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I have recently started some summer tuition for a student who has failed in mathematics at UK key stage 3 (age 14), and trying to plumb the depths of his knowledge. Having engaged him in some simple exercises in simultaneous equations in 2 variables, with a view to presenting these equations as lines in a Cartesian plane and showing him where they cross, I have identified a particular problem which concerns me.

He can (just about) understand the concept of $(x, y)$ being points on a plane, and how an equation $a x + b y = k$ can be converted into an equation of the form $y = m x + c$, but when I came to giving him some simple examples I pulled off the top of my head, I found that the inevitable fractions that came his way as a result were completely beyond his ken.

I found myself having to explain how to add fractions, and how to multiply fractions, and just about how to divide fractions, most of which he vaguely remembers from many years ago; however, the concept of the reciprocal, and the technique of turning the fraction upside down, was completely alien to him, and it frightened him.

The question I have: is it expected that he ought to be fluent with manipulating fractions? He keeps converting to decimal everywhere he goes, particularly in applications where he has to work out, for example, how far he can go on 18 litres if 15 litres takes him n miles -- which is all very well till first thing he does is he divides by 15 and chops off the recurring 1/3 at the first digit.

I am putting a set of exercises in his way to allow him to manipulate algebraic expressions. I believe it may be necessary to insist he does some algebraic exercises every day. Does anyone know of any good works which I may use to support this student? Most of my library is either way too advanced or positively antiquarian.

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    $\begingroup$ I don't know what KS3 is, and probably your question is not really a good fit here (both the question and most any direct answer to it are probably too dependent on specific local requirements and conventions to be a good fit here), but (+1) (maybe make what follows explicit in your question) a natural question is to what extent should you separate the learning of factions and equation solving: (A) Allow equations to be solved by decimals and work on fractions separately, with the goal of combining them later, or (B) try now to integrate fraction background work directly into equation solving. $\endgroup$ Aug 7 at 11:30
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    $\begingroup$ @DaveLRenfro Sorry, KS3 = "key stage" 3, which is how it is organised in the UK: certain skills are required at certain levels. My student is 14. How this translates into an equivalent level in other countries may differ, which is why I specified the KS rather than the age group. British respondents, particularly teachers at exactly that level, may be able to help. $\endgroup$ Aug 7 at 12:12
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    $\begingroup$ @DaveLRenfro So the question remains: what is the forum into which this is a better fit? I would have thought that "Mathematics Educators" would have been perfect. I just need to know what I can reasonably "expect" of a student, so I know what to get him to revise, rather than forcing stuff into his head that his assigned level in school has not prepared him for. I am many years behind the curve of what is being taught in school nowadays. $\endgroup$ Aug 7 at 12:14
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    $\begingroup$ @PrimeMover: When someone asks for clarification of a question on SE, the thing to do is edit the question to clarify it, not provide the clarification in comments. $\endgroup$
    – user507
    Aug 7 at 12:46
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    $\begingroup$ @DaveLRenfro We do have Common core as a tag and several questions about it. That, too, is a national thing, so it seems to be of roughly the same scope. $\endgroup$
    – Tommi
    Aug 7 at 18:03
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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:

  1. The student having "forgot" how to add, subtract, multiply and divide by fractions
  2. OR The student adding fractions incorrectly, like 1/2 + 1/3 = 2/5 because you add the numerator and denominator
  3. OR The student is good at fractions in general, but struggles with mixed fractions
  4. 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.

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    $\begingroup$ Indebted to you for the recommendation of the catch-up books. Good old rainforest are delivering these for me tomorrow. $\endgroup$ Aug 7 at 18:49
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    $\begingroup$ Mixed fractions are a scourge that should never have been invented or taught. It is rarely used outside elementary math, does not have a decent symbolic representation and is all together non useful. Just use a bloody plus sign. $\endgroup$
    – DRF
    Aug 8 at 9:20
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    $\begingroup$ @AdamRubinson They need to know that $5+\frac{1}{3}=\frac{16}{3}$, that's not mixed fractions, that's addition. Mixed fractions are $5\frac{1}{3}=\frac{16}{3}$ which is ambiguous confusing and stupid IMO. $\endgroup$
    – DRF
    Aug 8 at 18:31
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    $\begingroup$ Mixed fractions are useful in the real world. A recipe will say $ 2 \frac 13 $ cups, not $\frac 73$ cups; but if all I have is a 1/3 cup measure, I'll need to be able to convert that. $\endgroup$ Aug 9 at 3:34
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    $\begingroup$ “Who uses times? Adjacency is multiplication.” Ok, so you think that $23$ is $6$? I’ve always thought $23$ was twenty-three. Adjacency as multiplication versus concatenation depends on context, and usually it is clear from the context. No one I’ve known or taught uses $5\frac{1}{3}$ to mean $\frac{5}{3}.$ You are the first! Also, I wouldn’t write $\frac{15\cdot 5}{7},\ $ as to me, the dot could easily get misinterpreted as a decimal point. $\endgroup$ Aug 9 at 11:07
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We DON'T need to know the intricacies of UK standards to say yes students are expected to know fractions before algebra. However, your puzzlement over it (like a new concept to you that a kid who fails algebra might be weak on arithmetic!) is concerning. And similarly, you are not the first, by any means, "good at math, but weak at teaching it" person to be flummoxed by such a common occurrence. But, there have been many discussions on this forum where people note the difficulty in teaching weaker kids at lower level colleges, with fractions being a notable hurdle. So it is something that experienced TEACHERS as opposed to mathematicians are not surprised by.

I'm not sure that starting with two equations, two unknowns is the right spot either. I don't know for sure, but would Bayesian bet that this is one of the later topic in the kid's year of math. They are one of the harder parts of algebra 1. A little more looking at the kid's overall ability and deficits is also in order. Also, you ought to get some texts and skim them to understand what topics are taught when. Not just plunge ahead. Even a very short perusal would bring your knowledge (of the customary teaching of the topic, not of math itself!) up hugely.

In terms of practical work, I wouldn't give up. And wouldn't just go all the way back to arithmetic either. But mix in a little bit of that stuff along the way (but separated from the algebra or at least with easier topics, like single equation in x). Oh...and try to assign problems with simpler arithmetic. It is just some extra work by you to design such problems that don't have the annoying fractions. (Or if they do, start with 1/2--look up the training concept of "progression".) The problem comes when you assign what (for the kid) are harder topics (systems of linear equations) along with hardest arithmetic. For some reason us meat computers slip a gear when you combine a couple things that are medium hard. Unlike silicon where either it knows the subroutine or doesn't. And you can just pile as many on as you want.

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Yes, he will have learnt how to do arithmetic with fractions, equivalent fractions, fractional/decimal/percentage equivalence, reciprocals, and how to solve problems involving fractions in Key Stage 3 - it is part of the National Curriculum, which can be found here.

Whether he ought to be fluent in manipulating fractions really depends on whether he will be taking Higher Tier or Foundation Tier GCSE. A Foundation Tier student will likely spend some time consolidating previous work on fractions, whereas higher tier students should be fluent in working with fractions by the end of Key Stage 3. It is not expected that students should know anything about algebraic fractions yet - that is usually covered in Key Stage 4.

For resources, Maths Genie and Corbett Maths both have plenty of practice questions in the style of UK exams. You might also like to look at GCSE papers for more difficult questions - almost everything that is in the Key Stage 3 curriculum can also be tested at GCSE.

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