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Q&A #18523


Cross multiplying

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From: Ian (for Teacher2Teacher Service)
Date: May 28, 2007 at 11:00:29
Subject: Re: Cross multiplying

Hi Margaret,

>We have been working on early algebra.  I am  trying to teach my son
>about doing the opposite operation in equations to isolate the variable.
>
>He solves by inspection frequently and can often just see things in one
>and even two step equations.

That's not necessarily a bad thing, since he'll actually want to be able to
do that when he gets past linear equations.  For example, given something like

  0 = 6x^2 + 17x + 5

he won't be able to 'isolate the variable' in the same way as with a linear
equation.  In fact, he'll need to factor it into binomials,

  0 = 6x^2 + 17x + 5

    = (3x + 1)(2x + 5)

and then he'll want to be able to look at each binomial and see, by
inspection, when it will be equal to zero:

  0 = 6x^2 + 17x + 5

    = (3x + 1)(2x + 5)
          |       |______ Equals zero when x = -5/2
          |
          |______________ Equals zero then x = -1/3

It's just my opinion, but it seems to me that undue emphasis on "isolating
the variable" in linear equations leads a lot of students into confusion
later on, when that method no longer works, and attempting to use it seems to
lead you around in circles.

>He has been doing what I used to call cross
>multiplication to solve equations involving fractions or division. He saw
>the pattern very quickly himself. I never taught him this, he just worked
>out that was how to get the answer.  His view is that he doesn't really
>need to understand why it works because he has found a method.

The quickest way to disabuse him of that notion would be to give him some
equations where his method doesn't work.  :^D

But before you do that, take a moment to think about what it is he's supposed
to be learning in his math classes.  Is it how to solve a linear equation by
following certain steps?  Not really.  What he's supposed to be learning is
how to think his way through a problem... and it sounds like he's already
doing that.

>I get a bit nervous because I wonder whether or not his chosen method
>(not just in this but in other maths subjects- eg: he multiplies normal
>numbers algebraically too) will ever disadvantage him in higher maths.

If anything, it sounds like he's got an advantage over other students who
wait to be told exactly what to do.

>It feels to me like he is using a different system for each operation. I
>try to be as flexible as I can be because I know in maths if you get
>something one way and people try to teach you an alternative, it becomes
>very frustrating.
>
>I know most kids at schools just learn methods and don't really know or
>care why they are doing things that way. So what am I really asking?  I'm
>not really sure.  Maybe just seeking some reassurance from the creative
>teachers here that it's okay to go with the flow and also that I won't be
>disadvantaging him if he only uses cross multiplication, rather than
>actually understanding what you are doing is an inverse operation.

Using inverses is part of what's going on, but as I said, that doesn't work
so well when you get to other kinds of problems.  The deeper issue is that
you want to change any problem in front of you into an easier problem.
Often, this _does_ entail using different methods on problems that seem, on
the surface, to be very similar, because you're taking advantage of what the
problem offers you.

And it sounds like he has a good intuitive grasp of that.  When he starts
running into problems where his invented methods don't work, I believe he'll
be open to learning about new additions to his box of tools.  But until that
happens, the tools you invent for yourself are almost always better than the
ones someone else just gives you.

To summarize:  I think you can relax.  :^D

 -Ian, for the T2T service

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