I. Basic Geometric Concepts and Spatial Visualization
A. Out of sight, Out of mind
Make a figure on an overhead geoboard. Flash the overhead on for a few
seconds. Students have to try to reproduce the figure on their geoboards.
B. Introduce terminology from a student viewpoint
Have the students build a figure on their geoboards using just one rubber
band. Look around the room for examples of polygons as well as
non-examples. Pick several of each and then tell students "These
are polygons. These are non-polygons." Have students attempt to
develop the appropriate definitions of a polygon. Use other examples
around the room to test the definitions.
Using only examples of polygons, sort them into those that are convex
and those that are concave. Again, have students generate the meanings
of these terms.
C. Building specific figures
Have students build different triangles. Have students classify the
triangles according to their angle measures (right, acute, or obtuse)
or according to the lengths of their sides (isosceles, equilateral,
or scalene). Have students record the different figures they construct
on dot paper. This is an important skill as sometimes students have
some initial difficulty at placing their figure in the proper place
on their dot paper.
Have students build a hexagon, a parallelogram, an obtuse triangle,
etc. Emphasize the vocabulary that you know students are having trouble
using. The nice thing about building the figures on the geoboard is
that you have an immediate means of assessing the students' understanding.
Have one student build a figure out of sight of a second student using
a specified number of rubber bands. Then have the student provide oral
directions so that the second student builds an exact duplicate copy.
Use an answering machine analogy when describing the activity to solve
the issue of questions being asked and answered.
In journals, draw a sketch of a figure on dot paper. On another page,
write a set of directions so that someone in your class can build your
figure. Have students read written directions and then check their answers
with the "student generator" of the picture.
II. Perimeter and Area Relationships
A. Finding perimeters and areas
Identify the unit of length as the horizontal or vertical distance between
two consecutive pegs. Identify the unit of area as one square unit enclosed
by four pegs.
Have students build a figure consisting of only right angles and find
its perimeter. Find the area of the figure you created.
Have students build a figure of a specified area. Suppose you restrict
the figures to those with only right angles. Find the perimeters of
the figures built. Can two figures have the same area but different
perimeters? How do you know?
Now have the students build a figure of a specified area but with no
other restrictions. (You have to be careful about asking for perimeters.
The diagonal distance from one peg to another is not an integer but
is irrational.) Have students share their solutions and explain how
they know that the area is the specified number.
Have students reproduce some figure you have created and find its area.
Include odd aspects of figures as well as triangular regions.
B. Deriving area formulas
Have students build rectangles with the specified lengths and widths.
Find the area and perimeter of the rectangles. Have the students observe
the data in their tables and attempt to derive their own formulas for
finding the area and perimeter of a rectangle.
Have students build a parallelogram and find its area. Move the rubber
band to change the parallelogram into a related rectangle. Use this
transformation to help derive the formula for the area of a parallelogram.
Build a parallelogram on the geoboard. Have the students use another
rubber band to create one of the diagonals. Use this transformation
to help derive the formula for the area of a triangle.
Have students build a trapezoid on the geoboard. Use another rubber
band to draw in one of the diagonals. How does this transformation help
you derive the formula for the area of a trapezoid?
A. Derive the formula for the sum of the angle measures in a convex
Build a quadrilateral, pentagon, hexagon, octagon, etc. Construct all
the diagonals from one vertex. Use this to develop a formula for finding
the sum of the angle measures of the interior angles.
Charles, Linda Holden and Micaelia Randolph Brummett. Connections:
Linking Mathematics with Manipulatives. Sunnyvale, CA: Creative
Marilyn Burns Manipulative Videos (Geoboards)
Addenda books from NCTM (There is one book for each grade K-6,
as well as books by strand: Patterns, Making Sense of Data, Number
Sense and Operations, Geometry and Spatial Sense.)