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Finding Lines of Symmetry

Maggie Williams
Location: unknown

Students identify lines of symmetry and congruent figures. They explore these concepts with paper cutting and modeling on the geoboard.

To assess students' prior knowledge, ask students to fold a piece of colored paper in half and cut a heart shape, leaving the fold attached as they would do to make Valentines to share with classmates. When they are finished, direct students to open their folded piece of paper to see that both sides are equal and are a reflection of the other. It may be helpful to have mirrors available so that students can see how dividing a figure with a line of symmetry creates a mirror image of the opposite side.

An excellent way to connect seasonal holidays with mathematics is to have students cut a shamrock, heart, Pilgrim’s hat, snowflake, or other holiday symbol by folding paper and cutting symmetrical figures. This activity engages students in problem solving that requires spatial and visual thinking. The symmetrical figures can be used as the border on a class bulletin board.

To begin the lesson, distribute one geoboard strung with rubber bands to each student. Ask students to use one rubber band to create a figure and use a second rubber band to divide it into two equal parts.

If you do not define how to divide the figure, students may create many different lines of symmetry, which allows for a rich discussion. The Rotations and Lines of Symmetry Teacher Resource Sheet provides a reference.

It is important to note that some figures will not be able to be divided into congruent shapes. As students are attempting to divide their shapes, walk around the classroom to identify examples of shapes which have line symmetry. Ask those students to share their examples. Also talk about which examples do not have line symmetry and why. This will allow for a rich discussion.

Next, have students compare their figure with those of a partner and discuss how the figures are the same and how they are different. They should be alike because they are equally divided and one side is the reflection of the other.

Have a pair of students show their figures and share the content of their conversation. Listen for students’ understanding of the meaning of symmetry. Ask pairs of students to repeat this process in order to hear the ideas of multiple students. This will help you understand what students know as you begin the lesson so that you can make adjustments.

Discuss with the class the meaning of symmetry. It is important to model line and reflectional symmetry and rotational symmetry. Samples can be found in the Paper Quilts Unit.

Unit Icon Paper Quilts Unit 

You may wish to project the Lines of Symmetry Overhead for students to discuss lines of symmetry in common geometric figures.

overhead Lines of Symmetry Overhead 

1800 lines of symmetry

Encourage students to discuss what they learned by cutting figures that were different from the figures on the geoboard. Discuss flips, turns, and slides.

Next, give each student a copy of the Creating Lines of Symmetry Activity Sheet.

pdficon Creating Lines of Symmetry Activity Sheet 

Ask students to use a pencil to draw as many lines of symmetry as possible for each figure. Have students discuss their product. Then place a transparent copy of the student learning guide on the overhead and have students volunteer to draw lines of symmetry on each figure.

Have students cut out a shape of their choice from the student learning guide. Ask them to cut along one of the lines of symmetry and try to fit the two pieces one on top of the other. The pieces should fit exactly. Flipping and rotating may be necessary to align the sides and angles.

Tell students that when two figures are the same size and shape, they are said to be congruent. Repeat the activity once or twice.

Model how to make a rectangle with one rubber band using the Virtual Geoboard E-Example.

appicon Virtual Geoboard E-Example 

Use a second rubber band to create a line of symmetry. Depending on the experience of your students, it may be best to begin with obvious and frequently presented lines of symmetry. This enables you to accommodate the varying levels of knowledge of your students during whole class instruction.

Now that students have experimented with creating symmetrical figures with paper, distribute to each student a geoboard strung with five rubber bands. Direct students to duplicate the same figures on their geoboard that appear on the Creating Lines of Symmetry Activity Sheet. Ask them to use rubber bands to show lines of symmetry. Provide Dot Paper so that students may draw the figure as it appears on the geoboard. Keep these as a record of the student’s work.

pdficon Dot Paper 

Make notes about the level of understanding students demonstrate on their recording. Use this information to determine the next instructional activity that is appropriate for the students as individuals or as a group.

Assessment Options 

1. At this stage of the unit, students should know how to:

  • make symmetrical shapes
  • make similar and congruent shapes
  • record shapes on dot paper

2. To assess students’ understanding of the above objectives, distribute sheets of dot paper to each student and ask them to draw a figure that can have only one line of symmetry. Then, ask them to draw figures that can have only two lines of symmetry and those that have multiples lines of symmetry. Once students have drawn the figures, have them draw the line or lines of symmetry.


Move on to the next lesson, Who Was Wassily Kandinsky?

Questions for Students 

1. How would you define symmetry to someone who did not know what it was?

[Student responses may vary, but students may say a shape has symmetry "if you can fold it on top of itself" or something similar.]

2. In what ways is symmetry used in everyday life?

[Student responses may vary but could include butterflies, snowflakes, and other examples in nature.]

3. Which figures were the most challenging to find and draw lines of symmetry for? What properties do these figures have in common?

[Student responses may vary.]

4. Describe a line of symmetry. Draw a figure that has more than one line of symmetry.

[Student responses may vary, but a student may describe a line of symmetry as a mirror.]

5. What does congruent mean? What does similar mean?

[Congruent means same size and same shape; similar means same shape but proportional side lengths.]

6. Are all similar figures congruent? Are all congruent figures similar?

[No; yes.]

Teacher Reflection 

  • Which students demonstrated an understanding of the objectives of this lesson? What were the indicators that they did understand the objectives?
  • Which students had difficulty with the activities in this lesson? What additional instruction do they need?
  • What strategies were most effective in assessing each student’s learning in this lesson?

How else can I accommodate students who moved quickly through each aspect of the lesson? Is it important to note the extensions that worked well?


Working with Shapes

Students review basic geometric terms related to triangles. They explore these terms and other geometric concepts by modeling them on the geoboard.

More Work with Shapes

Students continue to explore geometric concepts by modeling on the geoboard.  Communication is the Process Standard emphasized in this lesson.

Who Was Wassily Kandinsky?

This lesson provides students with an exploration of the geometric figures Wassily Kandinsky used in his art. Students participate in a scavenger hunt to become familiar with Kandinsky’s works and the geometric figures used in his paintings.

Seeing Geometry in Art

Students use paintings studied in the previous lesson to connect their knowledge of geometric shapes and terms with Kandinsky’s use of geometric figures.

Mirroring Kandinsky

This lesson allows students to apply what they have learned in previous lessons by designing their own art. Students use Kandinsky’s style of art and their own creativity to make paintings that reflect their understanding of geometry.

Learning Objectives

Students will:

  • Make symmetrical shapes.
  • Make similar and congruent shapes.
  • Record shapes on dot paper.

NCTM Standards and Expectations

  • Describe location and movement using common language and geometric vocabulary.
  • Build and draw geometric objects.
  • Create and describe mental images of objects, patterns, and paths.
  • Recognize geometric ideas and relationships and apply them to other disciplines and to problems that arise in the classroom or in everyday life.

Common Core State Standards – Mathematics

Grade 4, Geometry

  • CCSS.Math.Content.4.G.A.3
    Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.