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INTRODUCTION
This resource file is an attempt to promote the learning of geometry through the use of
dynamic geometry computer software. Cabri II is used as a tool for students to construct
their own mathematics. Through activities of geometrical construction and investigations,
students are led to discover important geometric theorems and properties at their own paces.
Students are initiated into a concept through guided geometrical constructions on the
computer screen. Investigation and exploration begin via “drag-mode geometry” that is
characteristic of dynamical geometry software. After some guided activities, students are
ready to express their findings in their own words and formulate conjectures. They are asked
to prove their discovered results when possible. This fosters the internalization of the
mathematical knowledge gained. Further activities are given to apply what students have
learned in a new (or modified) situation. Thus after students have gone through a sequence of
worksheets, they complete a cycle of a learning process. In this way, mathematical
knowledge is being discovered by students themselves rather than “transmitted” symbolically
by the teacher. Students come away with a higher level of mathematical awareness and
criticism. The teacher does not really teach in the traditional sense but acts as a guide or a
facilitator. His/Her role is to maintain students‟ interest in working out the tasks and provide
technical assistance in using the software and hardware properly. Once a while, the teacher
needs to summarize students‟ findings and provide explanations or clarifications when
necessary. Two to three students are assigned to a computer. They work as a group to
establish a cooperative learning environment.
The resource file is divided into three units. Students are introduced to Cabri II prior to
the use of the worksheets. They need not however be experienced users. Only rudimentary
familiarity with commands to do basic constructions, to create macro, to trace out locus and
to animate is enough to perform the tasks. Unit one is about standard plane Euclidean
geometry. Unit two introduces the ellipse, and hence the other conic sections, as the locus of
certain geometric construction. Unit three explores the properties of reflection as an example
of geometric transformations.
During the second term of the 1997 – 98 school year, all the units were tried out with a
form 4 science class. Evaluation of each unit is attached at the end of that unit. There is also a
collection of students‟ comment on using Cabri II as a tool for learning mathematics at the
end of the file. These comments were made after students have gone through Unit 2.
UNIT ONE
The purpose of this unit is two-folded. Students learn to use interactive geometry
software as a tool like a ruler and compasses, while at the same time they are learning some
new mathematics. After their acquaintance with Cabri II, this is the first time students use the
software as a learning tool. The different tasks in this unit give students an e xposure to the
step by step investigative and explorative approach in learning mathematical concepts.
Students also learn how to work together in a small group. This is an important aspect in a
computer-aided learning environment because there need to be a balance in the interaction
with the computer and with other people. Students ought not feel inferior in front of the
computer and should be reminded that the computer is only a tool. It is people who are doing
the thinking.
In TASK 1 and TASK 2, students are led gradually to devise their own method of
constructing of the circumcircle of a triangle. An important part of the activities is that
whenever feasible, students are asked to give a formal proof of their construction. This gives
students an opportunity to appreciate the different perspectives that formal deductive
reasoning and interactive, dynamic geometry hold.
Building on the experience gained in TASK 1 and TASK 2, TASK 3 continues with an
exploration of the possibility of drawing a circumcircle to a quadrilateral with the purpose of
leading students to discover a defining condition for cyclic quadrilateral. Students are asked
to make their own conjecture and prove it when possible. TASK 4 carries the same spirit as in
TASK 3 and leads students to discover a theorem concerning Napoleon triangle. TASK 5
poses a problem for students to explore on their own.
TASK 1 P.1
CONSTRUCTION AND INVESTIGATION
1. Construct two points A and B.
2. Devise a procedure to construct a circle that passes through A and B . Describe
the procedure. Can you justify your construction using known geometric
theorems?
Procedure: ________________________________
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B
Proof: _____________________________________
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3. Perform the construction procedure in Cabri II.
4. How many circles can you draw that pass through A and B? Explain.
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TASK 2 P.2
CONSTRUCTION AND INVESTIGATION ONE
1. Construct three non-collinear points A , B and C .
2. Join A , B and C with line segments to form a triangle ABC.
3. Devise a procedure to construct a circle that passes through the vertices
of ABC .
Procedure: _______________________________ A
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B
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4. Perform the construction procedure in Cabri II. Diagram
5. How many circles can you draw that pass through A , B and C?
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The circle that passes through the vertices of a triangle
is called the Circumcircle of the triangle. The centre of
the circle is called the Circumcentre.
CONSTRUCTION AND INVESTIGATION TWO
1. Create a macro that can locate the circumcentre of any given triangle. Name it
Circumcentre.
2. Construct a triangle ABC.
3. Mark and measure the three interior angles of ABC.
4. Use the Circumcentre macro that you have created to locate the circumcentre
of ABC.
5. Drag a vertex of ABC to different positions and observe the location of the
circumcentre. When is the circumcentre inside, outside or on ABC ? Describe
what you see.
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TASK 3 P.3
CONSTRUCTION AND INVESTIGATION
1. Use your experience in TASK 1 and TASK 2, investigate in Cabri II the possibility
of drawing a circumcircle of a given quadrilateral. You may begin by considering
special quadrilaterals like rectangle, parallelogram and symmetric
trapezium. Record your findings.
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2. Does any quadrilateral have a circumcircle? Investigate. If yes, how many?
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3. Hypothesize condition(s) under which a circle can be drawn through
four non-collinear points. Write down your claims in the CONJECTURE box on
the next page. You can draw a picture to illustrate your claims.
P.4
CONJECTURE
4. Prove your conjecture using known geometric theorems.
Proof: ___________________________________
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A quadrilateral that can be inscribed in a circle is
called a Cyclic Quadrilateral.
TASK 4 P.5
CONSTRUCTION ONE
1. Construct a line segment AB .
2. Devise a procedure to construct an equilateral triangle with AB
as one of its side.
Procedure: ____________________________
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_____________________________________ A
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_____________________________________ B
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3. Perform your construction in Cabri II. Diagram
4. Create a macro called Equilateral that can construct an equilateral triangle
on any given line segment with the line segment as one of the sides.
CONSTRUCTION TWO
1. Construct a triangle ABC.
2. On each side of ABC , use the macro Equilateral to construct an equilateral
triangle that points outward.
3. Use the macro Circumcentre created in TASK 2 to locate the circumcentre of
each outer equilateral triangle. Label the circumcenters D, E and F .
4. Join the three circumcenters with line segments to form a new triangle DEF.
You have just constructed something like this:
A
D F
C
B
E
DEF is called a Napoleon Triangle of ABC
INVESTIGATION P.6
1. Drag the vertices of ABC and observe how DEF changes.
2. Mark and measure the interior angles of DEF.
3. Repeat 1 above and observe what happens.
4. Describe what you saw.
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5. Can you state what you have just discovered in the form of a theorem?
THEOREM
6. Prove the theorem you have just stated.
Proof: _______________________________________
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E
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7. Is the result still true if the equilateral triangles on the sides are all pointing
inward? Investigate.
TASK 5 P.7
With the experience that you have gained from the previous tasks, you can now devise
your own plan of investigation using Cabri II. Try the following problem.
The Simson line
The circumcircle of a triangle ABC is drawn. A free point P is placed on the circumcircle and
perpendiculars are dropped from P to the sides of the triangle (extended if necessary) meeting
the sides at D, E and F. (D lies on the side that is opposite to the vertex A, E lies on the side
that is opposite to the vertex B and F lies on the side that is opposite to the vertex C.)
QUESTIONS FOR INVESTIGATION
1. What happens to D, E and F as P is moved around the circumcircle? Is DEF always
a straight line?
2. Is D always between E and F?
3. What are the loci of D, E and F? (Use the Locus command to investigate.)
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EVALUATION ONE
TASKS 1 to 3 were developed out of an actual class lesson where students used
compasses and rulers instead of computer. The lesson was quite successful and the whole
lesson was repeated in the computer room using Cabri geometry the week after. Thus the
worksheets for these three tasks can be modified for use in a regular classroom.
Students were genuinely impressed by the investigative learning/teaching approach.
They were serious in working out all the activities using their rulers and compasses, and later,
computer. TASK 1 and TASK 2 were carried out smoothly and students obtained the
expected results. They were also able to explain and prove their constructions. The climax of
the whole learning activity was the discovering of cyclic quadrilateral in TASK 3. In this
task, students did not know what to expect and the learning atmosphere in the classroom
became inquisitive and explorative. They were advised to use special quadrilaterals like
rectangle and parallelogram to start the investigation. However, when arbitrary quadrilaterals
were used afterwards, students were at a loss in formulating a workable procedure to
construct a circumcircle. Near the end of the lesson, students were asked to make some
hypotheses on the conditions under which a circumcircle can be drawn for arbitrary
quadrilateral. Three students came up with three different conditions:
(1) The quadrilateral must be symmetric in some ways, like a rectangle or a parallelogram.
(2) The quadrilateral has diagonals of equal length.
(3) The quadrilateral ABCD must satisfy a condition like this:
A B
D AX DC and XC AB .
X
C
The third condition was especially interesting because it was a creative guessing. The lesson
ended at this point, suspended in an open question. A student commented after class, “This
was really interesting.” Even though students did not yet realize the condition that a cyclic
quadrilateral must satisfied, they had already learned the process of mathematics discovery.
They just needed some careful guidance from the teacher to finish up the learning episode.
(This was done during the next class.) The activities outlined in TASK 1 to TASK 3
successfully introduced students to a new way of learning mathematics.
TASK 4 was given to students in a lesson conducted in a computer room. It follows the
same theme as in the previous tasks. Students also had a chance to further explore the
capability of Cabri II. They had no trouble doing the construction and creating the macro.
The investigation part of the task pointed to an obvious result and students could state the
theorem without hesitation. Students were impressed that they were able to “see” a
mathematical theorem using drag- mode geometry. This was not quite feasible using rulers
and compasses. The proving part was not so easy this time but it could be left as homework.
TASK 5 was designed as a problem rather than a guided worksheet and it was optional.
Some students did it enthusiastically and found the experience enjoyable rather than
burdensome. They were successful in doing their own investigation and one student asked for
more tasks to do.
STUDENTS’ COMMENTS P.1
Students found computer-aided learning fun and interesting. A few students preferred
the traditional classroom learning and their main concern was that computer-aided learning
could not help them to get good grades in examinations. One student regarded Cabri II as a
computer game with no academic value. The rest of the students gave high rating for the
investigative and explorative approach of learning through the use of computer.
The following is a collection of students‟ comments (directly quoted) which they gave
after they had finished Unit 2. These are only half of all the students‟ comments but they are
representative of students‟ reaction. The first four carry negative sentiment while the rest are
positive.
“Computer-assisted learning may be helpful if we have a lot of time to prepare the CE
syllabus. But we are a bit short of time to go thru the 2- year syllabus already. Using computer
to investigate different mathematical concepts is not practical in this case, we prefer to do
more exercise in the classroom rather than playing with a computer. …… The most important
thing is that playing with a computer software cannot help us to get a good result in HKCE.
You may say we shouldn‟t aim at examination result, but it‟s the qualification that we need in
our future.”
“It‟s interesting but I don‟t like to do these worksheets. I think these worksheets don‟t have
much use. Also, I think playing with this don‟t help me much with doing the A. Maths
problems in textbook. It is only for fun. But I still think that if there is enough time, it is good
to play with this game.”
“Using computer is more easy to understand. It is more interesting. But I prefer learning in
classroom."
“I think using computer to study the relationship between of circle, triangle … etc. is fun. But
I like the old way of learning rather than pressing a few buttons, the efficiency of doing a
problem by my own (not computer) is higher. I‟m not confident in using computer.”
“Learning these are interesting and meaningful, also being more active than just using
blackboard. By using computer, I can see locus of some point very easily. When doing
questions in textbooks, it is very difficult to figure the locus out (abstract). Computer helps
me a lot! Also, it is very convenient (just press button and the thing you want will appear on
screen).”
“It is interesting since the studying atmosphere is more active and lively. A. Maths this
subject usually consist of things which are hard to understand and is quite ambigious, though
this, it visualize the matters and it is more easy to understand.”
“I think this idea is quite good, but sometimes I think I won‟t learn anything I need because
I‟m not calculating. But it‟s quite fun. … Sometimes I prefer this kind of way of learning.”
P.2
“This is an interesting program. I have learned a lot.”
“Playing Cabri has more fun compared with Math/A. Math lessons. Geometry is kind of
abstract sometimes but Cabri turns something abstract to a more realistic thing. It is
especially useful in the locus part because it‟s sometimes difficult to imagine the locus of the
point but Cabri gives you the exact movement of the point. …. Cabri is undoubtedly a very
useful teaching tool which helps students to learn geometry more easily. I suggest using it
twice every month. Ranking: (out of 5 )”
“I like using the computer in finding out equations and things so on. It is interesting when
you find out that what U learn can actually apply to real situations, it is really “fun”! ….
Moreover, if there are steps on how to find out locus & the equation of it in the program will
be better. I hope that not only Geometry has this kind of practise. That is even like
simultaneous equations, etc. can use this sorts of programme. Because I want things that I
learn can be applied to daily situations.”
“Using the computers to learn mathematics is more relaxing and interesting. We discuss and
can always find interesting points to raise when observing the pictures in the computer. We
can discover new things about that topic.”
“It‟s quite interesting. I like it. If we have time, using computer to learn A. Maths (or Maths)
is wonderful. I hope that we have more chances to have our lesson in the computer room.
However, the computers are not enough, some of my classmates cannot have their own
computer to the work.”
“It‟s more fun than have a lesson in classroom. Easier to learn and more clear because we can
draw the picture by ourselves. I like to have lesson here because I know more about locus
than I look at my text book.”
“I think it is easier to understand the abstract ideas. For example, I cannot imagine or portray
how the „locus‟ of a point moves in some problems. They seem to vary to different shapes.
But after working on the program, I can see the continuous movement of a locus, which I
start to understand more about what a locus really is. For accurate drawing and measurement,
using this program certainly helps a lot. I can investigate the properties of locus more
thoroughly. One criticism is the operation is sometimes quite complicated. I am sometimes
struck in the operation problems; but perhaps it‟s only my problem. Anyway …”
“The lessons are very interesting indeed. The programme is very useful. I‟ve a much more
clear idea of what locus is. I hope we can have more chances to use the computer to solve
some of the problems in Maths or A. Maths. It‟s much fun to have lessons in this way.”
“It‟s really interesting to learn from the other media besides the textbook. Using the computer
is more fun and by using it, more discoveries can be found. We can especially find out how
will the locus moves by using the animation function of the Cabri software. It is more lively
to use the computer!”
P.3
“It is more interesting to learn A. Math. We can observe the change of the picture and
understand more about it. The mind is more clear. …. We know more about the name of each
thing, e.g. perpendicular bisector, locus.”
“It is more interesting for us to learn in computer room than in the boring classroom. Also,
there are several adv. 1 convenient 2 more accurate for caculation 3 for locus, I don‟t really
understand what it is since it is so abstract. But using the computer, it is very clear. However,
this software has a time limit. It makes us feel astonished.”
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