Amy Mirra National Council of Teachers of Mathematics

Geometric Thinking in Grades 3-5

Amy MirraNational Council of Teachers of Mathematics

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Today’s Agenda

� Introduction of Workshop Format� Themes of the Geometry Standard in

Principles and Standards for School Mathematics

� A Sampling of Geometry Activities for Grades 3-5

� Next Steps for You. Questions.

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Quick Warm-Up!

� Fold a paper square (a square sticky note works well) along a diagonal to divide the square into two triangular regions

� Divide the triangle in half again to divide the square into four triangular regions

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Quick Warm-Up!

� Cut along each fold line to make four congruent triangles.

� Arrange the triangles to create as many polygons as possible.

� Record each solution and identify the polygons created.

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Shape Cutter Applet

http://illuminations.nctm.org/ActivityDetail.aspx?I D=72

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Today’s Agenda





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NCTM Vision for School Mathematics

The full text of the document is available to

NCTM members at http://standards.nctm.org

Non-members can register for 90-day free access

Principles and Standards for School Mathematics,pages 396-397, orPSSM Quick Reference Guide: Table ofStandards and Expectations

http://standards.nctm.org

Standards

Principles and Standards for School Mathematics,pages 396-397, orPSSM Quick Reference Guide: Table ofStandards and Expectations

http://standards.nctm.org

Standards

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Why Study Geometry?

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Van Hiele ModelLevels of Geometric Thinking

� Level 0: Visualization� Level 1: Analysis� Level 2: Informal Deduction� Level 3: Deduction� Level 4: Rigor

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Characteristics of van Hiele Levels

� Levels are sequential� Levels are not age-dependent� Geometric experience is key� Instruction must match student’s level of

thought

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Level 0: Visualization

� Figures are judged by appearance� Grouping of shapes that seem to be

“alike”� A square is a square “because it looks like a

square”� Rotated square is not a square to this level

of thinker

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Level 1: Analysis

� Descriptive level

� Properties of shapes � A square is a square because it is a rectangle with

four congruent sides.

� Relationships among properties not developed� Won’t see the relationship between squares,

rectangles, and parallelograms

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Level 2: Informal Deduction

� Relationships among properties of geometric ideas developed

� Able to justify relationships based on properties� “If all four angles are right angles, the shape

must be a rectangle. If it is a square, all angles are right angles. If it is a square, it must be a rectangle.”

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Van Hiele ModelLevels of Geometric Thinking

� Level 0: Visualization� Level 1: Analysis

�Level 2: Informal Deduction� Level 3: Deduction� Level 4: Rigor

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Today’s Agenda





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Major Themes of Geometry that Start in Pre K and Go all the Way to Grade 12

• Analyze characteristics and properties of Shapes

• Specify Locations and describe spatial relationships

• Apply Transformations and symmetry

• Use Visualization , spatial reasoning, and geometric modeling

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Analyzing characteristics and properties of Shapes

Pre-K-2 Experiences:

� Building and drawing shapes� Comparing shapes by attributes� Putting together and taking apart shapes� Identifying shapes in world � Examples and non-examples of shapes

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Analyzing characteristics and properties of Shapes

3–5 Experiences:� Describing shapes focusing on properties� More complex two- and three dimensional

shapes� Generalizations about classes of shapes such

as prisms or parallelograms� Parallelism and perpendicularity� Area and perimeter� Congruence and similarity

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Build What I’ve Created Activity

� Work with a partner

� One partner create a design using less than 10 pattern blocks without your partner seeing it

� Give directions (orally or written) to your partner to re-create the design

Navigating Through GeometryIn Grades 3-5, Page 11

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Illuminations Web site

illuminations.nctm.org

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Build What I’ve Created Activity

Class Discussion Questions:� What words or phrases helped you re-

create the design?� What words or phrases confused you?� Can you think of better ways to explain

the directions for making the design?

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Pattern Patch Applet

http://illuminations.nctm.org/ActivityDetail.aspx?I D=27

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How are these shapes alike and different?

Alike Different

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Mathematics Assessment Samplers

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NCTM E-examples

standards.nctm.org

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Exploring Dynamic Rectangles and Parallelograms

E-example 5.3

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Exploring Dynamic Rectangles and Parallelograms

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Specify Location and describe spatial relationships

� Where is it? (location)� How far is it? (distance)� Which way is it? (direction)� How is it oriented? (position)

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Pre-K-2 Experiences:� Learn concepts such as above, below, in

front, behind, between, to the left, to the right, next toKatie’s desk is between my desk and Alex’s desk.

� Adding concept of distance and direction“walk three steps forward”

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3-5 Experiences:� More precise in measurements and begin to

examine multiple routes or shortest route

� Understanding some positional representations are relative (e.g., left or right),subjective (e.g., near or far), or fixed (e.g., north or west)

� By grade 5, experiences with the coordinate plane and the origin (0,0)

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Destination Space Station

� Locate and mark the asteroids on your grid

� The asteroids are located at (3,2), (7,2), (8,4), (3,6,), (6,8), (10,6), (10,9), (6,3), (1,7), and (5,5)

� Plan a safe path from the Earth (0,0) through the asteroids, to the Space Station located at (7,7) Mission Math II: Grades 3-5

1 8 9 106 74 532

1

2

3

4

5

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7

8

9

10

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Apply Transformations and symmetry

Pre-K-2 Experiences:

Children intuitively use geometric motions when they solve puzzles – they turn the pieces, flip them over, and slidethem into place.

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Early Discoveries:

A triangle is still the same triangle even if it is flipped or slid across the table.

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3-5 Experiences:� More formal and precise

� “translate” the square 10 cm to the right� “reflect” the triangle horizontally

� “rotate” the square 90 degrees clockwise

� Line and rotational symmetry become ways to describe geometric properties

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Use Visualization , spatial reasoning, and geometric modeling

Pre-K-2 Experiences:� Recall and describe hidden objects� Imagine a shape cut in two� Imagine a shape turned � Describe how an object would look from

a different side

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Use Visualization , spatial reasoning, and geometric modeling

3-5 Experiences:� Creating mental images of two and three

dimensional objects� Constructing three-dimensional shapes

from two-dimensional representations� Visualize how objects appear from

different perspectives� Predict the result of one or more

transformations

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Tangram Challenge

E-example 4.4

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Tangram Challenge Questions

� Use the small tangram pieces to create one of the larger tangram pieces (i.e., Create the large triangle)

� Assemble shapes to create another shape (i.e., Can you use two pieces to create a parallelogram?)\

� Can you use all seven tangram pieces to create a square?

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Tangram Challenge Questions

� Try to create a square using one tangram piece, two tangram pieces, three tangram pieces, and so on.

� Which of these figures can you make using all seven tangram pieces?� A trapezoid� A rectangle that is not a square� A parallelogram that is not a square� A triangle

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Today’s Agenda



� An Overview of Geometry Ideas that can be presented in Grades PreK - 8


� Next Steps for You.

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WHAT’S NEXT? What do you do before our follow-up session?

Try the activities explored in this E-Workshop with your students!

We will discuss your experiences at the follow-up session.

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Follow-Up Session: Jan. 18, 2007

Remember: Your registration fee included participation in a

Follow-Up Session!

If you choose to participate, you will need to register for the Follow-up Session using the link provided in the post-workshop e-mail message you will receive soon after the

conclusion of the E-Workshop.

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Preparing for the Follow-Up Session

1) Try a lesson using something from this E-Workshop with at least one group of students.

2) Share your reflections, questions, comments, and concerns in advance or during the follow-up session.

3) Make electronic copies of student work, activities that you create, and insights generated by your students. E-mail them to [email protected] by January 11, 2007. (Please type “follow-up session” in the subject line of your e-mail message.)

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What evidence shows students are learning with understanding?

What type of student responses demonstrate a lack of understanding?

What questions do you ask to deepen understanding?

Reflecting on Teaching and Learning

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How can students’conjectures and classroom interaction reveal their higher order thinking skills and reasoning ability?

How do you build new student knowledge from classroom evidence?

Reflecting on Teaching and Learning

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Available Resources

A resource that demonstrates how some of the fundamental ideas of Geometry can be introduced, developed, and extended.

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Contact Information

[email protected]

Tonight’s E-Workshop Leader: Amy MirraTonight’s E-Workshop Moderator: Amanda Ross

1906 Association Drive, Reston, VA 20191-1502TEL: (800) 235-7566 | FAX: (703) 476-2970

www.nctm.org

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Evaluation

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development offerings. Thank you!

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Thanks!

Thank you for all you do in providing

More and Better Mathematics

forAll Students.

Documents

Amy Mirra National Council of Teachers of Mathematics