An Introduction to Pi and the Area of a Circle

Edwina R. Justice              Gunsaulus Scholastic Academy
                               4420 South Sacramento Ave.
                               Chicago IL   60632
                               (312) 535-7215

Objectives (Staff):

     * Demonstrate a phenomenological approach to teaching mathematics

     * Inspire others to use the approach

Objectives (Grades 5-7):

     * Observe and discuss the relationship between circumference & diameter and 
       how that relationship, called pi, is used in the formula for the area of a 
       circle.

Materials:

     round container lids with varying circumferences
     4-column math table (label: circumference, diameter, c/d, & lid #)
     graph (label - horizontal axis: diameter; vertical axis: circumference)
     small circle drawn on centimeter grid
     small circles
     metric tape measures
     calculators
     glue

Recommended Strategy:

     * Count square centimeters inside circle and estimate the area.

     * Draw a square outside the circle.  Calculate the area of the square.

     * Draw a square inside the circle.  Calculate the area of the square.

     * Estimate the area of the circle by relating it to areas of the outer and 
       inner circles.

     * Cut a small circle into 16 equal pie-shaped pieces.  Arrange these 
       pieces to form a parallelogram and glue them on centimeter grid.

     * Calculate the area of the parallelogram made with the pie-shaped pieces.

     * Measure circumference and diameter of lids and record on 4-column math 
       table.

     * Divide circumference by diameter and record.

     * Plot ordered pairs (diameter, circumference).

     * Discuss graph.

     * Discuss results of C/D.

     * Roll large lid or trundle wheel on board and mark circumference.  Show 
       how diameter relates to it.

     * Show how the area of the parallelogram, made from 16 pieces, is equal to
       (pi)r2:

             Area = base x height                   Note:  c/d = (pi)

                  = 1/2 circumference x radius               c = (pi) x d

                  = 1/2 [(pi) x 2r] r                        d = 2 x r
                              
                  = (pi)r2                                   c = (pi) x 2r

     * Use formula to calculate area of initial circle.  Compare to estimates.

     * Estimate areas of other circles and then calculate actual areas and 
       compare to estimates.

Performance Assessment:

     This is an introductory lesson.  It is not necessary to assess usage of 
     area of circle formula at this time.

     Ask the following question:

          "What mathematical relationship does pi represent?"

     Students should write responses on paper.  Collect, read, and assign a 
     rating to each.  

     Expected responses:

          The circumference of a circle is 3.14 times its diameter.  This 
          relationship is called pi.

          Pi represents the circumference of a circle divided by its diameter.

          Pi = c/d.

Also see the file guests/edwina1.html