Determine symmetries of reflection, symmetries of rotation and symmetries of translation of a geometric figure.


Clarification 1: Instruction includes determining the order of each symmetry.

Clarification 2: Instruction includes the connection between tessellations of the plane and symmetries of translations.

General Information
Subject Area: Mathematics (B.E.S.T.)
Grade: 912
Strand: Geometric Reasoning
Status: State Board Approved

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment


Terms from the K-12 Glossary

  • Reflection 
  • Rotation 
  • Translation

Vertical Alignment

Previous Benchmarks

Next Benchmarks


Purpose and Instructional Strategies

Symmetries of reflection were introduced in the elementary grades through lines of symmetry. In Geometry, students studied other types of symmetries coming from rigid transformations that map a polygon onto itself, and they determined the number of times such a transformation must be applied before each point in the polygon is mapped to itself. In Math for College Liberal Arts, students determine symmetries of reflection, symmetries of rotation and symmetries of translation of a geometric figure. 
  • Instruction includes multiple opportunities for students to explore symmetries using both physical exploration (transparencies, mirrors or patty paper) and virtual exploration, when possible. 
  • Instruction includes using a variety of shapes (mathematical and real world) to explore the reflection symmetry and rotational symmetry of the shapes. Include identifying the lines of symmetry, the order of symmetry and the angle of rotation that will map the figure onto itself. 
  • The order of symmetry is the smallest (nonzero) number of times that you must apply the corresponding transformation to map each point of the figure onto itself. 
  • The order of rotational symmetry is the number of times the figure maps onto itself as it rotates through 360° about the figure’s center. Instruction includes identifying the angles of rotation when determining symmetries of rotation. 
    • For example, the order of rotational symmetry for a regular hexagon is 6 with the angle of rotation of 60°. 
    • For example, the order of rotational symmetry for an isosceles trapezoid is 0. 
    • For example, the figure below has an order of 4 with angle of rotation of 90°. 
      plus sign
  • The order of a translational symmetry is infinite because no matter how many times one applies it, no point gets mapped onto itself. Translational symmetry results from mapping a figure onto itself by moving it a certain distance in a certain direction. Show students tessellations (covering of a plane using one or more geometric shapes with no overlaps and no gaps), or have them create them, and discuss the translational symmetry in the tessellation (MTR.5.1). 
  • Depending on a student’s pathway, instruction includes making the connection to symmetry as a key feature of the graphs of polynomial and trigonometric functions and the importance of applying such functions in the real world. 
  • Problem types include using symmetries to classify geometric figures. 
    • For example, given that a two-dimensional figure has one line of symmetry through midpoints of the two parallel sides, no rotational symmetry and no point symmetry, students can deduce that this figure is a trapezoid.

Common Misconceptions or Errors

  • Students may not make the connections to the idea of an infinite pattern (wallpaper pattern, border pattern, etc.) when working with only a portion of that pattern. If students arrive at an order for a translational symmetry that is not infinite, ask them if their answer would be different if the pattern continued. 
  • Students may have difficulty distinguishing between mapping a figure onto itself and mapping every point of the figure onto itself. To help address this misconception, have students highlight a particular point and observe how it is affected by one application of a transformation that maps the figure onto itself.

Instructional Tasks

Instructional Task 1 (MTR.4.1, MTR.5.1
  • Use the figures below to answer the questions. 

    • Part A. Draw the lines of symmetry, if any, on each figure. 
    • Part B. What is the order of symmetry for reflections? 
    • Part C. Determine the order of rotational symmetry for each figure. Discuss with a partner how you determined each. 

Instructional Task 2 (MTR.3.1, MTR.5.1,) 
  • Given Quadrilateral ABCD and EF below, where would points G and H need to lie so that Quadrilateral ABCD is congruent to Quadrilateral EFGH?


Instructional Items

Instructional Item 1 
  • Part A. Which capital letters that are vowels of the English alphabet have a line of symmetry? 
  • Part B. Do any of the vowels identified from Part A also have rotational symmetry?

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

Related Courses

This benchmark is part of these courses.
1206320: Geometry Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912070: Access Mathematics for Liberal Arts (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 - 2023, 2023 and beyond (current))
1207350: Mathematics for College Liberal Arts (Specifically in versions: 2022 and beyond (current))
1212300: Discrete Mathematics Honors (Specifically in versions: 2022 and beyond (current))

Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.

Related Resources

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Lesson Plans

Regular Polygon Transformation Investigation:

This is an introductory lesson on regular polygon transformation for congruency using a hands-on approach.

Type: Lesson Plan

Rotations of Regular Polygons:

This lesson guides students through the development of a formula to find the first angle of rotation of any regular polygon to map onto itself. Free rotation simulation tools such as GeoGebra, are used.

Type: Lesson Plan

How to Land Your Spaceship:

Teach your students to maneuver a "spaceship" through a sequence of transformations that will successfully carry it onto its landing pad. GeoGebra directons are provided.

Type: Lesson Plan

How do your Air Jordans move?:

In this inquiry lesson, students are moving their individually designed Air Jordans around the room to explore rigid transformations on their shoes. They will Predict-Observe-Explain the transformations and then have to explain their successes/failures to other students.

Type: Lesson Plan

I Am Still Me Transformed.:

Students explore ways of applying, identifying, and describing reflection and rotation symmetry for both geometric and real-world objects, for them to develop a better understanding of symmetries in transformational geometry.

Type: Lesson Plan

Reflections Hands On:

Students will use a protractor/ruler to construct reflections and a composite of reflections. They will create transformations using paper cut-outs and a coordinate plane. For independent practice, students will predict and verify sequences of transformations. The teacher will need an LCD Projector and document camera.

Type: Lesson Plan

A Transformation's Adventure with Patty Paper: Exploring Translations, Reflections and Rotations.:

Students are introduced to isometric transformations using patty paper. Translations, reflections, and rotations will be explained and practiced, emphasizing the properties preserved during those transformations and, without sacrificing precision, allowing students to differentiate between these isometries. The lesson can also be taught using GeoGebra free software.

Type: Lesson Plan

Flipping Fours:

Students will translate, rotate and reflect quadrilaterals (Parallelogram, Rectangle, Square, Kite, Trapezoid, and Rhombus) using a coordinate grid created on the classroom floor and on graph paper. This activity should be used following guided lessons on transformations.

Type: Lesson Plan

Perspectives Video: Professional/Enthusiasts

Reflections, Rotations, and Translations with Additive Printing:

Transform your understanding of 3D modeling when you learn about how shapes are manipulated to arrive at a final 3D printed form!

Type: Perspectives Video: Professional/Enthusiast

Bacteriophage Geometry and Structure:

Viruses aren't alive but they still need to stay in shape! Learn more about the geometric forms of bacteriophages!

Type: Perspectives Video: Professional/Enthusiast

Student Resources

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Parent Resources

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