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Correcting colleagues

Discussion in 'Mathematics' started by DeborahCarol, Nov 24, 2011.

  1. DeborahCarol

    DeborahCarol New commenter

    To save others reading it, this is hilarious. The writer advises others on the pitfalls of correcting the grammar of others. There are at least ten spelling, punctuation and grammar mistakes in the article, so I suppose it's a warning to others that 'people in glasshouses...'! Too right. I just hope people realise it's a spoof. (It IS, isn't it? She says, nervously...)
     
  2. DeborahCarol

    DeborahCarol New commenter

    Oh blimey. I can't be the only one wondering about this, so I'll put up my hand....
    You'll be relieved to hear I only teach KS2/KS3, but why isn't the square root of 36 'plus or minus 6'? I would have said it was!
    I think I do vaguely remember learning that the square root symbol only refers to a positive square root, but the symbol isn't being used here.
    Have just double-checked in a GCSE textbook (as the lesson was given to Year 10), and it says 'Because -4 x -4 = 16, there are always two square roots of every positive number.
    And anything to the power of 1/2 equals its square root, doesn't it?
    Could one of you explain to me (and other lurkers, I'm sure!) why the teacher was incorrect, so that I can put right my misconception?
    With many thanks.

     
  3. DM

    DM New commenter

    It doesn't matter that an index is used here rather than a square root symbol.
    If a square root could be both positive and negative we wouldn't need the plus or minus sign in the quadratic formula would we?
     
  4. DeborahCarol

    DeborahCarol New commenter

    So is the statement in the textbook incorrect? Doesn't 'two roots' mean a positive and a negative? Sorry if being dense, but can't be the only one. Need to be sure, as if ever covering-for-GCSE-class-despite-not-being-qualified, don't want to impart incorrect information and don't want to look a twit!
     
  5. DeborahCarol

    DeborahCarol New commenter

    Also, the quadratic formula uses the square root symbol, which would equate to what I think I was told once - that the symbol refers to a positive square root only.
    But I still don't understand why the square root of 36 isn't +/- 6. As the GCSE teacher observed didn't either, need one of you maths-degree people to spell it out in very simple terms for (some of) us. We'll be the richer for it!
     
  6. Both +6 and -6 are square roots of 36. The expression sqrt(36) or, equivalently, 36^(1/2), represents the positive square root of 36.

     
  7. DeborahCarol

    DeborahCarol New commenter

    OK - well, that makes sense to me, algebraist - thanks. So fractional powers refer to positive square roots only. It's just that Brookes implied with his/her post that the square root of 36 wasn't +/-6 and, as I would teach children that it is, had me worried!
     
  8. Does it?
    http://www.ucl.ac.uk/Mathematics/geomath/powsnb/pow5.html

    "
    "So x1/n means the nth root of x, i.e. the number that, when multiplied by itself n times, gives x."
    "This shows that if we square our expression "x to the power a half" then we get simply x. Therefore "x to the power a half" must be the square root of x.

    The square root of a quantity is the number which when you square it gives the original quantity.
    Similarly the cube root of a quantity is the number which you have to cube to get the original quantity.

    For example the square roots of 16 are 4 and -4, since 4 squared is 16 and (-4) squared is 16.

    We have found out above that we can write the square root function as the power of a half, so we can rewrite the last sentence as:
    161/2 = 4 or -4.
    Similarly,
    251/2=5 or -5,
    since 5 squared is 25."
    That's what I thought but correct me if I'm wrong.
     
  9. As someone said on page 1 (I think), the square root (whether in index form or "ticky" symbol) is, by definition the positive root. When it comes to equations, x^2=16 there are two solutions. Again, as someone said, why else would we have the "plus or minus" symbol.
     
  10. DM

    DM New commenter

    If you graph y = x^(1/2) you only get the positive branch of the parabola y^2 = x. You cannot simply square both sides and assume it will make no difference.

     
  11. I'm confused so you'll have to indulge me.
    I was always under the impression that the square root of a number was a number that when multiplied by itself gave the original number. So that would include positive and negative numbers.
    I would have thought the reason the quadratic equation includes +/- is because people are likely to forget you have a negative solution as well.
    Nothing I have read indicates otherwise and I would have thought the UCL link I gave was a reliable link.
    The graph of y = sqrt x would have 2 possible solutions for Y for each X wouldn't it?
    I have also read that a function f (x) can only be a function if f(x) only has 1 solution. Therefore f(x) = sqr root (x) is not a function.
    Am I correct?
     
  12. [​IMG]
    Now that's from the BBC!!
     
  13. This is just to reiterate what others have said before.
    It is a matter of convention, the use of terminology and
    correct use of these..
    My calculator only gives the answer 6 when I key in
    36^(0.5) or squarerootsign(36). My computer tells me that
    the solution to x^2 = 36 is plus/minus 6.
    If I ask for THE square root of 36, I'd not worry too much if I got
    the answer plus/minus 6, but would have to warn the class to
    listen more carefully to my question. If I asked for the square
    rootS of 36, I'd be disappointed if I just got 6.
    I don't think it is a big deal, but I know some examiners are
    terribly pedantic, so better get it right.

     
  14. I've just had a look at the UCL link. It is a load of garbage!
    It is not even consistent. For example it claims that 81^(0.25) = 3,
    which is correct, but contradicts the way it talked about square
    roots earlier. What about -3?
    Don't believe everything you read in print or on the internet or by the
    BBC.
     
  15. DM

    DM New commenter

    No. It is there because the number that follows the +/- sign is positive.
    No. It has one. This is the graph.
    [​IMG]
    f(x) = sqrt x is a function because it IS one-to-one.
     
  16. DeborahCarol

    DeborahCarol New commenter

    So, er...it seems that some people think the teacher was wrong because she said 36 to the power of 1/2 was +/- 6, which has been much discussed. But others think she was fine saying that, but she should have also written that 64 to the 6th root could be +/-. So perhaps what we can say is that she was being inconsistent?!
     
  17. No, she was wrong.

    When it comes to definitions we need an authority. Here's one I trust: mathworld
    I'd rather trust Stephen Wofram than atics or Stephap.
     
  18. DeborahCarol

    DeborahCarol New commenter

    There does seem to be a lot of disagreement on this thread.
    Wanting so much to be clear on this, all I've managed to establish so far is that the use of the 'square root symbol' means positive square root only, which I knew already, ie that it is fine to say 'the square roots of 36 are 6 and -6', but 'square root symbol' 36 is 6 only.
    But there isn't consensus here on whether x^1/2 has a positive solution only, or a positive and a negative.
    Have just found this on the UCL website:
    'We have found out above that we can write the square root function as the power of a half, so we can rewrite the last sentence as:
    161/2 = 4 or -4.
    Similarly,
    251/2=5 or -5'.

    On the other hand, BBC GCSE Bitesize defines x^1/2 as 'square root symbol' x, which would of course give a positive solution only.

    Null, many thanks for the link, but I'll confess as a KS2/3 teacher I find some of it a bit hard to follow!
    When you say the teacher in your opening post was 'wrong', could you clarify what error you felt the teacher was making? Was the error that she'd said 36^1/2 was +/-6, and that she should have written a positive root only? Or, as one poster suggested, was the error that she hadn't written that 64^1/2 was also +/- and should have?
    I'm assuming you mean the first, but, if so, what about the UCL definition? Shouldn't they know what they're talking about?
    Many thanks for starting the thread by the way. Confusing, but comforting as well, that experienced GCSE maths teachers can't seem to agree on this one.
     
  19. mmmmmaths

    mmmmmaths New commenter

    Taught this recently to GCSE foundation resit students. Two square roots but the symbol means the positive square root. They seemed quite happy with it. They actually asked what the symbol was for the negative root! I am sure the specification makes it all clear.

    Retaught it in Sept to AS many of whom had been taught incorrectly at GCSE. using the symbol and power half to mean the positive root. They seemed ok with the idea that it is a convention.

    I do like the math world link, hadn't seen that before; makes it all very clear.
     
  20. My position would (as usual) be pretty close to polecat's.
    If the context is the reals, then sqrt(x) >=0 makes sense. (And more generally, over the positive reals it makes sense to define x^a = exp(a log x)).
    Once you work in the complex field, things get more, um, complex. It's quite common to have a situation where "1 = sqrt(1) = -1" (consider what happens with z = e^it as t goes from 0 to 2pi).
     

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