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Private: Learning Math: Geometry
Private: What Is Geometry? Part D: Basic Objects (20 minutes)
Session 1, Part D
In Part C, you used dots on paper to represent points, segments drawn by a pen, and folds to represent ideal line segments and even lines.
Think about these “basic objects” of geometry:
Problem D1
What is a point? How is it different from a dot on a page?
Problem D2
What is a line? A segment? A ray? How are they different from the representations you were using?
Problem D3
What is a plane? How is it different from a sheet of paper?
Problem D4
What is a circle? Why is it impossible to draw a true circle?
Problem D5: Write and Reflect
Reflect on your learning of geometry in the past. What is geometry “all about”? What is important in becoming a successful learner of geometry?
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Video SegmentIn this video segment, participants discuss some of the vocabulary of geometry. Watch this segment after you have completed Problems D1-D3 and compare your thinking with that of the onscreen participants. What are some of the difficulties the participants ran into while trying to define a point and a line? How are your descriptions similar to or different from those of the onscreen participants? You can find this segment on the session video approximately 20 minutes and 48 seconds after the Annenberg Media logo. |
Solutions
Problem D1
A point is an exact location. It differs from a dot in that it has no dimensions — i.e., no length, width, mass, etc.
Problem D2
A line is an object that has length but no breadth or depth. A ray is a half-line in the sense that it extends indefinitely in one direction only, and a segment is a subset of a line with finite length. Lines, rays, and segments do not have thickness, while our representations for them do. Also, lines and rays extend indefinitely, while our representations for them do not.
Problem D3
A plane is a flat, two-dimensional surface with no thickness and that extends indefinitely in all directions. We often use a piece of paper, a blackboard, or the top of a desk to represent a plane. In fact, none of these is actually a plane, because a plane must continue infinitely in all directions and have no thickness at all. A plane can be defined by two intersecting lines or by three non-collinear points.
Problem D4
A circle is a set of points, all of which are the same distance away from a fixed point (the center). It is a one-dimensional object and therefore has no thickness. In reality we can never draw a circle, since our representation is bound to have a thickness.
Problem D5
Answers will vary. Willingness to experiment, conjecture, and think rigorously all help in learning geometry.
