Download Geometry

Geometry
Geometry
© Copyright, 2004
It is illegal to duplicate the contents of this manual under any circumstances.
Authorization to use excerpts of the presentations and lectures for use in academic
papers may be requested by writing to [email protected]
The Institute for Montessori Education
37 Rockinghorse Road
Christchurch 8062
New Zealand
(643) 382-2023 Phone
© Copyright, 2004
Page 2
Geometry
Table of Contents
I.
Introduction .............................................................................................................. 6
Stories...................................................................................................................... 7
Why Study Geometry?............................................................................................. 9
II.
Initial Presentations ............................................................................................... 13
Preliminary............................................................................................................. 14
“Geometry Cabinet” ............................................................................................... 15
Constructive Triangles – First Series..................................................................... 17
III.
Congruency, Similarity and Equivalence ............................................................... 22
Congruency ........................................................................................................... 23
Similarity ................................................................................................................ 25
Equivalence ........................................................................................................... 27
Constructive Triangles – Second Series: Triangular and Two Hexagonal Boxes . 29
Equivalence with Metal Insets ............................................................................... 38
Equivalence with Pythagorean Insets.................................................................... 46
Pythagorean Theorem with Constructive Triangles............................................... 49
Pythagoras Three (Euclidean Logic) ..................................................................... 52
IV.
Geometry Nomenclature........................................................................................ 57
Classified Nomenclatures ...................................................................................... 58
A)
Fundamental Concepts ................................................................................. 60
B)
Study of Lines ............................................................................................... 64
Positions of a Straight Line (space) ....................................................................... 66
Positions of a Straight Line on a Plane.................................................................. 68
Parts of a Straight Line .......................................................................................... 69
Parallel, Convergent & Divergent Lines................................................................. 70
Oblique and Perpendicular Lines........................................................................... 72
C)
Study of Angles ............................................................................................. 73
Measuring Angles .................................................................................................. 75
Adjacent Angles..................................................................................................... 77
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Geometry
Vertical Angles....................................................................................................... 78
Complementary Angles ......................................................................................... 80
Supplementary Angles........................................................................................... 81
Two Non-Parallel Lines Cut by a Transversal ....................................................... 82
Parallel Lines Cut by a Transversal ....................................................................... 83
D)
Polygons ....................................................................................................... 86
E)
Study of Triangles ......................................................................................... 88
Seven Triangles of Reality..................................................................................... 91
Triangle Nomenclature .......................................................................................... 92
Altitudes/Heights of Triangles ................................................................................ 93
Other Triangle Exercises ....................................................................................... 94
Nomenclature for the Right Angle Triangle............................................................ 96
F)
Study of Quadrilaterals ................................................................................. 97
The Six Quadrilaterals of Reality ........................................................................... 98
The Trapezoid...................................................................................................... 101
G)
Study of Polygons ....................................................................................... 102
Apothem .............................................................................................................. 103
Sum of the Angles of Plane Figures .................................................................... 104
Sum of the Angles of a Polygon Chart................................................................. 106
H)
V.
Study of The Circle ..................................................................................... 107
Area ..................................................................................................................... 112
Introduction .......................................................................................................... 113
Common Parallelogram ....................................................................................... 115
Triangles .............................................................................................................. 116
Right Angled Triangles ........................................................................................ 118
Obtuse Angled Triangles ..................................................................................... 120
Square ................................................................................................................. 122
Rhombus ............................................................................................................. 123
Trapezoid............................................................................................................. 126
Polygons .............................................................................................................. 128
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Page 4
Geometry
The Circle ............................................................................................................ 133
The Area of the Circle.......................................................................................... 137
The Area of the Sector......................................................................................... 140
Area of a Segment............................................................................................... 142
The Ellipsis (Ellipse) ............................................................................................ 144
The Tiling Game .................................................................................................. 148
VI.
Volume................................................................................................................. 153
Volume................................................................................................................. 154
Volume of Other Figures –“Blue Solids” .............................................................. 158
Prisms.................................................................................................................. 159
Pyramids.............................................................................................................. 163
Solids of Rotation................................................................................................. 167
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Geometry
I. Introduction
© Copyright, 2004
Page 6
Geometry
Stories
A series of stories may be told to the six year old to generate interest.
Source: “The Makers of Mathematics”, Chapter II “The Birth of Geometry and the Golden
Age of Greek Mathematics.”
How Geometry Got Its Name
Picture a scene in ancient Egypt thousands of years ago. Life centered around the river
Nile. The Nile River has been called the Mother of Geometry as well as the Mother of all
Mathematics. For centuries the Nile overflowed its banks year by year and the
floodwaters washed down the dark fertile mud of the nearby Abyssinian Mountains.
The name Egypt is a Coptic word meaning “black earth”.
So this black earth came down year after year covering up landmarks and forcing the
Egyptians to make out their landholdings over and over again.
Explain to students that the Egyptians were not interested in why it worked -- only that it
produced the desired results.
Presentation
Demonstrate “Rope Measurement”.
I need 3 slaves. “We are going to mark out our land also that
it is the right size.” DEMONSTRATE. Ah! “This is where our
land markings were!”
The man who was in charge of the slaves was called a
surveyor or harpedonapta. He was responsible for earth
measurements in ancient Egypt.
Thousands of years…
See Thales pg 32 – 33
Some thirty-six years…
See Pythagoras pg 37 –
42
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Geometry
Our story of geometry…
See pg 44 – 45
We have looked at four
possible stories that
could be told to the six
year olds. There are
others. Read the
material and make up
your own.
© Copyright, 2004
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Geometry
Why Study Geometry?
The Historical Approach
The teacher is searching for stories that create interest.
There is an assumption that the child’s development follows the historical pattern of
species’ development.
Early humans through exploration discover how to satisfy their needs. Later they search
for reason and rules that are based on their accumulated practical experiences and
knowledge. Children must also follow this path.
6 – 12
Children from 6 – 12 are interested in analyzing and discovering relationships relevant to
their factual knowledge. Hopefully the work in 3 – 6 has given them a love for geometric
investigation.
Children take great joy in intellectual activity. The Geometry materials at this level are
creative in nature because they give ample opportunity for the children to create their
own abstraction.
The formal laws that govern geometry will principally be given at the secondary school
with the theorems, but we work with the younger children to create demonstrations of
these formal theorems – not using the formal wording, but preparing for the
understanding of formal wording.
Geometric knowledge is not being presented for its own sake but rather for the purpose
of providing a stimulus for intellectual development…
experience with logical reasoning
experience with deduction
experience with forming abstractions
As we proceed through the presentations we see that manual activity continues to serve
children’s intellectual development.
One does not have to follow the same sequence with each child. All areas need not be
covered – just as long as we keep in mind the public school curricula (minimum)
standards.
Review of the 3-6 year olds' experiences
Everything in the environment contains lines, slopes, and angles. So children really
experience Geometry from the moment of birth – these are impressions from the
environment, stored in the mind.
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Geometry
Parents of three to six year olds are encouraged to use geometric terms (nomenclature)
in the home environment. This forms the basis of a “remote” preparation before children
enter the 3 – 6 class. But even if children are not given the terminology, they have
“seen” geometry.
The sensorial material helps children classify, clarify and make concrete the impressions
already received. Children use the Geometry Cabinet quite early. Initially to trace the
shapes.
What are the advantages of tracing?
It provides muscular memory at the sensorial level of the different shapes. (For
example, 4 equal sides, 4 equal angles become engrained if the very young child
experiences the tracing.) However, older children do not come to that level because
their understanding is more visual.
Somewhere between three and four years of age the children
start working with the names. They start with the basics:
triangle, circle, and square.
Contrasting selections
Why do we start with these three?
“Square” – for measuring area
“Circle” – for measuring angles
“Triangle” – for universal constructor
Subsequently we give the names for the other figures:
Some of the regular polygons
parallelograms
trapezoid
rhombus
oval, ellipse, curvilinear triangle, quatrefoil, common
quadrilateral, etc.
Hopefully the children will be exposed to the language of the
Geometry Cabinet in 3 – 6 class. As they move through the
cabinet they work with the cards – generalizing the
experience.
More attuned to small
differences between
shapes
Real Square
Tracing
Solid Picture
Thick Lines
Thin Tracing
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Geometry
Children are gradually led from a recognition using muscular
memory to a point where they can recognize visually
Geometry Cabinet also
is an indirect preparation
for “similarity”.
With the pink tower, broad stair and geometric solids the child
gains a beginning understanding of solid geometry—matches
solids to base outlines to understand connections to plane
figures.
The constructive triangles enable the child to discover the
constructive nature of the triangle.
Although not exactly sensorial material the large metal insets
are a sensorial preparation. The children draw around the
inset, constructing the shapes themselves. When they
superimpose insets they get the idea of proportion and the
relation of shapes.
There is also the work with the small metal insets used for
making designs. Children take assorted pieces to their table
and create designs at random. Later the children may trace,
cut out and paste designs.
If the children are left
free to do these designs
they come to see the
difference between
shapes, e.g., can’t
inscribe pentagon in
triangle.
In the Geometric
Cabinet the shape is
made for the children
whereas with the metal
insets they draw around
them and make for
themselves.
They may also work with the classified and relevant
nomenclature at the 3 – 6 level.
Gives language for parts
of plane figures e.g.
PARTS OF A SQUARE.
Encourage children to
make their own
booklets.
Children who have been in a Montessori school since an
early age may have had some experience with a measuring
ruler and compass to construct shapes. Geometric
construction is a wonderful way to foster handwriting
development.
Making and labeling
angles, etc.
Some 3 – 6 children may be exposed to the knowledge of
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Geometry
similarity, congruency and equivalence.
The above experiences for the 3-6 year old emerge from a
fully developed class.
Hopefully the child upon entering the 6-9 or 6-12 class will
have experienced:
1. Sensorially based work.
Familiarity with geometry cabinet and cards.
Knowledge of basic language.
2. Work with geometric solids and names.
3. Work with geometric solids and bases.
A factual relationship
between plane figures
and solid shapes.
4. Work with Constructive triangles (6 – 9 work is a direct
outgrowth)
5. Some designs: superimposed geometric figures, large
metal insets, small metal insets
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Geometry
II. Initial Presentations
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Page 13
Geometry
Preliminary
No matter how little the child’s exposure may have been at
the 3–6 level it is important to see the 6-12 work as a
continuation of knowledge for the child.
Children have been seeing geometric shapes all their life – a
remote preparation.
The first task is to find out how much the new children know.
It is done informally—perhaps by playing some games
relating to the shapes in the environment.
For example: “Close your eyes. Think of somewhere in the
room where there is a square. Are you ready? Open your
eyes and go find it.”
Why do we teach the
names of geometric
shapes? To give the
child access keys to
language and maths.
Not by testing or
confrontation!
Use any games that will allow one to point out the shapes.
Other activities:
•
•
•
•
Group design with small metal insets
Talk about names of shapes
Look in magazines for shapes
Use shapes for art work, collage
Informal not
“TEACHING” names
Start only with shapes found in the Geometry Cabinet.
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Geometry
“Geometry Cabinet”
Presentation
1. This is a scalene triangle. Do you know what the word
scalene means? It comes from the Latin “in the shape of
a ladder”. What do you suppose is the relationship
between a ladder and this scalene triangle? This is a
ladder that farmers used to collect fruit from trees. It is
what the Greeks called scalene. So the scalene triangle
is formed by three of the steps of this ladder (3 of unequal
length). We can say that the scalene triangle encloses
our human
experiences.
In children’s house the
geometry cabinet is
given for aiding
movement (eye). In 6-9
not for movement but to
strike the imagination.
Also necessary to give,
not just the word, but its
history and derivation.
In 3-6 the child traced
with concentration. In 69 the child would move
quickly; no sensitivity to
‘touch’.
In 6-9 we give the name
followed by a modified
three period lesson.
2. This is an isosceles triangle. Do you know what is
means? It means, “having equal legs”. But, can a
triangle have legs? If I said you have equal legs, how
many legs would I be talking about? “2”. So, having equal
legs means having two equal legs. Which two sides are
equal in this triangle? Therefore we say an isosceles
triangle has two equal sides, but we should say two equal
legs.
THESE ARE SAMPLES
OF SOME OF THE
WORD DERIVATIONS.
3. Repeat highpoints of 1 and 2 followed by three period
lesson.
Must have for all
shapes!
4. The word trapezoid in Greek means “small table”. A
Greek farmer had a table that looked like this. The legs
were in this special position to give stability.
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Geometry
5. Showing the rhombus, we say: this is called a rhombus.
What does rhombus mean? Lets see what happens
when we rotate it on one point. It spins like a top. It
comes from the Greek “anything that can be spun
around”. Greek priests created the top as a sacred
symbol. So we have here part of the culture of the past.
6. Here is an ellipse. What does it mean? From the Latin
meaning “that to which something is missing”. i.e. the
circle.
7. Continue exploring the etymology of all the words.
8. Match the reading labels to the figures and/or loose
figures on cards.
9. Commands
Dramatize the command for polygons and bicycle
demonstration…. Aim of which is to show that the circle is
the limit of all polygons and that the more sides there are,
the shorter they will be (within the same base).
© Copyright, 2004
Available through
several mail order
services.
Page 16
Geometry
Constructive Triangles – First Series
Material Description “First Box”
Right Angle Scalene Triangles
•
•
•
•
2 each yellow with black line on minor leg
2 each green with black line along major leg
2 each gray with black line along hypotenuse
1 each red with black line along major leg
Right Angle Isosceles Triangles
•
•
2 each yellow with black line on one of the = sides
2 each green with black line along hypotenuse
Equilateral Triangles
•
2 each yellow with black line along one side
Obtuse Angle Triangle
•
1 each red with black line along longest side (single
triangle-companion of other red)
Presentation 1st Box
…for six year old
1. Holding the 2 yellow equilateral triangles… What is
this? “An equilateral triangle” superimposing the
child says they are “equal”. Placing one of the
triangles on the table we point out the black line.
Ask the child to unite the 2 triangles along their
common black line by holding one in place and
bringing the other next to it. What figure does it
form?
A rhombus
2. Same procedure for 2 yellow right-angle isosceles
triangles. What is it? Unite it!
A parallelogram
3. Same procedure for 2 green right-angle isosceles
triangles.
A square
Constructive triangles called
so because the “construct”
new figures.
4. Now the 2 yellow right-angle scalene triangles.
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Geometry
A parallelogram
5. Now the 2 gray right angle scalene triangles.
A rectangle
6. Now the 2 green right angle scalene triangles.
A parallelogram
7. So, from the isosceles triangles we formed a
rhombus or square. From the scalene triangles we
formed the common parallelogram and rectangle.
Child Can: Cut figures out of colored paper, paste
and label them. Repeat what was done above.
8. After a period of time:
Introduce 2 red triangles. What are they? Unite
along black lines….
A trapezoid
Aim: Give children the concept of construction of plane
figures.
Children will discover that by
uniting 2 triangles, a
quadrilateral can be formed.
Material Description “Second Box”
2 blue equilateral triangles (no black line)
2 blue isosceles right-angel triangles
2 blue right angle scalene triangles
2 triangles with different shapes (similar to 2 red triangles in
first box)
Presentation 2nd Box
1. Display 2 blue equilateral triangles and repeat first
part of lesson as in previous presentations. Name
of figure? What kind? Then ask the children to
put them together. How? As you want to! What is
formed? A rhombus. Lets try to slide one of the
figures around to see if we can form something
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Point out characteristics of each
triangle.
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Geometry
else. Still a rhombus but on a different side. To
emphasis, tell child to hold the other triangle and
move this one around it. Same result!
Indirect preparation for
equivalence and area
calculations.
2. Display two right angle isosceles triangles.
Introduce as before … Then ask children to unite
in any way. We want to form as many
quadrilaterals as possible. Rotating one figure
around the other the children discover they have
formed a square and some parallelograms.
3. Display the two blue right angle scalene triangles.
Introduce as before… Then ask children to unite
in any way, of course, forming quadrilaterals. A
rectangle and two common parallelograms.
4. Of course we have three sides in the equilateral
triangle, but how many different lengths are
there? “One”. How many quadrilaterals did we
form? “One”
5. How many different lengths are there in the
isosceles triangle? “Two”. How many
quadrilaterals did we form? !!!! Was it two or
three? Lets look closely. The first one is definitely
a square. But what is your opinion about these
other two? “They are identical because one is the
reverse of the other; turn them over and see”. So
we really only formed two different quadrilaterals.
6. And with the scalene triangles? We had three
different lengths and formed three different
quadrilaterals.
Repeat: 1 length
1 quadrilateral
2 lengths
2 quadrilaterals
3 lengths
3 quadrilaterals
7. Now lets looks at these last two scalene triangles
of different sizes. What can be formed? A
trapezoid and (if formed) a concave quadrilateral.
© Copyright, 2004
Page 19
Geometry
Material Description “Third Box”
12 equal scalene triangles, very special in measurement and
construction.
Group Presentation I “Lets Construct the Stars”
1. Ask children to analyze triangle! Size? “Scalene” Angle?
“Right angle”. Both together? “Right angle scalene
triangle”. Demonstrate what happens when two are put
together (one turned over). “An equilateral triangle”. We
show that all these 12 triangles are half of the equilateral
triangle.
Lets call this angle the big angle (right); this one the
medium size angle (60°); this one the small angle (30°).
Small, medium and large angles…
This is the long side; this is the short side and this is the
medium size side.
Now, we lay one triangle down on the table. Point out the
small angle. Lets construct a star putting all the small
angles together… How many points does it have? “12”
How many triangles did we use? “12”
Conclusion: We have constructed a star using all 12
triangles and it has 12 points.
2. Now lets form a star with the medium size angles. How
many points does it have? “6”. How many triangles
used? “6”. So we can construct another six-pointed star
with the remaining triangles.
3. Now form a star with the big angles. How many points
does it have? “4”. How many triangles used? “4”. So with
the remaining triangles we can construct two more stars
just like this one.
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Page 20
Geometry
Presentation Group II “Lets Construct
Diaphragms”
1. Starting with the first star constructed of 12 triangles,
demonstrate what happens when we “open” the centre of
the star and form a diaphragm (not for child: concentric
dodecagons). This opening has a certain form! It has 12
sides as well as an outer edge.
2. Starting with the second star constructed of six triangles
demonstrate what happens when we “open” the centre of
the star and form a diaphragm (concentric hexagons).
This opening has six sides, as does the outer edge!
Minor leg forms outer perimeter.
Major leg forms outer perimeter.
3. Starting with the third star constructed of four triangles
demonstrate what happens when we “open” the centre of
the star forming a diaphragm (concentric squares). This
opening has four sides, as does the outer edge!
Children should cut out triangles making stars and
diaphragms… pasting and coloring.
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Page 21
Geometry
III. Congruency, Similarity and
Equivalence
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Page 22
Geometry
Congruency
Material
Insets of the square and triangle. Need to describe to
children.
•
•
•
•
•
•
•
•
•
Square divided into 2 parts (midpoints)
Square divided into 4 parts (midpoints)
Square divided into 8 parts (midpoints)
Square divided into 16 parts (midpoints)
Square divided into 2 parts (diagonal)
Square divided into 4 parts (diagonal)
Square divided into 8 parts (diagonals and
midpoints)
Square divided into 16 parts (diagonals and
midpoints)
Triangles divided into 2, 3 and 4 parts
Presentation
1. Show material to children (3)
2. Today we are going to learn about shapes that are equal.
Point out triangles to be
used for follow up.
3. Remove two (1/4) triangles from square inset. “Are they
exactly the same?”
4. When we have two figures that are exactly the same size
and shape we call them congruent shapes.
Congruent from the
Latin “to meet together”.
5. The children find congruent shapes. The teacher asks
why they are congruent. Return pieces.
6. Prepare label “Congruent”. Ask children to cover their
eyes. Give a shape to each child. They find a congruent
shape.
How did you know they were congruent?
7. Children take turns giving each other shapes and finding
congruent counterpart.
Again:
© Copyright, 2004
We went from sensorial
to naming shapes. Most
children can go right
on/this is pretty simple
Page 23
Geometry
How did you know they were congruent?
With most children you
will go right on to next
lesson on Similarity.
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Page 24
Geometry
Similarity
Material
Same material as with congruency.
•
•
•
•
•
•
•
•
•
Square divided into 2 parts (midpoints)
Square divided into 4 parts (midpoints)
Square divided into 8 parts (midpoints)
Square divided into 16 parts (midpoints)
Square divided into 2 parts (diagonal)
Square divided into 4 parts (diagonal)
Square divided into 8 parts (diagonals and midpoints)
Square divided into 16 parts (diagonals and
midpoints)
Triangles divided into 2, 3 and 4 parts
Presentation
1. Invite three children. Remove two different squares. (E.g.
1/4 and 1/16)
2. Are they congruent? “No”
3. When we have two figures that are a different size but
have the same shape they are called Similar.
4. Ask children to find similar shapes. What are they?
“Name the shape” Are they the same size?
5. Prepare a ticket “Similar”. Children also have “Congruent”
label.
Children find shapes and determine if they are similar or
congruent and place under appropriate label.
Again: Question “How did you know?”
Inspire follow up work: What can children do?
•
•
•
•
My booklet of similar shapes
My booklet of congruent shapes
Make constructions with similar figures
Try to get children to extend the work – designing with
© Copyright, 2004
Handwork needed after
lesson to solidify
concepts.
The material limits the
choice to make it work.
It is helpful to get
specific names from the
children so they don’t
get the idea that all
triangles are similar.
This material is limited
which is why rectangles
work.
Writing the labels is
important because it
gives the children a
starting point.
Page 25
Geometry
•
similar shapes
Figures drawn at random on sheet of paper
Find the congruent pairs.
Find the similar pairs.
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Page 26
Geometry
Equivalence
Note
After presenting congruency and similarity you can tell how
much the child has understood. This is six-year-old work. –
proportions are not yet understood.
Equivalence is the most important concept. It is used
throughout the work we do. e.g. Pythagorean theorem, study
of area.
Children really should have worked at the introductory level
with fractions before this lesson (numerator, denominator,
sensorial exchanging)
Review with children
what has been done.
Don’t get upset if they
don’t get it right away.
Materials limit the
concept of similarity.
Also a more interesting
concept: Find out what
children know – not by
testing – subtly.
If children have not had
fraction work it is still
possible to do this
lesson.
Presentation
1. Invite three children and ask them to find congruent
shapes.
2. Now ask them to find similar shapes. Now ask them to
find equal shapes. Why did you choose these? Etc.
3. Teacher removes inset of whole and replaces it with two
halves. Do they fit? “Yes”
Now try with the other halves. Do they fit? “Yes”
4. This is worth half of the square! (The rectangle)
This is worth half of the square! (The triangle)
Both have the same value
When two figures have the same value but are different
shapes they are called EQUIVALENT – from Latin “equal”
“value” (Try another equivalence – ask why they are)
5. Demonstrate their equivalence. Superimpose.
Show that the left over part is the 1/8 triangle and
converts the triangle to an equivalent rectangle.
© Copyright, 2004
Informal Review
We could say that each
piece is worth how much
of the square?
Children cut out of paper
and discover for
themselves.
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Geometry
6. Ask children to find equivalent shapes:
Are they congruent?
Do they look the same?
7. Give children a figure and ask them to find an equivalent
shape.
Children may do more of the above work or move into work
with three labels.
Congruent figure
Similar figure
Equivalent figure
If the children have not
had fractions we can
say: This is one piece
out of four and this also
is one piece out of four.
Write labels and children find
The children may
discover fractional
equivalence.
1/2
=
1/4 +
1/8 + 1/16 + 1/16
Children may continue
to work on their own –
more exploration.
Game
I would like to find a square that is equivalent to this rectangle
(1/2).
Use two half squares by
diagonals
Extensions
“My book of equivalent shapes”: trace, or trace, cut and
paste.
Equivalence with Non Geometric Shapes
This “tree” is equivalent to this teapot.
© Copyright, 2004
If you have not found
anything that interests
them.
Page 28
Geometry
Constructive Triangles – Second Series: Triangular and Two
Hexagonal Boxes
Note:
This work does not necessarily follow right after the similar, equivalent, equal/congruent
work with insets of square. The Classified Nomenclature is a concurrent activity.
Another set of materials that we use to give concepts of equal, similar and congruent are
the constructive triangles. In general if the children are working on their own with a
particular concept it is not necessary to give new material with the same concept.
Material
Triangular Box
Small Hexagonal Box
Large Hexagonal Box
Presentation Triangular Box (T)
1. Children remove triangles from box. Do you remember
these? Name some of them. Are there any here that
are congruent?
2. How did you know they were congruent?
(Or equal)
We are reviewing with
children.
3. Can you find some that are similar?
4. Can you find some that are equivalent?
•
•
•
Place 3/3 over gray triangle and remove
Place 4/4 over gray triangle and remove
Place 2/2 over gray triangle and remove
We also know that 1/2 = 2/4 (If they have studied
fractions)
© Copyright, 2004
Page 29
Geometry
Try more arrangements:
Don’t exhaust all
possibilities. Leave some
for children to discover.
Are these equivalent? How do you know? Can you make
some more equivalent shapes?
Exercise
It is not our purpose to
show every possibility.
Follow what children
initiate. Give only
examples.
Children may draw some. However most of the time, just
move on.
Presentation Small Hexagonal Box (H1)
1. Remove contents from box with three invited children. Identify
triangles.
2. Are there any congruent shapes? (Superimpose)
Are there any similar shapes?
Are there any equivalent figures?
(Note to teacher: The
yellow triangle and the
six red obtuse triangles
are new for 6-12; not
used in 3-6.
Underscore importance
- going from rhombus.
3. Now the gray equilateral and the red isosceles are equivalent
because they are both 1/2 of the equivalent rhombi.
4. Build hexagon with six gray triangles. Make equivalent figures
with six red triangles or three red obtuse triangles and large
yellow.
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Geometry
5. What can be done with 1/2 of each shape? Split hexagon into
two trapezoids. Is this trapezoid equivalent to the red triangle?
(Yes) Because it is half of two equivalent shapes.
6. Demonstrate equivalence between yellow triangle and green
trapezoid. (Use red obtuse triangle as mediator.)
7. How about this parallelogram and hexagon?
Note pattern:
Find congruencies
Find similarities
Find equivalence
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Geometry
Presentation Large Hexagonal Box (H2)
Note:
1. Three children are invited and contents are removed.
What are they?
2. Lets find congruent figures. Now find similar figures. Are
there any equivalent shapes?
Not as much to do with
this box as with others.
Review with students
procedure
1. Congruent
2. Similar
3. Equivalencies
(Whole hexagon;
half hexagon; red
parallelogram equals
yellow rhombus.
We are not teaching; we
are letting the children
explore.
The yellow hexagon is found to be equivalent to
•
•
•
Yellow/red/gray hexagon
Yellow parallelogram
Three rhombi
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Geometry
Presentation Triangular and Small
Hexagonal (T and H1)
1. Remove all pieces.
2. Find congruent shapes. Demonstrate all
the congruent rhombi made with red
obtuse triangles and equilateral triangles.
Show that they are not the same as the
yellow rhombus.
3. Find similar triangles.
4.
Lets find out the relationship between
the large yellow triangle (T) and large gray
triangle (H1). How much smaller is the
yellow? Superimpose two green halves over
gray. Change green halves into deltoid
(below) -- superimpose large yellow and red
obtuse to show equivalence.
Note:
Interesting discoveries! The children
notice (when shown similarity) that the
yellow and gray triangles are not
equivalent. We want them to find a
measurable difference between the two.
Large Gray Triangle = Yellow Triangle + Red
Obtuse Triangle
Demonstrate how above red obtuse triangle
is equivalent to red equilateral triangle by
showing how each is half of equivalent
rhombi.
Therefore:
Large Gray Triangle = Yellow Triangle + Red
Equilateral Triangle. (Illustration on next
page)
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Geometry
5. Searching for the exact difference!
Substitute four gray equilateral for the
large gray equilateral triangle. Substitute
three red obtuse isosceles triangles for
yellow triangle.
Now: It takes three red obtuse triangles to
make yellow triangle. The red equilateral
triangle is equivalent to the red obtuse
triangle (Step 4).
But look the red equilateral is 1/4 of the
large gray triangle and 1/3 of the yellow.
(Through intermediary of red obtuse)
How can 1/4 be the same as a 1/3?
Three of the fourths of the large triangle
can make the yellow triangle. Therefore
the yellow triangle is 3/4 gray triangle
and, it will take four of these thirds to
make the gray triangle. Therefore the
gray triangle is 4/3 yellow triangle.
Note: Practice this on your own.
6. Double the three equilateral triangles as
illustrated below:
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Geometry
Replace above small gray equilateral
triangles with three yellow obtuse triangles
(right illustration) and transform to hexagon
(below).
Replace one of the small equilateral
triangles above with red obtuse triangle
(equivalence already shown) and
demonstrate equivalence to the right
scalene.
Note: There is a need to play with
relationship between equilateral red and
obtuse red.
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Geometry
Presentation Triangular and Large
Hexagonal (T and H2)
1. Remove all pieces. Find congruent shapes. Find
similar shapes.
2. Equivalent shapes. Look at all the possibilities for
large equilateral triangles…
Then:
Set all them out
Whatever we make out of two of
these will be equivalent to what
we make out of the other two. Try
it!
This can be converted to hexagon (below) by
replacing red equilateral with three obtuse isosceles
triangles (right column).
3. Add in a third triangle. Also: Various trapezoids
with obtuse isosceles triangles.
Try it!
4. Some other composite equivalences:
Keep in mind we are not trying
get children to memorize but to
explore and reason – some talk
about fractional parts.
How to get children to work?
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Geometry
Suggest some hand work if they
are not exploring templates.
Manipulation first, handwork if not
exploring significantly enough to
reinforce concepts of similar,
equivalent and specifically equal
or congruent.
Remember:
•
•
© Copyright, 2004
Explore relationships with
both hexagonal boxes with
children.
Explore relationships with
all three boxes with
children.
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Geometry
Equivalence with Metal Insets
Material Description and Explanations.
Some people number these plates for their albums.
The first one is used by itself and shows the triangle on one
side and rectangle on the other. The newer material shows a
“whole” triangle -- thus both sides are filled.
Briefly describe these
insets to Students. Not
for children!!!
The next group is for rhombi. The new materials come with
an extra plate the size of (4) with the whole rectangle.
Later this one is added (5)
Next is the trapezoid (6)
Regular Polygons – Decagon
We do this work in two stages. First is the sensorial. How do
you think this is done? “By replacement” The second stage
brings awareness about the relationships between various
lines in these figures. …bases, heights and other names that
are pertinent to the development of area formulae…
major/minor base, major/minor diagonal, apothem, perimeter.
Generally start by having the first two sets of plates ready on
the table.
Presentation “Triangle – Plate 1”
1. What have we got? “Triangle and rectangle”
2. I wonder if they are equivalent. Try it out.
3. Where is the base of the triangle?
Where is the base of the rectangle?
What is the height of the triangle?
What is the height of the rectangle?
Try to find a way to make the
material demonstrate what
you want to show.
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Geometry
Continuation of Plates 1
1. We know these figures (1) are
equivalent. Identify base of
triangle and rectangle.
2. What do you know about their
length? “Equal”
What do you know about their
height?
Demonstrate that the height
of the triangle is twice the
height of the rectangle.
Demonstrate
To students: Pull this out!
3. Try to summarize: We found
out that a triangle and a
rectangle are equivalent when
their bases are equal and the
height of the triangle is twice
that of the rectangle.
Presentation “Rhombus –
Plates 2, 3 and 4”
1. Identify all the shapes. Are
they equivalent? Exchange
wherever you can to see.
Note: You could do 1 – 3 above
and then this one and stop. If
children are ready to go ahead.
Do until children feel confident that all figures are
equivalent?
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Geometry
Bring out Plate 5
1. Are these figures here (5)
equivalent to these others (2),
(3) and (4).
Note: Here you get an idea of
children’s visual perception
and deductive logic.
2. Now try to put the pieces
back in their original spaces.
3. Let’s look at the relationship
between the lines using this
figure (5).
Trial and error
4. Show me base of rectangle
and rhombi. What about
them? “They are all equal”
(demonstrate)
5. Where is the height of the
rectangle? Place the half of
the rectangle into the rhombi
to demonstrate. What do we
find?
Heights and bases of the
rectangle equal heights and
bases of rhombi”.
6. Therefore the rectangle and
rhombi are equivalent.
”Rectangle and rhombus that
have equal bases and heights
are equivalent”.
7. I am going to take pieces of
the rectangle and transform
them into these two triangles.
8. Is each of these pieces
equivalent? “Yes” Why?
“Because they are 1/2 of
equivalent figures”
9. What do we know about
equivalent triangles? “They
have equal bases and equal
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Geometry
heights”
”Triangles that have equal
bases and equal heights are
equivalent.”
Extension: Draw a triangle and
rectangle that are equivalent.
Could be constructed and could
be done sensorial.
To students: All the answers are in the material.
We have not only dealt with the rectangle and
rhombus but the triangle as well.
For the rest of the plates we will
do the sensorial stage followed
immediately by the relationship
between lines.
Presentation “Trapezoid – Plate 6”
1. Identify trapezoid. How many bases? “2”
What are they called? “Minor base and major
base”
2. Are they equivalent? Child exchanges: “yes”
3. Are the trapezoid and the rectangle
equivalent? Child exchanges: “yes”
4. Let’s think about the relationship between the
base and height of the trapezoid and the
rectangle.
5. The trapezoid has a major base – show and
place in rectangle opening.
It also has a minor base – show and place in
rectangle opening in special way.
6. What about the base of the rectangle? “It is
equal to the major base + the minor base of
the trapezoid.
7. What about their heights? Demonstrate by
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Geometry
aligning small triangles in trapezoid to show
height; then remove one triangle and show
that it is the height of the rectangle.
“Therefore the height of the trapezoid is twice
the height of the rectangle.”
8. What do we know about equivalent
trapezoids and rectangles? “They are
equivalent when the base of the rectangle
equals the sum of the major and minor bases
of the trapezoid and the height of the
rectangle is half the height of the trapezoid.
Pull it out. As you work with the
materials try to find smooth wording
that works for you.
Presentation “Pentagon” – Plates 11
and 12”
1. Rotate (9) to demonstrate “regular”
pentagon.
2. Demonstrate equivalence between (9) and
(10)
3. What does inset 12 tell us? Any regular
polygon may be divided into as many
triangles as it has sides
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Geometry
Presentation “Decagon – Plates 7, 8, 9 and
10”
1. Demonstrate equivalence between (7) and (8)
2. (From 9) Show how two halves of this large
rectangle are equivalent by superimposing.
3. Is the rectangle equivalent to the decagon?
Demonstrate by moving the divided rectangle to
inset (7). Put them back.
Or, remove rectangle and slide pieces to left.
4. (Holding rectangle from 10) lets find out about this
rectangle and these pieces. Superimpose and
demonstrate equivalence.
5. I wonder if this rectangle is equivalent to the
decagon. Move the divided rectangle into the
decagon inset (7)… put them back.
6. Go over what equivalences have been found:
(8) to (7)
(9) to (7)
(10) to (7)
7. Are the rectangles from 9 and 10 equivalent? “Yes”.
Why? “Because they are both equivalent to the
decagon”.
You want the pieces of the
perimeter to be along the
base.
Relationship between lines
8. Identify base and height in rectangle from (9).
Identify perimeter and apothem in decagon.
9. The base of the rectangle equals how much of the
perimeter? Demonstrate ½ by superimposing base
of (9) over base composed of five triangles (half of
them) from (8) or other part of (9).
10. Let’s look at the height of the rectangle (9). “The
height of the rectangle is equal to the apothem.”
DEMONSTRATE
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Geometry
11. The decagon and rectangle are equivalent when the
base of the rectangle equals half of the perimeter
and the height of the rectangle equals the apothem.
12. We also know that the decagon is equivalent to this
rectangle (10). How?
This time the base equals (count: 1, 2, 3…10) the
whole perimeter and the height of the rectangle
equals half of the apothem. Demonstrate.
The reason we found these
specific relationships is to
prepare for the study of area
formulae.
Presentation
“Rhombus\Rectangle” – Plate 13
1. Interchange B and C to confirm
congruency. Return to original
frames.
Objective: Triangles having the same
altitude and base are equivalent.
A
C
B
2. Take one part of A, B and C and
classify: A/2 is a right-angled triangle;
B/2 is an acute angled triangle; C/2 –
obtuse angled triangle.
3. Take A/2, B/2 and C/2 and place in
Frame D. The altitudes of the three
triangles are the same.
4. How about their bases? Place A/2 in
B and C; Place B/2 in A and C. All
three triangles have the same base.
5. The three triangles are equivalent
because they are 1/2 of equivalent
figures
© Copyright, 2004
D
Draw this out from the children.
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Geometry
EXTENSIONS OF THE WORK WITH
INSETS OF EQUIVALENCE.
1. Children individually repeat the
presentation given by teacher.
2. Teacher prepares triangles,
rhombi, parallelograms,
trapezoids and polygons with
more than four sides. Ask children
to prepare a rectangle equivalent
to each.
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Geometry
Equivalence with Pythagorean Insets
Remember we have already mentioned Pythagoras’ name when we talked about the
history of geometry, the rope. We have also explored right-angled triangles with the box
of sticks: right isosceles with the neutral sticks and right scalene with special 3-4-5
combinations.
Presentation “Pythagorean Plate I
or Sensorial Plate”
1. Ask children to identify the white
triangle: …right isosceles. Identify
other shapes … squares.
2. We are going to discuss something
about the relationship between these
two (yellow and blue) whole squares
and this whole one (red).
3. The sides of these two squares are the
same as the legs of the triangle.
4. The length of the side of the large
square is the same as the hypotenuse.
5. Demonstrate equivalences sensorial.
Red triangles to red square and
replace.
Yellow triangles to yellow square and
replace.
Blue triangles to blue square and
replace
May have to review “leg”
6. Now let’s look at these divided squares
… something interesting!
Remove two red triangles –
interchange with two blue triangles.
7. Remove other two red triangles –
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Geometry
interchange with yellow triangles
Now what can we say about this red
square?
It is equal to the blue plus the yellow!
Also, the blue equals half the red and
the yellow equals half the red.
Extension:
Children can trace in their notebooks
After some time of independent work try to
draw out the Pythagorean Theorem:
This is an early experience because it is
demonstrated sensorially.
In a right triangle (remove the white
triangle) the sum of the squares built on
the legs is equal to the square constructed
on the hypotenuse.
Explain to the children that this is the same
as the early rope experiences of the
Egyptians.
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Geometry
Presentation “Plate II – Numerical”
The material: triangle 3cm
leg – 9
4cm leg – 16
5cm leg - 25
1. Here we have another demonstration of the theorem
of Pythagoras. In this inset all the squares are divided
up so that we can use numbers of squares.
2. We are going to try to prove that this square plus this
one equals this one… that the squares built on the
legs equal to the square built on the hypotenuse.
3. The children interchange pieces.
4. Show the numerical value:
32 = 9
42 =16
52 = 25
Children may discover the proportions of the triangle. In
that case you might introduce the idea of the Pythagorean
triples. This is also a good time for work problems. For
example:
Given: length of two legs
Find: the hypotenuse
Given: length of leg and hypotenuse
Find: length of other leg.
If children can do this second example you know you
have finished the work because they have internalized
the process.
It is your job to make sure
that the children are ready
to discard the material and
work abstractly.
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Geometry
Pythagorean Theorem with Constructive Triangles
Presentation… with plane figures
other than squares
Material: Constructive Triangles: hexagonal
boxes H1 and H2, and triangle box T
1. Remove green right scalene from triangle
box. Identify: leg, leg, hypotenuse.
2. What do we already know about the
relationship between the length of the
hypotenuse and the length of the legs.
”The sum of the squares is equal to the
length of the hypotenuse” SHOW
PYTHAGOREAN TEMPLATE 1
The square built on the legs is equal
to the square built on the
hypotenuse.
3. This theorem is usually expressed in terms of
squares. Have you ever wondered since the
square is a regular polygon, if there is also
was a relationship with other regular
polygons built on the legs of a right angle
triangle?
Place gray equilateral (T box) on hypotenuse
Small red equilateral on shorter leg
Yellow equilateral (H1 box) on longer leg
4. We already know something about the
relationship between these equilateral
triangles.
Demonstrate equivalences you may have
done before:
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Geometry
5. Exchange four gray equilaterals for the large
one.
Exchange three red obtuse isosceles for the
yellow equilateral.
6. All these little triangles are equivalent.
7. Remember we said that maybe these would
equal this one? Let’s see… Here we have
one and three, which equals four, and here
we also have four!!
8. So it also works for equilateral triangles.
9. I wonder what else we could make. What if
we doubled the equilateral triangles? Do it!
We get another series of regular polygons …
rhombi
2+6=8
10. What if we added a third equilateral triangle?
(add wholes) … trapezoids
3 + 9 =12
11. Doubling this (with paper yellow equilaterals)
makes hexagons:
6 + 18 = 24
12. Another way to look at these is to treat the
large gray equilateral triangle as the unit of
measure.
Then in the first case we would have:
1/4 + 3/4 = 4/4
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Geometry
In the case of the rhombi:
2/4 + 6/4 = 8/4
The trapezoids:
3/4 + 9/4 = 12/4
The hexagons:
(6 x 1/4) + (18 x 1/4) = 24/4
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Geometry
Pythagoras Three (Euclidean Logic)
The Pythagoras three plate
Note:
The children have already worked with the Pythagorean
theorem, which states that the sum of the squares built on the
legs of a right triangle equal the square built on the
hypotenuse.
Presentation
1. Introduce the plate. Do you remember the
Pythagorean theorem? “Yes: The sum of the
squares built on the legs of a right angled
triangle equals the square built on the
hypotenuse.” We are going to use this plate
and try to prove that these red rectangles
equal the blue square plus the yellow square.
1
2
2. Remove the red rectangles. Slide the white
triangle down and place the yellow and blue
parallelograms in the space. Replace pieces
as in 1.
3
3. The sum of the blue and yellow parallelograms
is equivalent to the sum of the two red
rectangles – (that form the square built on the
hypotenuse). Return pieces as in 1.
4
4. Remove the yellow square and slide white
triangle up as shown in (4). Replace space
with yellow parallelogram. Does this
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Geometry
demonstrate that the yellow parallelogram is
equivalent to the yellow square? Return pieces
as in 1.
5. Remove the blue square and slide the triangle
up as shown in 5. Replace space with blue
parallelogram. Does this demonstrate that the
blue parallelogram is equivalent to the blue
square? Return pieces as in 1.
5
6. Therefore, we can see that the yellow
parallelogram is equal to the yellow square
and the blue paralegal is equal to the blue
square. (6).
6
7. Take the small red rectangle and the blue
parallelogram. Considering the longer sides as
base, identify the altitude and base of each
figure… sensorially show that they are the
same.
8. These two figures are equivalent because b=b
and h=h.
Now take the small red rectangle and place in
the frame (You can turn plate vertically) as
shown. Note that the altitudes are equal and
that the length of the whole opening will hold
both pieces perfectly!
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Geometry
9. Take large red rectangle and yellow
parallelogram. Demonstrate that b=b and h=h
and are therefore equivalent to each other!
Place large red rectangle in frame as shown
we did in (1) above. Note that the altitudes are
equal and that the length of the whole opening
will hold both pieces perfectly!
10. ANOTHER WAY.
Take blue parallelogram and place in frame as
shown; also place yellow parallelogram in
frame as shown. Note that the bases are the
short sides!
Show that the blue square is equivalent to the
blue parallelogram because b=b and h=h.
Similarly show equivalence of yellow square
and yellow parallelogram.
11. We have shown that
small red rectangle = blue parallelogram
and
blue parallelogram = blue square
12. We have also shown that
large red rectangle = yellow parallelogram
and
yellow parallelogram = yellow square
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Geometry
13. Therefore:
This could not be shown directly
because their measurements are
not commensurable… cannot be
measured with same unit!
The study of areas and metric
system will fall between these two
ages. Also in that time frame will be
the arithmetic demonstration of the
extensions of Pythagorean Theorem
will also be after the areas.
Summarize: We have show that the square built
on the shorter leg is equivalent to the smaller
rectangle, which makes up the square of the
hypotenuse; the square built on the longer leg
makes up the larger rectangle, which forms part of
the square of the hypotenuse. The sum of the
squares of the legs is equal to the square of the
hypotenuse.
Ages
For presentation of first two Pythagorean Frames
8 1/2 +
For third (Euclidean) Frame 11 1/2
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Geometry
Algebraic Demonstration of Euclidean
Theorem
Using the third Box of Constructive Triangles from
Series1:
1. Child takes one triangle, identifies it: right-angle
scalene… hypotenuse, major leg, minor leg.
2. Take three more of the triangles and show
congruency.
3. Isolate one of the triangles and apply the letter
“a” to the major leg, “b” to the minor leg, and “c”
to the hypotenuse. Show that the same
nomenclature applies to all four triangles.
The area of the whole square = c2
The area of each triangle is 1/2ab; of the 4, =
2ab
The inner square is = (a-b)2
Therefore
c2 = (a-b)2 + 4 ab/2
c2 = (a2-2ab+b2) + 2ab
c2 = a2 + b2
Another demonstration of the same procedure…
(a + b)2 – 4 ab/2 = c2
or, (a + b)2 = 4 ab/2 + c2
a2 + 2ab + b2 = 2ab + c2
a2 + b2 = c2
Age: 12
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Geometry
IV. Geometry Nomenclature
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Geometry
Classified Nomenclatures
The classified nomenclatures are also a work for reading.
Material Description
Series of eight nomenclatures A through H
A) Basic Ideas … point, line, surface/beginning of the whole
geometrical world. They constitute plane and solid
geometry. (Folder is Gold)
B) Study of lines
C) Study of Angles
D) Plane figures … to define, it is necessary to have
concepts learned in B and C above. Nomenclatures
divided into two sections: those limited by curves and
those limited by straight-line segments.
E) Triangles
F) Quadrilaterals
G) Regular Polygons
H) Circle
Each series contains the following:
1. Folder containing picture cards with no words.
Corresponding reading labels (at level of words).
Definitions without subject (at sentence reading level)
2. Wall chart with names on each picture: This is the ‘control’
used at level of word reads.
3. Booklets with pictures and definitions: Control for
sentence reading level.
Typical activity with geometry nomenclature…
Children read definitions and place label on dotted lines
checking their work with control book.
Matches labels to pictures. Read definitions and matches to
label and picture by removing the label and placing it on the
definition…reconstruction control book!
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Geometry
Children can write definitions, cut them up, and try to
reconstruct.
Definition cards can be
used with or without
pictures but always with
labels.
Other Materials
•
•
•
A box of sticks. There are 10 different sizes of sticks
that are designed to attract two sensorial stimuli: color
and length. All are in increments of 2cm beginning
with 2cm and ending with 20cm.
Another series of sticks. 10 “neutral” sticks
representing the hypotenuse of each isosceles
triangle formed with the other sticks.
Wooden Tack Board covered by several pieces of
paper representing the “plane”.
© Copyright, 2004
Also used for diagonals
of some regular
polygons.
Page 59
Geometry
A)
Fundamental Concepts
Material for 1st Presentation “Fundamental
Concepts”
•
•
•
•
•
•
•
•
Any box
A ball
Plane insets from Geometry Cabinet
Series of small geometric solids
Paper
Sharpened pencil
Sharpener
Decimal system material: 10 units, 10 tens, 10
hundreds, 1 thousand cube
Presentation “Fundamental Concepts”
1. Ask children to place rectangular box on table. Give them
the ball and say “put it in the same place”. “But I did not
say to move the box, or put it on top of the box or next to
it; I asked you to do it right there where the box is”.
“Impossible”. “You can see that every THING occupies a
space!” “When I go out of the room and you are in my
way, I must ask you to move!”
2. Let’s go into a more precise analysis. Picking up the
sphere from the small solids: This was my ball! (remove
the original box and ball). This was my box (holding the
quadrangular prism). Now look at this object; it has a
curved surface like the sphere and a plane surface like
the prism. It is called a cone. Now this one; it has a
curved surface and two plane surfaces and is called a
cylinder. Here we have another prism (triangular). And a
cube, a pyramid, and another pyramid. This one looks
like and egg and is called an ovoid; notice how it is all
curved. This one is also curved but it looks different and
is called and ellipsoid.
© Copyright, 2004
Two opposite objects:
One limited by a curved
surface and the other
by a straight surface.
Use of the word “thing”
as opposed to the
technically correct
“body” is easier to
understand, at this
level.
Sphere and prism
represent opposite
extremes. (Line up as
shown)
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Geometry
3. Let’s use some more refined words for each solid:
cube
OK
st
1 prism
quadrangular prism
nd
2 prism
triangular prism
1st pyramid
triangular pyramid
nd
2 pyramid
quadrangular pyramid
With this presentation
we have given the
concept of ‘solid”, and
they occupy space, can
have curved, plane or a
combination of curved
and plane surfaces.
4. Bring back ball and box. Let’s see how this box is limited:
It has plane surfaces and is stable. The ball: It has curved
surfaces and is not stable. These two objects are limited
by a straight plane surface and a curved surface! What is
a surface? It is like a very very thin layer of paint.
Teacher takes figure(s) from geometry cabinet. The
surface would be this thin layer of paint that covers this
shape.
5. Taking the plane square inset: Is this a surface, and how
is it limited? ”by a line”. Holding the circle inset: How is it
limited? “by a line; a curved line”.
6. Look! We can draw a line. Take a piece of paper and red
pencil. This can be the image of a line; but it actually is
too thick, it must be
much thinner!
Note how the concepts
of very thin surface and
very thin line are
emphasized.
We start with solid and
work back to point!
7. Teacher gives concept of ‘point’ by marking a dot with the
sharpened pencil. This is a point, but it is really much
smaller than this. The point is like the mark left by a very
sharp pencil!
© Copyright, 2004
We have given the
concept of surface, line
and point at the
sensorial level.
Page 61
Geometry
First nomenclature
Material
Inside the folder:
•
•
•
•
a red (folded cube)
a red square on paper
a red line on paper
a red point on paper
1. Form the cube! It is a solid that occupies space. This is a
surface; it goes on and one in all directions. The line also
goes on and on….
Children match labels
Three period lesson
2. Try to form definitions with children.
’A solid’ It is what occupies space.
‘A surface’ It is like a layer of paint; it can be straight or
curved.
‘A line’ It is like a very thin hair and can be straight or
curved.
’A point’ It is like a dot left by a very sharp point.
Bend picture of line to
show curved line!
Solid: 3 dimensions
Surface: 2 dimensions
Line: 1 dimension
Point: none
Note: The point, line
and surface can only
be described with terms
prefaced by ‘like’. The
solid can be defined
accurately.
To emphasize the basic ideas of geometry we
use the decimal system material to illustrate
the four concepts
1. This is a bead, a bar, a square, and a cube. Nothing
new! Now, this is a point, a line, a surface and a
solid.
2. Now lay out 10 beads forming a “line” of beads. As
© Copyright, 2004
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Geometry
we add more “points” we make a line! We can say
that the point is the constructor of the line. [Alternative
would be to “move” one bead in a path forming a line.
By moving a bead (a point) we create a line.]
Take a ten bar and say “Moving as line in any way
creates a surface.” The line is the constructor of the
surface.
(Flashlight is dark room)
In reality, the point is
the constructor of
everything.
Can be demonstrated
with paintbrush.
Now, what happens when I move a surface? It
creates a solid.
3. Now we display folder with pictures. Point …
line….surface… solid. Hold up and show to children
how we have a point; then by moving the line we form
the square; and the square forms the cube!
Age: 6 years
Aim: Give “so called” basic ideas based on reality as we
use common objects and then the decimal system
material.
© Copyright, 2004
Page 63
Geometry
B)
Study of Lines
Presentation One
1. Take out all (or some) of the Geometry Cabinet drawers
and have children follow the contour of the shapes.
Review that the figures are limited by lines. Straight and
curved line terminology should be given to the children.
2. Now take all of the insets.
Form two columns. First column for those figures limited
by straight lines and the other limited by curved lines.
Children can make lists of each series.
The explorations of
curved and straight
lines linked to objects.
It is necessary to go
from object to concept.
Material
•
Two spools with one string attached
Presentation Two
1. Holding spool of string (inside hands so the
spools can not be seen by children) with
clenched fists, teacher pulls hands apart while
still grasping string demonstrating a ‘line’ by
keeping the string taught. “This is a line”
(moving arms in all directions maintaining
tightness). “ This is a line.” “This is a line.”
“This is a straight line.” Laying string down on
table forming a curve (but still holding spool in
fists): “This is a curved line.” Repeating
several variations of curved and straight lines
and getting response from children.
At first we use the word ‘line’ along,
adding ‘straight’ and ‘curved’
adjectives later.
2. Three period lesson: Show me a straight line!
A curved line! What is it when it is like this
(bunched up)… curved. Like this? Like this?
© Copyright, 2004
Page 64
Geometry
Game:
Ask children to look for straight and curved lines in
the environment.
3. It is evident that the line stopped where the
string ended (our hands limited it) but it could
go on and on forever. Ask children to draw a
line on the blackboard… At both ends we draw
three dots or points, which means these lines,
go on to infinity. When we identify this concept
of infinity we have to imagine that it has no
end.
We give the concepts of infinity of
a line in this way:
4. Ask children to grammatically analyze these
expressions:
”the line”;
”the straight line”;
”the curved line”.
Match with grammar symbols.
© Copyright, 2004
Page 65
Geometry
Positions of a Straight Line (space)
Material
•
•
•
•
•
•
•
•
Box of sticks and supplies
Two beakers
Red coloring
Spoon
Piece of cloth
Globe
Water
Plumb Line
Presentation
1. Take teaspoon of red coloring and add to opaque pitcher
of water. Pour an equal amount into both beakers. Shake
one of the beakers and announce that we must wait for
the second one to come “to rest”.
2. We say that the position of the water is horizontal. (from
the word horizon…imaginary line dividing the earth from
the sky). Taking red stick from box: This is a straight line.
Let’s see what position this straight line takes. Placing it
carefully on the surface of the water in beaker two… Now
we can say that the position of this stick is horizontal. The
horizontal straight line is lying on the surface of the water
forming part of its surface.
3. Holding stick outside the beaker match to water level on
horizontal plane: I know that the surface of the water
gives me the horizontal position.
4. Ask children to take the plumb line and not to move it.
When it is perfectly still: It is the image of a straight line in
vertical position. With the weight of the plumb line resting
on table match red stick to plumb line string: this straight
line follows the vertical line of the string and it is a vertical
line!
Game:
Take globe and ask children to find where they live. The
plumb line is the imaginary line that passes through centre of
© Copyright, 2004
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Geometry
the Earth from any point to another on its surface.
5. The horizontal is only one position as is the vertical, but
there are many many more. Demonstrate the stick in
horizontal position (compared to water in beaker) and
slowly move it up to vertical position (compared to plumb
line. “One horizontal and one vertical and in between an
infinite number of positions”. When a line is not vertical or
horizontal it is oblique.
Game:
Say the positions as you are moving the stick… horizontal,
oblique, oblique, oblique, oblique, oblique, oblique, vertical,
oblique, oblique, oblique, oblique, oblique, horizontal! (next to
the water)
© Copyright, 2004
First the two opposite
concepts of vertical and
horizontal. Now the
gradations in between.
Children love this.
Page 67
Geometry
Positions of a Straight Line on a Plane
Material
•
Geometry tack board covered in paper
Presentation
1. Take three sticks that are the same and pin the first on
the board. This is a straight line; look at its position.
Tell me what it is! “Horizontal”. Teacher moves entire
board to various positions and children respond:
“oblique, oblique, oblique, vertical, oblique, oblique, etc.
When we look at something that is vertical we look up
and down. When it is in the position of still water,
horizontal, we look left and right or right and left. When
it is not vertical or horizontal it is oblique.
2. Place the other two sticks on the board. One in the
oblique position and one in the vertical position.
Three period lesson. Which is vertical? Horizontal?
Oblique?
3. Take nomenclature for horizontal, vertical and oblique
and review with children. Children can now draw
horizontal, vertical and oblique lines on paper.
Games:
Look for straight lines in the environment with the different
positions.
In relation to space they
are all horizontal but in
relation to plane they are
different.
4. Ask children to grammatically analyze these
expressions:
”The straight horizontal line”; “The straight vertical line”;
“The straight oblique line”. Place grammar symbols
next to words. We can leave out the word “straight” and
it will still be understood!
© Copyright, 2004
Page 68
Geometry
Parts of a Straight Line
Material
•
Same plus two pairs of scissors
Presentation
1. As before, take spools of string and pull apart to form
straight line! Take a red felt pen and mark one point on
this string: This is a straight line (cut it at the point). Now
it is in two parts. This is a “ray” and so is the other part. It
has a beginning on one side and no end on the other.
The beginning is called the origin.
2. Now we mark two points with the red felt tip pen on a
straight line. Two children cut string simultaneously at
both points. This is a line segment. It has a beginning
and an end:
_________________________
segment
__ __ __ _________________________
ray
_________________________ __ __ __
ray
Let child cut it
A third child says “1, 2,
3 cut”
3. All these sticks in the box are line segments because they
have a beginning and an end. Now we take the string
and show how it can form an arc. Show with semicircle
from stick box.
4. Children draw straight line, ray, and segment. Review the
differences.
© Copyright, 2004
Use classified
nomenclature.
Page 69
Geometry
Parallel, Convergent & Divergent Lines
Material
•
•
•
•
Two plumb lines
Two strings on two spools each
Geometry accessory box
Happy, sad and indifferent children’s pictures
Presentation One
Using plumb lines, dramatize the meaning of parallel.
Parallel comes from the Greek word parallelus meaning
beside one another.
Play games with positions of string asking if parallel.
Then two children hold two strings in fists on floor. One
holds hands steady on top of each other while the other
spreads hands (arms) apart while pulling the string and
forms divergent lines. From the Latin “to bend apart”.
Opposite procedure for convergent lines. From the Latin “to
bed together”.
Three period lesson.
Presentation Two
Take two sticks of same size and place one on the plane
setting the other one aside. Now take two more sticks (as
shown) and place them perpendicular to the one already
pinned down. Take the second stick that you had set aside
and place it perpendicular to the two others (as shown) –
demonstrating these two original sticks are the exact same
distance apart. Remove the “spacer” sticks. Take two
“indifferent” children and demonstrate how they walk the line
not caring if they will ever meet. Replace them with red
arrows in both directions.
© Copyright, 2004
Page 70
Geometry
Form two convergent lines with sticks. (Use two different
sizes of sticks to separate the lines.) Take two “smiling
children” and show how they walk together towards one
another. Then replace the children with red arrows.
Form two divergent lines with sticks. Take two “sad” children
and show how they walk apart. Then replace the children with
red arrows.
© Copyright, 2004
Page 71
Geometry
Oblique and Perpendicular Lines
Material
•
Two balls of yarn divided into two balls each.
Presentation
Children form a circle sitting on floor. Take one ball of year
and roll it across so that two children on opposite sides are
holding each end. Take the second ball and do the same
with two different children. While the first string is held
stationary, rotate the second piece of yarn by having the
children pass it in clockwise direction. Do this very very
slowly noting how the groups in the different sections of the
circle get smaller and larger. Then notice when all the
groups seem to be the same size.
When they are the same size we have four equal groups
and the lines are called perpendicular. Perpendicular
comes from the Latin meaning “plumb line”. A plumb line
that hangs true against the horizon is called perpendicular.
When they are not perpendicular, they are oblique.
Three period lesson.
Form perpendicular and oblique lines on a plane by fixing
one stick and rotating its bisector using the measuring
angle.
© Copyright, 2004
Page 72
Geometry
C)
Study of Angles
Presentation: Angles
1. Work on the plane by requesting two sticks -- the second
one longer and with holes all along. Fasten the second
one thru the first at one end to from the vertex on the
angle. Now: This is a whole angle because we have made
a whole (trip around the circle is made by placing a pencil
in the last hole as if it were a compass). Slowly opening
the angle we say… “angle, angle, angle, straight angle.”
2. Go back to beginning and rotate top stick again saying,
“angle angle, right angle. We will call this the measuring
angle! (Introduce it to children.) It is our point of
reference.” Now… angle, angle, angle, acute angle. This
is acute because it is smaller than the measuring angle.
Same procedure for obtuse angle.
3. Demonstrate that two “measuring angle” equal one
straight angle; and four equal one whole angle.
4. Give the nomenclature: SIDES of an angle. SIZE of and
angle. VERTEX of an angle. Three period lesson.
5. Demonstrate that the acute angle is less than the right
and that the obtuse is greater than…. “Acute, acute,
acute, right… Obtuse, obtuse, straight.”
© Copyright, 2004
Size is also amplitude
Exercise Take all the
figures in the Geometry
Cabinet and using the
measuring angle,
classify all the angles.
Why?... Children will
realize that as the
number of sides
increase the size of the
angle increases.
Page 73
Geometry
Presentation Convex Angles / Reflex Angles
As in angle presentation above, superimpose two sticks.
Rotate saying acute, acute… right, obtuse, obtuse…straight,
greater than straight…whole. “We have seen many acute
angles, one right angle, many obtuse angles, one straight
angle etc. Now: rotate again, this time as children repeat the
same nomenclature but say “convex, convex…straight angle”
(coinciding with children’s straight angle). Continuing past the
straight angle, “reflex, reflex,…whole angle” (to coincide with
children saying "whole angle”).
We now point out that the acute, right and obtuse angles are
CONVEX. That the straight angle is neither convex nor
reflex. That the angles greater than the straight angle are
REFLEX (but less than the whole angle).
Teacher asks children to form any obtuse angle. The
children then color one angle in red and other in blue (any
color). The obtuse angle is red; the reflex angle is blue. Now
we can classify these angles according to their size
(amplitude). The convex angle is less than 180°; the reflex
angle is greater than 180°.
Etymology
Concave – Latin meaning “hollow”
Convex – Latin meaning “harsh”
Reflex – Latin meaning “to reflect” or “to think back”
Comment: A spoon represents both convex and reflex.
© Copyright, 2004
If they have had the
lesson on measuring
angles.
Page 74
Geometry
Measuring Angles
Materials
•
•
•
•
•
•
•
Fraction Insets
Geometry Cabinet
Two boxes of geometric surfaces and outlines
Measuring Angle
Montessori & Regular Protractor
A stick
A compass
Presentation
Teacher dramatizes task by saying we are going to measure
angles and tries to do so with a regular ruler. First we
measure this side, then this one and now the angle??? We
must use a circle to measure and angle.
The Sumerians were interested in Astronomy and studied the
solar system. They were very good at measuring and kept
records of everything in the form of pictography (picture
writing). On a particular day they were studying one star. So
they observed it here (draw dot on board) in the night sky—
and every night they marked where the star was until it
returned to its original position. They ended up with 360
marks and concluded that the star must have gone around in
a circle. They divided the circle in these 360 parts. We call
them degrees, symbolized by a little °. The stars walk around
the sky. Degree comes from the Latin word meaning “step”.
To measure an angle we draw an arc representing the
movement of the star and write the ° representing degree. If
we took this whole circle inset, we could mark 360 lines
representing the star’s movement. But we use a special
instrument to measure these 360 divisions called a protractor.
It comes from the Latin word meaning “to substitute for, or
trace”. Describe Montessori Protractor to children. 20°
intervals; red dot in centre called vertex; vertical line called
side.
Show Montessori
protractor.
Show how the “whole” fraction fits in the protractor. Taking
the measuring angle show how many measuring angles are
© Copyright, 2004
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Geometry
in the circle protractor. “4” This measuring angle was used to
measure right angles and tell us about acute and obtuse
angles, but with this protractor we can be more precise.
Organize fraction insets as follows:
1
/3
1
/6
1
/9
1
/4
1
/5
1
/8
1
/2
1
1
/7
1
/10
(Grouping determined
by increasing difficulty
is reading. We start
with the circle because
there is only one angle
making it simple)
Place each fractional piece in Montessori protractor: … read
the degree.
Holding the 1/4 fraction: It can be called 1/4 or now we know it
is 90°.
Teacher measures a square inset in the protractor and reads
90°. With the 1/16 square inset it is necessary to take a stick to
prolong the side in order to read the angle size.
Activities:
Children can measure the insets and any figures with straight
sides.
Now we take the regular circular protractor. Ask children to
find vertex and side. Identify divisions in 10° increments.
Now one can measure the surfaces and outlines of the
geometric shapes.
Operations with Protractor
Addition is accomplished by placing two insets in the
Montessori Protractor and seeing the total. Multiplication,
similarly. Subtraction is accomplished as follows: 120° - 30°.
Place 120° inset in protractor and move it 30° past 0 to the
left and read “what is left” from the original place.
AGE (for protractor operations): 7 1/2 +
© Copyright, 2004
Division can be
demonstrated by
teaching how to bisect
an angle with compass.
(See any geometry
book to learn how to do
this with a compass
and a ruler.)
Page 76
Geometry
Adjacent Angles
Presentation
Child selects four sticks – two must be the same length.
Create two angles making sure that the stick that is the same
length is in each one. Repeat nomenclature: sides, size,
vertex.
Unite the two angles on the common side. Since the common
sides are the same we can eliminate one of them. Give
nomenclature: side, common side, side, vertex. Adjacent
angles have one vertex and one side in common.
Activities: Children may construct their own adjacent angles;
find adjacent angles in the environment; exercises with the
geometry nomenclature
Adjacent means “near
or close”. It comes from
the Latin “to adjoin.”
© Copyright, 2004
Page 77
Geometry
Vertical Angles
Presentation
Child selects four sticks and forms two angles. Unite the
angles in such a way that the vertices are common and
each side is the prolongation of the other.
Remove one of the pins in the vertices and unite all four
sticks with one pin. One stick is now the prolongation of
the other.
Nomenclature: sides, vertex, vertex, angle, angle. But
these two other vertex angles (obtuse) are also angles.
We have actually formed four angles that are opposite
from each other.
Mark each angle with a colored tack. We call each angle
that is opposite from each other vertical angles.
Two angles are vertical if the prolongation of the sides of
one angle forms the sides of the other.
Show this by replacing the individual sides with two sticks.
Ask child to select one angle and find its opposite
(vertical).
Presentation: Vertical Angles are equal
Form vertical angles on board over a letter size or A4 slip of
paper. Ask child to select a pair of vertical angles and color
them in. Remove the paper and cut apart in such a way that
the colored angles may be superimposed.
Both angles are equal. Therefore opposite vertical angles are
equal.
© Copyright, 2004
Page 78
Geometry
© Copyright, 2004
Page 79
Geometry
Complementary Angles
Presentation
Child selects four sticks and forms two angles. Join the
angles and eliminate the “extra” common side.
Fix one side and its common side to the board leaving
the other side free to move. Child moves the remaining
side until it forms a right angle – using the measuring
angle to make sure.
When two angles together form an angle equal to a
right angle and therefore equal to 90°, they are called
complementary angles.
Complement means to “complete.” What are they
completing?
© Copyright, 2004
Page 80
Geometry
Supplementary Angles
Presentation
Ask children to select four sticks and form two right angles
using the measuring angle. Fasten one to the board and
move the other along side it so that one side forms the
prolongation of the other. Eliminate the common side and
replace the two sticks with one stick.
Take the measuring angle and show that we have formed two
right angles, which is a straight angle. Then move the
common side in either direction and show that these other
angles also form a straight angle (measuring angle can be
flopped at vertex). When two angles form a straight angle,
and therefore equal to 180°, they are called
supplementary.
Supplementary comes form the Latin that means “that which
is made full”.
© Copyright, 2004
Page 81
Geometry
Two Non-Parallel Lines Cut by a Transversal
Material
•
•
•
Board
Box of sticks
Geometry supply box
Presentation “Concepts and Terminology”
Take two straight lines and attach to board. We know
that these are not parallel and they go on to infinity on
in both direction. This plane is subdivided by the two
lines into three parts. The parts above and below are
called the “external” parts and are colored in red. The
part that lies between both lines is the “interior” and we
color it blue.
Take a third straight line (choosing stick with many
holes). Fix it to the board in such a way that it “cuts”
through the first two sticks. This line is called a
transversal. How many angles are formed when two
straight lines are cut by a third one? Lets count:
1,2,….8. Teacher places small tacks in each angle.
Therefore, two straight lines cut by a transversal form
eight angles.
Which are the interior and exterior angles? Remember,
the red area was the exterior side and the blue was the
interior side! Lets count them. 1, …4 external angles.
This is an interior angle. There are 1, 2, …4.
Therefore, two non-parallel straight lines cut by a
transversal form four exterior and four interior angles.
© Copyright, 2004
Page 82
Geometry
Parallel Lines Cut by a Transversal
Material
•
•
•
•
Two crayons
Paper/newsprint
Box of sticks
Different colored push pins
Preparation for
theorems having to do
with intersecting lines.
Presentation
1. Set out two long parallel lines with sticks. We have
interior space. We have exterior space.
2. Color interior one color and exterior, another.
3. Intersect both with a transversal (as shown.)
4. We have angles here and here. Some are interior, some
are exterior.
5. Show me interior angles, exterior angles.
Give derivation of “alternate”.
From the Latin alternus "to
do first one thing, then the
other."
6. I am going to mark alternate pairs of interior angles (Each
pair in a different color). Teacher makes one pair;
children make the other pair. Why are they alternate?
7. We can also mark alternate pairs of exterior angles (do
same as above)
8. Remove all push pins and ask children to repeat: Find
exterior and interior alternate angles and mark with pins.
One at a time – What have you made? Children ask each
other.
Follow up
a) Show and mark angles in notebook
b) Geometry nomenclature
Another Day “Corresponding Angles”
1. Set up parallel lines but by a transversal.
© Copyright, 2004
Page 83
Geometry
2. There are other relationships. Lets mark two
angles on the same side of the transversal – one
interior and the other exterior (e.g. yellow tacks).
These are called corresponding angles. Are
there other corresponding angles? Who can find
the corresponding angle to this?
or
Teacher can set up corresponding angles and
ask children – “How would you describe these
angles?” ...then name them.
Presentation: Exercises with Sizes of
Angles
1. Draw two parallel lines across entire paper.
Color exterior space one color and interior
space another.
2. Draw transversal to edge of paper.
3. Lets find some alternate angles (or
corresponding angles). Using two colored
crayons mark either alternate interior/exterior or
corresponding angles – generate the questions
so that desired result is obtained.
4. Cut along transversal with scissors and remove
first angle with dramatic “snip”.
Can you find other places that it fits?
Cut further and remove the next angle.
See if it fits anywhere.
Repeat with next angle. Repeat with next angle.
5. Place all cutout angles on uncut side of
transversal.
What do we know about all the angles? – Only
two sizes.
© Copyright, 2004
If they draw the transversal
Page 84
Geometry
6. Children can make their own drawings.
© Copyright, 2004
perpendicular– another discovery.
Page 85
Geometry
D)
Polygons
Material
•
•
•
•
•
Board
Box of sticks
Geometry supply box
Piece of red string
Last drawer of the Geometry Cabinet
Presentation I: Introduction
1. Holding the red string: Remember this was a line
segment. Here are the two end points of the line
segment. Teacher places string on the “plane”. It
is no longer a line segment but a curved line.
2. Take any three sticks and unite them. We have
three line segments and each is separate and not
the continuation of the other. Now we bend one
segment as if we had broken one segment of a
straight line.
Now lets see if this curved line (string) is “open” or
“closed”. I can go from the outside to the inside so
it is opened. Now how about this broken line? It
is also open.
3. Now we take this string and unite the end points
(tie). How is it now? External, external, external…
no way for me to get to the inside. Take the
broken line and close it. External, external … no
way to get inside!
Show “as if” you are breaking
it.
Therefore we say that when we close our curve
we obtain a simple curved closed region. And
when we close our broken line we obtain a figure
called a polygon.
4. If we chose two segments we could not make a
polygon (of course one wouldn’t either). We
choose three because it is the minimum possible
number to form a polygon.
Error!
5. Teacher takes red paper: My hand now
Recall formation of geometric
© Copyright, 2004
Page 86
Geometry
represents the red string! Cut out the simple
curved closed region with scissors. This is a
region limited by a curved line.
figures with chains from 1 – 9.
The three chain formed first
polygon.
Now we cut the red paper to form the three-sided
figure. This region limited by a broken line is
called a polygon.
6. Exercises with geometry nomenclature.
Presentation II - The Structure of Polygons
1. Ask children to choose a stick and then tell them to close
a region with it. “impossible” Choose two new sticks and
unite them. Do they form a region?
Choose three other sticks and unite them. I have finally
limited a region! All regions limited by three sticks (sides)
are called Triangles.
Now we choose any four sticks of any length. We can
make another closed region limited by four sides. We say
that all figures limited by four sides are called
quadrilaterals. Continue with 5, 6, 7, 8, and 9 sticks.
2. Review what one stick forms, two sticks, three,
four…nine.
3. Remove one stick and the two sticks: All these we
constructed have a last name “polygon” and a first name,
triangle, quadrilateral, pentagon etc.
© Copyright, 2004
Page 87
Geometry
E)
Study of Triangles
Triangles According to Sides
Material
•
Box of Sticks
Presentation
1. Ask child to give you three sticks of different lengths.
Then ask for three more, two of which must be equal
to (same color) one of the other three.
Then ask for three more, all of which are the same
as the two equal ones just obtained.
2. Ask children to construct the triangles!
This triangle is scalene because…
This triangle is isosceles because…
This triangle is equilateral because…
3. Review the derivation of the words followed by a
Three Period Lesson.
Triangles According to Angles
Material
•
•
Box of Sticks
Measuring triangle
Presentation
Lets construct triangles according to their angles.
Teacher chooses the 6 cm and 8 cm sticks to make an
angle exactly like this angle here, in red. (Measuring
triangle).
Tell children to unite the first two sticks. Then we take
© Copyright, 2004
Page 88
Geometry
the measuring triangle and adjust the sticks in order to
form the right angle by matching. Place the black
colored stick in such a way to complete the triangle. (10
cm)
Now we chose two more sticks that we know will form
an obtuse angle triangle when ‘closed’. Use the
measuring triangle to form the obtuse angle and have
children find the third stick that will close the triangle.
In the same way construct an acute angled triangle.
Now: What is this triangle? “Right angle”. How many
right angles “one” and it has two other smaller acute
angles. But the most important aspect is that it has one
right angle.
This triangle has three acute angles -- we can verify it
with the measuring triangle. It is an acute triangle
because it has all three acute angles. This triangle is
obtuse angle triangle because is has one obtuse angle.
Ask questions relevant to all three triangles. How many
right angles, obtuse angles, etc. Try to form.
Repeat:
“This is a right angle triangle because….”
“This is an obtuse angle triangle because….”
“This is an acute angle triangle because….”
We have three triangles formed by sides and three
triangles formed by angles.
Take the triangle drawer from the Geometry Cabinet
and match the plane figures to the stick triangles.
Scalene-isosceles-equilateral
right angle-acute angle-obtuse angle
© Copyright, 2004
Page 89
Geometry
Note characteristics:
One acute angle
Two acute angles
Three acute angles
Union of the Two Characteristics
Presentation
Classify all previously made six triangles by sides and
angles writing labels for each one. Exact the correct
nomenclature in the process and give the name of acute
angle isosceles equilateral triangle to the equilateral
triangle.
“I as the teacher know there is one missing”. Now we
construct the right angle isosceles using a neutral stick.
Label it.
This will vary due to the nature
of the children’s selections.
Now we take the geometry cabinet drawer with the plane
triangle figures and match them to the prepared stick
triangles. Have the children identify them.
We have proven beyond a shadow of a doubt that there are
only seven triangles humans can construct.
Exercise: Have children construct their own “Seven
Triangles of Reality” and label.
© Copyright, 2004
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Geometry
Seven Triangles of Reality
Note:
The above is the ideal “seven”. The children in selecting their own sticks will probably
not construct these same triangles. Number one could be an acute angles scalene.
Number two could be an obtuse angle isosceles. Number 6 could be an acute angle
Isosceles.
We may find that it is necessary to construct other triangles to “complete” the
presentation and clarify the learning experience.
© Copyright, 2004
Page 91
Geometry
Triangle Nomenclature
Material
•
Box of Sticks
Presentation
Ask child to choose any triangle from the Geometry Cabinet.
The painted area is called the “surface” of the triangle.
This is a vertex of the triangle. And this is a vertex of the
triangle. And this is a vertex of the triangle. How many
vertices does a triangle have?
These are the sides of the triangle.
Let’s learn another new element. Watch as I change the
position of this triangle. Each time the bottom rests on the
table we call that the “base” of the triangle.
The distance around the sides of the triangle is called its
perimeter.
© Copyright, 2004
Note. This is the
nomenclature for the
triangle as appears in
the Geometry
Nomenclature. Every
nomenclature may not
be the same. We use a
similar process for
naming the parts of
quadrilaterals and
regular polygons.
Page 92
Geometry
Altitudes/Heights of Triangles
Material
•
•
•
Plane triangles form geometry cabinet
Triangle Box of Constructive Triangles
Altitude Stand
Presentation I – “Constructive Triangles”
1. Place large grey equilateral triangle in stand.
2. Hold plumb line. What kind of line? “Vertical”
3. Move plumb line in front of table so that the line
coincides with triangles vertex. We call this line
the “height”
”a vertical lines connecting the vertex with the
base”
4. Try with the right scalene triangle. Move the
triangle to show the different bases and
corresponding heights.
5. Try with obtuse isosceles triangle. Identify special
case of the external height and show how dotted
line is drawn.
Presentation II – “Triangles from the Geometry
Cabinet”
Take a few triangles and explore the various altitudes of
each when the base changes.
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Page 93
Geometry
Other Triangle Exercises
Presentation– “Orthocenter”
1. Prepare multiple paper copies of the seven triangles of
reality.
2. Take one of the triangles and draw the three altitudes.
3. Where they meet is called the orthocenter. Color the
intersection in red.
Exercises for “Orthocenter”:
Ask children to take the drawings of the seven triangles of
reality and find the orthocenter for each.
Etymology:
orthos meaning straight
centre meaning point of concurrency
the point where all the straight things meet!
What straight things? altitudes
Concurrence of Medians
Median: A line segment from the vertex to the mid point of the opposite side. All the
medians of a triangle meet at a point called Centroid.
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Geometry
Concurrence of Axes
The perpendicular straight line drawn from the midpoint of a line segment (in this case,
each side of the triangle) is given the name axis of the line segment.
The point where all the axes of the sides of a triangle meet is called the circumcenter.
Concurrence of Angle Bisectors
A ray that divides an angle into two equal parts is called a bisector.
© Copyright, 2004
Page 95
Geometry
Nomenclature for the Right Angle Triangle
Presentation
Ask children to construct the two right-angled triangles of
reality with the sticks. They identify them: right-angled
isosceles and right angled scalene.
Lets add something new…. The names of the sides of a
right-angled triangle are special. The two sides constituting
the sides of the “measuring triangle” (demonstrate) are called
legs.
Leg, leg, leg, leg. When they are equal it is an isosceles
triangle. When they are not equal they are referred to as
major and minor legs of the scalene triangle.
The third side is the hypotenuse.
© Copyright, 2004
Page 96
Geometry
F)
Study of Quadrilaterals
Introduction
There are six quadrilaterals of reality. Here they are listed
from the most general to specific:
•
•
•
•
•
•
Common Quadrilateral
Trapezoid
Parallelogram
Rectangle
Rhombus
Square
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Geometry
The Six Quadrilaterals of Reality
Presentation
1. Ask children to choose four sticks and unite
them. We count the sides; 1, 2, 3, 4. It is a
common quadrilateral. We take another set of
the same sticks and unite them similarly. This
time we twist it slowly to form a trapezoid. Then
we compare to the first one showing it is the
same and then the difference. We repeat;
common quadrilateral, trapezoid! So, a trapezoid
is a common quadrilateral with two parallel sides.
2. Now ask the children to choose any two pairs of
sticks and unite. The figure formed has two pair
of parallel dies and is called the common
parallelogram. Take two more identical pairs
and unite. (Then “knock” slowly to form a
rectangle.) This time the sides are not only
parallel but they form four equal angles. That is,
four right angles. Check with measuring angle.
3. Ask children to take four equal sticks. What
figure? A Rhombus. Now “knock” a second set
of four equal sticks into position forming a
square… four equal sides, four right angles.
Check with measuring angle.
4. Count the figures and repeat terminology.
Then pose interpretive questions. What is this?
“square” But a rhombus also has four equal
sides! “The angles are all equal” etc.
5. Line up the figures according to their specificity.
Begin analysis of figures as they compare to
each other.
Any four sided figure – common quadrilateral
At least one pair of parallel sides – trapezoid
© Copyright, 2004
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Geometry
This one has two pairs of parallel sides and that
is why it is called a parallelogram.
Without looking at the angles this figure with four
equal sides is a rhombus.
A rectangle has four right angles.
A square has two main characteristics: equal
sides and equal angles.
6. What is this? “A common quadrilateral”. Is this a
common quadrilateral? Continue asking this
question down to the square.
What is this? “A trapezoid” Is the common
parallelogram a trapezoid? Yes, because it has
two pair of parallel sides some call it “twice a
trapezoid”
Is this a trapezoid? etc down to square
Is this rhombus a square? A rectangle?
Is this rectangle a rhombus? A square?
Is the square a rhombus? A rectangle?
”Give me” the quadrilaterals
”Give me” the trapezoid
”Give me” the parallelograms
”Give me” the rhombuses
square
”Give me” the rectangles
square
all
from here down
from here down
rhombus and
rectangle and
7. To prove that this Note was not so with triangles
we go back to the three boxes of constructive
triangles.
Ask the children what were these called?
“Constructive triangles”. Why called
“constructive”? “They construct other figures”.
The triangle constructs all other figures or reality.
The Diagonal
8. Take all six quadrilaterals and join two opposite
vertices on each in the order to obtain a “stable”
figure. What do we see? “Two triangles”.
© Copyright, 2004
Note: At the end of all this work
the children discover that the
quadrilateral figures are not
stable. It is possible to make
more than one figure with the
Page 99
Geometry
The line segment that joins the opposite vertices
is called a diagonal.
same sticks.
9. Now ask the children to take their drawings of
the seven triangles of reality. “I want to unite two
vertices”. They will see that there are already
lines (triangle’s sides) that unite the vertices.
The triangle does not have a diagonal. It is only
present in the figures starting with the
quadrilaterals.
10. Now looks at the six quadrilaterals, each with a
diagonal stick. Ask the children what they
observe: “each quadrilateral has been divided
into two triangles”.
11. Separate the square and the rhombus. Place
both diagonals on each. What do you notice?
“The diagonals of the square are the same size.”
The diagonals of the rhombus are different sizes.
12. What happens when we flatten out the rhombus?
”One diagonal becomes longer, the other
shorter.”
13. We have special names for the diagonals of a
rhombus”
longer diagonal: Major Diagonal
shorter diagonal: Minor Diagonal
© Copyright, 2004
Page 100
Geometry
The Trapezoid
Presentation
1. There are six quadrilaterals of reality. One is the
trapezoid that has four different shapes according to
the position of the sides and angles.
2. Nomenclature; major base, minor base, oblique
side, oblique side, etc. We will see this is a scalene
trapezoid. When we used the word scalene to
describe the triangle we said that all the sides were
different lengths. The trapezoid is scalene when
the oblique sides (which can never be the bases)
are different.
3. Form an isosceles trapezoid. The two opposite
sides (non-parallel) are equal and this is why it is
called isosceles. Take the plane inset and rotate it
in its frame to prove. We have used the sides to
give the name to this trapezoid.
4. Using the measuring angle, construct a right-angled
trapezoid. It is a right-angled trapezoid because is
has one right angle.
5. Construct an obtuse angle trapezoid. Point out that
the first two trapezoids had two obtuse angles but
that they were next to each other on the same side.
But this time the obtuse angles must be opposite!
We call this an obtuse angled trapezoid because it
has two obtuse angles that are opposite.
6. Children match reading labels to figures; draw the
trapezoids and writes their name.
© Copyright, 2004
Page 101
Geometry
G)
Study of Polygons
From Irregular to Regular Polygon
Presentation
1. Choose any five sticks and fasten together. Since none
of the sides and angles are equal we call this an
irregular pentagon.
2. Choose six sticks of the same size and pin to board
making sure angles are equal. (Use specially prepared
cardboard angle). This is a regular hexagon.
3. Have the children construct other regular polygons.
4. Name all the regular polygons. Match them to the plane
figures from the Geometry Cabinet.
5. Name the parts of the polygons.
Presentation
6. Construct the diagonals for the different polygons in
notebooks.
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Page 102
Geometry
Apothem
Presentation
Demonstrate with children that a line from the centre of a
regular polygon to the midpoint of one of the sides is called
the Apothem.
The Radius is the distance from the centre to one of the
vertices.
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Page 103
Geometry
Sum of the Angles of Plane Figures
Material
•
•
•
•
Selection of plane figures; triangles, quadrilaterals,
polygons
Envelope 1: Different triangles with angles marked in
red. Note: All the angles of a particular triangle must
have the same radius marking.
Envelope 2: Quadrilaterals similarly marked.
Envelope 3: Regular and irregular polygons similarly
marked.
OR, children can make
their own by placing
shape on paper and
putting a dot at the
vertices – then
connecting them. If
they make their own
they need to color in
the size of the angles
with the same arc.
Presentation
1. Choose a triangle. I am going to try to find out what the
sum of the angles of this triangle is.
2. “RIP” off angles and place them together. What kind of
angle did I make? “Straight angle” How many degrees?
180°
Some child might say
they know –
“protractor”
What will the children
discover? “Always get
180°
Ripping as opposed to
cutting emphasizes
angles, not triangles.
They can write in their notebook: “The sum of the
angles of this triangle is a straight angle or 180°.”
Repeat for as many triangles as they like.
© Copyright, 2004
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Geometry
3. What about four sided figures? Take one out and RIP.
What did we get? Two straight angles… one whole
angle 360°
In their notebook: “The sum of the angles of this
quadrilateral is equal to a whole angle or 360°.”
4. They can do the same for regular polygons with more
than four sides and make their own discoveries.
5. Some children may want to go further with this
exploration by working on the Table that follows this
presentation.
Through this kind of research children may come to this
conclusion:
6. “The sum of the angles of a plane figure is equal to the
number of sides minus 2 times 180.”
N = (s-2)*180
© Copyright, 2004
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Geometry
Sum of the Angles of a Polygon Chart
n
…
…
10
9
8
7
6
5
4
3
1. How many sides have
your polygon?
2. How many diagonals can
you draw from one vertex?
3. How many triangles have
you formed?
4. How many straight angles
does your polygon
contain?
5. How did you obtain the
number of straight angles?
6. How many degrees make
up the sum of the interior
angles of your polygon?
7. How many degrees does
each interior angle of your
regular polygon have?
8. Is this interior angle of your
regular polygon contained
exactly in a whole angle?
(yes/no)
9. With the corresponding tile
why is it possible or is it
not possible to cover a
surface?
© Copyright, 2004
Page 106
Geometry
H)
Study of The Circle
In three subdivisions
a)
Nomenclature and Properties
b)
Reciprocal Relationships between a straight line and a circle
c)
Relationships between the position of two circles
a)
Nomenclature of Circle and its Properties
Material
•
•
•
Board
Sticks
Fraction insets
Note: Children already know that a circle has no end and no
beginning. They have also identified circles according to size
in the sixth drawer.
Presentation
1. Choose any stick from box. Fasten it to the board
using a red tack, which will represent the centre.
Rotate, marking the circle with a red pencil. “This is
a circle!” “A circular region” Teacher colors the
interior of the circle red: “The circle is that part of
the plane colored in red”. Compare to metal inset.
2. The center of this circle is my red tack and the
radius is the stick. The circular red line that I made
by rotation the radius is called the circumference.
3. Lets think in terms of distances. The radius is the
distance from the centre to any point on the closed
curved line… the circumference.
What about the centre? It is an interior point equidistant from all the points on the circumference.
To define center we need
the concept of
circumference. To define
circumference we need
concept of centre!
4. Ask children for another stick the same as the first
one. Superimpose at the centre point forming a
prolongation. This is called the diameter. Rotating
© Copyright, 2004
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Geometry
both sticks together: “all these are diameters”.
What is a diameter? A line segment that unites two
points on the circumference and passes through the
centre. Two equal sticks form this diameter. That
means D = 2 R.
5. Take a different colored pencil and draw an arc on
the circumference (pencil goes through both radii
superimposed). This is an arc. And so is the other
part of the circumference an arc.
What is and arc? A part of a circumference.
6. Unite the end points of the arc. We say this line
segment is called a chord.
The arc designated by the diameter is called the
semi-circumference. It divides the circle in two
semicircles.
7. Now we must give a name to that part of the circle
cut by the chord. It is called the segment of the
circle. Each part of the circle divided by the chord is
a segment of the circle.
8. The part of the circle enclosed by an arc and two
radii is called the sector of a circle. Not only this
part, but also this other part is also a sector.
Three period lesson – reinforced with classified
nomenclatures
This special inset can be
used to describe
segments of a circle.
Game
Take four metal insets from fraction material:
”whole”
What is this?
circle
” 1/3”
What is this?
sector
” 1/2“
What is this?
semicircle
”segment”
What is this?
segment
© Copyright, 2004
Page 108
Geometry
b)
Reciprocal Relationship between a Straight Line and Centre of a
Circle
Material
•
•
Sticks
Wooden ‘circumference’
Presentation
Level 1
Case 1: “External”
Stick and circle on plane. Move the stick to various external
positions. Then move the circle to various external positions.
They are external because they do not touch. The straight
line is “external” to the circle and vice versa.
Case 2: “Tangent”
External, external, external…..tangent! First moving the stick
towards the circle and then moving the circle towards the
stick. They are tangent because they are touching, or better,
because they have one point in common.
Case 3: “Secant”
External, external, external, tangent, secant! First moving
stick towards circle and then moving the circle toward the
stick. They are secant because they have two points in
common.
Level 2
Case 1: “External”
Using a stick for a radius. Lets consider the distance
between the straight line and the centre of the circle. Is the
distance less than, equal to, or greater than the radius?
“Greater” Now we can say that the position of a straight line
is external when the distance is greater than the radius.
If r=radius and d=distance then: d > r
Case 2: “Tangent”
Using a stick for a radius and the measuring angle. First we
show that when they are tangent the radius is perpendicular
© Copyright, 2004
Page 109
Geometry
to the straight line (use measuring angle). They are tangent
when d = r.
Case 3: “Secant”
The line is a secant when the distance between it and the
circle is less than the radius. d < r
© Copyright, 2004
Page 110
Geometry
c)
Relationships between the Positions of Two Circles
Material
•
•
•
Two circumferences with different diameters
Box of sticks
New measuring angle (2)
Presentation
Level 1
Case 1: “External”
Both circles are on plane. Move one of the other showing the
external positions. How are these circles? “External”. They
are external because they have no points in common.
Case 2: “Internal”
Place the small circle inside the large circle. They are internal
because one is internal to the other and they have no points
in common.
Case 3: “Externally Tangent”
External, external, external… tangent! They are external
tangent when one is outside the other and they have one
point in common.
Case 4: “Internally Tangent”
Now we flip the smaller circle inside the larger. They are
internally tangent when one is inside the other and they have
one point in common.
Case 5: “Secant” Intersecting
External, external, external, tangent, secant! They are secant
when they have two points in common.
Case 6: “Concentric”
Move one circle inside the other so that they have a common
centre. They are concentric when they have no points in
common and have the same centre.
© Copyright, 2004
Page 111
Geometry
V. Area
© Copyright, 2004
Page 112
Geometry
Introduction
“The difference between concept of Surface and concept of Area”
Area is the not the same as surface. A surface is that part of the plane limited by a
closed line… straight or curved. In this sense all the plane insets represent surfaces.
The area is the measurement of surface. You cannot calculate “surface”, but only the
area of the surface.
The children are well prepared to face this subject and its concepts. They have had
direct and indirect preparations for both surface and area concepts. (Red cardboard, 100
square...for surface concept; preparation through multiplication for area, insets of
equivalence). When, for example, we say 9*3 it means the 9 bar taken 3 times and the
answer is represented by 27 (two golden 10 bars plus 7). The surface is 9*3; the area is
the result 27.
Material
•
•
“The Yellow Material”
Box with 20 pieces that form:
4 rectangles
2 parallelograms
3 acute angled triangles
2 right angled triangles
2 obtuse angled triangles
Segment
The Rectangles
Presentation “How to measure Area of a
Surface”
1. What is this? “Rectangle” Nomenclature: sides, perimeter,
surface, base, and altitude or base altitude.
2. How can we measure surface? We must establish a unit
of surface! Draw a short segment on piece of paper,
(Equal to the distance between the lines on the first
rectangle.) Cut paper off at segment.
Transfer this unit of measurement to two consecutive
sides of this rectangle (demonstrate on reverse side of
appropriate rectangle). We obtain 10 divisions on this
© Copyright, 2004
Remember that in the
chapter on equivalence
the rectangle was the
last term of
comparison.
Segment
Page 113
Geometry
side and five divisions on this. Prolong the marks on the
long side and we get this (show next rectangle).
But this small rectangle is still not my unit of measure.
Show first rectangle again and explain what would
happen if we prolonged the segments marked on the
short side. We obtain five long rectangles… show third
rectangle. These long rectangles are still not the unit of
measure.
3. Lets create a unit of measure. Superimpose the second
and third rectangle. Imagine that they are transparent.
The long and short rectangles will result in this rectangle
with a series of squares. Each square may be a unit of
measure.
4. The unit of measure is the square. It does not have to be
this particular square but any square. To measure this
rectangle we could say lets count the number of squares
contained in it. “50”
But counting one by one takes too long! If we take five
bars of ten, or ten bars of five we reach the result 50 not
by adding each individual bead. In this case the factors
that form 50 are 5*10 or 10*5.
5. So, we can compute the area like this … There are 10
divisions on the long side of the rectangle and five
divisions on the short side: 5 * 10 = 50. To find the area of
the surface we multiply the divisions on one side by the
divisions on the consecutive side.
If we consider the base of the rectangle as its longest side,
and the altitude as its shortest, we can say that the area of
the rectangle is found by measuring the measure of the base
times the measure of the altitude!
Make labels representing: A = area; b = base; h = altitude
Match the labels to the rectangle. At this point we can
formulate the formula A = b × h
A = bh
simplified
© Copyright, 2004
€
We could also say:
A= ba (Little a =
altitude)
Page 114
Geometry
Common Parallelogram
Presentation
1. Ask children to identify common parallelogram.
Now lets count the squares. But not all are
complete. In order to count them what must we
do?
Take the parallelogram (in two pieces… as shown)
and superimpose over first parallelogram. They
are equivalent! Remove first one. Move small
triangular piece to form rectangle.
Now count the squares. Same as first rectangle.
Remove the two-piece rectangle. Take the original
parallelogram and the first rectangle and state that
they equivalent.
2. The area of the parallelograms surface is
calculated in the same way as the rectangles
because their bases and altitudes are the same.
3. Derive the formula: A = bh
4. Form the inverse relationships: b = A h
h=Ab
5. Note that the presentation
sequence for this whole
section will follow a general
scheme:
(1) Identity figure; counting the
squares; sensorial experience of
equality.
(2) Verbal organization of rule.
(3) Derivation of formula
© Copyright, 2004
Page 115
Geometry
Triangles
Acute Angled Triangle
Method I
1. Identify the figures. Can we count the squares? “No”
Need a mediator. Take two “half” triangles and place
over original triangle to verify congruency.
Triangles classified
according to angles.
Now place these halves in such a way that we form a
square with the three triangles. Now we can count the
squares…. 100.
The original triangle is equivalent to 1/2 the square,
having the same base and same altitude.
2. What is the area? The area of the square would be equal
to its base times its altitude. But we only want half of it
since we are looking at the triangle that represents half
the square.
3. We can write: A = b h 2
Method II
4. Demonstrate that triangle formed of “two halves” is
congruent to whole triangle. Substitute the two halves.
Can we count the squares?
By flipping-over right hand triangle we can form a
rectangle with these two halves. This rectangle is
equivalent to the original rectangle.
5. The area of this new rectangle is therefore the same as
the original one. Note that the base is equal to half the
base of the whole triangle and the altitudes are the same.
6. We can write: A =
b
h
2
7. Inverse rules; (see section after Method III)
© Copyright, 2004
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Geometry
Method III
8. Select mediator as illustrated and demonstrate
congruency to whole triangle. Substitute mediator.
Explain construction of mediator.
Now we can count the squares? “No” flip over two smaller
triangles as shown and form rectangle. This rectangle is
equivalent to our original rectangle.
9. The area is therefore the same. Note that the base of this
rectangle is equal to the base of the triangle while its
altitude is half of the triangles altitude.
10. We can write: A = b
© Copyright, 2004
h
2
Page 117
Geometry
Right Angled Triangles
Method I
1. Take right-angled triangle and count squares…. Not
possible. Take another right-angled isosceles triangle
and demonstrate congruence.
Combine both triangles in such a way that they form
halves of the resultant square. Now we can see the
resultant area is 100 (10*10).
2. We double the area of the triangle and obtained an area
of 100. This 100 is equal to the height times the base. If
we only are concerned with the triangle’s area, it must be
half of that.
3. Therefore… A = b h 2
Method II
4. Again the right-angled triangle. Can we count the
squares? “No”. Need a mediator. Take mediator and
demonstrate congruency… substitute.
Explain how mediator is formed… obtaining trapezoid and
similar triangle.
Move smaller part of mediator in such a way that we form
a rectangle equivalent to our original rectangle.
Demonstrate.
5. Note that this rectangle was formed by “taking” 1/2 the
base of the triangle. Its height is the same as that of the
triangle.
6. Since the areas of the two equivalent rectangles are the
same, we can now express the area in our formula.
7.
A=
h
b/2
b
h
2
© Copyright, 2004
Page 118
Geometry
Method III
Same pattern as before with result being: A = b
© Copyright, 2004
h
2
Page 119
Geometry
Obtuse Angled Triangles
Method I
1. Count squares of obtuse angled triangle? “No”.
Take mediator and demonstrate congruency.
Then place it as illustrated to form parallelogram.
Can we count the squares now? “No”. But we
already know how to compute area of
parallelogram.
2. Since the area of the parallelogram is equal to its
base times its height, then the area of one of these
triangles must be half of that.
3.
A = bh 2
Method II
4. Count squares of triangle? “No”. Take mediator and
demonstrate congruency. Substitute.
Place mediator as shown in illustration, locate altitude.
Now, move outer-part of triangle in such a way that we
have formed a parallelogram.
5. Area of this parallelogram is A = bh Q.E.D. Now we
can see that see that this parallelogram, which is
equivalent to our obtuse-angled triangle has an altitude
equal to the triangle’s and a base equal to half of the
triangle’s.
6.
A=
b
h
2
Note: It is unnecessary to go “one-step” further and make the
rectangle from the PG.
© Copyright, 2004
Page 120
Geometry
Method III
Same procedure as above following the illustrations.
Resultant formula will be:
A=b
h
2
Exercises/Activities
Children construct figures on graph paper (squares not equal
to material). Children construct figures of different sizes to
prove formulas.
Direct Aims: Calculate area of some plane figures. Concept
that the square is a unit of measure… Noting that the “triangle
is the “constructor; the square is the “measure”.
Indirect Aim: Preparation for calculation of lateral and total
surface solids.
© Copyright, 2004
Page 121
Geometry
Square
Material
•
•
Square metal inset
Yellow “area” rectangle
Presentation
1. Identification of square; side, side, side, side, (turn,
rotate). Like this … base, height.
Cover “half” of the rectangle with the square inset. How
many squares are covered? “25” How can this 25 be
obtained? 5 x 5
2. We could consider one five the base and the other the
height. But the base and height are both equal to the side
of the square “s”.
Then, in place of “b” we could write “s” and in place of “h”
we could write “s”. In fact, we can say side (s) times side
(s) instead of base (b) times height (h).
A =b×h
3. A = s × s
A = s2
4. Inverse Rule: If we know “A” how can we find “s”?
€
A = s 2 ,…. s 2 = A , … s = ? By removing the exponent we
have divided the side by itself s × s s … What do we do
with A?
© Copyright, 2004
s=
Application of principle
used for
comprehension of
square root.
A
Page 122
Geometry
Rhombus
Method I
Demonstrate with insets of equivalence Frame 4. By
showing that the rhombus is equivalent to the rectangle
we can say that its area is computed the same way QED.
A = bh
Method II
Take rectangular piece of paper and fold in fourths. Draw
thick lines in each fold as shown. Green lines join the red
and blue lines and form the outline of the rhombus.
Cut out rhombus and reconstruct rectangle as shown.
1. Identify the long diagonal D, and the short diagonal d.
Observe that the base of the rectangle is the same as D
and the altitude is the same as d.
Remove the 4 “outer” triangles and form another
congruent rhombus. Now, if the area of the rectangle is b
x h, we can say that, in this case D x d will give us the
area of the rectangle. Furthermore, since the rhombus is
1/2 of the rectangle, the area of the rhombus can be
expressed as below.
2.
A = Dd / 2
© Copyright, 2004
Page 123
Geometry
Method III – 1 “Using Paper Illustrations”
3. Take only the “interior rhombus from Method II and cut
along the shorter diagonal. Then cut out half of the half
along the red line.
4. Position pieces of rhombus as illustrated to form
rectangle. We can see that the altitude of this rectangle is
equal to half the longer diagonal, while its base is equal to
the shorter diagonal. We know the area of the rectangle
is equal to base times height. Since the rectangle is
equivalent to the rhombus, we can substitute “D” and “d”
as follows…
5.
A=
D
d
2
Method IV – 1 “Using Paper Illustrations”
6. Take rhombus, which was used in Method II and cut the
other half in half, positioning pieces as illustrated.
7. Reposition pieces to form rectangle whose altitude is
equal to half of the shorter diagonal and whose base is
equal to the longer diagonal of the rhombus.
8.
A=D
d
2
Method V – 1 “Use of Insets of
Equivalence”
Remove one of the equilateral triangles from Frame
3 and replace it with equivalent halves from Frame
4. Identify the long and short diagonals. Remove
the two halves and place around equilateral triangle
as shown (to form rectangle).
© Copyright, 2004
Page 124
Geometry
This demonstrates that… A =
D
d
2
Method V – 2 “Use of Insets of Equivalence”
Place Frame 3 in alternate position as shown. Take the first
figure of Frame 13 and remove the bottom half and place in
Frame 3.
Now take the two smaller pieces from Frame 4 and complete
the rhombus in Frame 2. Moving the two smaller pieces
around, we obtain a rectangle. By examining this rectangle
we can demonstrate…
A=D
d
2
© Copyright, 2004
Page 125
Geometry
Trapezoid
There are two approaches to solve for computation of the area of the trapezoid. One
would transform the trapezoid to the rectangle through the FRAME of Equivalence
insets. The second, preferred by Dr Montessori uses the triangle as mediator.
Specifically, Dr Montessori demonstrates the “triangle” approach by using the blue
triangles in the second box of the first series of constructive triangles. That is, the two
blue triangles that will form a trapezoid.
Method I
1. Take the two blue triangles from the second box, first
series, of constructive triangles and form trapezoid.
(Older material required “flipping” over the smaller
triangle)
2. We can see that the base of the triangle is equal to
the sum of the bases of the trapezoid and that the
altitude of the triangle is equal to the altitude of the
trapezoid.
We have seen that this triangle is equivalent to the
trapezoid. Since the area of the triangle is 1/2bh, we
can substitute (B + b) for b, and obtain the formula
for the trapezoid.
3.
A=
( B + b) h
2
© Copyright, 2004
Page 126
Geometry
Method II
4. Draw any trapezoid. Prolong the major base
and mark off a distance equal to the minor
base. Join point “x” to point “y”. The part of
the trapezoid that is above the line xy is
cutout and placed in position shown to form
a triangle.
5. We can see that the area of this triangle is
equivalent to the trapezoid. The base of the
triangle is now the sum of the major and
minor bases of the trapezoid and it s altitude
is equal to the trapezoid’s.
6.
A=
( B + b) h
2
© Copyright, 2004
Page 127
Geometry
Polygons
For demonstrating equivalence Montessori transforms the polygon (decagon) into a
rectangle. But in working with areas, the Montessori solution is to use the triangle as
mediator. Why? Because it is not necessary to have the rectangle in order to “count” the
squares. The triangle formulas have been proven and are much more suitable for the
area demonstration.
Note for teacher… In all regular polygons there is always a constant, irrational number
that enables us to calculate the area by knowing one side. This is an advance study.
Method I
Regular Polygon
Conclusion from work with the insets of
equivalence: The regular decagon,
Frame 8, is equivalent to the rectangle,
Frames 9 and 10. Make labels for
Perimeter (P), half perimeter (P/2),
altitude (a) and half altitude (a/2). One
rectangle demonstrates that the base is
P/2 and altitude as “a”. The other
rectangle shows the base as “b” and the
altitude as “a/2”.
A=
P
a
2
A=P
€
a
2
Method II The Apothem
Material
•
•
•
•
Regular Polygons from Geometry Cabinet
Frame/inset… Triangle inscribed in circle
Frame/inset… Square in fourths, by diagonals
Largest Circle form Geometry Cabinet
© Copyright, 2004
Page 128
Geometry
Presentation
1. Remove inscribed triangle. Remove two triangles of the
large square to form a smaller square inscribable in 10 cm
circle.
2. Verify that all the polygons, triangle thru decagon are
inscribable in the 10cm Frame of circle. The apothem is
the perpendicular line joining the centre (knob) of each
inset and the base of the polygon -- the radius of the
inscribed circle.
3. There is also an apothem in a square and triangle, which
can be seen by drawing the “inscribed” circle and joining
the mid point and the point tangent to the base.
4. Children trace all the regular polygons and draw the
apothem in red. They identify the side of the polygon that
will be considered the base in “blue”.
5. Children compare the two lines in each polygon and
identify which is shorter, or longer.
This triangle is used
because it is
inscribable in a 10cm
circle.
We are heading
towards a ratio
between the length of
the side and the
apothem.
This is done with all polygons through the decagon.
as
as
as
as
as
as
Statements concerning the above comparison:
From the triangle to the hexagon a  s
From the heptagon to the decagon, ad infinitum… a  s
© Copyright, 2004
Page 129
Geometry
In advanced geometry we can calculate the exact ratios for
computation of area.
© Copyright, 2004
Page 130
Geometry
Method III “Used by Mathematicians”
Take any regular polygon (e.g. pentagon).
Divide into triangles and locate apothem as illustrated.
Separate pentagon so that each side is adjacent as shown in (1) below.
(1)
(2)
Draw “triangle” line and show relocation of pieces with colored dots indicated where “cut
off” triangle would fit. (Note: Line may be drawn from any vertex. e.g. One could draw
the dotted line from the center triangle to each end of the base.)
© Copyright, 2004
Page 131
Geometry
(3)
h = apothem
b = perimeter of pentagon
Therefore: A( triangle ) =
bh
ap
= A( pentagon ) =
2
2
(4)
€
All five triangles above are equivalent because all have the same base and height.
© Copyright, 2004
Page 132
Geometry
The Circle
The circle as the limit of regular polygons
Material
•
•
•
•
Circle Drawer
Regular Polygon Drawer
Inscribed Equilateral Triangle
Square Inset + 4ths
Presentation
1. Take frame of largest circle from geometry cabinet and
place the equilateral triangle in it. Point out the three
uncovered spaces. Repeat with all the regular polygons
form the square to the decagon.
2. Children will recognize that as the number of sides of
the polygon increases, the spaces get smaller.
If we had a regular polygon of twenty sides, 100, or 1000
sides the spaces would continue to get smaller but their
number would increase.
3. Place the circle inset in its frame. Ah! How many sides!
They can’t be counted… infinite. The circle is a regular
polygon of an infinite number of sides. Each point on the
circumference is a side of the circle.
© Copyright, 2004
Aim: This exercise has
as its aim to identify
regular polygons to the
circle.
Page 133
Geometry
Transfer of Polygon Nomenclature to the Circle
Material
Circle and Decagon insets
•
Presentation
List the nomenclature of both simultaneously:
Decagon
•
•
•
•
Has side
Group of
sides/perimeter
Has a centre
Line segment from
its centre to side is
apothem
Circle
Has no side, just point
Group of
sides/circumference
• Has a centre
• Line segment from
centre to
circumference… radius
Therefore the circle also has a
perimeter and an apothem…
just different names.
•
•
Measuring the Circumference
1. We know the perimeter of this decagon is equal to the
number of sides “n”, times their measure. What about the
circle? Perhaps we could transfer the measure with a
string! But that would apply for just this one circle.
2. Trace a line on board. Take largest inset of circle, and
mark a starting point on its edge. Also make a starting
point on line. Join both starting points and “roll” circle until
the mark touches the line again. Identify that point on the
line. This line segment represents the circumference.
A most exciting
discovery for children
and adults.
Carefully so that circle
does not slip.
3. On the same line, mark off the number of diameters that
are contained in that line segment. “Three plus a small
fraction”.
Do the same with the five remaining circles in the draw
© Copyright, 2004
Page 134
Geometry
(9,8,7,6, and 5cm). Also do it with circles of 4,3,2, and
1cm diameters.
The same result! Each diameter is contained in its
circumference three times plus a fraction.
Two ways to determine that Fractional Part
4. Take slip of paper and mark off that left over fractional
part. Note that it is contained in the line segment a little
tiny bit more than seven times because there are seven of
these little fractional parts in this segment (left after the
three diameters have been measured out). It has a value
of about 1/7 the diameter.
5. Take a circle with a diameter of 10 centimeters and divide
its circumference into 100 parts. The left over fractional
part will correspond to 14 millimeters. Hence, the
diameter is 3.14 times the circumference.
Good time for History of
(You can use the
centesimal circle.)
π:
π
The Greeks have a name to this number. “P” in Greek is
.
Instead of writing 3.14 we could write
. Emphasize that it
is an irrational number. We have only identified two decimal
places. There are many more 3.141589….
π
€
(100-page book!)
How can we calculate with circumference? We must know
the diameter. Since the diameter is a line segment we can
measure it, and we can use this
.
€
€
π
π
The circumference is contained in the diameter
times. If I
know the diameter, how can I find the circumference? We
must repeat the diameter,
times.
π
€
Put out all the circles in order from 1 – 10 cm:
π
π
In this circle, d = 1….. C =
In this circle, d = 2…..C = 2
€
etcetera through 10 cm circle.
€
Now, if d = 1, and π = 3.14 then C = 3.14
If d = 2, and π = 3.14 then C = 6.28 etcetera through to
d=10.
€
€
€
€
© Copyright, 2004
Now have built a set of
fixed numbers that are
multiples of Π .
Page 135
Geometry
Organization of Rule and Formula
The circumference is equal to the diameter times the constant
. Substituting C, d and
we get: C = d
π
π
π
Instead of the diameter we could express this in terms of the
radius: C = 2rπ , or C = 2πr
€
€
€
€
€
Exercises
Children calculate the circumference of all the circles
available in the material.
Children do the same with circles that they make themselves.
© Copyright, 2004
Page 136
Geometry
The Area of the Circle
Material
•
•
•
Yellow and green cardboard circles
Sectors of each circle
Rectangles
Presentation
1. We know the formula for the regular decagon:
A=
p×a
If we substitute “C” for “p” and “r” for “a” then
2
we would probably obtain the formula for the circle:
A=
C×r
Now we must prove it!
2
2. Take the green and yellow circles and their corresponding
sectors and demonstrate by superimposition that the
union of the sectors is congruent to its respective circle.
Remove the yellow and green “wholes” and retain the
yellow and green circles comprised of ten tenths.
3. Arrange the yellow sectors as follows:
4. The resultant figure is equivalent to the yellow circle. Now
take the green sectors and arrange them as shown.
5. The resultant figure is “more or less” a rectangle. It is not
exact because its base is made of arcs which when
placed all together form the circumference.
Suppose that the sectors were smaller, then the arcs
© Copyright, 2004
Page 137
Geometry
would also be smaller approximating a straight line.
6. Superimpose the green rectangle over the “sort of”
rectangle above. This new rectangle is more accurate. Its
base is underlined in black corresponding to the
circumference of the circle. Its height is equal to the
radius.
But this rectangle is equal to two circles. We have here
the area of two circles. We must divide the product of its
base x height (or, as previously indicated… its
circumference x radius) by 2.
First way A = C × r 2 Since C = 2πr,
(2πr) × r
= πr 2
2
€
Now to prove second way: A = πr 2
Second way A =
7.
We know that this rectangle is equivalent to two circles.
€
We can see that the green rectangle is the same size as
€ rectangles. Superimpose to prove.
two of the yellow
8. Now remove the green rectangle and one of the yellow
rectangles.
9. With one of the yellow sectors show that the height of the
remaining yellow rectangle is equal to “r”. Use the yellow
circle to show that the side of the sector is the same as
“r”:
Also demonstrate that the dimensions of the “squares”
contained in this rectangle are “r” x “r”. Therefore, each
square represents the square of the radius. How many
are there? Measuring with the sector we see that there
are 1, 2, 3, plus a fractional part and we know that this
fractional part is 1/7 or .14….
10. The rectangle is formed of 3.14… r2’s; or r2 taken π
© Copyright, 2004
€
Method One:
circumference
important
Page 138
Geometry
times.
Therefore: A = πr 2
Method Two: r2
important
€
© Copyright, 2004
Page 139
Geometry
The Area of the Sector
Material
•
•
•
Metal Fractional Circle insets
Frame 8 (Decagon) of Insets of Equivalence
Montessori Protractor (for the second proof)
Presentation
1. Select the circles divided into thirds and tenths. Remove
⅓ from inset of thirds and identify it as sector referring to
geometry nomenclature. We also can identify the
remaining 2/3 as a sector.
Children will work with
all.
2. Do the same with all the fractions from 1/4 thought 1/9.
Now demonstrate the same with the 10ths.
3. Take a 1/10 sector of the circle and a 1/10 from Frame 8…
isosceles triangle of the decagon.
Superimpose! They almost correspond because the
angle at the centre has the same measure. One is limited
by an arc and the other by a line segment.
We compare these two
because the circle is
the polygon of infinite
sides and the decagon
is the largest regular
polygon we have.
The triangle: Here is the base. The altitude passes
through the knob.
The sector: Where is the base? “Arc”. The height passes
through the knob.
It is evident that the “height” is equal to the radius, which
is equal to the sides of the sector.
The area of a triangle: b h/2…substitute “a” (arc) for base
and “r” for height sector area A =
ar
2
4. But how can we determine the length of the arc?
Montessori chooses the
10cm circle to make the
calculations simple.
Since we are using the 10 cm circle we know that the
circumference will be 3.14… Furthermore, the sector
© Copyright, 2004
Page 140
Geometry
represents 1/10 of the circle… therefore the arc must be
1
/10 of the circumference, or 3.14. We also know that the
radius of the circle is 5 cm.
Therefore, A =
3.14 × 5
= 7.85cm 2
2
Similarly repeat with other fractional parts.
© Copyright, 2004
Page 141
Geometry
Area of a Segment
Material
•
•
•
Equilateral Triangle inscribed in circle
Circle fraction inset of 3rds
Circle fraction inset of halves
Presentation
1. Remove one segment from the inset of the inscribed
equilateral triangle. Identify it as a segment referring to
classified nomenclature. Note that remaining part is also a
segment of the circle. One segment is smaller than 1/2 the
circle and the other is greater than 1/2 circle. Use the 1/2
circle to demonstrate.
Segment of Circle Smaller than 1/2 Circle
2. Remove 1/3 from the (fraction) circle frame of thirds and
place in the segment from inscribed equilateral triangle inset,
filling in the empty space.
3. This segment is equal to the removed sector (1/3) less the
“exposed” isosceles triangle. The base of the triangle
corresponds to a chord of the circle and the altitude of the
triangle unites the centre of the circle with the midpoint of the
chord.
Area of the sector =
a×r
2
Area of the triangle =
Area of Segment =
b×h
Substituting “K” (chord) for “b”….
2
ar Kh ar − Kh
−
=
2
2
2
4. This is the area of the segment when it is less than 1/2 the
© Copyright, 2004
Page 142
Geometry
circle.
Segment of Circle Greater than ½ Circle
5. It will be equal to the Area of the “large” sector plus the
area of the equilateral triangle.
Area of Segment =
ar + Kh
(“a” will be the length of the
2
large arc)
Therefore: A =
ar ± Kh
2
Area of the Ring
A = πr 2 − πr 2 = π (R 2 − r 2 )
€
© Copyright, 2004
Page 143
Geometry
The Ellipsis (Ellipse)
Material
•
•
•
•
Cylinder Frame/inset from Geometry Cabinet
Circle Frame/inset from Geometry Cabinet
Ice cream cone
Toilet paper cylinder
Presentation
1. This is a cylinder. Identify its base and its height. Holding
a sheet of paper. This paper is a plane. First show the
plane parallel to the base and then not parallel.
Toilet paper cylinder
2. Cut the cylinder through a plane not parallel to the base
and place the “plane” in between cutout.
3. Take the cone and identify it. Similarly show the “plane”
parallel to the cone’s one base and then not parallel to the
base. Cut the cone as we have the cylinder on a plane not
parallel to the base and put plane in between cutout.
4. With the two parts of the cylinder: Follow the cut part with
your finger. This is an ellipse. And so is the other part of
the cutout.
Shows child how ellipse originated!
By varying the angle of the “cuttingplane, as we approach 90°, we
approach the circle.
Create a supplementary six-page
nomenclature for study of the ellipse.
With the two parts of the cone, repeat same process.
© Copyright, 2004
Page 144
Geometry
5. This figure is called an ellipse or ellipsis. Ellipsis means
“something missing”. But what is missing? Something is
missing compared to another! What are we comparing it
to? The circle.
Take the inset frame of the circle and place the ellipsis
inside. What is it that is missing? The uncovered parts.
6. With the large 10cm frame circumscribed about the ellipse
we see that its diameter is the same as the axis of
symmetry (major). Now take the 6 cm circle from its frame
and place it in the frame of the ellipsis. Now we see the
minor axis of symmetry equals the diameter of this circle.
(In this illustration we might also point out the derivation of
the technical name for the ellipse: Prolate Circle.)
7. Construct drawing as seen in illustration. We can clearly
see the major and minor axis of symmetry as they
correspond to the diameters of the 10 and 6 cm circles.
The conic closed curved line is called an ellipse.
The circular closed curved lines whose points are equidistant from the centre is called a circle.
Take the drawing of the ellipse and the drawing of either
circle and compare the two:
Major axis of symmetry: diameter of circle
Minor axis of symmetry: diameter of circle
Centre of symmetry: centre of circle
8. Radius of circle, we can call it “a”.
Other radius, we can call it “b”.
Now, we know that the circle is the limit of regions closed
by a curved line. So, the axes of symmetry in an ellipse are
different than the diameters of a circle because they have
different lengths in an ellipse.
9. Identify the major axis of symmetry as “2a”; its semi axis
will be “a”. Identify the minor axis of symmetry as “2b”; its
© Copyright, 2004
Page 145
Geometry
semi axis will be “b”.
We can call the horizontal axis of the circle “a” and the
vertical axis of the circle “b”. Then,
a =r
b=r
and ,
a=b
We can say: A = πr × r Substituting “a” for “r” and “b” for
“r” we get…
A = πab ; π times the length of the semi axis.
€
€
€
How to Construct an Ellipse
10. Draw a line segment. Locate two points with
stars and call them foci. The foci lie on the
major axis of symmetry.
11. The green line represents the distance between
the foci and we identify it as “2c”… Then, “c” will
be ½ distance between the foci.
Keppler’s law: The orbit followed
by the earth is an ellipsis where
the sun occupies one of the two
foci.
Identify the foci as F and F1. 2c=FF1
If “O” is centre of symmetry, c = O =OF1 = FO
12. Take string and tie knot on either end so that
they are “FF1” apart. With thumbtacks in F and
© Copyright, 2004
Page 146
Geometry
F1 place loop around each and scribe circle.
The loop represents the distance from the foci to
the vertex of the ellipse. In our example we
have formed an ellipse with the string that is four
times the size of the one in the material.
Area of the Ellipse inset
= AΠab = Π × 5 × 3 = 15Π
The area of the ellipse we drew will be 16 times
that!!
© Copyright, 2004
Page 147
Geometry
The Tiling Game
Material
Board to places tiles… represents a “surface”. The following
shapes (six of each):
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Regular Pentagon
Regular nonagon
Regular decagon
Regular octagon
Circle
Regular hexagon
Regular heptagon
Flower
Other flower
Square
Equilateral triangle
Rectangle (½square)
Rhombus
Other figures
Green
brown
orange
green
blue
green
pink
gray
orange
yellow
red
green
Level 1 How can we Tile a Surface?
With equilateral triangles?
with squares?
with rectangles?
with rhombi?
with pentagon?
with hexagon?
with heptagon?
with octagon?
with nonagon?
with decagon?
with circle?
with flower?
with flower?
© Copyright, 2004
Yes
Yes
Yes
Yes
No
Yes
No
No
No
No
No
No
No
Page 148
Geometry
Examine those Cases of Possibility
Analyzing the equilateral triangle, square, rectangle, rhombi,
and hexagon we find that each ahs an underlying
characteristic that permits us to place them on a surface in
such a way that there are no “spaces” in between them.
This characteristic is the fact that each of their interior angles
is contained in 360° an exact number of times.
In other words, they can form a whole angle!
Level 2 Examining the Impossible Cases
Basic reason is because their angles are not sub-multiples of the whole
angle. If we want to sue these to cover our surface we must employ
mediators.
Regular Pentagon.
Rhombus whose acute angle = 360° - 3(108°)
Obtuse angle = 360° - 2(108°)
Heptagon:
”Bow-Tie”, eight sided figure, or two irregular pentagons
© Copyright, 2004
Page 149
Geometry
Octagon:
Square with sides = sides of octagon
Nonagon:
Irregular dodecagon “can be split” into two equivalent figures.
Decagon:
© Copyright, 2004
Page 150
Geometry
Irregular concave hexagon. A butterfly. Two trapezoids.
Circle:
Internal curvilinear square; Internal curvilinear triangle
Quatrefoil:
© Copyright, 2004
Page 151
Geometry
Second flower: Curvilinear square similar to the one made with circles.
© Copyright, 2004
Page 152
Geometry
VI. Volume
© Copyright, 2004
Page 153
Geometry
Volume
Material
•
•
•
•
•
•
•
•
•
Red rods
Brown stairs
Pink tower
Series of Solid Insets
Series of small geometrical solids
Box of 250 cubes
Rectangles of the yellow material
Series of cubes from the cabinet of powers
Ten 100 squares
Introduction
This work will follow similar pattern as work with areas… the
starting point is the same.
The Solids
Concept of Solid: The children have already had this concept
in children’s house. Through the education of the visual
senses there is a perfecting of the concept of size. This has
been developed in children’s house with:
Blocks (stairs, rods, cubes)
Series of Solid Insets
Small geometrical solids.
Ellipsoid
Concept of Volume: This is the measurement of the solid.
We have led the way with the first presentation of the decimal
system with the golden bead material. We first explained that
“this is 100”... then we superimposed more of the same
squares increasing the thickness of the “body” until we
reached 1000.
Cube (regular
hexahedron)
We must point out to the children that all things in reality are
three-dimensional.
Ovoid
Sphere
Right Circular cone
Equilateral cylinder
Regular hexagonal
right prism
Regular triangular
pyramid
Right rhombic
Parallelepiped
Regular Parallelepiped
(rectangular solid)
Square-based
rectangular solid
Right circular cylinder
Right square pyramid
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Geometry
Presentation
1. Take 12 small cubes from the box containing 250
cubes. Line them up in a row (a) noting that there are
only two positions that are different since one face is a
square.
(a)
2. Form figure (b). Note there are three different
positions this solid can take.
(b)
3. Form figure (c). Only two positions.
(c)
4. Form figure (d). Again, three positions are possible.
5. All four are equivalent figures because they are
formed by the same number of units (cubes).
(d)
Volume of the Square Based
Rectangular Solid
1. Take large brown stair. Compare it to the
large blue rectangular solid… they are the
same!
2. How can we measure this blue solid? We
note that two of the surfaces are squares.
Then we take our predetermined unit of
measure (2cm) and mark off five units on
the side of the square. If we cut this block
at the places where we marked it, we will
obtain five slices.
3. But these “slices” do not represent the unit
of measure that we are going to use for
measuring volume!
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Geometry
Take one of the slices and mark with our
“measuring slip” the short and long side of
one of the rectangular faces of one slice.
If we draw the lines as we did with the
area material we obtain the result seen in
the second slice.
The result of the individual cuts will be
cubes equal to the wooden cube, which
will be our unit of measure.
4. We want to know how many cubes of this
size are contained in this large squarebased yellow rectangular solid. This box
of wooden cubes has the number of
individual cubes that corresponds to the
yellow one.
5. Since each “slice” is composed of 50
cubes, the whole SBRS will be 250 cubes.
The cube is the unit of space… volume.
Any cube! Why do we analyze volume as
above? Children see that need for
decomposing the solid into cubes for
measuring volume?
At this point we deviate: Noting work of
Froebel on the sphere, cylinder and cube.
The cylinder is the mediator between
curved and plane surfaced figures. The
cube and sphere are conceptually the
“perfect” solids because they are the same
in all position as and because all spheres
of reality are similar as are all cubes of
reality.
6. Since we can’t always separate the solid
into cubes, just as we could not separate
all plane figures into square, we must find
another way!
Three methods
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Geometry
1. Isolate the three edges common to
one vertex as shown. (5x5x10)
2. Form square base and line of height
as shown. (5x5) x 10
3. The rectangular face (50) times
width (5). This method is not used
because we are considering the solids
in a particular position and are
recalling all the indirect preparations of
passing from the square (the base) to
the cube -- with decimal system
material.
7. In the first case we see that the volume is
the product of the three dimensions; in the
second case we see that the volume is the
product of the area of one side (base)
times the length of the third side (height).
8. Nomenclature: The three dimensions of
the solid are represented by a, b, and c.
V = abc or, V = (ab)c
Note that the height is always the third
dimension.
9. Identify the figure as a square based
rectangular solid.
This will be our point of reference at the
level of volume just as the rectangle
was at the level of area.
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Geometry
Volume of Other Figures –“Blue Solids”
Material
Box 1
•
•
•
•
•
Regular triangle right prism
Regular triangle right prism (2 pieces)
Right rhombic parallelepiped
Regular hexagonal right prism
Regular hexagonal right prism (3 pieces)
Box 2
•
•
•
•
•
•
•
•
•
Right triangular prisms (1 tall, 1 short)
Regular triangular pyramid
Square based rectangular solids (tall and short)
Right square pyramid
Right circular cylinder (tall and short)
Right circular cone
Ovoid
Ellipsoid
Sphere
Solids
A) Prisms
B) Pyramids
C) Solids of Rotation
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Page 158
Geometry
Prisms
Regular Triangle Right Prism
1. Identify regular triangle right prism oriented as shown.
Identify the two possible bases and the other three faces.
Where does the name come from?
”Regular” because it is equilateral triangle at bases.
“Right” because the edge that forms its height is
perpendicular to its base.
2. Compare this figure to the blue square based rectangular
prism. Sides are same but bases are different. The
secret lies in the base.
3. How can we calculate its area? We can’t count cubes, as
we have learned when we could not count squares in
area work. So, we need a mediator… Same figures
divided in half by slicing through the altitude of the
equilateral triangle. Join both figures as shown and
compare to blue square based rectangular solid.
4. We note that they are not the same. Why? Because we
have shown that their sides were equal, and when we
sliced it in half, the new side because the side of the new
figure. And, the new side we equal to the altitude of the
equilateral triangle… and altitude of an equilateral triangle
does not equal side.
It would have to have been an isosceles base for the two
figures to be the same.
5. In this way we can apply the formula: V = Ab × h QED
but, since we are seeking the volume of the triangular
prism we must take half of that… V =
Ab
h
2
6. Working with Mediator Alone:
Ask children to take both halves of the triangular prism
and to construct a solid whose cubes are countable.
They will discover that by inverting one of the halves they
can form a rectangular parallelepiped… all the other
figures are not divisible into cubes.
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Geometry
We know that the regular triangular right prism is
equivalent to this parallelepiped… whole volume will be
the Area of its base times the height.
But the base of this rectangular solid will have an area
that can be computed as ½ b h (as compared to the base
of the triangular prism).
We can further extend the formula as follows:
V =
Ab
h
2
All solids we examine
will have same height.
Therefore, the bases
will hold the ‘key’.
Right Rhombic Parallelepiped
1. Classify figures by examining base “right
rhombic parallelepiped”. We cannot count its
cubes so must have a mediator. Take the
mediator and the “whole” used to determine
the volume of the regular triangular right
prism.
2. Form the RRP with the mediator and
sensorially show equivalence. Se aside RRP
and work with three-part mediator.
We must form a solid that can be divided into
cubes. The resultant solid is the same as the
one we formed in the triangular right prism
proof above… rectangular parallelepiped.
We know its volume is equal to the product of
the area of its base and height.
3. We also know that the Area of a rhombus is
the product of its diagonals divided by 2 QED.
Substituting we obtain:
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Geometry
V =
d×D
h
2
Regular Hexagonal Right Prism
1. Study the base and identify the solid as
regular hexagonal right prism.
Demonstrate with 10 cm frame/inset of the
circle that the 10 cm circle is inscribed in
the hexagonal base.
With the hexagon inset demonstrate that it
coincides with the midpoints of the
hexagonal base.
2. To find the volume of this solid we must
divide it into cubes; so we need a
mediator. This mediator is constructed by
tracing a line through one vertex to the
next non-consecutive vertex obtaining an
isosceles obtuse angled triangle. Then
slicing through the solid we obtain a right
triangular prism. With this prism we draw
its only internal altitude and slice it.
3. Now the children must discover how to
manipulate these pieces in order to obtain
a “cubable” solid. The result is a
rectangular parallelepiped, which we have
sensorially demonstrated is equivalent to
the regular hexagonal right prism because
their bases are equivalent and they have
the same height.
4. Relationship of Lines. Now we take the
two small pieces and form an equilateral
triangle. By placing them on the
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Geometry
hexagonal solid we can see that its altitude
is the apothem of the hexagon. We also
note that this special altitude/apothem is
equal to ½ the special chord used to
determine the mediator. (We observe this
because the apothem is the radius of the
inscribed circle). In addition, the altitude
determines the base/2.
5. Reforming the rectangular parallelepiped
we can label its sides as follows:
1 side =2a
1 side: 1
2
12
6
the perimeter of the hexagon =
+ 1 × 1 (½ the base of the
2
6
equilateral triangle = ½ side of hexagon =
2
12
+ 1 = 3 = 1 …which is p/4 of the
12 12
4
hexagon.
6. We have previously encountered three
formulas for the area of the polygon. Now
we can construct a fourth:
1)
p
a
2
2) p a 2
3) p
a
2
p
2a substituting this in the volume
4
formula we obtain:€
4)
: V R. Hex. P = Ab × h =
© Copyright, 2004
p
2ah
4
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Geometry
Pyramids
Right Square Pyramid
1. What is a pyramid? From the Egyptians
“pyr” meaning fire -- sticks stacked for
burning!
Identify the figure by studying base… It is a
right square pyramid because a plumb line
form the vertex to the base will coincide
with the centre of the base and will be the
pyramid’s altitude.
This figure rests on only one side… all
pyramids are the same.
2. Identify lateral face, lateral face, and lateral
face.. base. Base is a square, therefore
right square pyramid.
3. Sensorial Demonstration: Take the square
based rectangular solid (prism) and show
that its base is the same as the base of the
right square pyramid. We also know that
their heights are the same by visual sight.
Substitute hollow rectangle solid. Substitute
hollow pyramid. Demonstrate that it takes
three loads of sand equivalent to the
pyramid to fill the rectangular solid.
Therefore the volume of the pyramid is ⅓
the volume of the square based rectangular
solid or, the rectangular solid is three times
the volume of the pyramid.
With the pyramid and prism in sight we can
say that they correspond to each other
because their bases are equal and their
heights are equal but the volume of the
pyramid is ⅓ volume of the prism.
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Geometry
Now we take the shorter square based
rectangular solid and its corresponding
“hollow”. Verify same base and altitude.
Sensorially demonstrate that the smaller
square-based prism (the solid one) is ⅓ the
taller one.
Now we take the hollow substitute for the
pyramid and the short prism. Fill one with
sand and empty into the other… their
volume is the same! They are equivalent
because they have the same base and the
prism is ⅓ the height of the pyramid.
4. Constructing the Formula: We have already
seen sensorially that the pyramid is
equivalent to the short prism. We want to
show this relationship by lines.
Take the largest pink cube (a special
parallelepiped) and demonstrate that it is
the same as another cube, which has been
divided into three parts. Remove the pink
cube.
Note that this cube is formed of three
equivalent pyramids. The base of each is
the same as the face of the cube, and the
height of each equals the height of the
cube.
We have seen that the three pyramids
make up the volume of the cube (prism).
Therefore the volume of one of the
pyramids is ⅓ of volume of the prism.
Why? Because they have the base and
height.
V =
Ab h
3
5. The height of the pyramid equals the edge
of the cube and corresponds to the side of
the square and forms the cube. (s)
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Geometry
This line corresponds to the diagonal of the
square face. (s 2 )
This line corresponds to the diagonal of the
cube. (s 3 )
Note that all three lines converge at one
vertex.
Regular Triangular Pyramid
1. It is regular because its base is an equilateral triangle.
Identify it as a regular triangular pyramid.
2. Form a “steeple” with short and tall triangular prisms and
the regular triangular pyramid on top.
All have the same base.
The tall prism and the pyramid have the same height.
The short prism’s height is ⅓ the tall prism’s height.
3. The volume of a pyramid whose height is three times a
corresponding prism having the same base can be
expressed as follows:
© Copyright, 2004
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Geometry
V =
Ab h
QED
3
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Geometry
Solids of Rotation
Cylinder
1. This is a cylinder. Means “surrounds all
around it”… also an object that rolls.
This is a base, and so is this. Its faces? One,
curved.
2. Superimpose regular right hexagonal prism
and cylinder. They look a little alike! How
many faces are there in the hexagonal solid?
“6” What if it had 1000 faces?... or even a
million.. we would approach an infinite
number of sides, and it would be like this
cylinder. “The limit of solid prisms.”
3. “Roll” the prism. The more sides, the easier it
will roll. Roll the cylinder... it rolls much
faster.
Identify its height and base.
4. We know the volume of the hexagonal solid is
given by Abh… In the cylinder the area of the
base is 11 r2. Therefore, the volume of the
cylinder will be;
V = πr 2 h
€
Cone
1. Meaning of cone? “The stone used to sharpen knives –
‘hone’”… it has the same form as that stone!
Identify base. Height is the line that passes through
vertex to centre of base.
2. Take the tall cylinder and recall that it is the limit of
prisms. The formula for the volume of the prism is
basically the same as that for the cylinder (only difference
is the expression of the area of the base).
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Geometry
Take triangular pyramid... three faces. Then the right
square pyramid… four faces. If we continued to add more
faces we would reach the limit of pyramids… Cone.
The volume of the pyramid was expressed in terms of its
relationship to the prism.. 1/3.
3. Consequently the volume of the cone shall be expressed
in terms of the cylinder!
Now, take the short cylinder along with the cone. Weight
them. The same weight… same volume.
V=
€
πr 2 H
3
Sphere
Note: In order to calculate the volume of the sphere it is
necessary to know the spherical surface. Archimedes
discovered the formula and it was later confirmed by a
disciple of Galileo, Bonavertura Cavalieri.
Theorem: The area of the surface of the sphere is equal
to four (4) times the area of its largest cross-section
circle… 4 πr 2
Now the area of these four circles would then determine
the area of anew circle. If the radius of each of the
€small circles was five, then the radius of the composite
new circle will be 10 (5 x 2… 25+25+25+25 = 100,
100 = 10)
Now we can prove this sensorially by constructing the
circles and then weighing them!
Constructing the Formula
1. Display the previously used “tall” cone; the short
cylinder, as a (teacher constructed) new “shorter”
cone. This new short cone has a base four times
that of the tall cone. (5cm radius versus 10cm
radius)
© Copyright, 2004
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Geometry
2. The radius of the tall cone is the same as the radius
of the sphere and it’s height is four times the radius
of the sphere (all material has a 20 cm height)
3. The base of the short cone has a radius equal to the
diameter of the sphere.
4. The height of the short cone is equal to the radius of
the sphere.
We can now say that the two cones are equivalent
because they have the same proportions… The
height of the short one is ¼ the tall one, while the
radius of the short one is twice the tall one – twice
being the inverse of the ¼ slice we are working with
the radius)
5. The short cone is equivalent to the sphere because
its base is the same area as the spherical surface
and its height equals the radius of the sphere.
6. How do we really know that the short cone is truly
the same as the sphere? The secret is in the
polyhedrons. If we “opened-up” an icosohedorn we
would see twenty right pyramids: The base of each
pyramid forms the surface of the icosohedron. The
height of each pyramid is the distance from the
centre of the icosohedron to the centre of the base.
If we can now imagine the polyhedron of 1000 or
1000000 pyramids we would begin to see that the
sphere is the limit of regular polyhedrons.
7. Now we would employ the method previously used
for deriving the area of a regular polygon (see
conversion of pentagon to one triangle by tracing
line form one vertex and repositioning cut-off
pieces).
Taking a series of cones we could demonstrate their
equivalence to the sphere.
© Copyright, 2004
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Geometry
Theory on the Formation of Solids
1. Take the metal inset of the square divided in halves
forming rectangles. Pour sand on paper and with ½ of the
inset in a vertical position rotate soma on paper as shown.
Identify axis. Explain “generatrix”... means “she who
brings forth” from Latin.
What does it bring forth, or generate? A Cylinder
2. Take ½ of the metal triangular inset and rotate as shown
in sand.
Identify axis. Locate the generatix. What does it bring
forth or generate? A Cone.
3. Take ½ of metal circle inset. Identify nomenclature:
diameter… semi-circumference. Place in sand and rotate.
Which is the axis? “Diameter”. Which is the generatrix?
“Semi-circumference”. What does it bring forth or
generate? A sphere.
4. Take the frame/inset of the ellipse. Trace an ellipse on
cardboard or paper with the frame. Cut it out and fold it in
half... cutout the half along the axis of symmetry. Rotate
this half in sand as shown.
Which is the axis? Which is the generatrix? What does it
generate? An Ellipsoid.
An ellipsoid is a relative of the sphere but something
happened?
5. Repeat the same experience with the oval, which will
determine and Ovoid.
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Geometry
Projections
Sphere:
Take the frame/inset of the largest circle and show
how the frame fits over the sphere. Therefore this
circle represents the maximum circle contained in
the sphere.
What will we obtain if we project a light on the
sphere? This circle.
Ellipsoid:
Place ellipsoid in frame of ellipse. The ellipse inset
will be the largest ellipse contained therein. A light
projected on the ellipsoid will produce the ellipse
inset.
But what if ellipse were in this other position? It
would produce the 6 cm circle image! So, we need
two pictures to obtain the ellipsoid... to represent
the two axes.
Ovoid:
Place ovoid in frame of oval. Its projection will be
the oval inset.
Demonstrate with several circle frames that it takes
many different circles to determine the ovoid. To
calculate the volume of the ovoid there is no fixed
rule because there are many possibilities, which
means that each oval and ovoid has its own rule.
© Copyright, 2004
You may have to prepare various
drawings.
Page 171