1. Making Sense of Geometry in Grades 3–5 Classrooms
Marcia Torgrude
K-12 Math Specialist
Technology and Innovation in Education
Penny Smith
SD Math Curriculum Director
SD Department of Education

2. Prior Knowledge When did you learn about geometry ?
What did you learn?
Do you remember working with shapes?
How and when do you teach geometry now?

3. van Hiele Levels of Geometric Thought

4. K-12 Geometry Standards Four Common Goals
Shapes and properties –study of the properties of shapes in both two and three dimensions, as well as a study of the relationships built on properties.
Transformation - study of translations, reflections, and rotations (slides, flips, and turns) and the study of symmetry.
Location –coordinate geometry or other ways of specifying how objects are located in the plane or in space.
Visualization –recognition of shapes in the environment, developing relationships between two and three dimensional objects, and the ability to draw and recognize objects from different perspectives.

5. Shapes Game Partners – determine who goes first
Place a pattern block in the shape
Continue taking turns placing the pattern blocks
The last one to fill the shape with a pattern block is the winner
What did you learn?
What strategy might you use the next time?

6. Online Shapes Activities
Patch Tool -
Shape Tool -
Geometric Solids -

7. Shapes Game Purpose
Increases spacial sense – moving from visualization to analysis (van Hiele levels)
Transformation - translations, reflections, and rotations
Visualization – the ability to draw and recognize objects from different perspectives

8. Grandfather Tang’s Story

9. Grandfather Tang’s Purpose
Increases spacial sense – moving from visualization to analysis (van Hiele levels)
Transformation - translations, reflections, and rotations
Visualization – the ability to draw and recognize objects from different perspectives
Connects Literature to Geometry

10. Trianquad Activity Work in pairs
Go through the cards one at a time, look at the diagrams, and try to determine the attributes of a trianquad, refining your definition as you work.
Note: a Trianquad is a fictional shape.
Defining a Trianquad:
Purpose of the activity:
• Geometry is a definition-heavy content area.
• Teachers often think they must provide definitions, or have students look up definitions, prior to
doing activities.
• Definitions can be developed through various hands-on activities (a “trianquad” isn’t an actual
geometric figure, but provides an example of how a definition can be developed, rather than “told.”)
Big ideas:
o Definitions do NOT have to come first
o Definitions can be derived from activities
o The teacher labels the students’ thinking with the correct word (definition): For example,
when students are discussing “corners” of polygons in the context of an activity, the teacher
might say, “Mathematicians call that a vertex.”
o Definitions need to be precise (cover all possibilities)

11. Trianquad Purpose Geometry is a definition-heavy content area.
Teachers often think they must provide definitions, or have students look up definitions, prior to doing activities.
Definitions can be developed through various hands-on activities –developing versus telling
Move students’ thoughts from analysis to informal deduction of van Hiele’s levels of geometric thought.

12. Trianquad Big Ideas Definitions do NOT have to come first
Definitions can be derived from activities
The teacher labels the students’ thinking with the correct word (definition): For example, when students are discussing “corners” of polygons in the context of an activity, the teacher might say, “Mathematicians call that a vertex.”
Definitions need to be precise (cover all possibilities)

13. Geoboard Activity Growing and Shrinking
Make this shape on your geoboard.
Keep the perimeter the same and make the area smaller.
Keep the perimeter the same and make the area larger.
Keep the area the same and make the perimeter smaller.
Keep the area the same and make the perimeter larger.
Online Geoboard
HINT: It is possible to do all of these with shapes made up entirely of squares. You may find it difficult to calculate perimeters if you allow diagonal edges.
What is the area and the perimeter of the original figure?
What is the area and the perimeter of each of the new diagrams?
What conclusions can you make about the relationship between area and perimeter?
Does this hold for any polygon your create?

14. Geoboard Activity Purpose
Area and perimeter are continually a sense of confusion for students
Finding a relationship between area and perimeter and deepen their understanding of each
Measurement and Geometry strands support one another to build understanding
Shape and Properties - Best device for drawing 2-dimensional shapes
Location – ways of specifying how objects are located in the plane or in space.
Visualization –recognition of shapes in the environment, developing relationships between two and three dimensional objects, and the ability to draw and recognize objects from different perspectives.
Junior Architects -

15. Revisit Common Goals Shapes and properties Transformation Location
Trianquad
Shapes Game
Grandfather Tang’s Story
Geoboard – Area and Perimeter
Transformation
Location
- Geoboard- Area and Perimeter
Visualization
- All of the activities

16. van Hiele Levels of Geometric Thought
Geoboard
Trianquad
Grandfather Tang
Shapes Game

17. Make Sense of Geometry in K-5
Give your students the experiences they need to succeed at the high school and college levels

18. Bibliography
Grandfather Tang’s Story, a Tale Told with Tangrams, Ann Tompert, Crown Publishers Inc. New York, 1990
Developing Number Concepts Books 1, 2, 3, and Planning Guide, Kathy Richardson1999
Developing Mathematics with Pattern Blocks, Dr. Paul Swan and Geoff White, Didax, 2006
Teaching Student-Centered Mathematics, John A. Van de Walle, 2006 and 2010