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Simplifying algebraic fractions add/sub

Discussion in 'Mathematics' started by DeborahCarol, Apr 30, 2012.

  1. DeborahCarol

    DeborahCarol New commenter

    The way I learned to do these was to find the lowest common denominator and then multiply the numerators appropriately.
    But I see textbooks nowadays (and some revision books) teach (eg for subtraction): a/b - c/d = ad-bc 'all over' bd.
    Problem here is that bd isn't necessarily going to give the lowest common denominator.
    Students then get confused when they see something like this:
    2x/x-1 minus 7x-3/x^2-1
    Applying 'bd', as they've been taught, they multiply the two denominators...and get a cubic expression. When that doesn't work, they factorise the x^2-1, but are still unsure how to proceed.
    But these problems wouldn't occur if they'd been taught to use 'lowest common denominator, and 'whatever you do to the bottom you have to do to the top' ' rather than the 'bd' and cross-multiplying method.
    Hope I've explained myself clearly, and if I'm being thick myself, please forgive, as I only officially teach lower secondary, but have been helping a friend's son with GCSE.
     
  2. PaulDG

    PaulDG Occasional commenter

    Ah, but you were also taught how to do fractions properly and no doubt did page upon page upon page of them at junior school.

    Today, we visit fractions for about 2 - 3 hours a year, the kids do perhaps a dozen examples in between "starters" and "plenaries" (necessary for "deep learning", don't you know) and then, around year 10, we throw algebra into the mix (bearing in mind we've kept algebra out of everything but "algebra lessons" in the meantime.

    So, at this point, they're both "lacking in confidence" about both fractions and algebra...

    And you want them to find the lowest common denominator?

    Ah, but they got at least a couple of marks in the question for doing that first step..

    Well, you know the issues then.. Outside of the top set (where there is time to actually teach finding the LCD as a shortcut because the kids generally won't be trying to throw stuff round the room while you're explaining it), do you do enough fraction work for them to apply LCD to numeric fractions?
     
  3. Anonymous

    Anonymous New commenter

    Do they know why they need to get a common denominator (regardless of it being the lowest one)?
    Once they appreciate you can't simply add 2/3 plus 1/5 together to get 3/8 [​IMG], then the idea of equivalent fractions and common denominators follows.
    I have the advantage of private tutoring so I can spend a lot of time doing this. Its fun with algebraic fractions.

     
  4. DeborahCarol

    DeborahCarol New commenter

    Paul - have to say I certainly do do loads of fractions work. I teach One to One to Yrs 7/8/9, and fractions is my 'default' topic! And I've also always insisted they find the LCM, not just any CM. I'd insisted on that for more efficient working, but, as per post, would make algebraic fractions easier.
     
  5. DeborahCarol

    DeborahCarol New commenter

    So...do people agree that it would be better to teach add/sub using lowest common denominator, ie the traditional way or the 'modern' way using 'the subtraction rule' (which, as per my post, has I think the disadvantage of being difficult to apply mechanically where the fractions have a denominator in common). And, as most GCSE students are going to apply that rule mechanically, surely they're going to get in hot water with certain problems, so...wouldn't it be better to teach them lowest common denominator from the start?
     
  6. PaulDG

    PaulDG Occasional commenter

    I think the right way is to get them familiar with both and ideally able to switch between the two.

    One thing I like to try to stress with classes I teach is the importance of "hmm.. that didn't work/look right, perhaps there's another way".

    That's particularly useful with algebra where looking for the common denominator might be the best way forward or leaving the denominator factorised might be a better way in other problems (often is as the factors often cancel after the addition/subtraction in GCSE questions).
     
  7. just recently tutored a student who had exactly this problem!

    they had something even more obvious eg 1/(x+1)(x+3) - x/(x+3)

    when i asked them to just try 1/5-1/10 they had no idea but to do 10-5/50.

    now if they were low ability i would not be so concerned but they are getting As in C1 C2 and B in C3. (yes,one of them unhappy with their score so retaking to get higher). They should get a grade A at A-level.
    Are fractions really that badly understood by students ?

    They also could not tell me what 2x(1/3) was as the 2 had no denominator to multiply the 3 with and as for tidying up things such as 1/(2/3) when we integrate.......

    And Paul,yes i did spend most of my primary school life doing fractions everyday so by the time i was doing O and A levels these skills were mastered.









     
  8. PaulDG

    PaulDG Occasional commenter

    Yes.
    But we shouldn't really be surprised by this - fractions have vanished from common use; with a decimal currency and decimal weights and measures, no one uses fractions outside the maths classroom.
    Even more scary in my experience is the number of kids who can't do decimals either!
    (I put this down to "move the number not the point" being taught in primary - so-called "understanding" being taught at a time when many do not have the mental development to "understand" and would be far, far, better off to have had a working algorithm ("move the point") drilled in instead.)
    Now most of their time in primary is spent doing "higher level" topics. Apparently this is progress.
     
  9. Students automatically cancel the x^2 in (x^2 + 6x)/(x^2 + 7x + 3)

    And never cancel the R in R sin x / R cos x
    It become R tan x


    But the one which gets me comes up quite often when students cancel a pi from pi over 2 pi to leave 1 over pi

    Because 2 pi minus pi equals 1 pi....
     

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