
xGeometry :: 69
:: Chapter Two 
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
 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 (reentrant)
 Closed curve regions/polygons
 convex/concave
 Polygons
 Triangles
 Triangles  Classification
 Triangles  Parts of the triangle
 Special nomenclature of the
rightangled 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
 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
[top]

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.
The classified nomenclature codes the concepts given in the presentations.
It is a bank of information to which the child will refer constantly.
Etymology continues to play a crucial role, as we take into consideration
the different psychological realities of the child from 3  6
years old, and the child from 6  12 years old. From 3  6, the
child has a drive to know "things", while from 6 
12, the child has a drive to know "the reasons of things".
Therefore at the level of language, the "thing" is
given by its name, and the "reason of the thing" is
given with the etymology of the name.
Materials:
Box of sticks
Accompanying box of supplies
Classified nomenclature
Wall chart of pictures and labels
Description of materials: Box of sticks
Eleven series of sticks, the first ten of ten different colors
and ten different lengths; the last is a series of varied natural
sticks.
dark brown  2cm light brown  12cm
violet  4cm green  14cm
orange  6cm pink  16cm (lengths are hole to hole)
red  8cm blue  18cm
black  10cm yellow  20cm
Eight of each of these ten series have holes at the extremities;
two of each have holes all along the length. The natural sticks
are of ten different lengths. These serve to construct rightangled
isosceles triangles whose hypotenuse are equal to s 2. Therefore
these sticks have the lengths of 2 2, 4 2, 6 2, 8 2 and so on.
Three groups of semicircumferences and three circles.
red circle  diameter of 10;
corresponding green semicircumference
silver circle  diameter of 7; corresponding orange semicircumference
white circle  diameter of 5; corresponding blue semicircumference
Box of supplies:
four different colors of thumbtacks
red upholstery nails
brads
three crayons  red, blue, black
Classified Nomenclature
for Geometry:
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.
Materials:
A box  parallelepiped, a cancylinder, a ball  sphere
The Geometric solids and the stands
Classified nomenclature
Decimal system materials: cube, squares, bars and beads
Pencil sharpener, pencil, paper
Presentation:
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.
We see that we cannot place an object in a place occupied by
another object. Everything that occupies a space is called a
solid. This box has flat surfaces. The ball has curved surfaces,
so it is the opposite. Since the can has flat surfaces and curved
surfaces, it can be placed between them. Examine the geometric
solids, naming them as they are divided into three groups: 1)
cube, squarebased parallelepiped, regular triangular prism,
squarebased pyramid, regular triangular pyramid; 2) cylinder,
cone; 3) sphere, ovoid (or ovaloid), ellipsoid.
Touching the objects lightly, the teacher compares the surface
to a very thin veil of paint, or to a piece of paper. With an
inset or an object, the teacher lightly runs her fingers over
the surface: this is the surface. We can touch anything in the
environment and label it a surface; table, wall, face, the globe.
The concept of surface is infinite. It goes on in all directions
to infinity.
The teacher runs a finger along the edge of the box. This is
a line, this is another line. continue identifying lines on other
objects in the room, on the geometric solids. In reality the
line as a concept has only one dimension, and it is infinite
in both directions. Let's try to draw a line. Sharpen the pencil
well. Invite the child to draw a line. That's too thick. Sharpen
the pencil more and try again. This is okay, but it's still too
thick. It should have no thickness at all. And it should be much
longer. It should to out the door, across the fields, into the
woods; and in the other direction as well, past the deck, through
the field, down the street.
On the corners of the box, identify a point, another point. What
is a point? It is nothing, but it is similar to a grain of sand,
a particle of dust, a grain of pollen.
Let's try to draw one. Be sure the pencil has a sharp point.
Invite the child to make a dot. That's too big, try again. It
is still too big. The concept of a point is that element that
has no dimension at all. Note: Only the solid has a definition,
the others can only be compared to things.
Bring out the special envelope for this series and construct
the cardboard cube. The teacher identifies the solid, and the
surface. Turn and bend the surface  it's still a surface. Identify
the line and change its position  it's still a line. Identify
the point. The child matches the labels and definitions using
the wall chart and control booklet for control.
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.
The teacher places a unit bead on the table. This is a bead,
a unit; it is a point. Add another point, still another point...
I've made a line. The point is the constructor of the line.
Replacing the beads with a bar of ten, the teacher identifies
the line. Other lines are added, making a column of bars. I've
made a surface. The line is the constructor of the surface.
Replacing the bars with a square, the teacher identifies the
surface. The squares are stacked; this is another surface, still
another surface... I've made a solid. The solid is made up of
many surfaces. The surface is the constructor of the solid. In
reality, the point, therefore, is the constructor of the solid.
Imagine that this point (the bead) is on fire, like the end of
an incense stick in a dark room or the phosphorescent plankton
at night. Roll the bead. When it moves I see a line. This moving
point has made a line before my eyes.
Imagine that this line (bead bar) is on fire. Hold it a one end
and quickly move it back and forth. This moving line determines
a surface.
Imagine that this surface is on fire. In moving the surfaces
I create a solid.
The point is the constructor of reality. Take an object from
the environment, a box or can. All things in reality are limited
by surfaces. The surfaces, in turn, contain lines. the lines
are made up of points, and the point is made up of nothing. Technically
the point has no dimensions.
The same identification can be made of unit beads, when placed
far away the cube itself appears to be no more than a unit bead.
The work with the classified nomenclature involves the picture
cards in the folder, with reading labels and the same definitions.
Direct Aim: To furnish the fundamental concepts: point, line,
surface, solid.
Indirect Aim: To furnish the concepts of all plane figures
(which are simply surfaces) and all geometric solids (which are
simply solids) in preparation for the study of area and volume.
Age: After 6 years
Note: Use the large dictionary
which is giltedged; 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 
A. Types of lines
Materials: A string attached to two small spools
Presentation: The teacher, unseen, places one spool
in each hand and closes his fists so that the spools cannot be
seen. The string passes between the fingers of each hand. Placing
the fists together, the children are invited to watch. The teacher
unrolls the spools as the hands are separated.. this is a line,
this is a line... As the line grows, the teacher changes its
position continuously  horizontal, vertical, oblique  still
identifying it only as a line.
Finally, with the string taut, this is a straight line; and with
it drooping, this is a curved line. "A line" cannot
exist by itself; it must be straight or curved or a combination
of both. Any line that I make will have one of these qualities.
The concept of a line goes infinitely in both directions.
Exercises: Classified nomenclature and commands
[top]
B. Parts of a straight line
Materials:
...Two strings attached to spools
...Two scissors, a red felt pen
Presentation: Again the teacher prepares the spools
in her hands and invites the child to watch as the line appears.
What is it? A line...a straight line... a straight line.... Invite
the child to find a point on this straight line and mark it in
red. The child then takes the scissors to cut the line at the
point... this is a straight line... (cut). Taking one spool and
the point, the teacher extends the string... this is a ray...
this is a ray. The teacher identifies the other ray in the same
manner. These two rays are equal. a ray starts from a point and
goes on to infinity. This red point is the origin. Therefore
a point divides a straight line into two rays.
With the other spool the situation is repeated, with child identifying
a straight line. This time the child is invited to make two red
points on the line, and to cut the line at the two points simultaneously.
Before the cut, the line is identified as a straight line. After
the cut the teacher takes on piece at a time. This is a ray.
This is another ray. this is a line segment. The two red points
on this line segment are called endpoints, because we can't tell
which is the beginning and which is the end. Therefore two points
divide a line into two rays and a line segment.
Exercises: classified nomenclature  after the
child has put the cards with the corresponding labels, have him
put them in order: origin, ray, endpoints, line segment.
[top]
C. Positions of a straight line
Materials:
...A transparent pitcher and vase
...Water and red dye
...A level, two plumb lines, with a red line (cord)
...A red stick
...A globe
Presentation: Dye the water in the pitcher red
(because red is always used to highlight the subject of a presentation)
and pour some into the vase, which is placed in the center of
the table. Observe the surface of this water and describe it:
it is still. This will be a point of reference.
Agitate the pitcher and place it next to the vase. Let's wait
without saying anything until the surface of this water becomes
like the other. When it is exactly like the point of reference,
the surface can be identified as horizontal. Place the red stick
alongside the vase so that it aligns perfectly with the water.
This stick represents a line which goes on in both directions.
When it has the same position as the surface of the still water,
it is a horizontal line.
Drop the stick in the water and wait until it is still (like
the point of reference). This stick represents one of the lines
that make up the surface of the water. This is a horizontal line.
Remove the stick.
A straight line is horizontal when it follows the direction of
still water (horizontal < horizon: Greek horizon <
horas, boundary, limit; thus the horizon is the boundary
of the visible earth in all directions, where it seems that the
sky touches the water. Bring the children up the hill to see
the horizon.
Hold the plumb line until it is still. This will be the point
of reference now. Get another plumb line and wait without touching
it, without a word, until it is exactly like the point of reference.
Place the red stick along the red cord, so that it coincides
just as the stick on the surface on the water. This straight
line which goes on in both directions infinitely is vertical,
because it follows the direction of the plumb line. This is a
vertical line (vertical: Latin verticalis < vertex,
whirlpool, vortex, crown of the head, summit, highest point,
<vertere, to turn; therefore vertex can be applied
to anything which turns like a whirlpool, or to the highest point,
like the crown of the head or the summit). A vertical line is
one which points to the vertex, that is, the topmost point in
the sky over our heads (zenith). It passes through the center
of the earth and on to the nadir (opposite of zenith). Take the
plumb line and hold it still again. Let's imagine that this is
a straight line which goes on in both directions  up to the
zenith and down through the center of the earth to the other
side. Use the globe to show that a vertical line is relative
to the position of the observer.
Place the points of reference for the two opposite elements in
front on the child. What is the median? Hold the red stick horizontally.
When a straight line follows the direction of the surface of
still water, what is it? A horizontal line. Hold the stick vertically.
When a straight line follows the direction of the plumb line
what is it? A vertical line. Hold the stick obliquely. Is this
straight line like the surface of water? The plumb line?
When a straight line is neither horizontal or vertical, it is
oblique. Turn the stick 360o identifying its position as it turns
 horizontal, oblique, oblique, oblique....vertical, oblique,
oblique, oblique, oblique, horizontal, oblique ...etc. (oblique:
Latin obliquus, slanting, sloping, not straight, not right,
devious) So what is straight, right, and normal? The horizontal
and vertical line. The oblique line runs contrary to the true,
contrary to vertical or horizontal.
Exercises: Classified nomenclature and commands.
Demonstrate use of the level for determining lines in the environment.
[top]
D. Horizontal line  curved or straight?
Materials:
...A globe
...Frame of smallest circle inset
...Knitting needle
Presentation: Invite the child to identify their
hometown on the globe. Place the inset frame on the globe so
that the town coincides with the center of the circle. This curve
(the rim of the frame) represents the horizon for everyone living
here in and around the town. Because we are standing outside
the earth we see the entire horizon as a circle, instead of as
an arcpart of the circumference.
With the chalk mark a point on the floor. Draw a circle around
this center point. Invite the child to stand at the center. What
do you see? Without turning the child can only see an arc, a
part of the circumference.
Reinforce the facts that in these demonstrations the child is
much bigger than the circle on the globe or on the floor, when
in reality it is the reverse. The child is a tiny, tiny point
in relation to the earth which is huge. The curvature of that
arc would be so slight that you would only be able to see a straight
line.
With the knitting needle, hold it so that it forms an arc on
a horizontal plane. Invite the child to identify what he sees
as he lowers his body..... a curved line.... a curved line...
a straight line. At eye level this curved line looks like a straight
line.
[top]
E. Straight line in a horizontal plane
Materials: Box of sticks, supplies, board
Presentation: The teacher tacks a stick onto the
board and identifies the board as a plane; a straight flat surface
that continues in all directions infinitely. Imagine that this
stick is a straight line that goes on infinitely in both directions.
Hold the plane vertically and ask the child to identify the position
of the line as the plane is rotated: 1800...horizontal..oblique...oblique...vertical...oblique...etc.
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.
Tack on two other sticks so that the three positions are represented:
, /, . If these three lines were considered in space they would
all be horizontal (hold the plane at eye level to show this).
Let's consider them on the plane surface. When a straight line
follows the direction of the viewer's body, it's vertical. When
a straight line doesn't form a cross or follow the same direction
of the viewer's body, it's oblique.
Exercises:
1. Leave only one stick on the board. As the plane is rotated
(always horizontally) the child identifies the position of the
line.
2. To understand that these positions are relative to the viewer,
seat two children so that a right angle is formed between their
bodies and the plane. As the plane is rotated, the children simultaneously
identify the line as they view it.
3. The child draws the lines on the blackboard. The criteria
hold true even if the plane is vertical (or oblique) like the
blackboard.
[top]
F. Two straight lines lying in a plane  coplanar lines
1. Parallel lines
Materials:
...Box of sticks, supplies, board
...Figures of children
...Red arrows
First Presentation: Identify the board as a plane. Place
a stick on the board and identify it as a line belonging to the
plane. We've already explored everything we can about this straight
line. Let's see what happens when we add another straight line.
Put another stick on the board so that it neither touches or
crosses the first. We have two coplanar straight lines; they
both belong to the same plane (coplanar: Latin con, together,
planus, plane; thus lying in the same plane).
One stick is fixed to the board horizontally. Taking two small
equal sticks, these are the key to the story. Place them perpendicularly
along one side of the first stick. Move the second stick toward
the first until it meets the guide sticks and fix it there. Remove
the guide sticks, but leave them nearby to remind the child of
their importance.
Place the two indifferent children on either sides of the lines
so that they are walking in the same direction. The expressions
on their faces show indifference. It is as though they don't
even know each other. Move the figures along to the end of the
line and turn them over; make them walk back. They are like two
people walking on opposite sides of the street. They don't care
to know each other. each one stays on his own sidewalk and they
will never have the chance to meet.
We can extend these straight lines to infinity (add sticks of
the same color, fixing them with the guide sticks until the lines
go off the board in both directions) but these two lines will
never meet. Substitute the red arrows for the two children. These
two lines are parallel. They never meet no matter how far we
follow them because they are always the same distance apart (parallel:
Greek parallelos < para, beside, and allelon,
of one another; thus one thing beside another)
Exercises: find parallel lines in the environment
 door frames, fence rails, telephone wires, rows in the garden,
railroad tracks.
Second Presentation (Parallel lines are parallel independently
of their position): Invite the child to construct two parallel
lines and then to identify their position: horizontal. Ask the
child to construct two vertical parallel lines and then two oblique
parallel lines using the same process as before. All are parallel
regardless of their position.
Remove two pairs of parallel lines. Rotate the plane in its horizontal
position as the child identifies the position...horizontal...oblique....vertical...etc.
Whenever we draw two parallel lines, the lines are also horizontal,
vertical, or oblique.
Construct a series of parallel lines, using the same guide sticks
or a pair of longer guide sticks. These are called "fascial
lines" because this was the symbol of fascism, first used
be Julius Caesar and later by Benito Mussolini.
[top]
G. Divergent and Convergent Lines
Materials: Box of sticks, supplies, board
Also 4 figures of children: 2 happy, two sad and 4 oneway red
arrows
Presentation: The two parallel sticks may be left
on the board for comparison. The teacher fixes one stick horizontally.
Two small, but different guide sticks are used to position the
second stick. The guide sticks are set aside.
Place the two unhappy children on the lines. These two children
are very sad. They used to get along very well, but as they went
along in life, the distance between them becomes greater and
greater. (move the figures along the lines) That's why they look
so unhappy.
Replace the figures with one way arrows. These lines go only
in one direction, the distance between the lines keeps increasing.
Place extra sticks at the wide end, showing that the guide sticks
would also need to increase in length. These are divergent lines.
(divergent < diverge: Latin diapart, separatelym
and vergo  to incline 0r  Latin divergare <
devergere, deopposite of con (together)
and vergare, to direct oneself; thus to move away from
each other). This term was coined in 1611 by Kepler to give the
opposite of convergere which means to direct towards each other.
[top]
Exercise: Find divergent lines in the environment
Presentation: Position one stick horizontally on
the board. As in the preceding presentation use two different
guide sticks to position the second stick. Fix the second stick
and set the guide sticks aside.
Place the two happy children at the wide end. As these two go
along, they become closer and closer and happier and happier,
knowing that in the end they will meet.
Replace the figures with oneway arrows. These lines go only
in one direction  toward each other. These are convergent lines.
(convergent: Latin con together, and vergera 
to incline) These lines come from two different points toward
each other to one point.
Love stories in geometry, like those in real life can change.
Place extra sticks at the narrow end to see how these line continue
in their one direction. What happens? After the point of convergency,
these two lines become divergent.
Exercise: Find convergent lines in the environment
[top]
H. Oblique and perpendicular lines
Materials:
...Box of sticks, supplies, board
...Measuring angle
Presentation: Take two pairs of sticks with holes
along the length and connect each pair with a brad at the center.
Let's see how two straight lines can meet. Two straight lines
can meet this way X or (rotate the second pair from an overlapping
position, through the position shown so that the child may see
that they are equal and then on to a perpendicular position)
two straight lines can meet this way + (Note: each pair started
form a horizontal position).
Invite the child to measure the four angles of the first pair
to see if they are right angles. None are right angles. In the
second pair, all are right angles.
When two straight lines meet and do not form angles that are
right angles, the tow straight lines are oblique to each other.
Review the meaning of oblique (deviated, slanting, not right).
When two straight lines meet and form all right angles they are
perpendicular to each other (perpendicular: Latin perpendicularis
< perpendiculum, a plumb line < per, through
and pendere, to hang). this perpendicular line hangs and
goes through the other Note: the Old English word for plumb line
is perpendicle.
Three period lesson with child constructing them.
Exercises:
1. Place a pair of overlapping sticks horizontally with the measuring
angle positioned at the vertex. Ask the child to identify how
the lines are in relation to one another as the top stick rotates
... oblique, oblique... perpendicular, oblique .... as they overlap
again  silence) .... oblique ... etc.
2. the child is asked to take three pairs of sticks and unite
them with brads in this way:
1st pair  both have hole along the length; united at the center
2nd pair  one has holes, the other is normal; united at the
center of the one with holes
3rd pair  both have only end holes; united at one end.
The sticks are lain overlapping in horizontal positions. Using
the measuring angle the child makes the first pair perpendicular
and counts the right angles formed (4). The number is written
on a piece of paper and is placed by the pair. The same procedure
is followed for the second and third pairs. When two lines meet
and are perpendicular to each other, they create four right angles,
or two right angles or one right angle. Invite the child to try
o arrange two perpendicular lines that create three right angles.
It is not possible.
The first pair are two straight lines; the second are a line
and a ray; the third are two rays.
3. With one pair of sticks with holes joined at the center and
placed horizontally on the board, the child is asked to make
the two line perpendicular, checking with the measuring angle.
These lines are perpendicular. The teacher turns the whole thing
450 and measures the angles to check. How are these lines in
relation to each other? Still perpendicular. Before the lines
were horizontal and vertical, now both are in an oblique position.
Do the same with the second and third pairs from the previous
exercises. With the measuring angle, show that right angles are
always formed, regardless of the position of the lines. Therefore
all of these lines are still perpendicular to each other because
the amplitude of the angle didn't change.
[top]
I. Two straight lines crossed by a transversal
Materials:
...Box of sticks, supplies
...Board covered with paper
Presentation: Place one, then another like stick
on the board, having the child identify the number of straight
lines on the plane. Then place a third stick (a different color
with holes along the length) so that it crosses the other two.
Now there are three straight lines on our plane; the third crosses
the other two.
Remove the sticks. Place one horizontally and tack it down reminding
the child that this straight line goes on in both directions
to infinity. Into how many parts does it divide the plane? Indicate
these two parts with a sweeping hand. Place the second stick
on the plane so that it is not parallel. Even this straight line
goes on to infinity. With a black crayon, draw lines to demonstrate
this. Identify the three parts into which the plane has been
divided. The part of the plane which is enclosed by the two straight
lines is called the internal part which we can shade in red.
Above and below the straight lines are the external parts of
the plane because they are not enclosed by these two lines.
Place the third stick across the other two and tack it down where
it intersects. This is a transversal (transversal < transverse:
Latin trans, across, and versus, turned; thus lying
crosswise). Two straight lines cut buy a transversal on a plane
will determine a certain number of angles  how many? Using nonred
or nonblue tacks, identify and count the angles. First conclusion:
Two straight lines cut by a transversal will form eight angles.
Some of these angles are lying in the internal part of the plane,
while others are lying in the external part. Remove the tacks.
Identify and count the angles in the internal part, using red
tacks (same color as the plane). These four angles are interior
angles because they lie in the internal part of the plane. Do
the same, identifying the exterior angles. The four angles are
exterior angles because they lie in the external part of the
plane. Second presentation: Two straight lines cut by a transversal
form four interior and four exterior angles.
We need to divide these eight angles according to different criteria.
Remove the red and blue tacks and identify two new groups using
two other colors: four angles formed by one straight line and
a transversal; and four angles formed by the other straight line
and a transversal. All of the work that we'll be doing involves
pairing an angle from one group with an angle from another group.
We won't be working with two angles from the same group because
that would mean only two straight lines were being considered,
not three. Let's examine these pairs.
Remove the tacks. Using two tacks of the same color, the teacher
identifies two angles. These two angles are a pair of alternate
angles. Recall the meaning of alternate. One is on one side;
the other is on the the other side of the transversal. On what
part of the plane are they? Internal, therefore they are also
interior angles. We combine these two characteristics into one
name: alternate interior angles. Invite the child to identify
the other pair with two tacks of a different color. The child
draws these and labels them.
Remove the tacks. The teacher identifies another pair of angles.
These are a pair of angles that lie on the same side of the transversal.
On what part of the plane do they lie? Internal, therefore they
are also interior angles. We can call these interior angles that
lie on the same side of the transversal. Invite the child to
identify another pair with two tacks of a different color. The
child draws the angles and labels them appropriately.
Remove the tacks. The teacher identifies another pair of angles.
These are alternate angles because they lie on on one side one
on the other side of the transversal. The child identifies in
what part of the plane they lie  external  and their corresponding
name  exterior. These are alternate exterior angles. Invite
the child to look for another pair and identify them with two
tacks of a different color. The child draws the situation and
labels it accordingly.
Remove the tacks. The teacher identifies two angles. These are
a pair of angles that lie on the same side of the transversal.
The child identifies in which part of the plane they lie  external
 and recalls their subsequent name  exterior. Therefore these
angles can be called exterior angles that lie on the same side
of the transversal. The child is invited to identify another
pair using two tacks of a different color. The child copies this
situation and labels it.
Remove the tacks. This time an exterior angle will be paired
in a relationship with an interior angle. The child chooses an
angle, identifying it with a tack. The other angle must be formed
by the other straight line, as you remember, so that three lines
will be involved. The teacher identifies the other angle of the
pair. These are corresponding angles, because they follow a certain
order. Both angles lie on the same side of the transversal, and
each angle lies above its straight line. Invite the child to
identify other pairs using different color tacks for each pair
of angles. All eight angles are used. The child copies the situation
and labels it accordingly. Note: These angles have only one quality,
since the pair is divided among the two different parts of the
plane.
Finish with classified nomenclature and commands. A command might
ask the child to identify the other member of a given pair of
angles.
[top]
J. Two parallel straight lines crossed by a transversal
Materials:
...Box of sticks, supplies, board
...Twelve xeroxed forms showing above
...Paper, pencil, and ruler
Presentation: With the sheet of paper from the
board and the sticks arranged as they were for the last series
of presentations, nearby, construct two parallel (horizontal)
lines crossed by a transversal. Recall construction of parallel
lines from before using the guide sticks. Position the transversal
and fix it with tacks.
Verify that the nomenclature used in the last situation is also
applicable here: internal and external parts of the plane (red
and blue shading is no longer necessary); interior and exterior
angles, and so on, until the four pairs of corresponding angles
have been identified. Since the same nomenclature is used, and
the same angles exist, the old plane and its lines can be removed.
What we must discover now is how the angles which constitute
a pair are related.
Invite the child to choose a pair of alternate angles, identifying
them with a pair of tacks which are the same color. The child
names the pair he has chosen: either alternate interior or alternate
exterior angles. The child then colors these angles on one of
the forms. Note: This is only a sensorial demonstration of congruency.
Cut the form along the transversal. This will always be the first
of the two cuts we must make. Choose one of the two resulting
parts and identify the sides of the angle to isolated. Cut the
form along the side of the angle. Superimpose this angle over
the other angle, sliding it along one side until the vertices
and other sides meet, showing that the two angles are congruent.
Repeat the procedure with the other pair of alternate exterior
angles, and the two pairs of alternate interior angles. Upon
completion, it will be evident that all possibilities for alternate
angles are exhausted, because the angles on the board will each
have a tack.
Use a new form for each pair of angles considered. After each
pair, make a conclusion about the congruency of that pair of
angles. At the end we can make this generalization: There are
four pairs of alternate angles. The angles of each pair are congruent
to each other.
Invite the child to choose a pair of corresponding angles, identifying
them with a pair of tacks. Remember that corresponding angles
are not differentiated by the name interior and exterior, because
one of the pair is lying in the internal part of the plane while
the other is in the external part of the plane. The teacher controls
the choice of angles; both angles lie on the same side of the
transversal; both have the same position with respect to their
line; i.e. they are both above their lines. The child transfers
this situation to a form by shading in the angles.
As before, make the first cut along the transversal. Notice that
this time both angles are still on the same part of the form.
Make the second cut along the line that will divide the two angles,
so that we will be able to superimpose them. Superimpose the
angles as before to demonstrate congruency. We can conclude that:
The first pair of corresponding angles are congruent.
Identify the second, third, and fourth pairs of corresponding
angles. Follow the same procedure, using a new form each time,
to demonstrate that the angles are congruent. Again, in the end
the angles will all be identified with tacks. Generalization:
There are four pairs of corresponding angles. The angles of each
pair are congruent to each other.
This time, instead of demonstrating that a pair of angles are
congruent, we must demonstrate that they are supplementary. This
time we will again have two pairs of interior angles and two
pairs of exterior angles. Invite the child to identify two angles
that lie on the same side of the transversal with a pair of tacks.
The child names the pair which he has chosen: interior or exterior
angles; and transfers them to a form. As before cut along the
transversal to find that both angles are still on the same piece
of paper. We must separate the angles by the second cut. Place
the two angles side by side so that their transversal sides are
adjacent and their nonadjacent sides form a straight line; thus
demonstrating that the two angles are supplementary. We can conclude
that the first pair of (i.e.) interior angles that lie on the
same side of the transversal are supplementary.
As before, repeat for the second, third, and fourth pair, making
a conclusion with each pair. In the end, all of the angles have
been identified with tacks. We can make this generalization:
These are four pairs of angles which lie on the same side of
the transversal. The angles of each pair are supplementary to
each other.
Exercises: The teacher prepares forms with only the parallel
lines. The child in his work completes the form by drawing the
transversal either way. The direction of the transversal cannot
be changed by changing the position of the form.
1. The child uses the forms to demonstrate that each pair of
angles is congruent or supplementary. For each pair he writes
the appropriate conclusion. At the end, he writes the generalization
(as in the presentation).
2. The child works from command cards. For example: Demonstrate
that the angles that constitute any pair of alternate angles
are congruent. Note: In order to do this, the child must have
realized that the two straight lines cut by the transversal must
be parallel. He may choose interior or exterior angles.
Age: After 9 years
Direct Aim: Knowledge of the theorems involved regarding
congruent angles and supplementary angles formed by two straight
parallel lines and a transversal
Indirect Aim: Preparation for a more advanced study of
parallelograms
[top]

xAngles 
A.Types of Angles and the Parts of an Angle
Materials:
...Box of sticks, supplies, board
...Red felt pen (with a long thin head)
...The measuring angle, in its envelope (both sides are colored)
Presentation: The teacher takes two different sticks
 a long one with holes along its length and a shorter one with
just two end holes. Placing the longer one on top of the other
at the center of the board the two sticks are fixed at the lower
end with a red upholstery tack. At this point both are mobile,
so place the shorter one in its vertical position and fix it
there. Now only one moves.
Place the red pen in the last hole of the top stick and begin
drawing a line in a clockwise direction... this is an angle,
this is an angle.... (indicate the angle with one hand moving
around between the sides of the angle) ... (at 3600) this is
a whole angle. It is called a whole angle because I went all
the way around to the same point again.
Watch where I stop this time. Repeat the same procedure. This
is an angle, this is an angle... (at 1800) this is a straight
angle. The line it has made is a straight line.
Repeat the procedure as before, starting from zero. This is an
angle, this is an angle ... (at 900) this is a right angle. Take
out the measuring angle and identify the red part which is the
"measuring angle". Note: this is the child's first
protractor). The measuring angle is a right angle. Demonstrate
that the angle just constructed is truly a right angle by sliding
the measuring angle along one stick until it meets the other
line... a perfect match.
Leave the measuring angle there. Repeat the procedure as before,
starting from zero. This is an angle, this is an angle ... (before
900) this is an acute angle. It is less than the measuring angle.
Leave the measuring angle in its place. Repeat the procedure
as before, starting from zero. This is an angle, this is an angle
... (after 900) this is an obtuse angle. It is more than the
measuring angle.
This red nail is the vertex of the angle. These two sticks are
the sides. They are rays which continue to infinity in one direction.
The inside part of the angle is the size of amplitude.
Using the measuring angle the child measures the size of each
of the angles constructed. The acute angle is less than the measuring
angle; the right angle equals the measuring angle; the obtuse
angle is more; the straight angle is twice the measuring angle;
the whole angle is four times the measuring angle.
To demonstrate the theorem that the size of an angle does not
vary with the length of its sides, place the pen in the second
hole and draw a right angle again. Measure it to see that it
is the same.
Exercises:
1. classified nomenclature
2. Draw angles and cut them out. Classify them using the measuring
angle.
3. Look for the various types of angles in the environment. The
child will notice that most angles are right angles.
Age: Seven years
Aim: To give the first concept of angle, to explore
the different types of angles, and to give nomenclature of the
types and parts of an angle.
[top]
B. Study of angles
Materials: Box of sticks, measuring angle
Presentation: Take four sticks, of which two are
alike. Separate the four into pairs having one each of the like
sticks. Join the pairs with brads to form two angles in such
a way that one of the like sticks is on top and one is on the
bottom.
These are two different angles; they have nothing in common.,
Place the angles so that the like strips are superimposed. Now
these two angles have one side in common. To emphasize that these
two angles have three sides, disjoin the angles, remove one of
the like sticks and rejoin the three with one brad. There are
two angles and three sides; they have one side in common, and
therefore a common vertex.
The two sides which are not in common are not opposite rays (opposite
rays form a straight angle).
These angles are called adjacent (consecutive) angles (adjacent:
Latin ad, to, near and jacere, to lie; thus to
lie near each other) (consecutive: Latin consequa, one
thing that comes after another). Put these angles aside, intact
for later use.
Take four sticks, two of which are alike. As before, form two
angles so that one like stick is on top; the other is on the
bottom. These are two angles; there are two vertices and each
angle has two sides. Superimpose the like sticks, then separate
the sticks and remove one like stick, then rejoin the three.
Now there are two angles, but they have one side in common and
thus a common vertex.
The two sides which are not in common are opposite rays. These
are adjacent angles. Put them aside for later use.
Take four different sticks and connect them to form two different
angles. The teacher then places them, side by side but not overlapping.
There are two angles, two vertices, and each angle has its own
two sides. They have one special characteristic (place the measuring
angle under the sticks to demonstrate), the sum of their angles
is equal to the measuring angle. These two angles are complementary
(complementary: Latin complementum, that which completes
or fills up the other; thus they complete a right angle).
Take four different sticks and join them to form two different
angles. Position the angles. there are two angles, each having
their own vertex and two sides. However there is one special
characteristic. Their sides form a straight angle. The sum of
their angles is 1800. Place the measuring angle to show that
the angle formed is the double of the measuring angle, and therefore
it is a straight angle.these are supplementary angles (supplementary:
Latin supplea, to fill up, more than). The etymology doesn't
help much. The angle formed is "more than" the measuring
angle, in fact it is the double.
Take the sample of complementary angles made previously. The
child identifies them and recalls their characteristics  sum
is equal to 90o. Take also the adjacent (consecutive) angles.
The child identifies them and recalls their principal characteristic
 one side in common.
Take four sticks, two which are alike and join them to form two
angles as before. Slide the two towards each other to make a
90 angle and verify this with the measuring angle. The child
identifies these angles: only complementary. Disconnect them,
remove one like stick and rejoin them. Position them again and
verify with the measuring angle. The child identifies them again
as having both characteristics: adjacent and complementary. These
two angles are complementary adjacent angles.
Take four sticks, two of which are alike. On the board, the supplementary
angles and adjacent (case 2) angle previously made, have been
placed for reference. The child identifies them and recalls their
principal characteristics.
As usual the four new sticks are joined to form two angles: one
like stick on top; one like stick o the bottom. These two angles
are only supplementary. Remove one like stick and rejoin the
angles. These two angles are supplementary (because their sum
is a straight angle) and adjacent (because they have one side
in common) They are supplementary adjacent angles.
Finish with a three period lesson. The child works with classified
nomenclature. They may draw, cut, paste and label the angles.
[top]
C. Vertical or Opposite Angles
Materials: Box of sticks, supplies
Presentation: The child brings four like sticks
and constructs two angles connecting them with brads. These two
angles have nothing in common. Let's give them a common vertex.
(simply superimpose them without disjoining and rejoining them).
Now let's move the sides of one angle. This side is the opposite
ray to this side. In the same way indicate that the other two
are opposite rays. Since this setup isn't too steady, we can
reconstruct them using two sticks with holes along their lengths.
Join them with a brad somewhere near the center and tack two
of the ends down. (note: they are oblique lines  not perpendicular)
Identify the four angles that have been formed using 4 tacks
of the same color from the box. This angle is on the opposite
side of the vertex from this angle. These two angles have a common
vertex. In the same way identify the other pair of angles as
being opposite the vertex from one another. The characteristics
of opposite angles have been identified: they have a common vertex
and their sides are opposite rays.
Note: At three different age
levels, there are three different demonstrations to show the
equality of opposite angles:
1. Sensorial demonstration (7 1/2 years) On a piece of paper
repeat the situation of the sticks, using these to trace two
lines in black which go off the page. This line represents one
stick; this line represents the other. Note the two angles indicated
by red thumbtacks and color them with crayon on the paper. Do
the same for the other two angles. Cut our plane (the paper)
along one of the black lines, thus dividing the plane into two
semiplanes. Take one of these and cut along the ray, dividing
the plane into three pieces.
We must satisfy that this (loose) angle, which is colored red
is equal to its opposite angle, shaded in the same color. Superimpose
the loose angle at the edge, matching the sides. Slide the piece
along that side until everything meets perfectly. In the same
way demonstrate that the other pair is equal.
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:
Note: Up until this point the child has only measured angles
with the measuring angle, and before that with reference to the
right angle (in the geometry cabinet drawer of triangles). Here
we will introduce the concept of degrees.
When we measure things we use certain units of measurement. For
water, we measure the length of an object in inches and feet.
In other parts of the world they use liters, grams and meters.
But when it comes to measuring angles there is a universal system
that was invented a long time ago by Babylonian priests.
"The Story of the Star"
In ancient times there were some Babylonian priests who were
very interested in astronomy and the calendar. These were priests
who were accustomed to sleeping during the day and staying awake
at night to look at the sky. They studied the paths of stars,
and the constellations.
They discovered that in order to see a star in the exact same
place as before, a certain number of days would pass. They counted
and counted again, and by trial and error they arrived at the
conclusion that 360 days would pass from the time a star was
visible in a certain place until it returned to that same place
again. They called this period of time a year.
They made a slight error in their calculations; they were off
by 5 days, 5 hours and 49 minutes. But considering their instruments
that they used for measuring, this wasn't much of a mistake.
This time period of a year was too large to be practical; they
needed a smaller space of time. They counted the dawns from the
appearance of this star to its reappearance and divided the year
into 360 days.
The priests thought that the paths followed by the stars made
a circle. So they divided the circle into 360 parts and called
each part a day. We call these parts degrees.
From that day onward no one has ever changed that measurement,
even though we have found out with our more modern instruments
that they had made a slight mistake.
From this path of the star in the sky, the Babylonian system
of numeration was developed. Their system of numeration was based
on 60. The priests discovered that the circle could be divided
into six parts. A regular hexagon could be inscribed in this
circle, then divided into six equilateral triangles( where two
sides are formed by radii, and the third by a cor equal to the
radius). Taking 1/6 of the path of the star (circumference),
they obtained 60 which is the base of their system.
The Babylonians also gave us their symbol for days which is our
symbol for degrees. Instead of writing the word degree(s) after
a number we simply use this symbol 0. A little circle to remind
us of the path of a star in the sky. Now instead of talking about
angles in terms of their size: being wide, big or small, we will
talk of the amplitude of an angle.
[top]
Materials:
...Montessori protractor and other protractors
...Circle, square and triangle fraction insets
...Ruler and compass
1. Presentation of the Montessori protractor
This is the instrument used to measure angles. Its rim, like
the path of a Babylonian star, is a circle. The Babylonians had
determined that it took 360 days for the star to go around and
come back to the same point; therefore we have divided this circle
with little lines into 3600 (degrees).
With the child start at zero, where the star started, and count
by ones up to 20, by 20's around to 340 and by ones to 360. But
360 is not written there, because we have reached the same place
from where we started zero. There's a line that runs from zero
to the center of the circle. The center represents the point
where the priests were standing in order to see this circle in
the sky.
[top]
2. Use of the Protractor
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.
Holding the knob of the piece, place the vertex on the red dot
which is the center. Place one side down along the black line
so that the side touches zero. Then place the inset piece flat,
so that the side will touch one of the degrees. From zero count
by 20's around to the other side  120. Therefore one third is
120 degrees. The child writes 1/3 = 1200.
Try with 1/9 placing the inset piece on the protractor as before
 verte, side along the black line, surface 1/9 = 400 Go on to
1/6 : 1/6 = 600. This is the subdivision that established their
system of numeration. The star had followed its path for 60 days
of the 360 days which is the whole angle.
With 1/2, identify for the child the angle and the vertex, and
the two sides of the angles. The vertex cannot be placed at the
center ( in the priests room) as before. However the shape helps
to line up the sides. We already know that this is a straight
angle. 1/2 = 1800
Examine a second group of fraction. Bring out the fourths, eigths,
fifths, and tenths. For each of these the second side meets a
little line that is not numbered. Following the circumference
of the circle from zero, count by 20's then by 10's, 5's or 1's
to reach the second side.
The last group consists of the whole and the sevenths.
Remove the whole from the frame. Identify the angle, which is
all of the interior, the vertex and the sides, which extend from
the vertex in all directions. When this inset is placed into
the frame, we cannot determine where the angle begins or end.
The important thing to note is that everything is covered. Therefore
the unit is 3600  the whole angle. It follows the complete path
of the star.
In measuring 1/7, we find that the side does not meet one exact
mark. We can say that 1/7 is approximately 510.
Examine the triangle fraction insets. Begin with the whole triangle.
Choose one angle, position its vertex on the center, one side
on zero and read the measure on the other side. Continue for
the other two angles. All angles are 600 therefore this equilateral
triangle is equiangular as well.
The 1/2 piece has angles of 300, 600, 900. 1/3 has 1200, 300,
300. 1/4 has all 600 angles, just like the whole.
Examine the square fractions insets which are triangles formed
by subdividing the square: 1/2, 1/4, 1/8, 1/16. All of the triangles
have angles of 450, 450, 900. With 1/16 use a ruler to prolong
the side,
Examine the rectangles and squares formed by subdividing the
square: 1, 1/2, 1/4, 1/8, 1/16. All have all 900 angles. Again
a ruler must be used to extend the sides of 1/8 and 1/16.
Ages: around 8 years
Aims: Measurement of angles
Indirect preparation for the sum of interior and exterior angles
of polygons
[top]
Operations with Angles
a. Addition
Materials:
...Circle fraction insets
...Montessori protractors
Presentation: The teacher proposes an example:
1/2 + 1/4. The child isolates the two pieces, measures them one
at a time and notes their measurements 1/2 = 1800, 1/4 = 900.
We could restate the addition in terms of degrees: 1800 + 900.
Place the 1/2 piece in the frame of the protractors and add the
1/4 piece. The result can be read on the protractor where the
nontangent side meets the frame. 1800 + 900 = 2700.
[top]
b. Subtraction
Materials:
...Circle fraction insets
...Montessori protractors
Presentation: The teacher proposes an example 1/3
 1/6. The child isolates the two pieces, measures them one at
a time and notes their measurements 1/3 = 1200, 1/6 = 600. Place
the 1/3 piece in the protractor. From this we must take away
1/6.
1st method: Slide the 1/3 piece counterclockwise the number
of degrees corresponding to 1/6. Read the result where the second
side of the angle meets the frame  600
2nd method: Place the two pieces in the frame  minuend first.
Slide the two pieces counterclockwise until the nontangent
side of the second angle meets the measurement of the first angle
 1200. Take away 1/6. Read the result where the second side
of the angle meets the frame  600.
[top]
c. Multiplication
Materials:
...Circle fraction insets
...Montessori protractor
Presentation: The teacher proposes an example:
1/10 x 8. The child isolates the piece 1/10, measures it and
notes its measurement. 360 x 8. Take the fraction 8 times placing
them in the frame starting at zero and reading the result where
the second side of the last fraction meets the frame.
Another example is 1/10 x 12 which is 360 x 12. Place all of
the 10 pieces on; the angle is 3600. Remove them and place only
two pieces on the protractor; the angle is 720. Add 3600 + 720
to obtain the results. In terms of fractions rather than angles,
this result would be a mixed number.
[top]
d. Division  Bisecting
an Angle
Materials:
...Full sheets of paper
...Compass, ruler
Exercise
1) An angle can be divided into two equal parts by folding the
paper so that the two sides of the angle (paper) meet. Recall
the nomenclature of an angle. This fold is called the bisector,
it divides the angle into two equal parts.
2) Draw an angle. With a compass, place the point on the vertex
and mark off two equidistant points on the sides. Open the compass
wider and with the compass point on one side point, then the
other, draw two arcs. A line drawn from the vertex to the intersection
of these two arcs is the angle bisector. Therefore the two angles
are equal.
Aim: Preparation for geometric constructions
[top]
e. Exercises in measurement and drawing of angles  other
protractors
Materials: Various protractors  circular protractors
(small enough to fit in Montessori protractor), those having
clockwise numeration, counterclockwise; also semicircular protractors
of various sizes
Presentation: Introduce the other protractors,
comparing the first to the Montessori protractor by placing it
inside the frame and aligning the degree marks. Show the others,
reinforcing the theorem that the amplitude of the angle does
not vary with the length of its sides.
Demonstrate the use of the protractor in measuring angles. Place
the center hole over the vertex of the angle. Align one of the
sides to zero, by sliding the compass around like a wheel. From
zero, read the numeration progressively until the place where
the second side corresponds to a degree mark. The child writes
the angle measurement inside the angle, with degree symbol in
red. the teacher can prepare many angles out of cardboard which
the child traces and measures.
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.
Age: About 8 years
Aim: Use of standard protractors, measuring and
constructing angles, and operations with angles, including constructing
a bisector
[top]
E. Convex and reflex angles
Materials: Box of sticks, supplies, board
Red pen (with a long narrow head)
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.
Presentation: (By size) Invite the child to choose
two sticks and fix them on the board as usual for making angles
(the longer stick on top of the shorter and secured at one common
end with the upholstery tack; the shorter is fixed at the other
end as well). This pair of sticks should be toward the left side
of the board, to leave room for the next pair.
Ask the child to make an acute angle or a right angle or an obtuse
angle using the red pen. Identify the angle that is constructed.
This angle is a convex angle. Color the angle. Because the child
was given the choice of three angles, we can say that all three
angles  acute, right, or obtuse are convex angles.
Choose two sticks identical to the first pair and fix them on
the board. This time draw an angle which is greater than a straight
angle, but less than a whole angle. This is a reflex angle. Color
the angle with a different color (convex: Latin convexus,
rounded or bent, curved, crooked) (reflex: Latin reflexus,
bending back). Do a threeperiod lesson.
Presentation: (By sides): Begin from where the
last lesson left off. Recall the nomenclature of the convex angle:
vertex, side, side. ask the child to take two sticks just like
the ones on the board and place them so that the sides of the
angles are extended (prolonged). The existing sticks represent
rays, therefore they extend the sides in the opposite direction.
Place the loose sticks in position without fixing them. This
green stick is a prolongation of this side of the angle, etc..,
for the reflex angle as well.
Consider the first angle. Indicate the colored region. What type
of angle is this? Convex. Do the prolongations of the sides fall
inside this angle? No. We can conclude that the prolongations
of the sides of a convex angle do not lie within the angle itself.
Consider the second angle with the same questions. Conclude that
the reflex angle does contain the prolongation of the sides.
Do a threeperiod lesson.
(Organization of the definitions) Combine the two viewpoints
of size and sides. An angle whose amplitude is less than that
of a straight angle is a convex angle. It does not contain the
prolongation of its sides. An angle whose amplitude is less than
a whole angle but greater than a straight angle is a reflex angle.
It does contain the prolongation of its sides. .
[top]
Classified nomenclature
Exercise: With only one pair of sticks fixed
on a clean surface, slowly construct the spectrum of all of the
angles while on child identifies them in the old way, the other
in the new way:
Child One: "Acute, acute
..... right, obtuse .... straightsilence ..whole
Child Two: "Convex, convex ...convex, convex .. silence,reflex
..silence
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]
F. Research in the environment
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.
Age: After nine years
Direct Aim: Knowledge of convex and reflex angles
Indirect Aim: Preparation for convex and concave
polygons
[top]
G. A new definition of an angle
Presentation: (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.
[top]

xPlane Figures 
Materials: Box of sticks, supplies, board Red cord Paper
and scissors, red pen Labels for writing
Presentation: The teacher places the red cord on
the board so that a simple open curve is formed. What is this?
It is a curve that has a beginning and an end. Which is the beginning
and which is the end? I can't tell, but what is important is
that my hand can go all around, inside and out without ever leaving
the plane. This is an open curve; it is like an open gate.
The teacher ties the two ends of the cord and places it on the
board again so that it forms a simple closed curve. Again the
teacher moves her hand around the figure on the plane: outside
the figure. The teacher lifts her hand and touches the plane
inside the figure and identifies it  inside the region of the
figure. Identify the internal and external parts of the plane.
This is a closed curve region. We can also add the quality "simple"
since it is not overlapping (demonstrate this) Thus, it is a
simple closed curve. Set it aside.
The teacher places a single stick on the board, moves her hand
all around it on the plane. A second stick is joined to this
one and they are placed on the board forming any angle. The teacher's
hand still can move all around the angle. A third stick is united
to these two and placed on the board; the teacher moves her hand
all around it. This is a broken line. It is open; it does not
form a region. Since it is straight in places, it is not a curve.
It is like a line that has been broken.
The three sticks are united to form a triangle. This time my
hand must go around the figure, but I must lift it to go inside.
Identify the internal and external regions. We can call this
figure a polygon, which is just a general name.
Place the simple closed curve region (cord) and the polygon (stick)
side by side on the board. The polygon is the opposite of the
closed curve region. These are the two big families with which
we will be working.
To reinforce the concepts just given, the teacher draws figures
in red on the paper and invites the child to cut them out. (remove
the cord and sticks)
We can classify all the figures into two groups.(place out a
label which says figures (or regions)) Beneath this heading place
the two subheadings  closed curve regions; and polygons. Sort
the four cutouts into these two groups.
All of these are figures (regions). They can be closed curve
regions, limited by a curved line. They can be polygons limited
by broken lines.
[top]
Classification Exercises
Materials: Geometry cabinet insets
Additional insets for the geometry cabinet
Two concave figures constructed previously with the child now
in red on laminated cardboard
Seven classification cards
A stick from the box of sticks
A. Classification of closed
curve regions and polygons
The teacher places out the label "Figures or Regions"
and below it: "Closed Curve Regions" and "Polygons".
The child is invited to classify all of the figures into these
two columns, emptying the cabinet, drawer by drawer, placing
the figures into the two groups. On a large sheet of paper the
child draws the result of this classification, drawing each figure.
B. A sensorial classification
of convex and concave (reentrant)
The teacher takes any convex polygon and the stick and passes
the stick over the surface of the inset saying  internal...
internal... internal. The internal region is not interrupted
by any part of the external region. This is a convex figure.
the teacher repeats the experience with any convex closed curve
figure and arrives at the same conclusions. Taking a concave
polygon, the experience is repeated  internal...internal....internal...
but, look... the stick touches an internal part, then an external
part and then an internal part again. This is a concave (reentrant)
figure. The experience is repeated with a concave closed curve
region, and the same conclusions are made. The teacher places
out the label "Figures (Regions)" and below it "convex"
and "concave (ReEntrant)". The child is invited to
classify these four figures. Notice that in each column there
is one closed curve figure and one polygon. the child continues
classifying all of the other figures. All of the figures will
end up in the convex group except the three flowers. These three
figures are a special case. It seems that we should classify
these flowers as concave, but concave flowers have a very different
shape (this will be shown later). these flowers are convex. The
child copies the situation on a large piece of paper.
Note: At a later age, these
figures will be classified according to the presence of convex
or reflex angles.
C. Closed curve regions/polygons
 convex/concave
The teacher places out the heading card  "Figures (Region)"
and below it  "Closed Curve Regions" and "Polygons".
Below each of these subheadings is placed "Convex"
and "Concave (ReEntrant)". the child classifies each
figure, making four columns. The child copies the situation on
a large piece of paper.
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 
A. Triangles  Classification
Materials: Box of sticks, supplies, board
Measuring angle (additional materials listed later)
Presentation: The teacher chooses three sticks
randomly  a, b, c. The second group of sticks containing a,
a, b. A third group of sticks are all equal  a, a, a. Invite
the child to unite the sticks of each group to form a polygon.
Notice the color of the sticks. These triangles are all related.
Identify each triangle. This is a scalene triangle, because all
the sides are different. This is an isosceles triangle, because
two sides are the same. This is an equilateral triangle, because
all three sides are the same.
Note: The child already knows the names and the etymology. The
new experience here is the construction of the triangles.
The isosceles triangle has two equal sides. The equilateral triangle
has two plus one... three equal sides. This means that the equilateral
triangle is also isosceles, but an isosceles triangle is not
equilateral. An equilateral triangle has two equal sides like
the isosceles triangle, but it has something more  the third
side is also equal, thus it is equilateral. Set the three triangles
aside.
The teacher takes two sticks: orange (6cm) and red (8cm). These
are united and placed on the board to form a right angle. Use
the measuring angle to verify this, and leave the measuring angle
in position. Let's find the stick that will join these two sticks,
without altering the position of the first two sticks, without
altering the angle. The stick that fits is the black (10cm).
Join the three sticks to form a triangle and place it on top
of the measuring angle.
Note: This 6, 8, 10 triangle is one of the Pythagorean triples
 a series of three numbers which satisfies the Pythagorean theorem:
a2 + b2 = c2. The Egyptians discovered the first triple: 3, 4,
5. With the box of sticks only 6, 8, 10 and 12, 16, 20 are possible
(others: 5, 12, 13 and 8, 15, 17).
The teacher takes two different sticks and unites them to form
an angle greater than the measuring angle. The third stick is
found to unite them. After checking the angle to be sure that
it is greater than the measuring angle, remove the measuring
angle.
The teacher takes three different sticks and unites them. With
the measuring angle and acute angle is formed, and the third
side is added and united. This time we must check the other two
angles to be sure that they are also acute angles. Measuring
just one angle is not enough. Since all three of the angles are
smaller than the measuring angle, that is, acute angles, this
is an acuteangled triangle.
Identify the others as well, stating the number of characteristic
angles. Note that the rightangled triangle and the obtuse angled
triangle each had two acute angles. An acuteangled triangle
however must have three acute angles.
[top]
Materials:
Geometry cabinet drawer of triangles, reading labels
Paper for making labels
Second level reading labels:
rightangled scalene triangle
obtuseangled scalene triangle
acuteangled scalene triangle
rightangled isosceles triangle
obtuseangled isosceles triangle
acuteangled isosceles triangle
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.
Every triangle by its nature has two qualities. One quality refers
to its sides and the other refers to its angles. If I ask you
to draw a rightangled triangle; the triangle you construct will
also be isosceles or scalene. Let's label these triangles as
we have always known them (using the geometry cabinet labels).
For each of these triangles we must add another quality. The
first triangles has been classified according to its sides (copy
the name onto a paper label). Let's classify it according to
its angles. Use the measuring angle to determine the classification.
Write a new label. Place the two labels on top of the figure.
Do the same for the other two figures. Use a protractor to measure
the angles to determine that all of them are equal. Therefore
it is also equiangular. Continue with the triangles that were
classified according to their angles. Classify them according
to their sides determining this by the sticks which are different
colors; thus the sides are different lengths. Write two labels
for each, as before. Now we no longer have some triangles classified
according to sides and others according to angles. All have been
classified by both characteristics.
Look over the triangles to see if there are any duplicates. remove
one acuteangled scalene triangle and its labels. There are no
other duplicates.
There is something missing from our series. The teacher constructs
an obtuseangled isosceles triangle. The child is invited to
classify it according to its sides (write a label) and then by
its sides (write another label). Be sure it is not a duplicate.
There is only one other triangle missing. Invite the child to
unite two like sticks and to form a right angle with the measuring
angle. Leave the measuring angle in its position as you try to
find a stick which will unite them. Allow the child to try a
second pair. It is impossible.
Introduce the neutral sticks. Invite the child to lay them in
a stair. This stick (indicating the first) will be used to close
a right angle formed by a pair of sticks of the first series.
Place a colored stick next to it. Go on up to ten.
Invite the child to try the neutral stick that corresponds to
his pair. Unite the sticks and classify the triangle. Be sure
it is not a duplicate.
There are no other triangles in reality. Arrange the triangles
in three columns  scalene, isosceles, and equilateral; then
in three rows  rightangled, obtuseangled, and acuteangled.
Isolate the first triangle and place its labels below it as you
read them. Ask the child to identify the functions of each word;
there are two different adjectives and two like nouns. Since
we only have one triangle we can eliminate one of these nouns.
Tear off the word "triangle" from "rightangled
triangle". Place both adjectives in front of the noun and
read the whole thing. Copy this onto one new long label. Remove
the old labels. Continue in the same way for the others. The
adjective describing the sides is always closest to the noun
because it is the most important.
With the equilateral triangle, isolate the three adjectives and
discard two of the nouns. Begin with "acute" and see
if that quality would precisely indicate this triangle. No, there
are other acuteangled triangles. With equiangular and equilateral,
each one by itself is sufficient to identify this triangle. We'll
use the more common one. We can remember that equiangular refers
to the same triangle.
Take the insets from the drawer of triangles and match them one
by one to the figures made with sticks. Identify each inset as
it was previously known and give it its second quality. By passing
the inset through its frame backwards, we can prove that it is
isosceles.
Two triangles of the geometry cabinet correspond to the same
triangle, so we can eliminate one inset and its frame. However,
there are two triangles which have no inset to match. Bring out
the additional insets. Invite the child to identify each using
the measuring angle. Use the surface cards as you would use the
frame to classify the sides.
These are all of the triangles of the 6  9 classroom. We also
have labels for the 6  9. Invite the child to match new labels
as he puts the triangle insets away. "Equilateral"
is an old label, because the name didn't change.
Exercise: Trace each inset, copy its name and
write the reason for its name.
[top]
B. Triangles  Parts of the triangle
Materials: Seven triangles constructed previously
Drawer of triangles, now including two additional inset All other
triangles in the environment Box of sticks, supplies, including
perpendicular angle Triangle stand
Presentation: The teacher takes the equilateral
triangle as a first example. Touching the surface, the teacher
says  surface. The layer paint gives the concept of the surface.
Using the corresponding triangle of sticks, the teacher runs
her finger around the perimeter. The sticks are the image of
the perimeter (perimeter: Greek peri, about, and metron,
measure; thus the distance around). the teacher indicates each
side, naming each "side". At the end of the plural
form is given "sides". The angles and vertices are
identified in the same way. The triangle inset is standing on
its side, perpendicular to the table as the teacher identifies
"base:; the triangle is turned to identify each new base.
The plural form is given "bases".
Note: This is to prepare for the conclusion  that any side can
function as a base.
Place the triangle inset upright in the stand with reverse side
closest to the groove and facing the children. Hang the plumb
line in the groove to that only the cord is visible. Move the
plumb line along until the cord meets the vertex opposite the
base. Holding the line at the vertex so that a line segment is
formed, the teacher says "height". This is the height
of the triangle in relation to this base. Indicate the base in
the groove.
Take out the perpendicular angle and identify it. Place it in
the groove at the center of the triangle so that one side will
coincide with the plumb line. Repeat the identification of the
height. since the height coincides, we can say that the height
is perpendicular to the base.
How many bases are there? Three. How many heights? Three, same
as the bases. How many sides? Three. We can conclude that the
number of heights is equal to the number of bases which are the
same as the sides. The common characteristics of the height is
that each is perpendicular to its relative base. Continue identifying
the parts: perpendicular bisector, median (medians), angle bisector
(angle bisectors).
Explore the nomenclature of other triangles. For two of the three
bases of the obtuseangled triangle, the height is external and
is perpendicular to the extension of the base. For two of the
three bases of the rightangled triangle, the height coincides
with a side.
Note: This is why the initial presentation should deal with an
acuteangled triangle.
[top]
C. Special nomenclature of the rightangled triangle
Presentation: Take the two rightangled triangles
and identify their parts as before. In identifying its sides,
we can give these sides particular names.
Rightangled scalene triangle  The side which is opposite the
rightangle is called the hypotenuse. The other two bear the
Greek name cathetus. One is longer (major cathetus); the other
is the shorter (minor cathetus). All of the other nomenclature
is the same.
Rightangled isosceles triangle  The longest side which is opposite
the right angle is called the hypotenuse. the other two sides
bear the name cathetus. since they are equal, there is no distinction
between them.
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 
A. Classification
Materials: Box of sticks, supplies
Presentation: The teacher constructs the first
figure using a yellow (20cm), a brown (12cm), a pink (8), and
an orange (6 cm), uniting them so that the yellow and orange
are not consecutive sides and are not parallel. This is a common
quadrilateral (trapezium). the child identifies its principal
characteristic; it has four sides (quadrilateral: Latin quadras
< quator, side, and lateris, side).
Invite the child to build a figure exactly like this one, of
sticks. Superimpose the second one on the first to show that
they are equal. Move the two sides back and forth like the arms
of a balance, stopping where the yellow and orange sticks are
parallel. Use two small sticks as guides to check if they are
parallel. This is a trapezoid. It is a quadrilateral, but it
has something more. It is a quadrilateral which has at least
one pair of parallel sides.
Note: It is important for the child to understand this concept;
a trapezoid is a quadrilateral, but a quadrilateral is not necessarily
a trapezoid.
Invite the child to choose two pairs of like sticks; join them
to form two angles, then unite them to form a quadrilateral in
such a way that two sticks of the same color do not touch. This
is a common parallelogram. It has two pairs of parallel sides
(indicate these pairs). The common parallelogram is a quadrilateral
with two pairs of parallel sides. Is it also a trapezoid? Yes,
but it has something more.
Invite the child to construct with the sticks the same figure
as before  the parallelogram. Superimpose this figure on the
other to show that they are equal. Stand the figure on end and
using the measuring angle as a guide, straighten the sides until
the side coincides with the measuring angle. If one angle of
this quadrilateral is a right angle, then all of the others will
also be right angles. This is a rectangle. It is a parallelogram
with all right angles. The previous figure was just a common
parallelogram, whereas this parallelogram is no longer common.
It has all right angles.
Is the rectangle a parallelogram? Yes, it has two pairs of parallel
sides. Is the rectangle a trapezoid? Yes, it has at least one
pair of parallel sides. Is it a quadrilateral? Yes. Is the common
parallelogram a rectangle? No.
Invite the child to unite four like sticks. This is a rhombus.
It is a parallelogram with equal sides. Is the rhombus also a
rectangle? No. Is it a parallelogram? Yes. Is it a common parallelogram?
No. Is it a trapezoid? Yes. Is it a quadrilateral? Yes. Is the
rectangle a rhombus? No.
Invite the child to reproduce the previous figure. Superimpose
one on the other to show that they are equal. As before, stand
the figure on end to straighten its sides, using the measuring
angle as a gauge. This is a square. It is a parallelogram ...
(is that all I need to say? No) .... with equal sides ... (is
that enough?) .... with right angles. Repeat the definition.
By saying that it has equal sides, the common parallelogram and
rectangle are excluded. In order to exclude the possibility of
being a rhombus, however, another quality must be added  right
angles. Questions similar to those for the rhombus may be posed.
These are all the quadrilaterals of reality.
[top]
B. Quadrilaterals  Types of trapezoids
Materials: Box of sticks, supplies Geometry
cabinet insets, reading labels "square", "rhombus",
"rectangle", "trapezoid" Additional insets:
common quadrilateral, common parallelogram, three trapezoids
(all except the equilateral) Second level reading labels  "common
quadrilateral", "common parallelogram", "isosceles
trapezoid", "scalene trapezoid", "rightangled
trapezoid", "scalene trapezoid", "rightangled
trapezoid", "obtuseangled trapezoid"
Presentation: Isolate the trapezoid previously
constructed with sticks. In order to extend our family of trapezoids,
we must first know its parts. The two sides which are parallel
are bases. Identify the other two as sides. Are the sides equal?
No. This is a scalene trapezoid. Recall the meaning of scalene
from the triangles.
Invite the child to take four sticks, two of which are equal.
Unite them so that the two sticks are not touching each other.
Arrange them so that the two bases are parallel. Identify the
two bases and the two sides. Are the two sides equal? Yes. This
is an isosceles trapezoid.
The teacher takes four sticks (yellow  20cm, black  10cm, orange
 6cm, brown 12cm) and unites them so that the yellow and black
sticks form the parallel sides. Identify the bases and the sides.
Use the measuring angle to show that a right angle is formed
by the perpendicular side. This is a rightangled trapezoid.
The teacher takes four sticks (yellow  20cm, green  14cm, red
 8cm, brown 12cm) and unites them so that the yellow and green
sticks form the parallel sides. Use the measuring angle to identify
the two obtuse angles. They are opposite each other. This is
an obtuseangled trapezoid.
Note: It is not so much the number of obtuse angles, but the
opposite position of the obtuse angles which is important. This
is the only trapezoid which can be divided into two obtuseangled
triangles.
Bring out all of the quadrilateral formed with sticks (common
quadrilateral, the trapezoids, common parallelogram, rectangle,
rhombus, square) and organize them as the child recalls the name
of each. Bring out the old labels and match them to the figures,
This old label "trapezoid" will be of no use anymore,
and discard it. Bring out the new labels and match them accordingly.
Each member of the family of trapezoids has its own characteristic;
therefore they each have a different name.
Match the insets of the geometry cabinet with the figures: square,
rectangle, rhombus. Before the trapezoid was known only as a
trapezoid. Identify its other characteristic: isosceles. Bring
out the additional insets and match them as well.
[top]
C. Quadrilaterals  Classification
according to set theory
Materials: All the quadrilateral insets (of
previous presentation)
Board covered with paper
Blank labels, pen
Presentation: Like the portrait gallery to be found
in the houses of noble families, the family of quadrilaterals
has its own gallery, which has six portraits of the six members
of the quadrilateral family. But instead of walking with our
feet, we have strolled with our minds through this gallery of
the quadrilateral. Now that we have examined these figures one
by one, face by face, we must look at this family of quadrilaterals
together, as a whole. We must look at the interrelationship of
the members of this family. As we stroll along, we notice similar
features in their faces. But in the family of quadrilaterals,
we see increasing perfection as we go through the generations.
This is like looking at the genealogical tree of the family.
With the last descendant of this family the square, we have the
perfect quadrilateral. It is the only quadrilateral which is
a regular polygon. In this family of geometry we see increasing
perfection in the last descendants. This is in contrast to some
noble families where we see degeneration in the last descendants.
Place all of the insets on the board in a random group. Draw
a circle around the figures. This is a gallery. All of these
portraits belong to the family of quadrilaterals. Write a label
"quadrilateral" and place it inside the circle. They
are all quadrilaterals because they all have four sides.
Isolate the common quadrilateral and the label to one side of
the circle, draw a circle around the remaining figures. These
are all trapezoids. Place a label inside the circle. "trapezoid"
All of these quadrilaterals have at least one pair of parallel
sides.
Isolate the four trapezoids and the label to one side of the
circle and draw a line around the remaining figures. These are
all parallelograms. Place a label in the circle. All have two
pairs of parallel sides.
Isolate the square and the rectangle and draw a circle around
them. These are rectangles. Place a label in the circle. They
are parallelograms with four right angles. Recall that a square
was included in the drawer of rectangles.
Place the rhombus and draw a circle in such a way that the rhombus
and the square are included in the circle. These are rhombi.
Place the label on the rhombus side of the circle. They are parallelograms
with four equal sides.
Place a label for the square in the intersection of these two
sets. This means that the intersection of the set of rhombi and
rectangles is the square.
It is interesting that at the end of our visit to this portrait
gallery, we see one family member who is perfect and marvellous.
His portrait is placed between the portraits of his father and
his mother. As it sometimes happens in nature, this child has
inherited the best qualities of both parents.
In the set of quadrilaterals, the child is the square. His father
and mother are the rectangle and the rhombus. The square has
the distinctive characteristics of the rhombus: equal sides.
The square is the perfect quadrilateral; it is the only regular
polygon among all of the quadrilaterals.
[top]
D. Parts of the Quadrilateral
Notes: 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.
Materials: Quadrilaterals made previously with
sticks
Insets of the quadrilaterals
Stand, small plumb line, perpendicular angle
Box of sticks, supplies
Presentation: (Square). Isolate the square made
of sticks and invite the child to name it and recall its characteristics.
Let's examine the parts of this figure. the teacher points to
each part as it is named: Surface ...perimeter... Where there
are more than one the teacher names and identifies each singularly,
then gives the plural ... side (sides), angle (angles), vertex,
(vertices), base (bases).. to identify the base, stand the figure
on its side. We can conclude that each side can serve as a base.
Repeat with the inset.
Place the figure in the stand and drop the plumb line into the
opening. Move the plumb line along until it coincides with one
of the sides ... "height" ... Continue moving the plumb
line along whispering ... "height, height, height"
..... when the plumb line coincides with the other side, say
aloud, "height". Place the perpendicular angle beside
the figure to determine that the height is perpendicular to the
base. The principle heights are those two on the sides. all the
segments in between are also heights but in reality they are
all the same height; they are all equal. Repeat the experience
with other bases.
Did you notice that I had to concentrate on keeping the square
erect in this stand? If I let it slant, it is no longer a square.
This figure needs a support. Line up the neutral sticks and find
the one that corresponds to the sticks used for this square.
Join it at two opposite vertices. This segment which connects
two nonconsecutive vertices is the diagonal (diagonal: Greek
dia, through, and gonia, angle; thus the line that
goes through the angle). We can say that for the quadrilaterals
the diagonal is one of the most important elements.
Note: Dr. Montessori suggests that at this point we introduce
the story of construction to show the importance of the diagonal.
It keeps the roof from opening up.
Notice that the diagonal divides the square into two triangles.
The triangle is the constructor of this figure. Place another
stick along the other diagonal to show that the square is divided
into four triangles. Again the triangle is the constructor.
(Rhombus). As before use the stick figure to identify ... surface,
perimeter, side (sides), angle (angles), vertex (vertices), base
(bases) ... In identifying the height, proceed as before this
time staying silent until the plumb line coincides with the first
top vertex ... "height". Continue, whispering "height,
height, height", until the plumb line coincides with the
second bottom vertex. silence prevails until the plumb line coincides
with the extreme top vertex ... "height". All of these
heights are equal. The first principle one is internal. The second
was external, and was perpendicular to the extension of the base.
These are relative to this base.
Repeat the experience with the other bases and those of the inset.
Lastly, identify the diagonals.
(Rectangle). Identify: surface, perimeter, side (sides), angle
(angles), vertex (vertices), base (bases) ... These four bases
are equal in pairs. Identify the height as for the square. Notice
that the heights are equal in pairs. If the base is long, the
height is short, and if the base is short, the height is long.
How many heights are there? Two. Identify the diagonals.
(Common Parallelogram). Identify the same parts as before. Notice
that the bases are equal in pairs. Identify the heights as for
the rhombus, and make the relative observations as for the rectangle.
How many different heights are there? Two. Identify the diagonals.
(Trapezoid). Begin with the most general: the scalene trapezoid.
Examine the parts as before: surface, perimeter, side (sides),
angle (angles), vertex (vertices), diagonal (diagonals). Identify
the two sides which can be bases. They are parallel. The other
two can never be bases. Notice that in all of the parallelograms
all of the sides served as bases. here, however, only two of
the four sides can serve as bases.
Identify the longer (larger) base and the shorter (smaller) base.
The other two sides which are not bases are called legs. (Recall
the etymology and the story of the Greek table)
Since the number of bases has decreased in relation to the number
of sides, the number of heights will also decreases. Identify
the height as before, saying "height" when the plumb
line coincides with the first top vertex, whispering "height,
height" as it moves along, saying "height" as
it coincides with the second top vertex and silence as it continues
along. Repeat with the other base. How many heights are there?
Only one, for they are all equal.
Identify median
 the line segment which connects the midpoints of the two legs;
and joining line of the midpoints of the parallel sides
 the line segment which connects the midpoints of the bases.
Identify the nomenclature of the isosceles trapezoid in the same
way. Notice that the legs are equal. The joining line of the
midpoints of the parallel sides corresponds to one of the heights.
Identify the nomenclature of the rightangled trapezoid. One
of the legs is perpendicular, thus it corresponds to the height.
The nomenclature of the obtuseangled trapezoid is the same as
before. Since one height is external, identify the heights for
the rhombus.
Note: For the following presentation, the common quadrilateral
was enlarged so that the longest side was 20cm. #1  plain, #2
 one diagonal drawn on each side, #3  cut along one diagonal
resulting in two triangles with the black line along one side
each.
(Common Quadrilateral). Identify each part as before: surface,
perimeter, side (sides), angle (angles), vertex (vertices), diagonal
(diagonals) (use #2). seeing that nothing is new, the teacher
asks after identifying each part  "Where's the difficulty"?
Stand up the figure. Where is the base? Is it this? Try all four
of them. We'll realize which is the base when we find the height.
Discover that each base has two different heights. It can't be!
The secret of this figure is that it has no base and therefore
it has no height.
(Solving for the area). When we calculate the area we must multiply
the base times the height. How can we calculate the area of this
figure which has no base and no height? This is where the triangle
becomes important. We must decompose the quadrilateral into triangles
by tracing the diagonal (use #2, then #3, superimpose the triangles
on the quadrilateral to show equivalence). With these triangles,
we have no problem calculating the area because each triangle
has as many heights as bases. Calculate the area of each and
add.
We can conclude that the common quadrilateral has neither a base,
nor a height, thus the diagonal has an important role. When we
must calculate the area of such a figure, we must divide the
figure along the diagonal.
[top]

xPolygons With More Than 4 Sides 
A. Parts of Polygons
Materials: Box of sticks, supplies "Polygons"
drawer of the geometry cabinet
Presentation: Invite the child to choose any five
sticks and write them. The teacher arranges the figures so that
it is convex. Ask the child to count the sides and identify the
figure  5, pentagon. Give the nomenclature as before: surface,
perimeter, side (sides),angle (angles), vertex (vertices). In
order to make this figure stable the teacher attaches one stick
and identifies it as a diagonal. Since it still is not stable,
the teacher adds a second diagonal.
Notice that a triangle did not met a diagonal. The quadrilateral,
having four sides (one more than the triangle) needed one diagonal
to make it stable. This divided the quadrilateral into two constructive
triangles. Here with a pentagon (having one more side than the
quadrilateral) two diagonals were needed. They divided the figure
into 3 constructive triangles. For a hexagon 3 diagonals will
be needed.
Identify one angle. Next to it is a successive angle. The next
angle is not successive, and there is a diagonal. The next angle
is also not successive, and there is the other diagonal. The
last angle is a succeeding angle. A line connecting this (first)
angle with a succeeding angle is merely a side. However, a line
connecting this angle with a nonsuccessive angle forms a diagonal.
Examine the other angles as a focal point. Each time it is possible
to make two diagonals, and three constructive triangles.
Take the polygon from its drawer. Ask the child to identify it
and its particular characteristic  pentagon, having equal sides
(turn it in its frame as proof) thus, a regular pentagon.
Repeat the nomenclature as before; everything is the same. There
are three other parts which only pertain to the regular polygon.
This knob indicates the center of the figure. A line form the
center to any vertex is a radius. Even a line from the center
to the midpoint of any side is also a radius, but it has a special
name: apothem. Three period lesson.
Examine the other figures as well.
Exercises:
classified nomenclature
command cards
[top]

xComparative Examination of Polygons 
(Triangle). What is the distinctive characteristic of the triangle?
what did it have? Everything. This is because the triangle is
the constructor. The only thing the triangle did not have was
a diagonal. It does not need a diagonal because it is stable
without it. All of the angles are successive.
(Quadrilaterals). The common most fundamental characteristic
of quadrilaterals is their diagonal. All, except the common quadrilateral,
had bases, and therefore they had heights. the common quadrilateral
has no base, thus no height. It had to be considered in terms
of the triangles formed by the diagonal.
(Polygons). The polygons have no base and therefore, no height.
The regular polygon had improved nomenclature  center, radius,
apothem. this is true also for the equilateral triangle and the
square, because they are also regular polygons.
A. Study of the apothem
Intuition of the apothem
This was given when the child studied the Parts of the Polygons.
The regular polygons had three elements of nomenclature that
was characteristic: the center, the radius, and a special radius
called the apothem. The presence of two different radii indicates
that two circles are also involved.
Identification of the apothem
Materials:
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.
Presentation: Recall the special nomenclature pertinent
to regular polygons: center, radius, and a special radius called
the apothem. Now we'll find out what's special about the apothem.
The child may remember these three elements of nomenclature,
but at this point he hasn't understood the reasoning.
Present the inset of the equilateral triangle inscribed in a
circle. remove the circle insets and invite the child to classify
the triangle. Verify that the sides are equal by tracing one
side on a piece of paper and matching the other sides. Isolate
the triangle at center stage left and get rid of the frame and
circle segments.
Bring out the square fraction inset. Lift our two opposite triangle
pieces and juxtapose to form a square. Take the red cardboard
square. Superimpose the fraction pieces to demonstrate congruency.
Place the cardboard square next to the triangle and get rid of
the inset, since these constructive triangles have served their
purpose.
Bring out the drawer of polygons and ask the child to line up
the polygons (in a row with the triangle and square) naming them
as he goes along.
We have eight regular polygons, all of which have a center, a
radius, and a special radius  the apothem. Since there are two
different radii, there must be two different circles. Bring out
the circle inset and frame from the presentation tray, and place
it above the series of polygons. This is the first of the two
circles. this series contains the second circle. Invite the child
to put the circles in order from smallest to largest. Place each
circle under a polygon. How interesting! The first circle is
common to all polygons, but each polygon has its own circle.
The triangle has this circle (indicate the circle inset) and
this smaller circle (indicate the corresponding small cardboard
circle) the square has ... and so on, for all polygons.
[top]
B. The polygon in relation to the first circle
Remove 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]
C. The polygon in relation
to the 2nd circle
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.
Because this line segment is perpendicular, it meets the midpoint
of the side. This circle is inscribed by the triangle; it is
contained within the triangle. Hold the triangle and circle as
before to show that the centers coincide.
This line segment which joins the center of the polygon with
the midpoint of one of the sides is that special radius called
the apothem. It is also the radius os the inscribed circle.
(apothem: from a Greek verb meaning to bring down, thus the line
segment is brought down (dropped) from the center of a regular
polygon to one of its sides)
Ask the child to identify the apothem without the aid of the
circle.
[top]
D. The polygon in relation
to the 1st and 2nd circles simultaneously
We've seen that the triangle is embraced by the first circle
and it embraces the second circle. Place the inset in the frame
wrong side up (extend off the edge of a table so the triangle
is flush with the frame) and place the small circle on the triangle
inset. We know that the three centers coincide; they have become
one center.
Repeat the same procedure with all of the other seven polygons.
After we can bring these facts to the child's attention: 1) Each
time the number of sides of the regular polygon increases, so
too, the size of the inscribed circle increases. 2) If the size
of the circle increases, the length of the radius increases.
The length of the radius varies from a minimum of 25 cm for the
equilateral triangle to a maximum of just under 5 cm for the
decagon. 3) As the number of sides of the regular polygon approaches
infinity, as the inscribed circle becomes larger, the radius
approached 5 cm the radius of the circumscribing circle. 4) When
the inscribed circle of the polygon coincides with the circumscribing
circle of the same polygon, the polygon no longer exists; it
is identified in the circles. 5) The two radii coincide, because
since there is no polygon; the circle are one. The radius of
a circle can be regarded as the apothem of a polygon having an
infinite number of sides,
Age: after 9 years
[top]

xFrom Irregular to Regular Polygons 
Materials: Box "regular and irregular polygons"
which contains:
Six special measuring angles: 1080, 1200, 1280 (approximately),
1350, 1400, 1440
Reading labels "equilateral and equiangular polygon",
non equilateral and nonequiangular polygons, equiangular but
non equilateral polygon, equilateral but nonequiangular polygon,
regular polygon, irregular polygon, non equilateral polygon,
nonequiangular polygon, equilateral polygon, equiangular polygon
Cardboard sticks (40 cm long) with a hole at one end only
Box of sticks and supplies, measuring angle
Geometry cabinet drawer of "polygons
Cards for 510, which have the greek roots on the other side
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.
Presentation: Invite the child to build an acuteangled
scalene triangle or any other triangle he'd like except the equilateral
triangle. Classify the triangle which was constructed by calling
attention to whether its sides or its angles are equal: Are the
sides equal? no, this is a nonequilateral triangle. (write the
label in black: nonequilateral triangle) Are the angles equal?
no, this is a nonequiangular triangle. (write another label)
Invite the child to construct with the sticks the triangle which
he was forbidden to build before. Ask the same questions and
write the two separate labels: equilateral triangle, equiangular
triangle. Now each triangle has two labels: one which refers
to the classification of its sides and one which refers to its
angles. Read them. We want to make one label for each. Tear off
"triangle" on one of them. Read what's left. We need
the word "and". Write "and" in red and place
it between the adjectives. Repeat the experience for the second.
Now we can make new labels for each one (no color distinction
for and anymore). Read the labels. "nonequilateral and
nonequiangular triangle", "equilateral and equiangular
triangle". All of the goodness is in one, while all the
badness is in the other. Two negative qualities give an irregular
triangle. Two positive qualities give a regular triangle. Set
these figures aside.
Invite the child to construct a common quadrilateral, any trapezoid,
or a common parallelogram. As before classify the figure by asking:
Are the sides equal? Are the angles equal? After each answer,
make the appropriate classification and write a label. Unite
the two adjectives with and in red as before. Read the parts
and rewrite a new label (all in one color): nonequilateral and
nonequiangular quadrilateral.
Invite the child to build a rectangle. Use the measuring angle
to verify that one angle is a right angle, therefore all are
right angles. Classify the figure as before writing two labels.
Notice that this is the first time a figure has one positive
quality and one negative quality. Tear off the adjectives and
eliminate one noun. Decide on which one shall come first. Read
the two adjectives; we need the work but this time. Write but
in red and place it between the adjectives. Read it and write
new label: equiangular but nonequilateral quadrilateral.
Note: this use of conjunctions
and, but will aid the development of set theory.
Invite the child to build a rhombus. Proceed as before. The label
reads equilateral but nonequiangular quadrilateral.
Lastly the child constructs a square. Proceed as before. Equilateral
and equiangular quadrilateral.
Align the four quadrilaterals and their labels in the order of
which they were presented. Two negative qualities produce an
irregular quadrilateral. One positive quality and one negative
quality still produce an irregular quadrilateral. Two positive
qualities produce a regular quadrilateral. Bring to the child's
attention the progression towards perfection.
Place the six figures just built  triangles and quadrilaterals
on the table with labels. In both families we have the two extremes.
Each family has two opposite figures: one has two negative qualities,
the other has two positive qualities. But only the family of
quadrilaterals has two intermediary figures which constitute
the passage from imperfect to imperfect.
At this point we can begin to make some generalizations: a polygon
is regular when it is equilateral and equiangular at the same
time.
Game: Examine the quadrilaterals. Why is this common quadrilateral
not regular? This rectangle seems to be regular for it has equal
angles. (mention this positive quality first to show the move
toward perfection) Have the child find the bad quality  the
sides are not equal. The sides are equal in pairs. That's not
enough. repeat the procedure with the rhombus. Finally the last
one is perfect; it is regular.
Ask the child to choose five sticks at random and join them.
Identify the figure: 5 sides, pentagon. As before classify the
figure, write the two separate labels. Unite the qualities with
and in red. Rewrite the label nonequilateral and nonequiangular
pentagon". Invite the child to unite five equal sticks and
arrange the figure so that it is not equilateral. Proceed as
before. The last label will read equilateral but nonequiangular
pentagon.
The third figure which must have the opposite qualities of the
last, will be more difficult to build. Present the special measuring
angles. Using the drawer of pentagons insets, we can find the
figure to go with each measuring angle. Lay out the numeral cards
510 in a row. Take any measuring angle and place it on the back
of an inset, matching the angles. The child copies this information
into his notebook.
Remove all of the measuring angles except that which pertains
to the pentagon. Ask the child to take these sticks: red(8),
black(10), brown(12), pink(16), blue(18); and unite them :redpinkbrownblackblue.
Check their angles with the pentagon measuring angle. Classify
this figure as before. equiangular but nonequilateral pentagon.
Ask the child to unite five equal sticks. Control the angles
with the appropriate measuring angle. Proceed as before. equilateral
and equiangular pentagon.
Observe that in the family of pentagons, the same thing happened
as with the family of quadrilaterals. There are two extremes
and two mediators. Identify the irregular and regular pentagons.
The two intermediary figures demonstrate the passage from imperfect
to perfect.
Proceed as before. In the examination of polygons having more
than four sides, we use one with an odd number of sides and one
with an even number of sides.
Notes: When the child is working alone, he may not remember the
colors of the sticks for the third figure, or their order. For
this reason the cardboard sticks are provided.
Unite two sticks and use the measuring angle to form the desired
angle. As each successive stick is added, check the angle formed
with the measuring angle. For the last side, try the sticks from
the box. If one cannot be found, measure off a cardboard stick.
Cut it, punch a hole and attach it. Check the angles with the
measuring angle.
[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)
First Exercise: Ask the child to read the four long
labels; observe that "polygon" has ben substituted
for the name of the family (triangle, quadrilateral, etc.)
Isolate two of these labels "equilateral and equiangular
polygon", nonequilateral and nonequiangular polygon".
Take all of the triangles (from cabinet and from the box of additional
insets) and classify them, making two groups under these two
headings. Since two equilateral triangles are identical, one
can be removed. Also one of the acuteangled scalene triangles
can be eliminated.
Bring out the two smaller reading labels  irregular polygons
and regular polygons and place them accordingly above the headings.
The child can copy this into his notebook, tracing the insets,
or substituting the reading labels and making two lists.
Proceed with the family of quadrilaterals. This time all four
long labels will be needed. Place them in order: negative, 2
mediators, positive. The child takes out all of the insets and
classifies them. (eliminate one square) Place the two small labels
above the headings. "Irregular polygons" includes the
first three columns.
Proceed with the pentagons. The four labels are needed again.
But there is only one inset. Classify it. Bring out the envelope
of Convex Polygons, and invite the child to find more pentagons.
Use the measuring angle to verify classification. Place the two
small labels above the headings. Proceed with the other polygons.
Second Exercise: (Triangles) Place the two circumferences
on the table side by side. Place the labels "equilateral
polygon", equiangular polygon" in one circle, and "nonequilateral
polygon", "nonequiangular" in the other. Invite
the child to place all of the triangle insets in their respective
places (again eliminating the duplicates). Only one is perfect.
Superimpose the two circumferences to make an area of intersection.
What triangle has the qualities of both sets? None. Why? Because
there are no mediators in the triangle family. Place the labels
"regular polygon", "irregular polygon" appropriately.
(Quadrilaterals) With the circumferences side by side, place
the labels "equilateral polygon", equiangular polygon",
one in each circle. Classify all of the quadrilaterals (excluding
the duplicate rectangles). Only a few of our quadrilaterals have
these characteristics. One figure is found in both sets  the
square. Superimpose the circles and place one square in the intersection;
eliminate the other. Place the labels "regular polygon"
 in the intersection; "irregular polygon"  so that
it touches both sets, though separately.
(Pentagons, etc...) Proceed as for quadrilaterals with other
polygons.
The child copies his work by tracing the insets or substituting
the names of the figures.
[top]

xConstruction of Polygons 
Note: This activity pertains to all polygons.
Materials: Different colored drinking straws,
scissors, yarn, Upholsterer's needle
Exercise: Following command cards, the child
constructs the figures, using diagonals as needed for stability.
For sides of equal length, the child should use straws of the
same color.
[top]

xCircle  Level One 
Materials: Box of sticks, supplies, board, red pen
Fraction insets of the circle; whole, half, one other
Inset of the triangle inscribed in a circle
Two wooden circumferences (painted embroidery hoops of two different
sizes, such that the sticks may serve as radii  large red hoop
20cm in diameter; small blue hoop 12cm in diameter)
[top]
A. Circle and its parts
Presentation: The teacher takes any stick and fixes
one end to the board. remember that when we constructed an angle,
two sticks were needed. Here, using one stick we'll construct
something different. Place the red pen in the hole and draw a
circle. Indicate the internal part: this is a circle. It is the
part of a plane enclosed by a very special closed curve. Identify
the center of the circle (tack) and the radius (radius: Latin
rod, spoke of a wheel) of the circle (stick). All of the points
that make up this red line are the same distance away from the
center. This is why this special closed curve line is called
the circumference (circumference: Latin circumferre, to
carry around, circum, around, ferre, bear).
Ask the child to take another stick identical to the first. Fix
them together at the center and arrange them so that they are
opposite rays. This is the diameter (diameter: Greek dia,
through, metron, measure).
Take the longest stick from the box and place it so that it touches
the circumference at its two ends. This is a cord (cord: Latin
chorda, cord, string).
Each of the two parts of the circumference divided by this cord
is an arc. This small one is an arc (arc: Latin arcus,
bow, arch); the large one is an arc.
The diameter is a special cord because it is the longest cord
possible, and it is the only cord which passes through the center
of the circle. The two arcs which result from the division of
the circumference by the diameter are special arcs. They are
called semicircumferences (semi: Latin half). Three period lesson.
Bring out the fraction insets and line them up left to right:
whole, some fraction (3/3), halves, and the inscribed triangle.
Take out the whole inset and place it on the table. This is a
circle. Take out the 1/3. This is a sector (sector: Latin a cutter)
of the circle. A sector is what we call each of the two parts
into which the circle is divided by the two radii. Any of our
fractions can serve as a sector. Perhaps the 1/2 could even be
considered as a special sector. Remove the 1/2 inset. This is
a semicircle. It is exactly onehalf of the circle.
Remove one of the "moon" pieces of the last inset.
This is called a segment (segment: Latin secare, to cut)
of the circle. A segment is the name given to each of the two
parts resulting from the subdivision of a circle by a cord. Perhaps
even the semicircle can be considered as a segment.
Redefine sector and segment a little more precisely. This is
a sector. It is formed by two radii, two radii which are not
prolongations of each other. Thus the semicircle cannot be called
a sector. The segment is formed by a cord, but by a cord which
does not pass through the center of the circle, that is a cord
which is not the diameter. Thus the semicircle cannot be called
a segment either, but it is the limit of these two figures.
The ring of a circle is the last part of the plane enclosed between
two circumferences having the same center. Three period lesson.
Classified nomenclature and commands.
Age: Seven and a half years
[top]
B. Relationship between
a circumference and a straight line
Presentation: Place a stick (straight line) and
the large wooden circle (circumference) on opposite sides of
the board. Move one toward the other, but do not let them touch.
The teacher says, "external, external, external..."
The straight line is external to the circumference and vice versa.
repeat the experience this time sopping when the stick touches
the circumference ... external, tangent (tangent: Latin tangere,
to touch). The line and circumference are touching each other
at one point.
Repeat the experience, this time placing the stick on top of
the circumference ... external, tangent, secant (secant: Latin
secans < seca, to cut). The line cuts the circumference
at two points. Identify the two points of intersection. Three
period lesson.
Note: Even if a short stick were used, the secant would intersect
the circumference at two points, because the stick represents
a straight line which goes on in both directions to infinity.
Exercises: Classified Nomenclature and commands
[top]
C. Relationship between
two circumferences
Presentation: Place the two circumferences on the
board and repeat the experiences of the previous presentation:
external  having on points in common
tangent  having one point in common
secant  having two points in common
Return the circumference to the tangent position. Ask the child:
is the one outside or inside the other? Flip one circumference
over to show that they would look like otherwise. So we can say
that this circumference is external; we can also say that it
is tangent. These two adjectives (external, tangent) refer to
the same circumference. When we combine the qualities, one adjective
becomes an adverb (externally, tangent).
Flip one circumference over. They are still tangent, but now
one is internal. Repeat the transition of the adjectives. These
circumferences are internally tangent.
Return to the first position and repeat the experience using
these new names ... external, externally tangent, internally
tangent. Move the inner circumference so that it is neither tangent
nor concentric. This circumference is internal. It is inside
the other, but there are no points in common.
Move the inner circle so that the circumferences are concentric.
Use two small sticks to check that one is equidistant from the
other all the way around. This is a particular type of internal,
called concentric (concentric: Latin con, with, together;
center). They have the same center. Concentric circles are a
subset of internal. Do a threeperiod lesson.
Exercises: Classified nomenclature and commands
Age: Eight years
[top]
D. Relationship between a circumference and a straight line
Materials:
...Box of sticks, board, measuring angle
...Circumference
Presentation: Place the circumference and a long
stick to serve as the straight line on the board. Allow the child
to experiment to find the stick which may serve as a radius for
this circle. The child chooses the one that looks right.
Move the straight line towards the circumference, external, external
... We will use the radius to put this straight line in relation
to the circumference. We'll choose a radius which is perpendicular
 use the measuring angle to check. Place one finger on the straight
line where a perpendicular would intersect, and place another
finger on the center. When the straight line is external, the
distance between the center and the straight line is greater
than the radius.
Move the straight line to the tangent position. Recall the first
level definition: they touch at one point. When the straight
line is tangent, the distance between the center and the straight
line is equal to the radius.
Repeat the experience ... external, tangent, ... secant. Recall
the first level definition. When the straight line is secant,
the distance between the center and the straight line is less
(shorter) than the radius.
Later we'll learn how to write the symbols for these positions.
Age: Nine years
[top]
E. Relationship between
two circumferences
Materials:
...Box of sticks, supplies and board
...Two circumferences
...Two charts  Internal and Secant
Presentation: Place the two circumferences on the
board and ask the child to find the two sticks which will serve
as radii; place these in them appropriately. Use a long stick
to align the radii in the same straight line. Remove the long
stick.
(External) Place a finger on each center, to show the distance
between the two centers. Is this distance equal to, greater than,
or less than the sum of the two radii? Place two sticks like
the radii, on the board end to end to allow the child to visualize
the sum of the radii. The distance is greater; by how much? find
an appropriate stick.
Two circumferences are external to each other when the distance
between the two centers is greater than the sum of the radii.
This can be expressed in symbols. The symbol /d/ will represent
the distance between the two centers (since d is used for diameter,
we'll use a small delta ). R will represent the longer radius
and r will represent the shorter radius.
Thus > (R + r), or we can consider the sum of the radii. Is
it equal to, greater than, or less than the distance between
the two centers? Less, so (R + r) < . The two statements say
the same thing, but different symbols reflect the different order
of the terms.
(Internal) Refer to the chart. Placing a finger on each center,
the distance between the centers is shown by the green line segment.
The radii are shown side by side on the chart, but the sticks
may be superimposed to leave a portion of the longer stick visible
as the difference. Look at the chart. Is the distance between
the centers equal to, greater than, or less than the difference
between the radii? Formulate a statement about internal circumference
and write the symbolized form. < (R  r) and (R  r) >
.
(Externally tangent) Ask the child to place the two circumferences
together again using the long stick to align the radii. Place
the two extra sticks end to end to show the sum of the radii.
Is the distance between the two centers equal to, greater than,
or less than the sum of the radii? Formulate a statement and
write the symbols. = (R + r) and (R +r) = .
(Internally tangent) Invite the child to position the circumference
accordingly; the radii should be superimposed. Place the two
extra sticks to demonstrate the difference between the radii.
Ask the question as usual, formulate a statement and write the
symbols. = (R  r) and (R  r) = .
(Secant) Refer to the chart. This time we need two series of
extra sticks. Place out a pair to represent the sum and a second
pair to represent the difference (Notice in all other cases we
used one or the other). Begin with the sum. Ask the question,
formulate a statement and write it in symbols in the two ways.
< (R + r)
Discard the sticks for the sum. (R + r) >
Consider the difference the same way. > (R  r)
Using these four statements we can combine them (R  r) <
to make one: (R  r) < < (R + r) or (R + r) > > (R
 r) .
(Concentric) Fix the two radii to the board with an upholstery
tack and position the circumferences concentrically. Invite the
child to use two fingers to show the distance between the two
radii. the two fingers are at the same place; there is no distance.
Recall the meaning of concentric. = 0, which means there is no
distance. Three period lesson. Classified nomenclature and command
cards.
Age: Nine years
[top]

x 

