Curriculum  /  Math  /  10th Grade  /  Unit 1: Constructions, Proof, and Rigid Motion  /  Lesson 16

Constructions, Proof, and Rigid Motion

Lesson 16

Math

Unit 1

10th Grade

Lesson 16 of 19

Objective

Perform rotations on a coordinate plane. Identify characteristics and algebraic rules for the rotation.

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.A.4 — Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
  • G.CO.A.5 — Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
  • 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.3

Criteria for Success

  1. Describe that rigid motions describe ways you can move a figure either on or off a coordinate plane without changing size, shape, angles, or relationship between any of the parts. 
  2. Describe rotations as a rigid motion that turn a figure around a point in a certain direction a certain number of degrees. To define a rotation, a degree, direction, and center of rotation is required. 
  3. Use an algebraic rule to show the rotation of a figure about a point and in a particular direction. Describe where the general rule is derived from. 
  4. Use notation for rotation of a figure around a point. For example, $${R_{(P,{90^{\circ}})}(\triangle {ABC})= \triangle A'B'C'}$$,  should be read as “The rotation of triangle $${ABC}$$ about point $$P$$ by $${90^{\circ}}$$ results in triangle $$A$$ prime, $$B$$ prime, $$C$$ prime.”
  5. Explain that rotations have one fixed point, the center of rotation. Describe that a center of rotation can be at the origin, on the figure, outside the figure, or inside the figure. 
  6. Rotate a figure on a coordinate plane—both when the center of rotation is on the figure, and when it is at the origin. 
  7. Use the understanding that the angle of rotation moves in a counter-clockwise direction to rotate figures from a description and an algebraic rule. 
  8. Identify approximate degrees of rotation of figures on the coordinate plane. 

Tips for Teachers

Rotation about a point that is not on the figure and not the origin will be done in the next lesson when rotations not on the coordinate plane using constructions is taught.  

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Anchor Problems

25-30 minutes

Problem 1

A rotation was performed to map $${\overline{AB}}$$ to $${\overline{FE}}$$. What information do you need to describe this rotation?

Guiding Questions

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Problem 2

What is the pattern of coordinate points that maps $${\overline {AB}}$$ to $${\overline {HG}}$$?

Guiding Questions

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Problem 3

Below are the coordinate points for the pre-image and the image. What is the direction of rotation? Degree? How do you know? 

$${\angle ABC}$$ has coordinates of $${A (1,5), B (3,3), C (1,2)}$$.
$${\angle A'B'C'}$$ has coordinates of $${A'(5,-1), B'(3,-3), C' (2,-1)}$$.

Guiding Questions

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Target Task

5-10 minutes

Rotate $${\angle RST}$$ 90° about the origin.

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

  • Include problems where students need to rotate line segments and angles on the coordinate plane using degree angles of 90° and 180°. 
  • Include some problems where students need to rotate a figure about a center of rotation that is not the origin. Encourage students to draw a temporary coordinate plane with this center as the “origin.” Consider using Anchor Problem #1 as the first introduction to this, and create “student work” that shows this process. 
  • Include “always, sometimes, never” problems with coordinate points and rotations. For example, determine if the statement is always, sometimes, or never true: “A rotation 180° will result in a transformation of $${(x,y) \rightarrow (-x,-y)}$$.”

Next

Use construction and patty paper to rotate a figure not on the coordinate plane. Describe the properties of a rotation.

Lesson 17
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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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