Stoopid Algebra Tricks

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This is a list of algebra mistakes made by our students. Hopefully a thorough study of these will allow us to figure out how to prevent more of the same. Perhaps our students will visit from time to time, to learn some tricks of the trades (one of the dangers of focusing on what one shouldn't do!).


Contents

Erors in Canceling

  • \frac{h^2+5h}{h}=h^2+5 where the student canceled the h's.
  • \frac{-2+6h+h^2}{h} = \frac{-2+(6+h)h}{h}=\frac{-2+(6+h)}{1}
  • \frac{n^2}{n^2+2n+1}=\frac{1}{2n+1}


Disturbutive Property

  •   \frac{1/t-1/2}{t-2}=\frac{(1/t-1/2)(t-2)}{(t-2)(t-2)}= \frac{1}{t^2-4t+4}
  • \frac{(2+h)^2+(2+h)-6}{h}=\frac{(4+4h+h^2)-12-6h}{h}
  • \frac{h^2+5h}{h}=\frac{h(h+4h)}{h}=h+4
  • \frac{6}{x+2}\cdot \frac{x+2}{x+2}=\frac{6x+2}{x^2+4x+4}


Divide (and be conquered)

  • \frac{1/t-1/2}{t-2}=(1/t-1/2)(1/t-1/2)
  •  \frac{1/t-1/2}{t-2}=\frac{2-t}{2t} \cdot \frac{t-2}{1}
  • \frac{1/t -1/2}{t-2} =\frac{(1/t -1/2)(t-2)}{(t-2)(t-2)}=\frac{t-2}{t^2-4t+4}
  • This one could fit under multiple headings. Can you find all the misteaks?
    \frac{\frac{2-t}{2t}}{t-2}=\frac{2-t}{2t} \frac{t-2}{1} =\frac{-t^2-4}{t-2}=\frac{(-t+2)(t-2)}{t-2}=t+2


Selective multiplication, aka multifrication

  • x=1+\frac{1}{t} \implies xt=1+1


Skwhere and skwhere route tricks

  • \sqrt{a^2+b^2} \ne a+b (in general!): for example, I had a student claim that

    \sqrt{(-2\sin(T)+2\sin(2T))^2+(2\cos(T)-2\cos(2T))^2}=(-2\sin(T)+2\sin(2T))+(2\cos(T)-2\cos(2T))

    or
    t^2=x+9 \implies t=\sqrt{x}+3

  • Similarly, (a+b)^2 \ne a^2+b^2 (in general!): for example, I had a student write

    \left(2(-\sin(T)+\sin(2T))\right)^2=4\left(\sin^2(T)+\sin^2(2T)\right)

  • How about this one? With x=9,
     \frac{1}{\sqrt{x}+9} = \frac{1}{\sqrt{81+9}}

    then
    \left(\frac{\sqrt{x}-9}{x-81}\right)^2 =\frac{\sqrt{x}+81}{x^2+6561}
    (at least they were consistent...)

Inequalities

  • How about this one?: 1 < x < 1 (maybe for narrow values of x around 1....)
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