CHAPTER 15 Geometry Tina Rye Sloan To accompany Helping Children Learn Math10e, Reys et al. ©2012 John Wiley & Sons  

Focus Questions Why should the elementary mathematics program include geometry? What should early childhood and elementary students learn about geometric shapes? How do the van Hiele levels guide the development of geometric experiences for elementary children? What types of explorations with geometry help build elementary children’s spatial reasoning and visualization skills? What roles do location and movement have to play in geometry? Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

The Geometry Standard Principles and Standards for School Mathematics National Council of Teachers of Mathematics 2000 Master 15-1: The Geometry Standard Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

 Instructional programs from pre-k through grade 12 should enable all students to: Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.  Specify locations and describe spatial relationships using coordinate geometry and other representational systems. Master 15-1: The Geometry Standard Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

 Instructional programs from pre-k through grade 12 should enable all students to: Apply transformations and use symmetry to analyze mathematical situations Use visualization, spatial reasoning, and geometric modeling to solve problems Master 15-1: The Geometry Standard Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Why Geometry? It is a mathematics topic that engages children differently both in performance and persistence. It is a natural site to include other skills such as following directions and for reasoning about shapes and their properties. Children can make and verify conjectures about geometric figures. It is also a topic that will help you teach many other mathematical topics. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Levels of Understanding Geometry Very young children often attend to only visual cues that are salient to them. Later, children focus on visual cues—it is a rectangle because it looks like a rectangle (Level 1, Visual). They then begin to recognize and more carefully describe properties of all rectangles and move to Level 2 (Descriptive/Analytic) . Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Levels of Understanding Geometry (cont’d) More abstract thought is at Level 3 (Abstract/Relational). At this level, students establish relationships among properties and among figures. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Three-Dimensional Shapes Describing and Sorting Constructing Exploring Discovering Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Describing and Sorting: Beginning The following questions ask students to describe and justify their answers when sorting shapes: Who am I? Who stacks? How are we alike or different? Who doesn’t belong? How many faces do I have? Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Describing and Sorting: Intermediate Edges, vertices, and faces: Who am I? Classifying Solids Searching for Solids. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Describing and Sorting: Advanced Parallel Faces Perpendicular Edges Right Prisms Constructing to Explore and Discover Are the solid shapes prisms? Why or why not? How are the solids named? Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Face Models In this activity, students investigate the relationship of the number of edges, faces, and vertices in open-ended prisms (called tubes). The formula for these tubes is V + F = E. Have your students look for other relationships between two of these variables. For example, they may see that the number of edges is always three times the number of faces (not counting the top and bottom). Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Properties of Shape Number of sides and corners Symmetry Lengths of sides Sizes of angles Parallel and perpendicular sides Convexity and concavity Altitude Classification schemes Master 15-2: Properties of Shape Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Locations and Associated Representations Map of Locations Coordinate Graphing Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Transformations Translation (slide) Reflection (flip) Rotation (turn) Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Congruence Congruence---same size and shape. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Similarity Two figures are similar if corresponding angles are equal and corresponding sides are in the same ratio. The shapes are also the same. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Visualization and Spatial Reasoning Using Geometric Physical and Pictorial Materials Using Mental Images Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Geoboard Quadrilaterals Mark off a 3 by 3 section on your geoboard. All your exploration will be done on these 9 pegs. How many non-congruent quadrilaterals can you form? Use your geoboard to explore and then record your discoveries on geoboard recording paper. Master 15-3: Geoboard Quadrilaterals Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012

Geoboard Quadrilaterals (cont’d) Compare your set of quadrilaterals with your peers to find those you may have missed. Can you categorize the quadrilaterals in any way? How? Master 15-2: Geoboard Quadrilaterals Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 10th Edition, © 2012