Learning by Logic - Total Surface Area

Boyd, Carolyne                              Bennett Elementary
510 Barrington Court                        821-2680
University Park, IL 60466
534-6486

Objectives:
1. The student will calculate the area of plane surfaces using the 
   formulas for the area of a square, rectangle and triangle.
2. The student will develop the formula for calculating the total 
   surface area of two geometric solids, the cube and rectangular 
   prism. 
3. The student will calculate the total surface area of cubes and 
   rectangular prisms.
4. The student will apply concepts to determine the total surface area 
   of a variety of classroom and household materials.

Apparatus Needed
geoboard
rubber bands
overhead projector
transparency film (clear, green, yellow)
markers
small boxes from household materials
yardstick
shoebox kit consisting of:
 plastic pieces of various sizes (poster board may be used)
   4 squares
   4 rectangles
 centimeter grid paper
 centimeter grid ruler
 pencils
 wooden cube*
 wooden rectangular prism* 
 construction paper
 small boxes from household products
 small plastic ziplock bags
 paper
 duplicated sheets
 scissors                     * milk cartons can be used to 
 tape                           construct a cube or rectangular prism

Recommended Strategy
Using a geoboard and overhead projector, lead students in a discussion 
of square units.  Develop the concept that squares are quadrilaterals 
with four equal sides, opposite sides parallel and each angle measuring 
90 degrees.  Develop the concept that rectangles are quadrilaterals 
with opposite sides parallel and each angle measures 90 degrees, the 
opposite sides are of equal length.  Develop the concept that a 
triangle is one-half of a quadrilateral. 
 Write formulas for each geometric figure discussed: 

              Square........ A = s x s  or A = s2
              Rectangle..... A = l x w
              Triangle...... A = 1/2 (b x h)   

Students will work in groups of four or five.  Each group will receive 
a kit containing the above listed materials.  Students will take from a 
plastic bag varied pieces of plastic. Examine each and look for  
similarities that would allow the pieces to be grouped.  Next arrange 
them into similar stacks.  Draw them onto the centimeter grid paper, 
arranging from largest to smallest.  Be sure to begin each figure even 
with a line on the grid paper.  Calculate the area of each plane 
figure, using the three formulas listed on the board. 

Given a duplicated sheet containing a variety of shapes, students are 
asked to divide and conquer.  Determine the surface area of the eight 
planar shapes.  The student will find the area of each planar region by 
adding the sum of the areas of its parts. 

Given a cube and rectangular prism, each student will wrap the 
geometric solid, to develop the concept that surface area means to 
surround.  Draw straight lines, cut out the six sections.  Tape the 
sections to a sheet of paper, look for similarities.  Develop a method 
for determining the total surface area of the cube.  Do the same for 
the rectangular prism. Use this information to develop a formula for 
finding the Total Surface Area of a cube and rectangular prism. 

The Total Surface Area of the cube is equal to the sum of the area of 
six equal sides:  

  Cube....................T. S. A. = 6 (s x s)   or 6 s2

The Total Surface Area of the rectangular prism is equal to the sum of 
six surfaces........
                     the front and back... (h x l)
                     both ends............ (h x w)
                     top and bottom..... ..(w x l)

  Rectangular Prism.......T. S. A. = 2 (h x l) + 2 (h x w) + 2 (w x l)

                    or....T. S. A. = 2 [ (hl) + (hw) + (wl) ]

Use the formulas developed to determine the T. S. A. of other cubes, 
rectangular prisms, a paper house, the floor and ceiling of the 
classroom, the painted surfaces of the classroom, household items and 
drawings on a worksheet. A centimeter ruler and yardstick are provided 
for convenience.  Calculators may be used for this activity. Make sure 
each answer contains the appropriate unit squared. 

Resources...
 Lund, Charles.  Dot paper Geometry with or Without a Geoboard
 Oregon State Math. Resource Project.  Geometry and Visualization
 Stokes, William T.  Gems of Geometry