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Topic: Easier and quicker methods for algebra - rote tricks? (was Re:

Replies: 3   Last Post: Oct 28, 2012 1:02 PM

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Paul A. Tanner III

Posts: 5,920
Registered: 12/6/04
Re: Easier and quicker methods for algebra - rote tricks? (was Re: Why?)
Posted: Oct 28, 2012 4:37 AM
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On Sat, Oct 27, 2012 at 10:40 PM, kirby urner <kirby.urner@gmail.com> wrote:
> On Thu, Oct 25, 2012 at 10:44 AM, Paul A. Tanner III <upprho@gmail.com>
> wrote:

>> Case in point: One of my favorite evaluations is to ask a person to solve
>> (you know what I mean - isolate the variable) for h in
>> (ab)/(cd) = (ef)/(gh)
>> and
>> (ab)/(cd) = (ef)/(g[h-i])
>> and get to the point of writing the final expression as quickly as
>> possible.

> What happens when a person does not evaluate well on a favorite evaluation I
> wonder.
> Have they lost all chance of being a favorite person? Hansen speaks of a
> "club" (the inner circle of those who "got it").

When I said "evaluation" here I meant as a pre-test to see what they know and are capable of already, to see who needs to learn what. (Some people already know these methods since they already discovered these methods on their own.) I don't see what the problem is supposed to be here.

>> h = (cdef)/(abg) + i.
>> A good manipulative to show this at least for the multiplication/division
>> context and practice visually solving for any randomly chosen of the 8
>> variables - especially with a group of people - is to use those children's
>> blocks with letters of the alphabet written on them and a device that has
>> the four positions of upper left, lower left, upper right, and lower right
>> to put the blocks on.

> I'm lost. You have letters a - i and you're looking at blocks with six
> sides each with colored letters on them... and we're supposed to do what
> again?

I said "the multiplication/division context and practice visually solving for any randomly chosen of the 8 variables" and this ends with h. When the "h + i" example holds we are now no longer in the the multiplication/division context only but also in the addition/subtraction context as well.

Have you not seen the blocks with each block having one letter of the alphabet?

The game is this:

The starting position of the blocks is that we have four places arranged like this (we have four spaces arranged the way a fraction equality looks - upper left, lower left, upper right, lower right, and letters representing the blocks):
ab ef
cd gh
where we pretend there is an equality symbol in the middle, and we see that we have left side with two levels and a right side with two levels.The rule is that we can move blocks diagonally only, and the the object of the game is to move the blocks as needed diagonally to get the desired finishing position of the blocks, which to isolate a given block in the upper level on one of the sides, this isolation meaning that all the blocks would be on the other side regardless of level. And we can do this isolation exercise 8 different ways since there are 8 blocks.

Once we get this game down, we can see that this "cross motion" way of thinking applies to the more general context of also including addition/subtraction.

But a manipulative for only the addition/subtraction context would be less involved, in that it would have only one level. One with four blacks would look like
a b | c d
where the "|" sign would be a divider placed where the equality symbol would be. The isolation game here is trivial - just slide blocks either to the left or right to isolate the chosen one on one side of the divider, where we could put a sticker on a block to represent a negative sign to put on a block that moves to the other side from the starting position.

One could of course expand these games to have a larger number of blocks, but the rules of the game are such that more blocks should not matter - and this is a positive feature of this way of looking at things, this feature being simplicity that is maintained even with many more variables (blocks).

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