessons boring, even if you find the maths basic and “obvious”.
But most importantly, if it’s been a few lessons and you are trying lots of different things to get them engaged but they still are not caring, then you should not blame yourself- you’re doing all you can and they are not caring. That’s not your fault. This sounds like the advice Robin Williams gives in that scene from “Good Will Hunting”, but it’s true. Leave the lesson, go home and at the end of the day say, “well I did my part”. Sometimes the kid just isn’t in the right place mentally to learn a bunch of maths. Maybe they have ADHD or some other learning difficulty. Maybe they’re just a kid and they are thinking about the crazy stuff kids think about. Maybe they just hate maths and they’re totally blocking learning anything related to maths out. Who knows. You’re not a psychiatrist (although sometimes it can feel like you are lol).
As a maths tutor, part of your job is to be a detective trying to figure out how you can help the student.
Obviously you should always be communicating with the parents. Be straightforward and tell the parents, “he/she was not engaging at all in today’s lesson...”.
But don’t spend 10 hours a week worrying about helping 1 student who isn’t being engaging or responsive or giving any real feedback. That’s their problem.
And by the way, loads of students (probably about half of my students)- even ones who do well in lessons - try to avoid doing homework - basically due to laziness. There’s not much you can do other than remind them (and the parents) it’s for their own benefit- which is certainly true.
I was actually thinking today- how much does a student improve when they do their homework compared to when they do no homework? I estimate around 3-10 times more (faster) improvement when they do their homework compared to when they do no homework.
So in conclusion- try your best, but don’t beat yourself up about it if what you try doesn’t work- you’re not going to be able to help everyone all the time. And it’s probably
(usually) not your fault. And you just have to accept this fact and move on.