Download Students Discovering Geometry Using Dynamic Geometry Software

The best way to learn anything is to discover it for
yourself.
Let them learn guessing. Let them learn proving.
Do not give away your whole secret at once­
let the students guess before you tell itlet them find out for themselves as much as
is feasible.
-George P61ya [2]
Many teachers believe this approach to learning by
P61ya is best; yet, often in the past it has been difficult to tum
students loose on their own, especially in geometry where
constructions can be unwieldy. Bertrand Russell also real­
ized the value of generating belief before attempting proof.
He wrote:
Students Discovering
Geometry Using
Dynamic Geometry
Software
Michael Keyton
What is best in mathematics deserves not merely to
be learnt as a task, but to be assimilated as a part of
daily thought, and brought again and again before
the mind, with ever-renewed encouragement. '"
In geometry, instead of the tedious apparatus of
fallacious proofs for obvious truisms which con­
stitutes the beginning of Euclid, the learner should
be allowed at first to assume the truth of everything
obvious, and should be instructed in the demon w
strations of theorems which are at once startling
and easily verifiable by actual drawing, such as
those in which it is shown that three or more lines
meet in a point. In this way belief is generated;
it is seen that reasoning may lead to startling con­
clusions, which nevertheless the facts will verify;
and thus the instinctive distrust of whatever is ab­
stract or rational is gradually overcome. Where
theorems are difficult, they should be first taught
as exercises in geometrical drawing. until the figw
ure has become thoroughly familiar; it will then
be an agreeable advance to be taught the logical
connections of the various lines or circles that oc­
cur. It is desirable also that the figure illustrating a
theorem should be drawn in all possible cases and
shapes, that so the abstract demonstrations should
form but a small part of the instruction, and should
be given when. by familiarity with concrete illus­
trations, they have come to be felt as the natural
embodiment of visible fact.
Bertrand Russell [3]
I think P61ya and Russell would enthusiastically em­
brace the opportunity to use interactive dynamic geometry
software such as The Geoml!ter's Sketchpad or Cobri Geom­
etry II which permits students to discover for themselves and
63
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GEOMETRY TURNED ON
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nGURE 1. The angle formed by two altitudes of a triangle is congruent to the third angle of the triangle.
to generate belief in a geometrical fact before attempting a
proof.
For the last three years, my Honors Geometry class
(grade 9) has been more exciting and more productive as a re­
sult of using Sketchpad (the first two years) and Cabri II (the
last year). Though this class did not use a formal textbook,
it covered the material from a classical elementary geometry
course. One of the major difficulties and struggles was get­
ting students to draw adequate diagrams for problems. The
class constantly verified that well-known, unstated postulate
of elementary geometry classes: all triangles drawn by stu­
dents are either isosceles or right However. with interactive
geometry software and its ability to alter a figure dynami­
cally to produce virtually hundreds of examples quickly. the
students were able to make significant discoveries beyond
anything 1 had seen during the eighteen previous years. As
a consequence, I have had to rethink the presentation of the
material for future classes.
The number of conjectures and new ideas that resulted
from the students using Sketchpad was not anticipated. Al­
most daily it seemed there was a new discovery. 1 had
planned a course of study that would develop the processes
of explaining why a feature of a drawing is true, which, with
adjustments, would lead to formal proof. I did not consider
a particular form of proof important, but during the first two
months devoted much discussion daily to the topic of how to
explain solutions well, until a natural format was accepted.
(I admit to a degree of student manipulation here; for bycare­
fully constructing questions, 1 obtained a format agreeable
to me. The students believed that the process was sensible
rather than arbitrary.)
By the end of the first quarter, the process ofdiscovering
conjectures, looking for counterexamples, and constructing
proofs became the normal activity. At this time, I expected
all students to know how to sequence both a direct and an
indirect proof, and how to construct a proof by contrapositive
and by mutatis mutandis. I expected them to understand the
distinctions between a definition, a postulate and a theorem,
to know and understand additional terms such as "lemma"
and "corollary:' and to realize the distinction and relationship
between a theorem and its converse.
Once these aspects had been exposed, discussed. and
used, each student began to explore with Sketchpad, both at
home and at school. On almost the first day, one student
returned with the conjecture that the angle formed by alti­
tudes from two angles of a triangle was congruent to the third
angle. The remarkable aspect of his investigation was that
he also found all three cases easily. (See Figure I, which
shows 2 cases. The third case, not shown, occurs in a right
triangle.) For many problems in which multiple cases must
be considered. the proof must be altered to account for each
one. In previous years, finding the multiple cases was one of
the more difficult aspects of proving a theorem completely.
Even Euclid had difficulty with this.[l] (One of the reasons
1do not like using most textbooks is that authors seem com­
pelled to tell a student when there are multiple cases for a
question. Why not put this information in a hint section
of the book and allow a good student to discover this extra
difficulty?)
In previous years, usually on at least one problem each
day I omitted the conclusion; later I would also omit one
or more of the hypotheses. This allowed me to discuss
with students the concept of improving a theorem by either
weakening a hypothesis or strengthening the conclusion. As
the class using Sketchpad progressed, I found that 1 could
expand this sort of exploration since the drawings were now
manageable. By the middle ofthe third quarter. enough ma­
terial was developed, enough proofs and counterexamples
generated, and sufficient ideas covered to give the students
several weeks to explore quadrilaterals on their own.
For this investigation, they were given standard def­
initions of eight basic quadrilaterals: cyclic quadrilateral,
trapezoid, isosceles trapezoid, parallelogram, rectangle, kite,
rhombus. and square. The definitions were changed slightly
from standard ones, for I preferred inclusion whenever pos­
sible. Thus a parallelogram is a trapezoid. a rectangle is an
Students DIscovering Geometry Using Dynamic Geometry Software
isosceles trapezoid, etc. They were also given definitions of
a few parts (e.g., a diagonal joins two opposite vertices, a
median joins midpoints of two opposite sides). They were
allowed to name new parts: if an object had a standard name,
it was then given and used; otherwise they could choose a
word that best described the phenomenon observed. On a
few occasions in the past, a word was so well chosen that
I used it in subsequent years. My favorite student creation
was "quord," a word chosen to denote a segment with end­
points on two sides of a quadrilateral (since it involves a
quadrilateral and somewhat resembles a chord of a circle).
Other favorites were "diacenter" (intersection of the diago­
nals) and the "medcenter" (intersection of the medians). I
found that students especially enjoyed discovering and nam­
ing objects. Sometimes they elected to name an object or
a theorem after themselves, but most of the time they tried
to find a name that connected the object linguistically to at
least one of its characteristics.
In previous years I had obtained an average of about
four different theorems per student per day with about eight
different theorems per class per day. At the end of the three­
week period. students had produced about 125 theorems.
Corollaries that were special cases were not counted; for in­
stance, one of the first theorems about a rectangle should be
that it is a parallelogram. so any theorem for a parallelogram
that transfers to a rectangle is excluded from their count. In
the first year with the use of Sketchpad, the number of theo­
rems increased to almost 20 per day for the class, with more
than 300 theorems produced for the whole investigation.
In the process of the investigation. several quadrilater­
als were studied in addition to the eight basic ones previ­
ously listed. A standard theorem states that the quadrilateral
formed by connecting consecutive midpoints of a quadrilat­
eral is always a parallelogram (students called these connect­
ing segments "midments".) This was apparently discovered
in the early 18th century by Pierre Varignon. Under certain
conditions. the varignon quadrilateral can be a rectangle. a
rhombus. or a square. The students were asked to investigate
the various converses of that theorem, that is. if the varignon
quadrilateral is a rectangle (rhombus. square). then ??'!??
This investigation produced two new quadrilaterals: one
with congruent diagonals (an "isodiagonal" quadrilateral)
and one with perpendicular diagonals (an "orthodiagonal"
quadrilateral). We made a rule that any new name for a part
of a quadrilateral or new type of quadrilateral must have at
least one theorem to support the definition; otherwise there
is no mathematics, only taxonomy.
I did not give students these new quadrilaterals, nor did
I supply any proofs for their conjectures, though I did work
on their conjectures for my own interest. Even when I knew
6S
the resolution of one of their conjectures, I did not give it to
them. I preferred to have them feel that the discoveries were
entirely theirs. My role was to collect their results. organize
them. and write up pages for their "textbook," Six of the
quadrilaterals they discovered and investigated were: (I) bi­
sectogram. (2) perpbisogram. (3) altagram, (4) exvarignon,
(5) midvexogram. and (6) tangentogram. In the last section
of this article I give a summary of the definitions of these
quadrilaterals and the results for them that the students found
and (with a few exceptions) proved.
In this class I found that students who struggled with
proof began to understand why proof is an essential part
of the process. Those who were excellent at mimicking
and thinking in small structures found they needed to reeval­
uate their position. to learn first to outline a proof. I also
found that more students became involved in the discovery
process, and that transferred to a desire to prove the results. A
few students who in the early stages of the course were very
good at proving theorems discovered that they lacked the
imagination to find new problems. while others who strug­
gled with proof could look at a few examples and quickly
discern relationships. More students became involved in the
course, and everyone discovered that there is considerably
more to mathematics than just proving existing results.
At the end of the year. many conjectures remained un­
proved. Some of the students continued their explorations
in succeeding years. and shared their results with me. One
of my greatest pleasures was to have a student continue to
explore mathematics and communicate results to me.
Student Discoveries. A "bisectogram" is a quadrilateral
formed by the bisectors of the angles ofa quadrilateral. (See
Figure 2). When the bisectogram degenerates to a point or a
line segment, we will say it does not exist. Students proved
the following theorems:
nGURE 2. Bisectogram EFGH of ABeD.
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66
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GEOMETRY TURNED ON
B I: If a diagonal of a quadrilateral bisects the angles at
its endpoints (kite. rhombus, square), then its bisec­
togram does not exist
B2: If the bisectogram of a quadrilateral exists, it is a
cyclic quadrilateral.
B3: The bisectogram of a parallelogram is a rectangle.
B4: The bisectogram of a rectangle is a square.
B5: The bisectogram of an isosceles trapezoid is a cyclic
kite (exactly two angles are right angles).
A "perpbisogram" is a quadrilateral formed by the per­
pendicular bisectors of the sides of a quadrilateral. (See
Figure 3). Students proved the following theorems:
PI: If a quadrilateral is cyclic, then its perpbisogram
does not exist (This takes care of the rectangle,
isosceles trapezoid, square, and cyclic kite.)
P2: The 'perpbisogram of a parallelogram is a parallelo­
gram.
P3: The diacenter of a parallelogram is the diacenter of
its perpbisogram.
P4.1: The perpbisogram of a trapezoid is a trapezoid.
P4.2: The angles ofa trapezoid are congruent to the angles
of its perpbisogram.
P5: The perpbisogram of a rhombus is a rhombus.
P6: The diagonals of a rhombus are collinear with the
diagonals of its perpbisogram.
P1: The perpbisograrn of a non-cyclic kite is a kite.
FIGURE 3. Perpbisogram I J K L of ABCD.
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FIGURE 4. Thealtagrams EFGH of qUadrilateral ABCD.
Left: Dextro-a1tagram E F G H. Right: Levo-a1tagram of ABC D.
P8: The main diagonal (line of symmetry) of a kite is
the same for its perpbisogram.
PlO: The angles of a kite are supplementary to the cor­
responding angles of its perpbisogram. (There is a
reversal of orientation of the perpbisogram that can
occur as the kite is dragged.)
PI I: The perpbisogram is non-convex if its quadrilateral
is non-convex.
An "altagram" is a quadrilateral formed by the altitudes
from the vertices. Since each vertex belongs to two sides of a
quadrilateral, there are two "opposite" sides for each vertex.
This means that for a given quadrilateral, there are some­
times two altagrams, and under some conditions none. To
distinguish between the two altagrams, I supplied the mod­
ifiers "dextro" for the altagram formed by going clockwise
around the quadrilateral and "Ievo" for the one formed going
counterclockwise (see Figure 4). Students discovered and
proved the following theorems:
AI: The angles of the altagrams of a parallelogram are
congruent to those of the parallelogram.
A2: The altagrams of a kite are congruent, and they in­
tersect the kite at the vertices of the congruent an­
gles. (They called these angles of a kite the iso­
angles; the others were called the trans-angles. The
symmetry diagonal was called the trans-diagonal,
and the other diagonal was called the vers-diagonal.)
A3: The a1tagrams of a rhombus are congruent parallel­
ograms.
A4: An altagram of a rectangle is the rectangle.
A5: The a1t&grams of an isosceles trapezoid are congru­
ent and homothetic (ratio = -I) with center on the
median between the parallel sides of the isosceles
trapezoid. (This median received different names:
transdicular, perpitude, transmedian, and altaver­
sal.)
A6: The altagrams of a trapezoid are trapezoids.
Students Discovering Geometry Using DynamIc Geometry Software
67
A
o
FlGURE 6. Left: Dextro-midvexogram lJ K L of ABC D.
Right: Levo-midvexogram I J K L of ABC D.
IGURE
5. Exvarignon EFGH of ABCD.
A7.1: The altagrams of a cyclic quadrilateral are cyclic.
A7.2: The cireumcircles of the allagrams of a cyclic
quadrilateral are congruent.
An "exvarignon" quadrilateral is formed by lines paral­
,I to a diagonal and passing through endpoints of the other
iagonal. (See Figure 5.) The following theorems were
iscovered and proved: .
Xl: For any quadrilateral, the ex varignon quadrilateral
is a parallelogram.
X2: If the diagonals of a quadrilateral are perpendicu­
lar, then the exvarignon quadrilateral is a rectangle.
(This takes care of the orthodiagonal quadrilateral,
which includes the kite. rhombus, and square.)
X3: If the diagonals of a quadrilateral are congruent,
then the exvarignon quadrilateral is a rectangle.
This takes care of the isodiagonal quadrilateral,
which includes the isosceles trapezoid. rectangle,
and square.
X4.1: For a quadrilateral, its exvarignon quadrilateraI and
2).
varignon quadrilateral are homothetic (ratio
(The first year. this theorem was stated as: the sides
ofthe varignon quadrilateral are parallel to the sides
of the exvarignon quadrilateral. The next year, I
started including homothetic figures as a basic ele­
ment of similarity.)
X4.2: For a quadrilateral, the diacen'ter of its varignon
quadrilateral is the midpoint of the diacenter of its
exvarignon quadrilateral and their homothetic cen­
ter.
=
A "midvexogram" is a quadrilateral formed by the in­
tersection of quords from a vertex to a midpoint of the op­
posite side (one year this segment was called a midvex).
As before, since each vertex of ABC D has two "opposite"
sides, there are two of these quadrilaterals called the dextro­
midvexogram and the levo-midvexogram (See Figure 6).
These quadrilaterals received more attention than all the oth­
ers combined. Students proved the following theorems:
M1: For a trapezoid, lines through corresponding vertices
of the midvexograms are parallel to the parallel sides
of the trapezoid.
M2: Ifeitherofthe midvexograms is a parallelogram, then
the other midvexogram is a parallelogram and the
quadrilateral is a parallelogram.
M3: For an isosceles trapezoid (or rectangle), the midvex­
ograms are congruent; in fact, the reflection of one
over the transmedian (median from the midpoints of
the parallel sides) is the other midvexogram. (The
discovery and proof of this theorem provided the
proofs for a large number of other theorems, many
of which had already been proved.)
M4: For a kite (or rhombus), the midvexograms are con­
gruent; in fact, the reflection of one over the trans­
diagonal (symmetry diagonal) is the other midvex­
ogram. (The discovery and proof of this theorem
also solved a number of other problems. With M3. it
connected kites with isosceles trapezoids.)
M5: For a parallelogram. the area of a midvexogram is
one-fifth the area of the parallelogram. (In general,
they are not congruent.)
M6: For a square, the midvexograms are squares.
The students also produced a conjecture that they could
not prove and two unsolved problems.
M7: (Unproved by the students) The area ofaquadrilateral is at least five times the area of a midvexogram. M8: (Unsolved) Under what conditions will be area of the midvexograms be equal (equiareal)?
68
GEOMETRY TURNED ON
A "tangentogram" is a quadrilateral formed by the per­
pendiculars to the diagonals at their endpoints. These lines .
correspond somewhat to tangents of a circle, perpendicUlar
to a radius at the point of intersection with the circle (see
Figure 7). The theorems for this quadrilateral are similar to
those of the varignon quadrilateral.
Tt: The tangentogram of a quadrilateral is a parallelo­
gram.
T2: The tangentogram of an orthodiagonal quadrilateral
is a rectangle.
T3: The diagonals of the tangentogram of a cyclic quadri­
lateral intersect at the center of the circumcircle. T4: The tangentogram of a quadrilateral with congruent diagonals is a rhombus. T5: The tangentogram of a square is a square. FIGURE 7. Tangentogram EFGH of ABeD.
References
They easily constructed examples with unequal areas.
They strongly believed that. if the areas of the midvexograms
are equal, then the quadrilateral is a parallelogram.
M9: (Unsolved) Under what conditions will the midvex­
ograms be congruent?
1. Heath, Sir Thomas L., Euclid: The Thineen Books of The
Elements. Dover Publications, Inc., 1956.
2. P61ya, George, Mathematical Discovery. John Wiley & Sons,
1%2.
3. Russell. Bertrand, "The Study of Mathematics," Mysticism
and Logic, Doubleday & Co., Inc., 1957.