xGeometry :: 6-9 :: Chapter Twoxxxxxxxxxxxxxxxxxxxxxxhome |

xTable of Contents: |

- Chapter One - Sensorial Exploration of Shapes
- Chapter Two
- Knowledge of Plane Figures and Details
- Introduction
- Fundamental Concepts
- Lines
- Types of lines
- Parts of a straight line
- Positions of a straight line
- Horizontal line - curved or straight?
- Straight line in a horizontal plane
- Two straight lines lying in a plane - coplanar lines
- Divergent and Convergent Lines
- Oblique and perpendicular lines
- Two straight lines crossed by a transversal
- Two parallel straight lines crossed by a transversal
- Angles
- Types of Angles and the Parts of an Angle
- Study of angles
- Vertical or Opposite Angles
- Measurement of Angles
- Presentation of the Montessori protractor
- Use of the Protractor
- Operations with Angles
- Addition
- Subtraction
- Multiplication
- Division - Bisecting an Angle
- Exercises in measurement and drawing of angles
- Operations with Angles
- Convex and reflex angles
- Research in the environment
- A new definition of an angle
- Plane
Figures - Difference Between Closed Curve Figures and Polygons
- Classification Exercises
- Classification of closed curve regions and polygons
- A sensorial classification of convex and concave (re-entrant)
- Closed curve regions/polygons - convex/concave
- Classification Exercises
- Polygons
- Triangles
- Triangles - Classification
- Triangles - Parts of the triangle
- Special nomenclature of the right-angled triangle
- Study
of Quadrilaterals
- Classification
- Quadrilaterals - Types of trapezoids
- Quadrilaterals - Classification according to set theory
- Parts of the Quadrilateral
- Polygons
With More Than 4 Sides
- Parts of Polygons
- Comparative
Examination of Polygons
- Study of the apothem
- The polygon in relation to the first circle
- The polygon in relation to the 2nd circle
- The polygon in relation to the 1st and 2nd circles simultaneously
- From Irregular to Regular Polygons
- Construction of Polygons
- Circle
- Level One
- Circle and its parts
- Relationship between a circumference and a straight line
- Relationship between two circumferences
- Relationship between a circumference and a straight line
- Relationship between two circumferences
- Chapter Three - Congruency, Similarity and Equivalence
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xIntroduction |

The two materials used in this
chapter are the box of sticks and the classified nomenclature
of geometry. The box of sticks is the most important instrument
used by the teacher for the presentations, and in the succeeding
work of the child.
red circle - diameter of 10;
corresponding green semi-circumference Box of supplies:
Series A is an exception since it also includes an envelope containing two white pieces of paper picturing a point and a line in red; a red square of paper (surface) and a cube constructed of red paper and dismantled to be stored in the envelope. Note: The line goes off the edges of paper to imply infinity. [top] |

xFundamental Concepts |

While most secondary schools present these concepts starting with the point, the most abstract, and progressing to the solid, reality, we will in the elementary school begin with reality; the concept of the body and go on to the surface, line and point. In the second presentation after the child has worked with these concepts we will present them in reverse order: point, line, surface, solid.
A. The teacher asks the child
to bring one of the three objects to place in the center of the
table. Now put another object in the place of that one. It can't
be bone unless the first object is moved. Try with the third
object. B. Note: The way in which the
quantities of the decimal system were presented is very important
now. The identification has already been made between the unit
and the point, the bar and line, the square and surface and the
cube and solid. The bead had to be held carefully for it is so
small it might roll away. the bar gave an idea of length like
a cane. The square covered the palm of your hand. The cube filled
up the hand so there was room for nothing else.
Note: Use the large dictionary which is gilt-edged; demonstrate sensorially the solid (the dictionary) is made up of many surfaces (pages). Isolate the gilt surface and show that it is made up of many line (edges of paper). One line is made up of many points (particles of gold powder). [top] |

xLines |

[top] B. Parts of a straight line
[top] C. Positions of a straight line
[top] D. Horizontal line - curved or straight?
[top] E. Straight line in a horizontal plane
Hold the plane obliquely. The
plane in space could be in any position, but to facilitate your
work, the plane will always be horizontal, like the surface of
your work table.
[top] F. Two straight lines lying in a plane - coplanar lines
1. Parallel lines
[top] G. Divergent and Convergent Lines
[top] Exercise: Find divergent lines in the environment
[top] H. Oblique and perpendicular lines
[top] I. Two straight lines crossed by a transversal
[top] J. Two parallel straight lines crossed by a transversal
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xAngles |

[top] B. Study of angles
[top] C. Vertical or Opposite Angles
Note: At three different age
levels, there are three different demonstrations to show the
equality of opposite angles: 2. (approx. 8 1/2 years) Take the envelope entitled "vertical angles" from the box of supplies which contain four cards on which 1 - 4 are written. Remove the tacks and number the angles. We must show that angle 1 = angle 3, and angle 2 = angle 4. When writing we can use this notation ^1 to say "angle 1". If we add ^1 and ^2, since they are adjacent and supplementary they will total 1800 (if the child has not learned how to use a protractor we say ^1 + ^2 = 2 measuring angles). Likewise, ^2 and ^3 form a straight angle. Placing a straight edge along the sticks isolates this characteristic, making it more visible to the child. Indicate the angles emphasizing the common angle - the common addend - angle 2. There for ^1 = ^3. Continue in the same way. 3. (approx. 11 years) given a diagram of the angles, we mark them with an arc to show they are equal; a double arc for the second pair. Simply state the textbook declaration: Angles 1 and 3 are both adjacent to angle 2. These are supplementary to the same angle, angle 2. therefore angle 1 and angle 3 are equal to each other. Finish the lesson with classified nomenclature and drawing of the angles. [top] D. Measurement of Angles
Introductory Presentation: [top]
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Bring out the circle fraction
insets of the thirds, ninths, sixths and halves. Taking 1/3,
identify the angle to be measured: the only true angle on the
piece. Recall the nomenclature and identify each part of the
angle: angle, vertex, 2 sides.
[top] s
Operations with Angle
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[top] c. Multiplication
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[top] e. Exercises in measurement and drawing of angles - other protractors
Demonstrate how to draw an angle. Make a point in red. this will be the vertex. From this point draw a ray; this will be one of the sides. Place the protractor so that the vertex corresponds to the center hole, and the side corresponds to zero. Make a mark at the number of degrees desired. Draw a line from the vertex through this mark to make the second side. Write its measure inside. The teacher can prepare command cards which tell the child to construct an angle of a stated number of degrees.
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E. Convex and reflex angle
Note: Up to this point an angle has been defined: "that part of a plane lying between two rays which have the same origin", and five angles were identified: acute, right, obtuse, straight and whole.
[top]
Child One: "Acute, acute
..... right, obtuse .... straight-silence ..whole Note: With this exercise the child realizes that the straight angle and the whole angle do not fit the definition of convex or reflex angles. [top]
Here always the classification will depend on one's point of view. which is more common in the environment; make a list convex: the corner of a table, shelf or cupboard reflex: a chair, sofa, a corner of the library or of the room Try to form different angles using your body.
[top] G. A new definition of an angle: (A new definition of an angle) Invite child
to draw an angle on a piece of paper, the sides reaching the
edge, effectively dividing the paper in half. Review definition
of a plane and on a slip of paper write, "An angle is",
and below it a label reading, "a part of a plane".
Planes go on for infinity but this plane is limited by what?
The angle. And what constructs this angle? Two rays. Write on
a new slip "limited by two rays" and place below previous
slips. Can these two rays exist anywhere on the plane? No, they
must share a common vertex. Write this on a final slip thus finishing
the new definition: "An angle is a part of a plane, limited
by two rays sharing a common point of origin". Stack sheets
of paper above and below to show the child that the angle exists
in that plane only.
Presentation [top] |

xPlane Figures |

[top] Classification Exercises
Note: At a later age, these figures will be classified according to the presence of convex or reflex angles.
At the end the teacher isolates the concave figures and their classification cards. We will only be interested in convex figures. When we talk about closed curve regions and polygons, we can assume from now on that they will be convex. One exception will be considered later. [top] |

xPolygons - Triangles |

[top]
The triangles just constructed
are lain out in two rows as they are in the drawer of triangles:
the top row is classified according to sides, and the second
row according to angles.
[top] B. Triangles - Parts of the triangle
[top] C. Special nomenclature of the right-angled triangle
Note: The presentation which should follow is "Points of Concurrency" which would be a study of the meeting point of the the three heights, that of the three medians, and the orthocenters. [top] |

xStudy of Quadrilaterals |

[top] B. Quadrilaterals - Types of trapezoids
[top]
[top] D. Parts of the QuadrilateralNotes: The nomenclature of quadrilaterals, and other polygons with more than four sides, will present some difficulties, because, unlike the triangles, not all quadrilaterals have the same nomenclature. Also the stick figures must be used in order to identify the diagonal.
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xPolygons With More Than 4 Sides |

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xComparative Examination of Polygons |

Geometry cabinet: presentation tray, drawer of polygons Inset of a triangle inscribed in a circle Fraction inset of the square divided into four triangles A special cardboard square (10cm diagonal and black dot in center Box, entitled "Apothem", containing white cardboard circles, each radius is drawn in red and each circle corresponds to a regular polygon: Polygon Sides Radius triangle 3 approx. 2.5cm square 4 approx. 3.5 pentagon 5 approx. 4.0 hexagon 6 approx. 4.3 heptagon 7 approx. 4.5 octagon 8 approx. 4.6 nonagon 9 approx. 4.7 decagon 10 approx. 4.8 Note: All of the measurements of the radii are irrational numbers with the exception of that which is relative to the equilateral triangle. In that case, the radius of the inscribed circle is half of the radius of the circumscribing circle. Theorem: The area of the circumscribing circle is four times the area of the inscribed circle. Note: The two new figures of the equilateral triangle and the square are introduced, because those same figures found in the presentation tray have sides of 10cm. Therefore it is impossible for the two original figures to be circumscribed by a circle which has a diameter of 10 cm.
[top] B. The polygon in relation to the first circleRemove the inset of the circle from its frame. Place the inset of the triangle in this frame. The circle holds it inside (circumscribe: Latin circum - around, scribere - write or draw).
Place the two insets back to back to see this in another way.
Hold the two insets back to back using two fingers of one hand
on the little knobs. We can see that the center of the triangle
coincides with the center of the circumscribing circle.Place the triangle inset in the circle frame. Now we can try to find the radius. It is the segment which joins the center to one of the vertices. This radius is also the radius of the circumscribing circle. The teacher identifies the radius of the figure, then removes the triangle from the frame, inviting the child to identify the radius without the help of the circle. [top]
Take the white cardboard circle
and superimpose it on the back of the triangle inset so that
it is inscribed. Then rotate the circle so that the radius is
perpendicular to the side of the triangle. [top]
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xFrom Irregular to Regular Polygons |

Note: For the child to understand the characteristics which determine regularity or irregularity of a polygon, we must examine more than one family of polygons, since the triangle will not be sufficient.
Note: this use of conjunctions
[top] Materials for Exercises: additional inset figures envelope containing cardboard figures "convex polygons" 3 irregular pentagons, hexagons, octagons, nonagons,decagons: one for each of the irregular classifications box "regular and irregular polygons" several cords or circumferences (for making sets)
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xConstruction of Polygons |

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xCircle - Level One |

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[top] D. Relationship between a circumference and a straight line
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