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Looking at
and
Geometrically
Eric Hegblom
The Mathematics Teacher
, October 1993, Volume 86, Number 7, pp. 584–587
Mathematics Teacher
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Article reprinted with permission from
Mathematics Teacher,
copyright October 1993 by
the National Council of Teachers of Mathematics. All rights reserved.
Eric Hegblom is a student at Cornell University majoring in applied engineering
physics. He resides at 8 Woodfield Road, Wellesley, MA 02181.
T
wo commonly taught algebraic sums are
The first equation has a short algebraic proof, and the second has a more intricate one.
An alternative approach is to evaluate the sums geometrically. A method for each sum
is presented here. Evaluating the first sum involves positioning squares and then
determining area. Evaluating the second sum involves arranging cubes then determining
volume. Although this article can use only twodimensional pictures to demonstrate the
second sum, manipulating (threedimensional) physical cubes adds considerable clarity
to a presentation.
To find the first sum, we position the squares with sides of length 1, as shown in
figure
1,
laying one square on the first row, two squares on the second row, three squares on
the third row, and so on up to the
n
th row, on which we lay
n
squares. Thus the area of
this shape equals the sum
1
1
2
1
3
1
4
1
???
1
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