Joseph A. Montagna
Using the narrative portion concerning Pythagoras’ life, the teacher should discuss with students the major points; the major sources of knowledge were temples, their teachings were usually secret, Pythagoras traveled to foreign lands to learn the secrets of other brotherhoods and Pythagoras established a school of his own.
Ask students for some reasons why people would want to keep their teachings and bodies of knowledge secret. Was there some advantage in doing so? Introduce the term “esoteric.” Its meaning, “knowledge that is known to a small group,” should offer some clues to the reasons behind these secret societies.
In modern times we can find evidence of this practice, although the “secrets” are not so secret. Consider, if you will, the carpenter, the plumber, or any skilled trade. People of these trades have devices and “tricks of the trade” which they are somewhat reluctant to share. The difference between the Egyptian architect, or the Pythagorean student and the skilled trades of modern times being the proliferation of, and the dissemination of knowledge. Education today is not limited to a chosen few, at least not in this country and a number of others.
Discussions of this nature are intended to give students a kind of foothold in understanding the ways things were in ancient times, and to help them relate it to their own period. If nothing else, students should come away with a clear understanding that our civilization provides us with numerous opportunities for self-fulfillment.
The remainder of the lesson and strategies portion is divided into the following sections:
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—immediately following are two “excursions” that take us back in time before Pythagoras. These excursions are intended to give students some information regarding what man’s earliest experiences with geometry were. There is much that is hidden by the first excursion, e.g. the scientific explanations behind the movement of the earth and the apparent sunrise and sunset. The teacher should take the time to explain this material. The second excursion takes the circle as being man’s primary experiences with shapes and the contemplation of same. For this excursion students will need a compass and straightedge.
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—the basic concepts of geometry.
I would suggest that students make a folder in which they should keep all of their work in this unit. Informing them at the beginning that a grade will given for the completeness of this folder is an incentive for them to get started on this right away. Opportunities for extra credit work are numerous throughout this course, and these assignments should be filed in their folders, as well. Finally, each student will need: a compass, a protractor and a straightedge.
What were man’s earliest experiences with geometry? Do we dare to call it geometry? There has been much speculation about the first question and, as we shall see, the answer to the second question is that it was not geometry, at least, not the geometry we know today. Nevertheless, he was engaged in the contemplation of objects he observed, giving rise to the rudimentary aspects of geometry. Consider the following excursion.
Some ancient people observed the sun and moon and regarded them in a mystical manner, i.e. the sun’s and moon’s movements are controlled by gods. A consequence of this reverence must have been close and painstaking observations of these heavenly bodies, and the recording of these movements. There is more than mere conjecture, since structures such as Stonehenge, the great circles of Avebury, Inca and Egyptian pyramids and other structures from ancient periods are concrete evidence that this occurred.
If a person established a site on high ground, a point from which the sunrise could be clearly observed each day of the year, and if that individual visited this site on a regular basis (every third day) to observe and record the point at which the sun peeked over the horizon what would he observe? Depending on the time of year and his position on the earth, a distinct difference can be discerned.
Let us put this person on top of East Rock on the first day of summer. On this day, the summer solstice, the arc of the sun across the sky is high and long. The sunrise and sunset are at their northernmost points at his time. If our observer continued to visit this site every third day to the first day of fall, the autumnal equinox, he would have observed the sun’s arc across the sky grow steadily lower and shorter, and he would have observed that the point at which the sun peeked over the horizon has moved steadily southward. Continuing through the year to the first day of winter, the winter solstice, our observer would notice this apparent movement toward the south, after which this movement would reverse direction through the summer solstice, completing the cycle.
Assuming that this observer recorded these changes, let us say by carving marks in a large slab of stone, perhaps the following would result:
(figure available in print form)
What did early man do with this knowledge after observing that this cycle repeats without end. This was his early experience with prediction of seasonal changes, time, calendar development and astronomy. This was also his first experience with angles and sighting along points to make a straight line, though he did not use these terms.
What can we do with this with students? We would find it impractical to visit a particular site at sunrise. There is a great deal of scientific knowledge that students of the seventh and eighth graders should know contained in the preceding page. Students should know why the seasons change, why we have periods of darkness and light, that the length of these periods change daily and the the earth is, indeed, closer to the sun in the winter than in the summer. If your students do not know the scientific explanations behind these and other related questions, then you should take the time to help them understand this. It would be a good opportunity for them to see the interdisciplinary nature of mathematics and science. Later, after they have learned how to measure angles, draw angles and determine what a plane is, it might be valuable to come back to this and learn about the plane of the earth’s orbit in relation to the sun, as well as those of the other planets in our solar system. Some valuable insights can develop from this, as well as some interesting lessons. I would also encourage students to research these topics independently.
A natural progression for early man was to contemplate the circle, the shape of the sun and moon. Man’s first compass was probably two sticks joined by a length of string, the length of which represented the radius of the circle to be drawn. Placing one stick down on a surface would serve as the center of the circle. The remaining stick stretching the string to its limit is free to pivot about the center, aiding one in drawing a circle. This method is adequate, yet our students can take advantage of commercially produced compasses. Of course, it would be interesting to allow students to experiment with the stick and string method at some point. Students need a lot of practice to become proficient in the use of a compass.
Have students draw a circle whose radius is 5 centimeters. More work on circles is included later in this unit, so it is sufficient, for the time being, that they learn only the terms radius and diameter. They should also know that the diameter is twice the radius, or the radius is one-half the diameter. Allow them the opportunity to discover this and other characteristics of circles, e.g. the boundary of the circle is exactly the same distance from the center. List these discoveries. Try to have students understand that we are attempting to learn about early man’s discoveries through our own.
The next step that probably followed was drawing a cross in the circle. Have students do this, making sure that the intersection of the cross is exactly over the center of the circle, and that the cross members touch the boundary of the circle at the 9, 12, 3 and 6 o’clock positions (later they will be introduced to these positions by their respective degrees of arc names). What observations can they make about this figure? List them on the blackboard. The following points should be covered: the circles is divided into four equal parts, the divisions of the boundary of the circle are equal, and the lengths of the cross members are equal. Some might even make the observation that each part of the circle is 90 degrees. Accept this answer but add that early man had no such term.
Now, let us add to this figure by connecting the ends of the line segments which form the cross. Draw straight lines with a straightedge. Label the points as in the figure below.
(figure available in print form)
List their observations of this figure. Make sure the following points are covered:
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—a square is formed (how do we know it is a square?)
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—four equal, or congruent, triangles are formed
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—line segments AB, BD, DE and EA are equal to one another (congruent)
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—line segments AC, BC, DC, and EC are each 5 centimeters long.
We know these things to be true. In fact, we can measure them to prove it. This is what ancient people did. Geometry, as it developed and exists today, seeks to prove these conclusions without actually measuring them. This is done through a process of logical reasoning called a proof.
The proof in geometry is an oftentimes painstaking process that seeks to arrive at a logical conclusion. We will take up this topic later in the unit, after we have learned some basic definitions. However, at this point it would be wise to introduce reasoning and logical thinking through “IF . . . THEN . . . ” statements. They are good exercises for any youngster, and ones which 7th and 8th graders should enjoy doing.
“If . . . then . . . ” statements are used by many of us daily. Below are a few examples. Point out that such statements are composed of two parts; the
hypothesis
and the
conclusion
.
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If Carol is 28 years old, then she is eligible to vote.
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If we win this last game, then we will be in the championship.
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If we live in Connecticut, then we must live in the United States.
All of these statements are true and are of the following form:
“IF A . . . ,THEN B”.
“A” is the hypothesis and “B” is the conclusion.
Have students to try their hands at writing this kind of statement. Share them. Discuss their validity.
POINTS
It is probably desirable to start out by discussing what a point is
not
. This (.) is not a point. Neither is this (¥).
Students have to understand that this concept is abstract, difficult to understand; as difficult as understanding the existence of atoms, even if one has never seen an atom. Perhaps a discussion of what other concepts are abstract will help i.e., justice, the existence of God. Some things have to be taken on faith. This and some other concepts in geometry fall into this category. We must have a point from which to begin.
Have students mark a point on a piece of graph paper. Explain that the dot they made represents a point, but it is not actually the point. The dot is, in fact, covering an amount of space that includes an infinite number of points.
Points are labeled with capital letters. Have students label their points on the graph paper.
LINES
Lines are also abstract ideas. A line is an extension of a point. Bring up the idea mentioned in the narrative that a line is formed by a point set into motion.
We sometimes think of lines as being a set of points existing alongside one another. A line contains an infinite number of points. No matter how close one represents two points on a line, by definition, at least one more point exists between them.
There are different kinds of lines:
(figure available in print form)
The shortest distance between two points is a
straight
line.
Lines that are in the same plane and maintain an equal distance between them are called
parallel
lines.
(figure available in print form)
Lines are named in the following manner:.
(figure available in print form)
AB is read “line segment AB”
(figure available in print form)
AB is read “ray AB”
(figure available in print form)
AB is read “line AB”
Each point on a line can be matched up with a real number:
(figure available in print form)
thus, the coordinate of point M is 1, the coordinate of point J is -2.
To find the length of any line segment, one need only to subtract the smaller coordinate from the larger. Thus, MP = 3 units.
ANGLES
Angles are formed when two lines have one point in common. The lines are called the sides, or legs, of the angle and the common point is called the vertex. It takes at least three points in the same plane to form an angle. Angles are named by these points, usually. The letter that represents the vertex point is always named in the middle. Thus, in the angle below the correct name for it is ABC, read as “angle ABC.”
Sometimes angles are named by the use of one number or one letter.
(figure available in print form)
There are several types of angles. They are classified according to their measure. The next lesson will give practice on how to measure and draw angles using a protractor. For now, students should learn the definitions of the classification of angles.
An
acute
angle is one whose measure is less than 90 degrees. An
obtuse
angle is one which is greater than 90 degrees. An angle that is exactly 90 degrees is called a
right
angle (this has nothing to do with the direction in which it faces, so the teacher should take care in clearing this up).
