Diagram for Factoring
- Thread starter FMMurphy
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At a guess (and please let me stress that this is a guess), you're supposed to draw a rectangle.
Out of one corner, mark off a square. Extend the sides of the square to meet the far sides of the rectangle. Label the sides of the square as "x", so the area of the square is "x<sup>2</sup>".
Your rectangle should now contain four sub-rectangles, one of which is actually a square and is labelled as "x<sup>2</sup>". Now look at the diagonally opposite rectangle. You are required to find some labelling of its dimensions such that its area will be "12", and that results in the areas of the two remaining rectangles summing to "7x".
If you label one side of this rectangle as "2" and the other dimension as "6", you'll get an area of "12", but (looking at the dimensions of the two remaining sub-rectangles) the other areas will be "2x" and "6x", summing to "8x", which is not what is required. However, if you use "3" and "4", you will (as you have already found algebraically) end up with "3x" and "4x", which sum to "7x".
Note: This "geometric" method is supposed to teach you "deep mathematics", but, in actuality, this method works only in very specialized cases, rather than in full generality. And that's assuming that I guessed the instructor's meaning correctly.
Good luck.
Eliz.
Out of one corner, mark off a square. Extend the sides of the square to meet the far sides of the rectangle. Label the sides of the square as "x", so the area of the square is "x<sup>2</sup>".
Your rectangle should now contain four sub-rectangles, one of which is actually a square and is labelled as "x<sup>2</sup>". Now look at the diagonally opposite rectangle. You are required to find some labelling of its dimensions such that its area will be "12", and that results in the areas of the two remaining rectangles summing to "7x".
If you label one side of this rectangle as "2" and the other dimension as "6", you'll get an area of "12", but (looking at the dimensions of the two remaining sub-rectangles) the other areas will be "2x" and "6x", summing to "8x", which is not what is required. However, if you use "3" and "4", you will (as you have already found algebraically) end up with "3x" and "4x", which sum to "7x".
Note: This "geometric" method is supposed to teach you "deep mathematics", but, in actuality, this method works only in very specialized cases, rather than in full generality. And that's assuming that I guessed the instructor's meaning correctly.
Good luck.
Eliz.
- Joined
- Feb 4, 2004
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Please don't think I thought this up myself. I've seen it discussed in "education" circles. I would never use this "method" myself. But it's one of the ways that educationists have found that allows them to avoid noticing the breakthrough power of abstraction, and instead spend valuable time using "tiles" or "blocks" in class.
Yes, they have college students play with blocks -- erm, "manipulatives". It helps them "get in touch with their inner mathematician" and "build their own math".
Hey; I'm not smart enough to make this stuff up! :roll:
Eliz.
Yes, they have college students play with blocks -- erm, "manipulatives". It helps them "get in touch with their inner mathematician" and "build their own math".
Hey; I'm not smart enough to make this stuff up! :roll:
Eliz.
