Geometry

Montessori Theory Foundation for the Instruction of Geometry

The child has been surrounded by geometry all of his or her life. Geometry, as a study, brings nomenclature and order to the sensorial lessons of the preschool where they learned the names of the simple shapes of triangle, circle, and rectangles. The study of geometry allows the child to manipulate the geometric materials in order to make observations that serve the six to twelve year old child’s need to know how and why things happen. For example, how did the engineers know that the tunnel they designed to pass under the English Channel would meet at the right place if they started digging from both sides? The answer is geometry.

Geometry provides an opportunity to exercise the intellect. There are comprehensive educational connections through the stories of famous mathematicians. Geometry is a whole new area for creativity and construction. The metal inset work which begins in the preschool classes continues into the elementary, but we adds more detail and more creativity. There is a need for activity. There are lots of things to do. It is a fertile field of activities for children to do. There are many things to do that do not require reading and writing; this is advantageous to the activities of the young first grader children.

The Geometry Program

We begin with the story of how geometry began. This story tells about the Egyptian wire stretchers who used large right-angle triangles to determine land ownership along the Nile after huge flood incidents. This story appeals to their imagination and gives them an integrated view of the study and application of Geometry. The child starts connecting the study of geometry to others areas of knowledge. There is a lot of work with lines and angles that lead to the study of polygons. The booklets allow the children to further their introductory lessons into specialized areas in which they are interested. There is an underlying work with lines, angles and polygons. This leads to the work with solids. There is separate, but related, work with the circles. There are an infinite number of polygons in a circle.

Congruence, Similarity, and Equivalence are a huge area of the geometry program. Equivalence is the most important, because it is the basis for our study of fractions, area, and volume. You need congruence to establish equivalence. Congruence is a special case of equivalence. So when we present we begin with the metal insets first. The constructive triangles are also brought in for this work. This begins with work that the child is familiar with and gives new dimension to that with which they are familiar. This sets in place the work with Pythagoras and triangles. We elaborate upon the many variations of triangles. We bring to their attention many elements about the Pythagorean theorem.

There is an element of measurement in geometry. We measure the geometric figures so that we can compare relationship of size, shape and volume. The lateral and surface area of solids synthesizes earlier work. Build into this work throughout is the need for reasoning. The child’s reasoning mind is called forth to make formulas for determining area and volume. We have at times mentioned the relationship to the three-period lesson; although they are not the three-period lesson of the primary, it is still and effective way to learn nomenclature. In the elementary we include more than three objects at a time. We include reading and writing as well. There is a connection to etymology and the study of the origins of our language. We can play games with the children to familiarize them with the etymology of the terms of geometry.

Three-period lessons are characterized by three distinct phases. The first is that the teacher gives the nomenclature. The second allows the child to show the teacher which element fits the term being given. The third period asks the child to name it. This is a testing phase. In the elementary, we add more new terms at a time than the primary class. However, the three-period lesson is still very effective in the elementary. If a child misses the correct term in either the second or third period, it is recommended to simply go immediately back to the first period and give the child the name of the term.

THE MONTESSORI SYLLABUS FOR GEOMETRY

Story of Geometry

The Story of how Geometry got its name

Other Interesting Information & stories

Congruency, Similarity, Equivalence

Introduction to the metal insets

(design without drawing or with drawing)

Congruency, similarity equivalence with the metal inset (conceptual)

Congruent Figures

Similar Figures

Equivalent Figures

Introduction of the signs =, ˜, ‗

Congruency, Similarity, equivalency

Congruency, Similarity, and Equivalence with the constructive triangles

Congruency – further exploration

Similarity– further exploration

Equivalence– further exploration

Using single figure

Using two figures

Combining boxes

Equivalent “pictures” (including drawings)

Equivalence of two key triangles(addition)

Box of Blue Triangles

Reasoning

Constructive Triangles and Metal Insets

Polygons

Type of plane geometric figures

Types of Regular Polygons according to the number of sides

Types of Planar simple closed curves

Parts of the triangle

Parts of the Quadrilateral

Parts of the Regular Polygon

Types of triangles according to the sides

Types of triangles according to angles

The Story of Pythagoras

Types of triangles according to sides and angles

Types of quadrilaterals

The family tree of quadrilaterals

Types of polygons

Diagonals of polygons

Sum of the angles of polygons

The Circle

Parts of a Circle

Relative Positions between a straight line and a circumference

Relative Positions between two circles

Circumference of the circle

Lines

What is a line? What is a straight line? What is a curved line?

Positions of a straight line (horizontal, vertical, oblique)

Parts of a straight line: ray, line, segment

Positions of two straight lines: parallel, convergent, divergent

Angles

What is an angle? What are the parts of an angle?: vertex, side

Variety of angles: right, acute, straight, obtuse

Complementary, supplementary, vertical angles:

Adjacent, Complementary Adjacent, Supplementary Adjacent (linear pair)

Angles made by a transversal

Interior, Exterior, Alternate Interior, Alternate Exterior, Corresponding

Measure of an angle in degrees

Adding, subtracting angles using the Montessori protractor

Measure of an angle with a standard protractor

Equivalent Figures with Metal Inset Plates

Equivalence of geometric figure to rectangle

Triangle equivalence to rectangle

Rhombus equivalent to rectangle

Trapezoid equivalent to rectangle

Pentagon equivalent to rectangle

Decagon equivalent to rectangle

Decagon equivalent to rectangles I and II

Equivalence with Metal Inset Plates: Theorems

All triangles having the same base and altitude are equivalent

Pythagorean Theorem

Sensorial Introduction: Plate I – isoceles triangle

Numerical Study: Plate II – scalene triangle; 4:3 sides in ratio

Sensorial Proof: Plate III – general triangle

(also called Euclid ’s Theorem)

Pythagorean Theorem applied to regular figures

other regular figures with constructive triangles

Area of Plane Figures

Area: Concept, Language, and Notation

Area of a Rectangle (given base, height)

Formulas for area base on transformation to a rectangle

Area of a parallelogram (given base, height)

Area of a triangle (double the area) (given base, height)

Acute isosceles; right isosceles; and obtuse scalene

Area of a triangle (bisect the base) (given base, height)

Acute isosceles; right isosceles

Area of a triangle (bisect the height) (given base, height)

Acute isosceles; right isosceles; obtuse scalene

Area of any triangle based on transformation to a parallelogram

More formulas for area based on transformation to a rectangle

Area of a trapezoid (given minor base, major base, height)

Area of a pentagon (given perimeter and apothem

Area of a rhombus (given minor diagonal, major diagonal)

Area of a decagon (given perimeter and apothem)

Area of a circle (given circumference and diameter/radius)

Solids

Nomenclature

Making solid figures

Basic concepts

Regular prisms, transformation into rectangular prisms

Polyhedrons

Lateral and total surface area of solids

Volume of Solid Figures

Concept of Volume

Volume of a right rectangular prism

Volume of non-rectangular prisms

Volume of a pyramid

Volume of a cylinder

Volume of a cone

Volume of a sphere

Story of Archimedes

Geometry Classified Nomenclature

General Purpose – to learn the names, to be used later for identification

And derivation of formulas

Description of materials (set 1, set 2, and control booklet)

Description of activities with nomenclature material

Geometry Charts

Design

Triangle set

Square set

Presentation of the charts

Child’s work with the charts

Geometry Activity Cards

Design

Child’s work with the cards

Sample activity cards

Geometric Design

With metal insets

Technique (how to hold inset; how to draw around)

Designs

Using one piece only (one sample)

Using two pieces only (one sample)

Using three pieces only (one sample)

Using more three pieces (one sample)

Geometric Constructions

With compass and ruler

Technique (how to zero ruler; how to draw a line; how to use a compass)

Designs: Using only straight lines (one sample)

Using only circles or arc of circles (one sample)

With compass and straight edge

Technique (may not measure; show all construction arcs)

Construction (one of each basic construction)