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Can 5th
Graders Derive AreaFormulae? Implications from
Japanese Curriculum Materials
Tad Watanabe!Kennesaw State [email protected]
!2008 Georgia Mathematics Conference!
The slides & handouts are available
at:!http://science.kennesaw.edu/~twatanab/!
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GSP M5M1 Students will extend their understanding
of area of fundamental geometric plane figures.!a.! Estimate the
area of fundamental geometric plane figures.!b. Derive the formula
for the area of a parallelogram (e.g.,
cut the parallelogram apart and rearrange it into a
rectangle of the same area).!c. Derive the formula for the area
of a triangle (e.g.
demonstrate and explain its relationship to the area of a
rectangle with the same base and height).!d.! Find the areas of
triangles and parallelograms using
formulae.!e.! ..!f.! Find the area of a polygon (regular and
irregular) by
dividing it into squares, rectangles, and/or triangles and
find
the sum of the areas of those shapes.!
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Can 5th grade students really
derive these formulae?!Through problem solving?!
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Typical Problem: Find the area of
this parallelogram.!A!
B! C!
D!
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Typical solution and conclusion:!
Therefore,Area = Base x Height!
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But, is this good enough?
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How about this parallelogram?!
C!
A! D!
B!
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M3M4. Students will understand and
measure the area of simple geometric
figures (squares and rectangles).
a.! Understand the meaning of the square unitand measurement in
area.!
b.! Model (by tiling) the area of a simplegeometric figure using
square units
(square inch, square foot, etc.).!c.! Determine the area of
squares and
rectangles by counting, addition, and
multiplication with models.!
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Find the area of L-shapeFind the area in as many different ways
as possible!
5 cm!
5 cm!
5 cm!
4 cm!10 cm!
9 cm!
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Area of L-Shape
! When we are given an unfamiliar shape, wemay be able to find
its area by making it
into a familiar shape (or a collection of
familiar shapes).!! Ways to make familiar shapes!
! Breaking the given shape apart!
!Cutting and re-arranging
!! Making it bigger!
! Add on, then subtract!! Double, then divide by 2
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GSP M5M1 Students will extend their understanding
of area of fundamental geometric plane figures.!a.! Estimate the
area of fundamental geometric plane figures.!b.! Derive the formula
for the area of a parallelogram (e.g., cut the parallelogram
apart and rearrange it into a rectangle of the same area).!c.!
Derive the formula for the area of a triangle (e.g. demonstrate and
explain its
relationship to the area of a rectangle with the same base and
height).!d.! Find the areas of triangles and parallelograms using
formulae.!e.! ..!f. Find the area of a polygon (regular and
irregular)
by dividing it into squares, rectangles, and/or
triangles and find the sum of the areas of thoseshapes.!
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Area of Parallelograms!
We can make a rectangle, whose width is the base ofthe
parallelogram and the length is the height of the
parallelogram.!Therefore,Area = Base x Height
!
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How about this parallelogram?!
C!
A! D!
B!
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Height stops here!!
C!
A! D!
B!
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Lets find the area using what we know?We know how to find the
area of nice parallelograms.!
C!
A! D!
B!
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Ashleys method!
C!
A! D!
B!
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Emmanuels method!
C!
A D!
B!
P!
Q!
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C!
A! D!
B! Q!
P!
R!
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Ummes method!
C!
A! D!
B!
P!Q!
height!
height!
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Joses method!
C!
A! D!
B!
P!
Q!
R!
L!
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Jahmirs method!
C!
A! D!
B!
P!
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Darlenes method!
C!
A! D!
B! P!
Q!
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We can still useBase xHeight!
C!
A! D!
B!
Base
Height
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We can still useBase xHeight!
C!
A! D!
B!
P!
Base
Height
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We can still useBase xHeight!
Base
Height
C!
A! D!
B! Q!
P!
R!
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These understanding applies to
other shapes:!
From Hironaka & Sugiyama (2006), 5A, p. 77!
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Even for circles!
From Hironaka & Sugiyama (2006), 5B, p. 88!
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From Hironaka & Sugiyama (2006), 5B, p. 91!
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Conclusions!! Yes, 5th graders can derive formulae.!! But, it
takes a careful planning by teachers.!! Students must understand
that the area of an
unfamiliar shape may be found by changingthe given shape into
something they arefamiliar.!
! Make general methods of changing shapesan explicit focus of
instruction.!
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Reference!Hironaka, H. & Sugiyama, Y. (2006).
Mathematics for Elementary School. Tokyo,
Japan: Tokyo Shoseki.!The series is available through Global
Education Resources!! !www.globaledresources.com!
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