Eiland, M.D. Investigation of Quadrilaterals
Investigation of Quadrilaterals
NCTM Standards:
Analyze characteristics – understand the relationships among the angles, side lengths,
perimeters, areas, and volumes of similar objects.
Specify locations – use coordinate geometry to examine special geometric shapes, such
as regular polygons or those with pairs of parallel or perpendicular sides.
Use visualization – use geometric models to represent and explain numerical and
algebraic relationships.
Overall Objective: To develop a quadrilateral hierarchy with definitions.
Quadrilaterals under investigation: Square, Rectangle, Trapezoid, Isosceles
Trapezoid, Parallelogram, Rhombus, and Kite.
Previous knowledge: Pythagorean theorem, calculating area of rectangles
and triangles.
Assignment 1: Area without formal equation (student_area.gsp)
Trapezoid, Isosceles Trapezoid, Parallelogram, Rhombus, & Kite
Using knowledge of rectangles and triangles, calculate the area of the above
quadrilaterals. Students should be encouraged to generalize properties of
quadrilaterals when possible.
Objective: students should use existing knowledge of rectangles and
triangles to calculate the area of semi-novel quadrilaterals. In a subsequent
lesson, students will derive the area formula for the same quadrilaterals from
the trapezoid equation.
Student Assignment Instructions: A new student isn’t confident in his ability
to calculate the area of the following quadrilaterals. Using your knowledge
of rectangles and triangles, demonstrate how the new student can calculate
the area of the quadrilaterals. The area of each figure is provided so that you
may check your answers. Use only the dimensions provided with the figure.
Write a brief explanation detailing your solution process including any
formulas used.
Assignment 2: Investigation of area equations (quadproof.gsp)
Trapezoid
0.5(b1+b2)h
Parallelogram
bh
Rhombus
0.5(d1+d2) [diagonals]
Kite
0.5(d1+d2)
Eiland, M.D. Investigation of Quadrilaterals
Using knowledge obtained in Assignment 1 encourage students to derive the
formula for the remaining quadrilaterals given the formula for a trapezoid.
Have students model their solutions using the Sketchpad file in Assignment
1.
Student Assignment Instructions: The new student in class is becoming more
comfortable calculating area of quadrilaterals because of your help. He is
unsure how the math book got the formulas for the quadrilaterals we used in
Assignment 1. Using your new knowledge of how quadrilaterals are related,
demonstrate how the area formula for a trapezoid, 0.5(b1+b2)h, can be used
to find the area formula for a kite, rhombus, parallelogram, rectangle and
square.
Assignment 3: Investigation of diagonals
Objective: students will investigate the properties of the diagonals from the
quadrilaterals used in Assignment 1 & Assignment 2.
Student Assignment Instructions: Using the Sketchpad quadrilaterals in the
previous two assignments, summarize in paragraph form the properties of its
diagonals. Place a check (√) in the flowchart to indicate the quadrilateral
has the listed diagonal properties.
Diagonal Properties
Diagonals bisect each other.
Parallelogram Rectangle Rhombus
X
Diagonals are congruent.
X
Kite
X
Isosceles
Square Trapezoid
X
X
X
Diagonals are perpendicular.
X
Diagonals bisect vertex angles.
X
X
X
X
X
X
Diagonals form 2 pairs of
congruent triangles.
Diagonals form 4 congruent
triangles.
X
X
X
X
X
Eiland, M.D. Investigation of Quadrilaterals
Assignment 4: Developing a hierarchy and flowchart for quadrilaterals
(Microsoft Word flowchart or Inspiration)
Objective: students should investigate the properties of quadrilaterals and
develop definitions based on the fewest amount possible. Using an
investigative approach student’s should agree on placement of quadrilaterals
in the hierarchy.
 Square: a rhombus with a right angle or an equilateral rectangle.
 Parallelogram: a quadrilateral with opposite sides parallel.
 Kite: possesses perpendicular diagonals.
 Rhombus: an equilateral parallelogram.
 Trapezoid: a quadrilateral with 1 pair of parallel sides.
 Isosceles Trapezoid: a trapezoid with congruent base angles congruent
left and right sides.
QUADRILATERALS
Opposite sides are
congruent and
parallel
1 pair of opposite
sides congruent
PARALLELOGRAM
1 right angle
TRAPEZOID
Congruent sides
RECTANGLE
RHOM BUS
Congruent sides
1 right angle
SQUARE
Student Assignment Instructions: With your partner, discuss the properties
you discovered concerning the quadrilaterals we have been investigating.
Try to place the remaining quadrilaterals (trapezoid, rhombus,
parallelogram, and rectangle) in the proper place on the flowchart. Describe
why you decided to place the quadrilateral in that particular position.
Assignment 5: Pencil and Graph paper exercise.
Eiland, M.D. Investigation of Quadrilaterals
Objective: beginning with a general quadrilateral, progress through the
hierarchy to create a trapezoid, parallelogram, rectangle, and square (Craine
& Rubenstein, 1993). [IN CLASS ASSIGNMENT]