The explorations in Kaleidoscopes, Hubcaps, and Mirrors help students to refine their knowledge of symmetry and use it to make mathematical arguments. Students explore transformations (reflections, rotations, and translations) that preserve angle and side length relationships of figures in the plane.

Goals of the Unit • Understand important properties of symmetry • Recognize and describe symmetries of figures • Use tools to examine symmetries and transformations • Make figures with specified symmetries • Identify a basic design element that can be used with a transformation to replicate a given design • Perform symmetry transformations of figures, including reflections, translations, and rotations • Examine and describe the symmetries of a design made from a figure and its image(s) under a symmetry transformation • Give precise mathematical directions for performing reflections, rotations, and translations in terms of the effect of the transformation on points of the original figure • Draw conclusions about a figure in terms of the effect of the transformation on points of the original figure based on what symmetry or symmetries the figure has • Understand that figures with the same shape and size are congruent • Use symmetry transformations to explore whether two figures are congruent • Give examples of minimum sets of measures of angles and sides that will guarantee that two triangles are congruent • Use congruence of triangles to explore congruence of two quadrilaterals • Use symmetry and congruence to deduce properties of figures • Write coordinate rules for specifying the image of a point under particular transformations • Appreciate the power of transformational geometry in the real world

Kaleidoscopes, Hubcaps, and MirrorsSymmetry and TransformationsThe explorations in Kaleidoscopes, Hubcaps, and Mirrors help students to refine their knowledge of symmetry and use it to make mathematical arguments. Students explore transformations (reflections, rotations, and translations) that preserve angle and side length relationships of figures in the plane.Goals of the Unit• Understand important properties of symmetry• Recognize and describe symmetries of figures• Use tools to examine symmetries andtransformations• Make figures with specified symmetries• Identify a basic design element that can be usedwith a transformation to replicate a given design• Perform symmetry transformations of figures,including reflections, translations, and rotations• Examine and describe the symmetries of adesign made from a figure and its image(s)under a symmetry transformation• Give precise mathematical directions forperforming reflections, rotations, and translationsin terms of the effect of the transformation onpoints of the original figure• Draw conclusions about a figure in terms of theeffect of the transformation on points of theoriginal figure based on what symmetry orsymmetries the figure has• Understand that figures with the same shapeand size are congruent• Use symmetry transformations to explorewhether two figures are congruent• Give examples of minimum sets of measures ofangles and sides that will guarantee that twotriangles are congruent• Use congruence of triangles to explorecongruence of two quadrilaterals• Use symmetry and congruence to deduceproperties of figures• Write coordinate rules for specifying the imageof a point under particular transformations• Appreciate the power of transformationalgeometry in the real worldKaleidoscopes, Hubcaps, and Mirrors