**ERIC Number:**EJ1110627

**Record Type:**Journal

**Publication Date:**2016-Jul

**Pages:**10

**Abstractor:**ERIC

**ISBN:**N/A

**ISSN:**ISSN-1072-0839

**EISSN:**N/A

Quilt Block Symmetries

Roscoe, Matt B.; Zephyrs, Joe

Mathematics Teaching in the Middle School, v22 n1 p18-27 Aug 2016

Geometric transformations have long been topics of middle school mathematics. Generations of middle school students have learned to reflect, rotate, and translate geometric objects. Historically, though, the mathematics of "movement" might have been considered a departure from other more central middle-grades geometric content areas, such as measurement, congruence, and similarity. In the era of the Common Core State Standards for Mathematics (CCSSI 2010), the study of reflection, rotation, and translation have been given special importance. Consider the introduction to the high school geometry domain in the Common Core State Standards for School Mathematics (CCSSM): The concepts of congruence, similarity, and symmetry can be understood from the perspective of geometric transformation. Fundamental are the rigid motions: translations, rotations, reflections, and combinations of these, all of which are here assumed to preserve distance and angles (and therefore shapes generally). Reflections and rotations each explain a particular type of symmetry, and the symmetries of an object offer insight into its attributes. (CCSSI 2010, p. 74) The document makes it even more explicit in the next paragraph, describing the approach to congruence, in which "two geometric figures are defined to be congruent if there is a sequence of rigid motions that carries one onto the other" (p. 74). Like congruence, the study of similarity in high school also rests on students understanding transformation by defining similarity as a sequence of rigid motions followed by a dilation, which "lead[s] to the criterion for [angle-angle] triangle similarity" (p. 74). These passages make it clear that one crucial element of students' success in high school geometry is a firm understanding of transformation, which must be acquired in the middle grades. One difficulty associated with teaching transformations in the middle grades is the facilitation of inquiry-based learning environments in which students explore and investigate properties of transformations in objects of their own creation. The study of transformation is a fertile setting in which students can make sense of problems, reason abstractly, construct viable arguments, and look for structure, all practices that are encouraged in the Common Core's Standards for Mathematical Practice (2010). In addition, the study of transformations has long been associated with tools of inquiry (witness the use of patty paper, the Mira™, and interactive geometry software). But few are the opportunities where student-generated examples are the focus of mathematical investigation. To this end, the authors created a series of mathematical tasks that provided students with the opportunity to construct understandings of transformation through an investigation of quilt block symmetries. This article provides a description of these tasks and a demonstration of their use in an eighth-grade classroom.

Descriptors: Common Core State Standards, Mathematics Education, Mathematics Instruction, Geometric Concepts, Geometry, Middle School Students, Teaching Methods, Grade 8, Mathematics Activities, Class Activities

National Council of Teachers of Mathematics. 1906 Association Drive, Reston, VA 20191. Tel: 800-235-7566; Tel: 703-620-9840; Fax: 703-476-2570; e-mail: NCTM@nctm.org; Web site: http://www.nctm.org/publications/mathematics-teaching-in-the-middle-school/

**Publication Type:**Journal Articles; Reports - Descriptive

**Education Level:**Middle Schools; Secondary Education; Junior High Schools; Grade 8; Elementary Education

**Audience:**N/A

**Language:**English

**Sponsor:**N/A

**Authoring Institution:**N/A

**Identifiers - Location:**Montana

**Grant or Contract Numbers:**N/A