Exploring Symmetry Core Mathematics Partnership Building Mathematical Knowledge and High-Leverage Instruction for Student Success 1:00 – 2:15 July 28, 2016

Learning Intention We are learning different types of symmetry and the relationship between symmetry and transformations. We will be successful when we can Describe what is meant by a symmetry of a shape or figure; Recognize different types of symmetry in real-world and mathematical situations.

Classification of Shapes Spatial Visualization Essential Understandings of Geometry Classification of Shapes Features or properties of geometric shapes can be analyzed and described to define and refine classification schemes with growing precision. Spatial Visualization Spatial relationships and spatial structuring involves developing, attending to, and learning how to work with imagery, as well as to specify locations. Geometry is the branch of mathematics that addresses spatial sense and geometric reasoning. Transformations Transformation involves working with geometric phenomena in ways that build on spatial intuition by explaining what does and does not change when moving and altering the objects and the space that they occupy.

Where in the World do we Find Symmetry? 01/30/11 Where in the World do we Find Symmetry? Create a list of items that have symmetry. What is your criteria for deciding if your items have symmetry?

Possible examples you might have shared.... 01/30/11 Symmetry exists all around us and many people see it as being a thing of beauty.

CCSSM Symmetry Standards Draw and identify lines and angles, and classify shapes by properties of their lines and angles. 4.G.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. 5.G.4 Classify two-dimensional figures in a hierarchy based on properties.

This photograph has 2 lines of symmetry. Can you find them? 01/30/11 This photograph has 2 lines of symmetry. Can you find them?

What is Symmetry? At your table create a definition of symmetry. 01/30/11 What is Symmetry? At your table create a definition of symmetry. Record it on an index card Put it away to return to later.

What is Symmetry? “Symmetry is an area shared by mathematics, the natural world, and art, so it offers opportunities for cross-disciplinary study... When we look at natural objects in the world around us, we find a mix of symmetry and asymmetry.” -Beckmann, 2005, p. 391 Repetition and patterns , music

Exploring Reflection Symmetry 01/30/11 Exploring Reflection Symmetry With Pattern Blocks Part 1: Fold a vertical line through the middle of a plain piece of paper. Use 6-8 pattern blocks to make a design on one side of the line. The design must touch the line in some way.

Exploring Reflection Symmetry 01/30/11 Exploring Reflection Symmetry With Pattern Blocks Part 2: Once your design is complete, stand up and move one table to the right. Find a seat at the new table. Make the mirror image of the design in front of you. What were you thinking about as you made the mirror image?

Exploring Reflection Symmetry 01/30/11 Exploring Reflection Symmetry With Dot Paper Draw a diagonal line on a piece of dot paper. Draw a shape or image on one side of the line. Draw the reflection of your shape or image on the other side of the line. Rigid motion preserves shapes/angles and area….eventhough the shape is orientated differently,

Exploring Reflection Symmetry 01/30/11 Exploring Reflection Symmetry With Dot Paper How was your thinking as you completed this task similar to your thinking in the previous one? How was it different? What is the underlying mathematics we need students to develop as they engage in this task? Rigid motion preserves shapes/angles and area….eventhough the shape is orientated differently,

Refining Your Definition of Symmetry 01/30/11 Refining Your Definition of Symmetry How do the tasks you have just completed support your definition of symmetry? What changes would you make to your index card?

Other Types of Symmetry? 01/30/11 Other Types of Symmetry? So far, we have talked about reflection symmetry But reflections are only one type of rigid motion Do the other types of rigid motion lead to other types of symmetry?

Checking for Symmetry For each design on the handout: 01/30/11 Checking for Symmetry For each design on the handout: Trace the design onto a piece of patty paper Find as many rigid motions as you can which leave the design (as a whole) unchanged

01/30/11 Rotation Symmetry A plane shape or figure has rotation symmetry if there is a rotation, of more then 0O but less than 360O, which leaves the figure as a whole in the same position as before the rotation

01/30/11 Translation Symmetry A plane shape or figure has translation symmetry if there is a translation, through some non-zero distance, which leaves the figure as a whole in the same position as before the translation

Glide Reflection Symmetry 01/30/11 Glide Reflection Symmetry A plane shape or figure has glide reflection symmetry if there is a glide reflection which leaves the figure as a whole in the same position as before the glide reflection

01/30/11 Where’s the Math? What are the critical skills students are developing as they engage in activities and conversations about symmetry?

CCSSM Congruence Standard Understand congruence and similarity using physical models, transparencies, or geometry software. 8.G.1 Verify experimentally the properties of rotations, reflections, and translations: a Lines are taken to lines, and line segments to line segments of the same length. b Angles are taken to angles of the same measure. c Parallel lines are taken to parallel lines. Where in our work this summer have we seen ideas related to this standard?

CCSSM Congruence Standard Understand congruence and similarity using physical models, transparencies, or geometry software. 8.G.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. Where in our work this summer have we seen ideas related to this standard?

Refining Your Definition of Symmetry 01/30/11 Refining Your Definition of Symmetry How has our discussion extended your conception of symmetry? What changes would you make to your index card?

01/30/11 What is Symmetry? A symmetry of a shape or object is a rigid motion which leaves the shape or object as a whole in the same position as before the rigid motion.

Learning Intention We are learning different types of symmetry and the relationship between symmetry and transformations. We will be successful when we can Describe what is meant by a symmetry of a shape or figure; Recognize different types of symmetry in real-world and mathematical situations.

PRR: Reading and Do Isosceles Equilateral Scalene Try the following: 01/30/11 PRR: Reading and Do Try the following: Make a triangle that has one line of symmetry Make a triangle that has more than one line of symmetry Make a triangle that has NO lines of symmetry How did you test for reflection symmetry? How do the lines of symmetry inform your understanding of those triangles and their properties? Isosceles Equilateral Scalene

Core Mathematics Partnership Project Disclaimer Core Mathematics Partnership Project University of Wisconsin-Milwaukee, 2013-2016   This material was developed for the Core Mathematics Partnership project through the University of Wisconsin-Milwaukee, Center for Mathematics and Science Education Research (CMSER). This material may be used by schools to support learning of teachers and staff provided appropriate attribution and acknowledgement of its source. Other use of this work without prior written permission is prohibited—including reproduction, modification, distribution, or re-publication and use by non-profit organizations and commercial vendors. This project was supported through a grant from the Wisconsin ESEA Title II, Part B, Mathematics and Science Partnerships.