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

2. 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

3. 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

4.  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

5.  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

6. 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

7. 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

8. 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

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

10. 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

11. 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

12. 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

13. 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

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

15. 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

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

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

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

19. 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

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

21. 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

22. 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