Transformations and Tessellations

Home , Geometric transformation, Rotational symmetry

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CHAPTER 7 Transformations and Tessellations

Content Summary Thinking about ideas from different perspectives can lead to deeper understanding. For example, geometric transformations can help students deepen their understanding of congruence and symmetry. You can think of a geometric transformation as a regular change of a figure in the plane. For example, a figure may be shifted to the right 5. Or, a figure may be enlarged to twice its original size. Chapter 7 focuses on transformations that don’t change the size or shape of figures. These transformations are called isometries. Expansions and contractions, called dilations, are studied in Chapter 11.

Isometries There are three major kinds of isometries: translations, reflections, and rotations. Translations are simply shifts. Students use translations when discussing tessellations, where a single shape is translated (shifted) repeatedly in different directions to cover the plane without any gaps or overlaps. Reflections flip a shape across a line to make a mirror image. If there’s a line through which a shape can be reflected to lay the image exactly on top of the original, then the figure has reflectional symmetry, as students saw in Chapter 0. Reflections can be used in designing figures that will tessellate the plane. They can also be used to help find the shortest path from one object to a line and then to another object. Rotations rotate an object around a point. If there’s a point around which a shape can be rotated through some angle (less than 360°) to get back to exactly the same shape, then the figure has rotational symmetry (also previewed in Chapter 0). Rotations can also be used in designing tessellations. This tile pattern has both Isometries give students a new way of thinking about congruence. Two figures 5-fold rotational symmetry are congruent if one can be transformed into the other using an isometry. and 5-fold reflectional symmetry. Compositions of Isometries One transformation followed by another is the composition of those transformations. This chapter considers compositions of two reflections: reflections across parallel lines (resulting in a translation) and reflections across nonparallel lines (resulting in a rotation). In the context of tessellations, this chapter also examines glide reflections, which are compositions of a translation and a reflection. (continued)

Chapter 7 • Transformations and Tessellations (continued)

Summary Problem Have your student trace this tessellation onto tracing paper or wax paper. Ask how it illustrates the concepts under consideration.

Questions you might ask in your role as student to your student: ● What isometries might be used to change one part of the figure to another? ● What is the underlying grid of this tessellation? ● Could this tessellation be made with just translations? ● What kinds of symmetry does the entire figure have?

Sample Answers The tessellation, based on a four-by-four grid of quadrilaterals, illustrates many transformations. Each of the dark figures can be translated to fit onto a dark figure two rows or columns away. Or, it can be rotated 180° to fit onto a light figure in the same row. To fit onto a light figure in an adjacent row, the dark figure must be reflected vertically, then translated. (All three transformations just described are also true for each of the light figures.) Any block of four figures that meet at one point can be translated to fill the entire plane.

Chapter 7 • Review Exercises

Name Period Date

1. (Lesson 7.1) Reflect the figure across PQ and rotate it 180° around point M. Does your answer change if you rotate it first and then reflect it?

2. (Lesson 7.2) Translate ABC using the rule (x, y) → (x 3, y 1).

4 B 3

A 1 x –4 –2 2 4 –1 C –2

3. (Lesson 7.3) Name the single rotation that can replace the composition of these two rotations around the same center of rotation: 36° and 120°.

4. (Lesson 7.1) What capital letters have horizontal but not vertical symmetry?

5. (Lesson 7.1) What capital letters have horizontal, vertical, and rotational symmetry?

SOLUTIONS TO CHAPTER 7 REVIEW EXERCISES

1. Reflection: 2. y

P 4 B 3

2 A M 1 x –4 –2 2 4 –1 C –2 Q

Rotation: 3. 36° 120° 156° 4. B, C, D, E, K P 5. H, I, O, X

No, the final answer is the same.