Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

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Teacher's Lab
Shape and Space in Geometry
IntroductionAbout ShapeAbout Space

Across the Grades

Symmetry is one of the most beautiful parts of mathematics. It's about what you can do to something and have it still be the same.

Here you explore symmetry spatially (the way we usually think of it), for example, looking at mirror symmetry. If you place a mirror on an object at the mirror-symmetry line, that object looks the same whether the mirror is there or not. You also explore rotational symmetry. Rotations of 180 degrees also have what we call point symmetry.

In this activity you identify what kind of symmetry a figure has and then use that knowledge to reproduce the figure most efficiently. We have invented names for the symmetries you will use here; they are based on letters of the alphabet (they have different official names that are not as mnemonic for young students). The M and B symmetries are really the same symmetry mathematically—there is a single line of mirror symmetry. In mathematics, it makes no difference whether the line is vertical (M) or horizontal (B).

Here's an excellent exercise for students: Ask them to classify all the letters by their symmetry properties.

across the grades

As with many activities related to symmetry, this one is accessible to young students if they solve the problem by guess and check. But even for older students, deciding on the kind of symmetry can be confusing. And once it is chosen, deciding on the square to use can also be a problem because of the way squares fit together to make stripes, diamonds, and chevrons.

Symmetry is important for a number of reasons. As students get older, they will recognize symmetry in various problems and use that symmetry to help solve the problems. This skill has practical applications in many fields, such as chemistry (where the symmetry properties of molecules help determine how they behave) and design (where, in a well-designed building, you can generally find the right restroom if you locate the wrong one).


Symmetry is a natural part of the visualization component included in the NCTM Geometry and Spatial Sense standard. For example, at grades 3–5, the current version of the Principles and Standards document lists as a focus area that students should "recognize the usefulness of transformations and symmetry in analyzing mathematical situations."

The Council then becomes more specific:

In grades 3–5, all students should

predict the results of sliding, flipping and turning two-dimensional figures;

describe a motion or series of motions that will show that two figures are congruent;

identify and describe line and rotational symmetry in various two-dimensional shapes.


In the Teaching Math K–4 video series, Elaine McAlear's first-grade class is doing a simpler, related activity with smaller quilt blocks and physical manipulatives. You can find resource material on page D-3 of the Teaching Math guidebook.

Back to Quilts Activity


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