Constructions, Proof, and Rigid Motion

Lesson 1

Math

Unit 1

10th Grade

Lesson 1 of 19

Objective


Describe the precise definition and notation for foundational geometric figures.

Common Core Standards


Core Standards

  • G.CO.A.1 — Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

Criteria for Success


  1. Describe geometric “undefined” notions and identify features, examples, and non-examples.
    1. A point is a position in space. A point has no length or width (no dimensions). 
    2. A line is an infinite number of points that are connected together without any curves and continues on in both directions infinitely. A line has no width and has an infinite length (one dimension).
    3. A plane has infinite length, infinite width, and zero thickness (two dimensions). 
  2. Describe basic geometric notions built from the “undefined” terms and identify features, examples, and non-examples. 
    1. A line segment is an infinite number of points that are connected together without any curves. Two points mark the ends of the line segment. A line segment has no width and has a finite length (one dimension). 
    2. A ray is an infinite number of points that are connected together without any curves, continues on in one direction infinitely, and has a point that marks the endpoint in the other direction. A ray has no width and has an infinite length (one dimension).
    3. A polygon is a closed figure that is formed by connecting endpoints of a particular number of line segments. A polygon is a two-dimensional figure. 
  3. Compare collinear and coplanar by knowing that “co” means with, and that the two terms define where points are in respect to a line and a plane.
  4. Describe that an intersection occurs when a part of one geometric figure meets another at exactly one point.
  5. Use appropriate notation to identify all geometric terms.

Tips for Teachers


  • A point and line are considered undefined notions. The Math Forum has an excellent description of why they are undefined.
  • Ensure that students use good mathematical definitions in their notes. Criteria for a good mathematical definition are (From Illustrative Mathematics “Defining Parallel Lines”:
    • It must be clearly and precisely stated with no ambiguity. 
    • It must capture all possible situations or scenarios. 
    • It should only use notions and prior knowledge which can be considered “more basic.”
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Anchor Problems


Problem 1

Which one doesn't belong? Name at least one reason why each quadrant does not belong with the rest of the quadrants. You should have at least one reason for each quadrant.

Guiding Questions

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Problem 2

Draw a diagram that shows the description below.

  • In the coordinate plane, line p contains point R and intersects $${\overrightarrow{TL}}$$ at point $$T$$.
  • Points $$L$$, $$T$$and $$R$$ are coplanar but not collinear.

Guiding Questions

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Problem 3

Without looking at any resources, define the following terms concisely and accurately.  

  • Line segment
  • Ray
  • Plane

Guiding Questions

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References

Continuous Everywhere But Differentiable Nowhere Attacks and Counterattacks in Geometry

Target Task


Problem 1

How are a point and a plane alike? Different?

Problem 2

Draw the geometric figure that is described below:

$${\overrightarrow{RM}}$$ interesects with $${\overleftrightarrow{SL}}$$ at a right angle.  $${\overline{MQ}}$$ is drawn such that points $$L$$ and $$Q$$ are collinear.

Is it possible to have a diagram that looks different than the one you drew above? How do you know? If possible, draw the variant.

Problem 3

Is this statement always, sometimes, or never true?

“Any two given points are collinear.”

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Lesson 2

Lesson Map

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Topic A: Constructions of Basic Geometric Figures

Topic B: Justification and Proof of Angle Measure

Topic C: Translations of Points, Line Segments, and Angles, and Parallel Line Relationships

Topic D: Reflections and Rotations of Points, Line Segments, and Angles

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