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Tuesday, July 07, 2015

Explore Geometry: The Circle

"The eyes is the first circle, the horizon which it forms is the second: and throughout nature this primary figure is repeated without end."
- Ralph Waldo Emerson (1803-1882, American essayist and poet)

In his book, A Beginner's Guide to Constructing the Universe, The Mathematical Archetypes of Nature, Art, and Science, Michael S. Schneider leads us in an exercise of seeing and understanding basic principles related to number and figure.

With each number, he couples geometric shapes with the math tools, explorations, history, and connections to nature to better understand both number and shape.

As we examine landmark numbers at the start of the fifth grade math year, we'll also explore the corresponding geometric figures to see a broader view of math and learn to use multiple math tools and processes.

So right after we explore the number one (also known as "the unit"), we'll explore the circle.

I'll begin with the question, What do you know about the circle? We'll list the ideas on a Google doc that is shown on the white board.

Next, I'll tell students that ancient mathematical philosophers considered the circle to be a symbol of the number one. I'll ask the students why they think that might be true?

I'll mention that people are drawn to the circle and that's why it's one of the earliest shapes that children draw and also a shape used a lot in advertising since it lures the human eye.

We'll look at the parts of a circle: radius, circumference, infinite points, center. Then we'll explore the classic tools of a geometer: straight edge, compass, and pencil. As Schneider suggests, I'll caution students to "Do nothing unconsciously. Be aware of each action you perform with them. No act in a geometric construction is trivial or without profound symbolism and correspondence to the world's creating process." Simply, when we construct in geometry, we often mirror the action of the natural world so pay attention to what is happening.

We'll then make lots of circles using our compasses. We'll then create lines with our straight edges and sharpened pencils from the center to points and from points to points. After that we'll use our plastic templates to create circles and place polygons inside to show that all geometric shapes fit into the circle. We'll later look at the area of a circle on a grid as compared to the area of figures with same perimeter to show that the circle has the greatest area when the perimeter is the same.

As I lead students through a number of playful explorations with these mathematical tools and constructing a large number of circles. I'll pose the following questions.
  • Where do we find circles in nature and nonliving objects? (water ripples, tree rings, wheels)
  • What is unique about the circle? (equal expansion from the center, infinite points, 0 dimension)
  • The circumference of the circle represents a cycle? What is a cycle? What cycles do you know about? Where are the rising and declining phases of those cycles? How do cycles magnify, diminish, and transfer energy?
  • A circle provides efficient space (greater area for same perimeter as other shapes). What objects in our world are made as circles to maximize efficient space? (plates, pizzas, round tables, cups/cans)
For home study, students will have the opportunity to choose one or more ways to explore this geometric figure and chart their exploration on a math journal page. The next day students will have the opportunity to share their explorations. 

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