Introducing Powers and Models.I
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Barrett, Sarah Mars Hill School
5916 W Lake Street
Chicago, Il 60619
(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
-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,
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.
Factoring Squares: to illustrate the power of 2 and that squares are
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.