Area, Arithmetic and Algebra

Larry Freeman                  Kenwood Academy
                               5015 South Blackstone Avenue
                               Chicago IL 60615
                               312-535-1409

Objectives:

To show how area of rectangles and squares can motivate the learning of 
multiplication rules for certain binomials,  specifically:

                 (x + y)2 = x2 + 2xy + y2 

                 (x + y)(x - y) = x2 - y2

To show those teaching upper-grade math and high school first year math how 
their students can perform arithmetic based on these identities.

To show by paper folding several applications of the "distributive law" of 
multiplication over addition/subtraction.

Materials:

Cardboard demonstration set for the teacher (with magnetic tape backing) and 
printed sets of squares and rectangles for students to examine and re-arrange at 
their desks.  Prepare packets of pre-cut squares and rectangles, one for each 
student.  

Student materials should be prepared on centimeter-ruled paper;  two cm = one 
unit. (In this fashion, area can be checked by simply counting squares).  The 
entire square will measure 16 x 16 units, with heavy horizontal and vertical 
lines partitioning it into an 8 x 8, 4 x 4, and two 8 x 4 rectangles.  All 
measurements start from the lower-left hand corner of the large square.  Enter 
dimensions on every edge;  in the interior of each rectangle should appear 
  "Area = ______________ square units". 

Two of each square will be needed for each student packet: one left whole and 
one cut along the heavy lines into the four rectangles.  The teacher will need a 
similar set of materials, very much enlarged and made of heavier paper 
("tagboard" is ideal).  In addition, prepare a set of the four cut rectangles 
very much enlarged.  They must be on heavier still cardboard and backed with 
magnetic strip material to adhere to typical metallic base chalkboards.  The 
teacher's uncut square should be pre-folded along the heavy vertical and 
horizontal lines. 

Strategy:

Review the area formula for a rectangle.  Immediately have students remove the 
four small rectangles and arrange them to form a large square.  How many 
different arrangements can they find -- rotations and reflections are 
"different" in this case?  Sketch each arrangement in a student's notebook.  
Find the area of every large square by adding up areas of the four components.  
Encourage students to confirm this identity: 

     (a + b)2 = a2 + 2ab + b2    [a = 8  and b = 4] 

Challenge:  Using as many of the small pieces as needed, ask students to create 
a rectangle whose measurements are 8 x 16.  Sketch the arrangement they 
discovered, and, as before, try to discover and sketch as many different 
arrangements as possible.  Calculate the total area by adding the components.  
The student will note that all but the small square were used in the second 
rectangle: 

     So 8 X 16 = 128  which also equals 122 - 42  [144 - 16 = 128].

The teacher should duplicate these arrangements with the large magnetized 
rectangles on the chalkboard.  The algebraic identity here demonstrated is: 

     (x - y)(x + y) = x2 - y2.     [Here x = 12  and y = 4].

Second challenge:  Have students take the larger square from their packet and 
fold it along the vertical line.  The left side is now a rectangle whose 
measurements are 8 x 12.  But it consists of two rectangles: 4 x 8 and 8 x 8.  
Thus they have shown that 8 X 12 = 8 X 4  +  8 X 8.  So they have proved that 

     8 X 12 = 8(4 + 8)   [an illustration of the distributive law]

Third challenge:  With new teacher-made packets -- identical to the originals 
except that variable names replace numerals for dimensions.  Another difference:  
These paper rectangles should not have centimeter ruling.  Now the student 
should follow every step above using variables instead of numerals.  The writing 
of the appropriate identities is left as an exercise for the teacher; answers 
available from the writer. 

Performance Assessment:

While students are working on this project, either individually or in pairs, the 
teacher circulates, assesses performance visually and gives hints, commendations 
or other encouragements (via adroit questions).  Later, notebooks themselves 
will be graded for completeness and accuracy. Ultimately knowledge will be 
"assessed"  via customary pencil and paper tests.  [Ruth Mitchell-type global 
assessment techniques do not seem cost effective for this unit.] 

References and credits:

This unit was inspired by a conference table discussion with Porter Johnson. In 
fact none of these techniques is really new; they are frequently re-discovered 
in many different places at widely different times by creative teachers inspired 
to improve textbook versions of "the method." 

Multicultural Dimensions:

Geometry is not the exclusive possession of any culture in any historical era 
ancient or modern.  All peoples who had concerns with land and its measurement 
or with calendars developed appropriate geometrical principles.  Similarly with 
arithmetic: Counting and rudimentary computing were known to all peoples, 
ancient or modern, no matter what geographical location.  None was limited by 
"culture" in matters of commerce;  rather our current awareness or ignorance is 
a function of the available historical record and its readability.  It is 
pointless to try to ascribe primacy or originality to any cultural group. 

Algebra was a European inheritance which came most directly from the 
Mediterranean Moorish (Muslim/Islamic) civilizations.  They, in turn certainly 
drew from the ancient Greek, Hindu and African civilizations.  Every  
civilization refines and improves what it inherits.