[tutorials & resource material arranged by topic
 
Class Table
Objective, Materials, Stuff To Review Before This Activities,
Procedure, Rectangle, Parallelogram, Triangle, Trapezoid, Circle,
On the Overhead


Objective
To prove each of the area formulas by paper folding and/or cutting.

Materials
  • a pair of scissors,
  • a ruler,
  • a pencil,
  • a sheet of rectangular paper for each of the following formulas: parallelogram, trapezoid, triangle,
  • a waste paper bag or basket,
  • a set of circluar papers for instructor use

Stuff To Review Before This Activities

      Point out that opposite sides of a rectangle are parallel.

      Review at least as much about area as is discussed in rectangle, below.


Procedure
      Review area, area of a rectangle, properties of a rectangle such as parallel opposite sides and right angles.
 
      These proofs need not be done at the same time. Each time one is done however, it is important to review the definition of area as the number of square units and to remind students that the reason "length times width" or "base times height" yields area is because it supplies the number of squares covered.
 
      The rectangle must be discussed before the parallelogram.
 
      The rectangle must be discussed before the triangle.
 
      The rectangle and the parallelogram must be done before the trapezoid.
 

Rectangle
      Understanding why the area of a rectangle is base time height is the key to understanding each of these paper-folding proofs of area.
      Area means number of square units.
 
      To compute the area, one must find the number of squares.
 
      The base is the number of squares across the "bottom".
 
      The height is the number of squares "high."
 
      The base times the height is the number of squares, the area.
 

Parallelogram
Make the Parallelogram
      Make a parallelogram from a rectangular sheet of paper in the following way:
 
1st: On one edge, mark an arbitrary point A and draw a line segment to the indicated opposite vertex.
 
2nd: Repeat this on the opposite edge: mark point A on the edge parallel to the edge with A already marked on it so that the distance from the edges are the same. Again draw a line segment from A to the indicated opposite vertex.
 
3rd: Cut the paper on the line segment.
 
4th: Throw away the extra pieces of paper (the shaded triangles of the picture above).
 
Demonstrate A = bh

Trapezoid
Make the Trapezoid
      Make a trapezoid from a rectangular sheet of paper in the following way:
 
1st: On one edge, mark an arbitrary points A and B and draw line segments to indicated opposite vertices.
 
2nd: Cut the paper on the line segment.
 
3rd: Throw away the extra pieces of paper (the shaded triangles of the picture above).
 
Demonstrate A = (½)h(b1 + b2)

Triangle

DON'T THROW ANYTHING AWAY!

     
  Your goal is to create one "center triangle" (shown in white) and two other triangular pieces (shown in gray). You need each of these triangular pieces to prove the area of a (white) triangle is half the area of a (original piece of paper) rectangle with the same base and height.
 
1st: On one edge, mark an arbitrary point A and draw a line segments to each of the opposite vertices.
 
2nd: Cut the paper on the line segments. SAVE EACH OF THE THREE TRIANGULAR PIECES.
 
 
Demonstrate A = (½)bh

On the Overhead
      Linked here is a template of the material I use to present areas of this page in a quicker format.
 
      Below is instructions on cutting and using clothing snaps to create the pivots needed.




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