I tutor maths as a full-time job, and many of the students I have tutored over the past few years are not fluent in their times tables up to $10$ or arithmetic with fractions, or both. And it's not like I select students with these low abilities on purpose.
And when I say they are not fluent, I mean that they are nowhere near as good as me at them (arithmetic with fractions). I mean, not so humble brag: I'm quite quick with arithmetic. But this isn't an excuse for students to be really bad or really slow. Some of my students are fine, but a lot of them are too slow or uncertain of their calculations when I begin tutoring them. With regards to fraction arithmetic in particular, this can be anything from:
- The student having "forgot" how to add, subtract, multiply and divide by fractions
- OR The student adding fractions incorrectly, like 1/2 + 1/3 = 2/5 because you add the numerator and denominator
- OR The student is good at fractions in general, but struggles with mixed fractions
- OR The student is too slow because they find calculating stressful or they are simply uncertain that their calculation methods are correct- for example, it takes them about 5 minutes to calculate 3/4 - 2/3, when it should take at most 30 seconds in my opinion.
OR something else...
It is my belief that students must become fluent at their $10$ by $10$ times tables first, then get good at fraction arithmetic as well as fraction problems, then you can move on to ratio/proportion and then algebra and so forth. Of course, you should not just be teaching in terms of numbers; use pictorial diagrams like pie charts and rectangles to help visual learners learn fractions. And number patterns e.g. $10 $ times something means you just add a $0$ on the end of the number...
But yeah, if you try to teach someone algebra when they are poor at fractions, then well... that will only get them so far in my experience. Perhaps other educators have experienced otherwise- but I don't see how this is possible.
This is not to say that arithmetic is the only thing you should be focusing on. If your student wants to go through what they are doing at school or are interested in another topic, then by all means spend a small amount of time going through that topic with them. But until they are just good enough at arithmetic to progress onto harder topics, you don't have much choice but to focus on arithmetic. Also, it should be mentioned that this process of getting them up to scratch at arithmetic shouldn't take a huge amount of time if the student puts the effort to improve in. If they don't put in the effort, then try to motivate them. If the student is trying hard to learn but simply cannot, and you have tried everything to help them learn the arithmetic, try to determine if they have learning difficulties.
To this end, here are some good "catch-up books" focusing on fractions, decimals and percentages for KS3. There are $5$ books in the series and the arithmetic gets progressively harder. I have only tried this with one student who particularly struggled with fractions. Previously he stagnated on fraction arithmetic for over a year. I would say by working his way through these books, he improved massively on fractions. I must say though, that it was through his own desire to improve and realisation that he must pass GCSE maths that motivated him to actually put some work in and improve. But it certainly is handy that those books exist, and they're relatively cheap, so if it's arithmetic you want your student to work on then buy the books and get them to work through them. One book per week or two weeks should be a fair pace for most students who are weak at arithmetic. During this process of them completing this homework, you can identify their weaknesses and help them improve.
Now it sounds like you're also looking for algebra books. But both CGP and Collins have lots of KS3/ GCSE level books. So try fish around on amazon to find the most relevant one to suit your student's needs.
This book is my go-to book for helping KS3/GCSE students who are good enough at the fundamentals to progress, as it has a lot of exercises for them to practice.
Someone else suggested mathsgenie.co.uk, and I second this website.