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Wednesday, July 08, 2015

Two: Birth of the Line

"Number proceeds from unity." - Aristotle (384-322 B.C.)

As students examine landmark numbers and figures, I will introduce them to enriching topics in simple ways. I won't expect students to understand all parts of this sophisticated mathematical understanding, but instead see this as opening the door to deeper understanding and exploration of geometry and number.

After we examine number one (unit/unity), the circle, and number two as part of our Examine Landmark Numbers and Figures unit and during an enrichment period, we'll use the geometer's tools, a compass, straight edge, and pencil, to create the vesica piscis. Students will connect the center points of each circle and create a line.

We'll then consider the line: one dimensional, no thickness, moving to infinity in either direction. We'll also consider the line segment: the part of a line that exists between two end points. I'll relate that the ancients considered the line (representative of two) and the point (representative of one) to be the parents of all geometric constructions.

We'll create a vesica piscis again and discuss the geometric figure. I'll tell students that we can find this figure used in art, architecture, religion, and myth throughout time. Then I'll show this short film:

Home study will include free exploration of lines, line segments, and vesica piscis through drawing, finding images online, in magazines, or in nature, and/or written explorations on a math journal page.

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During another enrichment period we'll discuss the ancient Greek words monad and dyad. The 
ancient Greeks term for the principles represented by the circle was Monad, from the root mention, "to be stable" and moans, or "oneness." These are representative of one, the unit. The principle of "twoness" or "otherness" was called Dyad by the Greek philosophers. We'll then think about words beginning with the prefixes mono (one), unit, di or du (two), and tw (two). We'll list those words and discuss the meaning and etymology or origin of the words. 

Later we'll notice the way that one represents unity while two represents polarity (the property of having two opposite poles, such as those possessed by a magnet, or having opposite properties or
characteristics. The direction or orientation of positivity relative to negativity*). I'll share this quote by Sir Isaac Newton, "For every action there is an equal but opposite reaction." Then we'll list common polarities or opposites such as day and night, light and dark, up and down. This is also a good point to explore our science standards related to the magnetic fields.
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Finally, I'll show students equation relationships that set one and two apart from other numbers:
  • 1 + 1 > 1 X 1
  • 2 + 2 = 2 X 2
  • 3 + 3 < 3 X 3, 4 + 4 < 4 X 4. . . and so we see that all numbers after one and two are greater when multiplied by itself than added to itself.
As seekers of patterns in number and figure in nature and on paper, exercises like the ones above will serve to open their minds to the greater world of math. These mind stretching exercises will help students to grasp math with greater interest, pattern identification, and connection.