Introducing Powers and Models.I

Barrett, Sarah            Mars Hill School
                           287-0025
                           5916 W Lake Street
                           Chicago, Il 60619 

Objectives:

(Adaptable to grades 4-10)
-To discover how to square numbers; that squares are areas; the 
meaning of terms: Exponent, to the power of, etc.
-To visualize "to the powers of" 2, and 3, using a plane grid and a 
series of models of geometric shapes, of plane figures, then of solids; 
to see the relationships and certain properties of various geometric 
figures. 
-To construct models after handling a series of models, and drawing 
them, to be able to describe squares, rectangles, cubes, triangles,
triangular pyramids, tetrahedrons, hexagons, prisms, dodecagons, et al.
-To enjoy manipulating puzzle pieces to form some of the above shapes
and figures by the correct assembly of pre-cut parts, to develop and
reinforce concepts absorbed, as well as visualization and creativity
in spatial relations. 
             
Apparatus Needed: (Materials for chalkboard need to be adapted.)
 
     Overhead Projector, acetate sheet with square centimeter grid pre-
drawn on it, colored marker pens, clear plastic protractor, several 
pre-cut clear or translucent squares, rectangles, right triangles, (at 
least two with the same side measurements as the square and cube to be 
overlaid on the acetate sheet grid, and pairs of right triangles with 
differing sides which can be matched to form squares or rectangles when 
their right angles are placed in opposition), equilateral and isosceles 
triangles with which to form trapezoids, rhombi, parallelograms, 
hexagons, etc. and three dimensional figures, e.g. cubes, pyramids, 
tetrahedrons, etc. 
   Scissors, tape, and grid (graph paper or same unit grid sheets used 
on the overhead) to distribute to students, with and without patterns 
for models to be cut and assembled. 
  (Optional: Precut enough small right triangles to give each student 
one to keep.) 
  Game pieces and pre-cut puzzle sets, e.g., the Pythagorean rectangle, 
the square within a square, the dodecagon. 

Recommended Strategy
   
   Factoring Squares: to illustrate the power of 2 and that squares are 
areas. 
   On the overhead projector lay out a grid sheet. Number across the 
top and down the left side (x and y axis) to the same ending number. 
Beginning with square 1, ask students for the products of each number 
on the x axis across the top multiplied by its match down the y axis. 
Fill in the intersecting squares, or if pre-done, expose each using a 
cover sheet until the entire series of squares is complete. Outline the 
right and bottom perpendicular boundaries of each square thus produced 
with bright markers. Have students count at least the  first few 
squares enclosed by each, to verify and visualize the meaning of area, 
(factors multiplied or squared). 
   As the diagonal bisecting each square develops down the grid, 
produced by filling in the products, ask students to identify it: the 
hypotenuse of the right triangles simultaneously produced. 
   Elicit observations of students about the intervals between the 
products of the factors squared. (Odd numbers; at higher grade levels, 
possible algebraic formulae to be derived, ...y = mx + b...). Write all 
valid statements on chalkboard, in addition to the "square facts," and 
terms illustrated, e.g. 3 x 3 = 9, = 32. 
   Collect the grids in sets of four and arrange them into the 
Cartesian Quadrants to display on a bulletin or window. 

   Models of precut squares can be positioned on the grid sheet on the 
overhead. Overlay right triangles bisecting the square diagonally. 
Derive the relationships of their angles and sides and their respective 
areas. Overlay examples of congruities and similarities. Move along in 
the same way with the cube model made by assembling six of the same 
squares used to start. Relate and show the other figures and shapes 
listed above, and others as desired. Follow this by identifying then 
distributing the models among the students to manipulate, draw, and 
tell/write observations about, as well as name. Be sure to reinforce by 
writing names, terms, observations on the chalkboard, and by repeated 
questions. Ask students to explain, demonstrate, show their model-
making, and to assemble the puzzle parts named above.