362
A Strategy for Discovering the Formulas for Finding the Area of
Polygons
Charles R. Neatrour
Department of Early and Middle Education
James Madison University
Harrisonburg, Virginia 22807
Frequently, teachers merely dictate area formulas to students
to demonstrate how to use them. With very little instruction,
they expect their students to remember the formulas and readily
use them with skill as well as to understand the concept of area
Figure 2.
width
i
of various polygons. As expected, the teacher ends up restating
the formula again and again or instructing the students to look
it up in their textbook. How can this practice be avoided with
students? By initially introducing such concepts through guided
discovery techniques, the teacher can give students insight into
how they can not only discover the formula but how they can
possibly rediscover it later.
The following strategy should allow time for students to
explore possible options and to verbalize these options.
Furthermore, the teacher might even go so far as to have
students write a paragraph or a step-by-step procedure which
logically leads to the desired formula.
Encourage students initially to find the area of rectangles by
both counting and using the formula. Students will check their
answers and reinforce their confidence in the more abstract
method.
Define Area
Area of Squares
A practical approach to defining area in the primary grades
would be to use a square as the unit. The process that then
evolves requires students to determine the number of these
square units that cover a specified space. Grid paper and
scissors are the media for implementing this strategy.
1. Have students draw a square on their grid paper. (The
students’ selected squares need not be the same size.)
Thus, the following formula is discovered:
Area of Rectangle = Length X Width
Figure 3.
Area of Rectangles
T
D area
1. Have students draw a rectangle on their grid paper.
(Students’ rectangles need not be the same size.)
Figure L
length >
(
width
A unit of
-length-^
1
2. Although students can initially determine the area by
counting the unit of area squares covering the larger square,
they can also use the formula for finding the area of a rectangle
if they know it. The latter process suggests:
D
A unit of area
Area of Rectangle = Length X Width
2. The primary student will initially determine the number
of units that cover the selected rectangle by counting them.
3. As soon as students have acquired multiplication skills
and the appropriate vocabulary, encourage them to discover a
rule or formula to replace the counting process.
3. Since the length and the width are the same measure for
squares,
Area of Square = Area of Rectangle.
Length and width are equated with the length of a side;
thus,
School Science and Mathematics
Area of Polygons
Length = Width = Side
363
finding the area of a rectangle.
Using substitution,
Figure 7.
Area of Rectangle = Length X Width
Area of Square = Length X Width
Area of Square = Side X Side
^
r^ i
\
width
Figure 4.
T
side
^- side
-»|
1
Area of Parallelograms
1. Have students draw a parallelogram on their grid paper.
(The size of the students’ parallelogram might vary.)
Figure 5.
4. Furthermore, they will agree that the area of the newly
formed rectangle is equal to the area of the original
parallelogram. This process of reforming the parallelogram
into a rectangle applies the conservation of area principle used
by Piaget in one of his experiments involving learning in
young children.
5. A little substitution at this point will readily assist
students to apply existing knowledge and discover the formula
for finding the area of a parallelogram.
Area of the Transformed Parallelogram = Area of Rectangle
Area of Rectangle = Length X Width
/
"y
/
’
vertical I
A~
7
^^^
i
2. Be sure they know the necessary vocabulary (base and
vertical height), and they can readily determine each.
3. Have students use a pair of scissors and cut out their
parallelogram. Very likely one of the students will have
suggested in his or her paragraph or step-by-step procedural
listing, the option of cutting off one of the triangular ends and
reshaping the parallelogram as a different polygon.
The length of the rectangle equals the base of the
parallelogram, and the width of the rectangle equals the
vertical height of the parallelogram. Then by substitution,
Area of Rectangle = Base X Vertical Height
Area of Parallelogram = Base X Vertical Height
Area of Triangles
1. Have students draw a variety of triangles on their grid
paper, preferably ones with an even number of units for the
vertical height (altitude).
Figure 8.
Figure 6.
cut
(
/
A~
;
i
;
Y
/
\
Assuming they reshaped the parallelogram to form a rectangle,
it is expected that students will already know the formula for
2. Be sure they know the appropriate vocabulary for a
triangle (base and height).
3. After exploring various options verbally or in writing,
have the students use a pair of scissors to cut out and cut up their
triangles. They then reshape the pieces to form a polygon for
which they already know the formula for finding its area.
Volume 91(8), December 1991
364Area of Polygons
4. First use the option where they reshape the triangle into
a parallelogram by cutting it into a trapezoid and a triangle, both
with a vertical height of one-half the height of the original
triangle.
Figure 11.
second cut
first cut-
Figure 9.
A
^
Lastly, the two small triangles are rotated 90 degrees to form
a rectangle.
Figure 12.
Now rotate the triangle 180 degrees clockwise or
counterclockwise and form a parallelogram. The area of the
/
Figure 10.
A
\/
\
newly formed parallelogram is equal to the area of the original
triangle. This process once again applies the conservation of the
area principle; however, it should be noted that to introduce this
option, it is necessary to teach first the formula for finding the
area of parallelograms.
5. A little substitution here will permit students to use the
formula for finding the area of a parallelogram and to discover
the formula for finding the area of a triangle. The process is:
Area of the Transformed Triangle = Area of Parallelogram
Area of Parallelogram = Base X Vertical Height
Since the vertical height of the parallelogram equals 1/2 the
height of the triangle, then
Area of Parallelogram = Base X 1/2 Height
Area of Triangle = 1/2 X Base X Height
6. Use the second option and reshape the triangle into a
rectangle. Here a student suggests to first cut off a smaller
triangle half the height of the original triangle. Then cut the
smaller triangle into two pieces along the height line.
/
"^
^
t
1/2 aid
..i.
The area of the newly formed rectangle remains equal to the
area of the original triangle. Using this option does not
necessitate the teaching of the formula for finding the area of
parallelograms first.
7. A little substitution at this point will assist students to
apply prior knowledge concerning the formula for finding the
area of a rectangle and to discover the formula for finding the
area of a triangle. The process is:
Area of the Transformed Triangle = Area of Rectangle
Area of Rectangle = Base X Height
Since the base of the rectangle equals the base of the
triangle and the height of the rectangle equals 1/2 the height of
the triangle, then the area ofthe rectangle equals the base times
1/2 the height and the area of the triangle equals 1/2 times the
base times the height.
8. There are two additional options available for discovering
the area of a triangle; however, they do not apply the conservation
of area principle and the use of grid paper is optional. One of
these models uses a right triangle. The right triangle and a
congruent mate can be used to form a rectangle. The fourth
model uses a nonright triangle. The nonright triangle and its
mate can be similarly used to form a parallelogram. In both of
the last two options, the area of the triangle is equal to one-half
the area of the rectangle or parallelogram formed.
Area of Trapezoid
1. Have students draw a trapezoid of their selection on grid
paper, preferably a trapezoid with an even number of units for
the vertical height.
School Science and Mathematics
365
Area of Polygons
Since the base of the parallelogram equals base 1 plus base
2 and vertical height equals 1/2 the height of the trapezoid,
then the area of the parallelogram equals:
Figure 13.
|t-base 2 «ij
(Base 1 + Base 2) X (1/2 Height of Trapezoid)
Area of Trapezoid = 1/2 X (Base 1 + Base 2) X Vertical Height
P^ vertical height
6. Use the second option and reshape the trapezoid as a
rectangle. This time students cut off:
Figure 16.
2. Be sure they know that the two parallel sides are bases
(base 1 and base 2), and that they can readily determine the
vertical height.
3. Once students have discussed their options and
accompanying procedures, have them use a pair of scissors to
cut out and cut up the trapezoid. They then reshape the pieces
to form a polygon for which they already know the formula for
finding its area. They have two optionsa parallelogram or a
rectangle.
4. First use the option where they reshape the trapezoid into
a parallelogram by cutting it into two trapezoids with vertical
h base 2
-^
heights one-half the height of the original trapezoid.
two small triangles from both ends and rotate each of the small
triangles 90 degrees to form a rectangle.
Figure 14.
Figure 17.
Now rotate one of the trapezoids 180 degrees and form a long
parallelogram.
Figure 15.
/
. base
^
1 + base 2
A/lffe
/oftrape
f
*
The area of the newly formed parallelogram is equal to the
area of the original trapezoid, and the conservation of the area
principle to derive the formula has been applied.
5. A little substitution at this point will readily assist the
students to apply earlier knowledge and to discover the formula
for finding the area of a trapezoid. The process is:
Area of Transformed Trapezoid = Area of Parallelogram
Area of Parallelogram = Base X Vertical Height
The area of the rectangle remains equal to the area of the
original trapezoid.
7. A little substitution at this point will readily assist the
students to apply earlier knowledge and to discover the formula
for finding the area of a trapezoid. The process is:
Area of the Transformed Trapezoid = Area of Rectangle
Area of Rectangle = Length X Width
Since the length of the rectangle equals 1/2 times (base 1
+ 2) and the width equals the vertical height, then:
Area of Rectangle = 1/2 X (Base 1 + Base 2) X Vertical Height
Area of Trapezoid = 1/2 X (Base 1 + Base 2) X Vertical Height
In conclusion, the use of grid paper provides a student-
Volume 91(8), December 1991
Area of Polygons
56
iented strategy for discovering the area formulas of basic
dygons, in particular for rectangles, squares, parallelograms,
angles, and trapezoids. Grid paper is not the only media that
nds itself as a strategy for discovering area formulas of basic
polygons. The use of paper-folding techniques, area
transformations with congruent polygons, and the geoboard
offer several additional options.
Curriculum and Evaluation Standards
October 1990 special issue on NCTM Standards
of School Science and Mathematics
Fifteen specially selected articles by leaders in mathematics education including these topics:
I. Background:
National Standards: A New Dimension
Evidence Which Supports NCTM’s Standards
II. Implementation:
Understanding and Implementing Standards in Grades K-4,5-8, and 9-12
Implementing the Evaluation Standards
Envisioning Change in Practice
III. Issues:
Mathematical Connections
Realizing the Curriculum for All Students
IV. Reactions:
A Science Educator’s Perspective
A Welcome Vision
A Business Perspective
The Socioeconomic Dimension
$5.00 prepaid - available from the SSMA Executive Office
School Science and Mathematics