Translations, Reflections, and Rotations

Abstract

This lesson is designed to introduce students to translations, reflections, and rotations.

Objectives

Upon completion of this lesson, students will:

  • have been introduced to the concepts of translation, reflection, and rotation.
  • have practiced translating, reflecting, and rotating two-dimensional objects on the coordinate plane.

Standards Addressed

Grade 3

  • Geometry

    • The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.
    • The student demonstrates understanding of position and direction.
    • The student demonstrates a conceptual understanding of geometric drawings or constructions.

Grade 4

  • Geometry

    • The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.
    • The student demonstrates understanding of position and direction.
    • The student demonstrates a conceptual understanding of geometric drawings or constructions.

Grade 5

  • Geometry

    • The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.
    • The student demonstrates understanding of position and direction.
    • The student demonstrates a conceptual understanding of geometric drawings or constructions.

Grade 6

  • Geometry

    • The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.
    • The student demonstrates understanding of position and direction.

Grade 7

  • Geometry

    • The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.
    • The student demonstrates understanding of position and direction.

Grade 8

  • Geometry

    • The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.
    • The student demonstrates understanding of position and direction.

Grade 9

  • Geometry

    • The student demonstrates an understanding of geometric relationships.
    • The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.
    • The student demonstrates understanding of position and direction when solving problems (including real-world situations).
    • The student demonstrates a conceptual understanding of geometric drawings or constructions.

Grade 10

  • Geometry

    • The student demonstrates an understanding of geometric relationships.
    • The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.
    • The student demonstrates understanding of position and direction when solving problems (including real-world situations).
    • The student demonstrates a conceptual understanding of geometric drawings or constructions.

Eighth Grade

  • Geometry

    • Understand congruence and similarity using physical models, trans- parencies, or geometry software.

Geometry

  • Congruence

    • Experiment with transformations in the plane
    • Understand congruence in terms of rigid motions

Grades 6-8

  • Geometry

    • Apply transformations and use symmetry to analyze mathematical situations

Grades 9-12

  • Geometry

    • Apply transformations and use symmetry to analyze mathematical situations
    • Use visualization, spatial reasoning, and geometric modeling to solve problems

Technical Mathematics I

  • Algebra

    • Competency Goal 3: The learner will describe the transformation of polygons in the coordinate plane algebraically.

  • Geometry and Measurement

    • Competency Goal 2: The learner will measure and apply geometric concepts to solve problems.

Technical Mathematics II

  • Geometry and Measurement

    • Competency Goal 1: The learner will use properties of geometric figures to solve problems.

Pre-Calculus

  • Number and Operations

    • Competency Goal 1: The learner will describe geometric figures in the coordinate plane algebraically.

3rd Grade

  • Geometry

    • The student will demonstrate through the mathematical processes an understanding of the connection between the identification of basic attributes and the classification of two-dimensional shapes.

6th Grade

  • Geometry

    • The student will demonstrate through the mathematical processes an understanding of shape, location, and movement within a coordinate system; similarity, complementary, and supplementary angles; and the relationship between line and rotational symmetry.

5th Grade

  • Geometry

    • The student will demonstrate through the mathematical processes an understanding of congruency, spatial relationships, and relationships among the properties of quadrilaterals.

4th Grade

  • Geometry

    • Standard 4-4: The student will demonstrate through the mathematical processes an understanding of the relationship between two- and three-dimensional shapes, the use of transformations to determine congruency, and the representation of location and movement within the first quadrant of a coordinate system.

Intermediate Algebra

  • Algebra

    • The student will demonstrate through the mathematical processes an understanding of functions, systems of equations, and systems of linear inequalities.

7th Grade

  • Geometry

    • 7.13 The student, given a polygon in the coordinate plane, will represent transformations — rotation and translation — by graphing the coordinates of the vertices of the transformed polygon and sketching the resulting figure.

4th Grade

  • Geometry

    • 4.17.c The student will investigate congruence of plane figures after geometric transformations such as reflection (flip), translation (slide) and rotation (turn), using mirrors, paper folding, and tracing.

5th Grade

  • Geometry

    • 5.15a The student, using two-dimensional (plane) figures (square, rectangle, triangle, parallelogram, rhombus, kite, and trapezoid) will recognize, identify, describe, and analyze their properties in order to develop definitions of these figures
    • 5.15e The student, using two-dimensional (plane) figures (square, rectangle, triangle, parallelogram, rhombus, kite, and trapezoid) will recognize the images of figures resulting from geometric transformations such as translation (slide), reflection (flip), or rotation (turn).

8th Grade

  • Geometry

    • 8.8 The student will apply transformations (rotate or turn, reflect or flip, translate or slide, and dilate or scale) to geometric figures represented on graph paper. The student will identify applications of transformations, such as tiling, fabric design, art, and scaling.

Textbooks Aligned

Grade Six

  • Ruins of Montarek

    • Investigation One: Building Plans

  • Shapes and Designs

    • Investigation Five: Side-Angle-Shape Connections

7th

  • Module 4 - The Art of Motion

    • Section 4: Rotations and Reflections
      • Reason for Alignment: The Translations, Reflections, and Rotations lesson contains a discussion of the basics of transformations, along with a good worksheet that could be used as a supplement to the text.

Student Prerequisites

  • Arithmetic: Students must be able to:
    • be able to identify the basic two-dimensional shapes of a square, a triangle, and a parallelogram.
    • have a small amount of knowledge about the cartesian coordinate system.
  • Technological: Students must be able to:
    • perform basic mouse manipulations such as point, click and drag
    • use a browser for experimenting with the activities

Teacher Preparation

Key Terms

Lesson Outline

  1. Focus and Review

    Remind students what has been learned in previous lessons that will be pertinent to this lesson and/or have them begin to think about the words and ideas of this lesson:

    • Can someone tell me where you might see a reflection in everyday life? Students may point out that we see our reflection in a mirror or in a still pond.
    • Can anyone tell me what it means to rotate an object? Students may describe this as turning an object.
    • Can anyone guess what it might mean to translate an object? Students may not have an answer to this question, in which case you may let them know that they will learn what it means to translate an object in today's lesson.

  2. Objectives

    Let the students know what it is they will be doing and learning today. Say something like this:

    • Today, class, we will learn what translations, reflections, and rotations are to a mathematician.
    • We are going to use the computers to learn about these three concepts, but please do not turn your computers on or go to this page until I ask you to. I want to talk about these ideas and show you a little about this program first.

  3. Teacher Input

    First, entertain a discussion about translations, reflections, and rotations with the class. You have a couple of options of how to do this:

    Explain to the students how to do the assignment. You should model or demonstrate it for the students, especially if they are not familiar with how to use the computer applets on the Project Interactivate site.
    • Open your browser to The TransmoGrapher in order to demonstrate this activity to the students.
    • Show the class how to choose the shape they wish to translate, rotate, or reflect using the buttons at the top of the applet.
    • Explain that they must pay close attention to the color of each side of the shape in order to see that the shape has been rotated, translated, or reflected.
    • Show the class how to enter a distance to translate, a degree by which to rotate, or a line of symmetry over which to reflect the object.

  4. Guided Practice

    • After answering all questions that the students might have regarding the use of The TransmoGrapher, pass out the Translations, Reflections, and Rotations Worksheet.
    • Walk the students through the first problem on the sheet. Help them by reminding them as you walk around the room what "rotate", "fourth quadrant", and "reflect" mean. Predict what they should see by drawing it on the board before the students try the steps.
    • If the students needed a lot of help with the first problem, walk them through the second problem on triangles as well.

  5. Independent Practice

    • Allow the students to work on their own and to complete the worksheet, should you choose to provide one. Monitor the room for questions and to be sure that the students are on the correct web site.
    • Have each student choose a figure and apply 2 transformations to it (noting what he or she did). Then have students change places and try to determine how to undo each transformation.

  6. Closure

    Allow students to explain the concepts of translation, reflection, and rotation. The students should share about the places where the activity was difficult. Ensure that all students understand the three concepts before moving on to another lesson.

Alternate Outline

This lesson can be rearranged in several ways.

  • When discussing and explaining the concepts of translation, reflection, and rotation, students may choose to physically act out the movements in order to understand them better.
  • You may invent your own way of using this lesson to suit the needs of your students.