Constructions, Proof, and Rigid Motion

Lesson 9

Math

Unit 1

10th Grade

Lesson 9 of 19

Objective


Describe rigid motions. Use algebraic rules to translate points and line segments and describe translations on the coordinate plane.

Common Core Standards


Core Standards

  • G.CO.A.2 — Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
  • G.CO.B.6 — Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

Foundational Standards

  • 8.G.A.1
  • 8.G.A.2
  • 8.G.A.3

Criteria for Success


  1. Describe translations as a rigid motion of individual points along a vector (direction) of a certain magnitude.
  2. Perform a translation on a coordinate plane described in words by shifting along horizontal and vertical vectors. 
  3. Use an algebraic rule to translate a figure in the horizontal and vertical directions a certain magnitude on the coordinate plane.
  4. Describe that the entire plane shifts with a translation. There are no fixed points.
  5. Use the coordinate plane to identify that a translation carries segments onto segments of equal length.

Tips for Teachers


  • A potential misconception for students is the difference between the algebraic rule for transforming coordinate points compared to the algebraic rule for function transformations. The algebraic rule for coordinate plane motion is intuitive- as opposed to function transformations where the horizontal transformations are “opposite” in the algebraic rule. 
  • Students will need notes that help them to organize all the features of a rigid transformation.  Ensure that students have the overarching rigid motions, and then detail the characteristics of each in terms of distance/angle preservation, and the necessary components to describe each transformation.
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Anchor Problems

25-30 minutes


Problem 1

Below are points $$A$$ and $$B$$. Translate each of these points two units right and three units down.

The notation $${(x,y) \rightarrow (x+2,y-3)}$$ is used to describe the translation you have just performed.  Explain this notation.

Guiding Questions

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Problem 2

Line segment $${\overline{CD}}$$ has endpoints of $${C (-1,-3)}$$ and $${D (-1,2)}$$.

Translate this segment according to the rule $${(x,y) \rightarrow (x-2, y-3)}$$.

Guiding Questions

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Target Task

5-10 minutes


Translate the line segment formed by the endpoints $${E(0, 2)}$$ $${F (1,-4)}$$ according to the rule $${(x,y) \rightarrow (x+5, y+4)}$$.

  • Find the endpoints of the translated image algebraically. 
  • Verify the translation on the coordinate plane. 
  • Show the translation vector on the coordinate plane.

Next

Translate points and line segments not on the coordinate plane using constructions. Describe properties of translations with respect to line segments and angles.

Lesson 10
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Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

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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