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The task is to plan a one hour lesson to teach 'one-on-one' someone (a 16 year old girl), who is at an F grade in maths, how to solve a simple equation like

$$ 0.24 = \frac{1}{x} $$

I have done this successfully many times (I have a PhD in maths), though there must be an agreed method which you guys think is the most clear and penetrating for a young girl who understands fractions, variables, multiplication and very elementary techniques. Any advice?

If by the end, we could tackle problems like solving

$$ 4x-1 = 2x + 4(x+4) - 5 $$

that would be a successful lesson I think. I'm looking for the optimal technique here, any discussion of why you think this works would be advantageous....

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    $\begingroup$ There is no single "optimal technique". The outcome depends on both ability and how information is received - which is not wholly captured by ability (which you emphasise with reference to the F grade). Are you able to furnish your question with an example of how you have approached it, or are thinking about approaching it? I think this would improve the focus of the question, reduce the possibility of answers telling you things you have already tried - and result in more useful and plentiful answers. $\endgroup$ – Rusan Kax Jan 7 '15 at 23:09
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    $\begingroup$ A similar question has been broached before on this site here. The answers there should provide a start for some of the issues that may arise. $\endgroup$ – Andrew Sanfratello Jan 7 '15 at 23:42
  • $\begingroup$ The issue is we don't know the source of the F. When dealing one on one with students, I try to get to the bottom of what they don't understand, and then try to fill in the gaps to full problem solving expertise. Your second example - would she know the first step can be distribute and combine like terms? There are a few skills needed for quick isolation of the variable. Tough to know where she's lost. $\endgroup$ – JoeTaxpayer Jan 8 '15 at 2:10
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    $\begingroup$ In my experience "who understands fractions" is a MUCH higher level skill than the essentially mechanical task that you're trying to accomplish. $\endgroup$ – Dave L Renfro Jan 8 '15 at 16:37
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It takes a few seconds to download Dragon Box 12+, then your job is just about finished. It will take her a few hours to go through, but it will teach her well. It is also a game, so it neatly sidesteps any maths phobia that your students has acquired.

I have used this program with students over the last few months. I have found that most of the Algebra you are looking at is taught in a way that requires very little tutor input, though I have actively reinforced good equation solving techniques in the last few levels. On the last level I have sometimes got them to repeat a problem to obtain the standard form of an answer that is not enforced by the program.

It does not cover polynomial factorisation or exponents, but it covers some fairly hairy problems up to that point. It also cleverly allows students to take common shortcuts in equation solving, but only after the "correct" way has been taught for a few levels (ie. subtracting from both sides vs. swapping the sign as you switch the side). It does not cover real numbers, but that should not be hard after she has learned how to do it with rationals.

I think I am a good tutor, but occasionally a new technology emerges that teaches something far better than I could, so I am happy to mostly sit back and let it do its job. Studies show that students are often happier to experiment and attempt more difficult problems with a computer tutor than a human. However, students often still require us humans to put things in context and make sure that their mental models are correct, so we are not out of a job yet!

Another similar program that is not gamified and that covers simple exponents is Algebra Touch, but Dragon Box 12+ (not to be confused with Dragon Box 5+) is currently the gold standard in algebra tutoring.

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