Geometry Elementary

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Virginia Department of Education

GeometryFor

Elementary School Teachers

A Staff Development Training ProgramTo Implement the

2001 Virginia Standards of LearningFebruary 2003

Office of Elementary Instructional ServicesVirginia Department of Education

P.O. Box 2120Richmond, Virginia 23218-2120


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Copyright 2003 by the


P.O. Box 2120

Richmond, Virginia 23218-2120www.pen.k12.va.us

All rights reserved. Reproduction of these materials for

instructional purposes in Virginia classrooms is permitted.

Superintendent of Public Instruction

Jo Lynne DeMary

Assistant Superintendent for Instruction

Patricia I. Wright

Office of Elementary Instructional Services

Linda Poorbaugh, Director

Karen Grass, Mathematics Specialist

Notice to Reader

In accordance with the requirements of the Civil Rights Act and other federal and

state laws and regulations, this document has been reviewed to ensure that it does notreflect stereotypes based on sex, race, or national origin.

The Virginia Department of Education does not unlawfully discriminate on the basicof sex, race, color, religion, handicapping conditions, or national origin in

employment or in its educational programs and activities.

The activity that is the subject of this report was supported in whole or in part by theU.S. Department of Education. However, the opinions expressed herein do not

necessarily reflect the position or policy of the U.S. Department of Education, and no

official endorsement by the U.S. Department of Education should be inferred.

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http://www.pen.k12.va.us/http://www.pen.k12.va.us/


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Acknowledgements

The Virginia Department of Education wishes to express sincere appreciation to the

following individuals who have contributed to the writing and editing of the activitiesin this document.

Dr. Margie Mason, Associate Professor

The College of William and MaryWilliamsburg, Virginia

Mr. Bruce Mason, ConsultantWilliamsburg, Virginia

Dr. Carol L. Rezba, Former Mathematics SpecialistVirginia Department of Education

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Introduction

The updated Geometry for Elementary School Teachers is a staff development

training program designed to assist teachers in implementing the 2001 Virginia

Standards of Learning for mathematics. This staff development program provides asample of meaningful and engaging activities correlated to the geometry strand of the

K-5 mathematics standards of learning.

The purpose of the staff development program is to enhance teachers' contentknowledge and their use of instructional strategies for teaching the geometry

Standards of Learning. Teachers will learn about the van Hiele model for the

development of geometric thought and how this can be used to guide instruction andclassroom assessment. Through explorations, problem-solving, and hands-on

experiences, teachers will engage in discussions and activities that address many of

the dimensions of geometry including spatial relationships, properties of geometric

figures, constructions, geometric modeling, geometric transformations, coordinategeometry, the geometry of measurement, informal geometric reasoning, and

geometric connections to the physical world. Teachers will explore two- and three-

dimensional shapes, paper folding and origami, tessellations and geometric designs,and the use of other manipulatives to develop geometric understanding. Through

these activities, it is anticipated that teachers will develop new techniques that are

sure to enhance student achievement in their classroom.

Designed to be presented by teacher trainers, this staff development program

includes directions for the trainer, as well as the black line masters for overhead

transparencies. An addendum to the module includes video segments of the van Hiele

levels. The video segments portray students engaged in assessment tasks with adiscussion of the students level of development of geometric thought. Assessment

tasks are included as blackline masters.

Trainers should adapt the materials to best fit the needs of their audience;

adding materials that may be more appropriate for their audience and eliminatingmaterials that have been used in previous training sessions. All materials in this

document may be duplicated and distributed as desired for use in Virginia.

The training program is organized into five three-hour modules that may be

offered by school divisions for recertification points or for a one-credit graduate

course, when university credit can be arranged.

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GLOSSARY

Acute Angle An angle that is smaller than a right angle.An acute angle measures less than 90 degrees.

Acute Triangle A triangle that has three acute angles.

Adjacent Sides Each pair of sides that have a common vertex.

Angle A geometric figure consisting of two rays or linesegments that have the same endpoint. The size of anangle measures the amount of rotation from one side toanother.

Arc Part of a curve. In particular, part of the circumference of a

circle.

Area The amount of surface in a region or enclosed within aboundary. Area is usually measured in square units suchas square feet or square centimeters.

Attribute A characteristic possessed by an object. Characteristicsinclude shape, color, size, length, weight, capacity, area,etc.

Base The bottom side or bottom surface of a shape.

Centimeter A metric unit of length equal to one-hundredth of one meter.

Circle A two-dimensional shape formed by a set of points thatare all the same distance from a fixed point called thecenter.

Circumference The boundary of a circle, or the length of that boundary.The circumference can be computed by multiplying thediameter bypi (), a number a little more that 3.14.

Concentric Circles Two or more circles that have the same center anddifferent radii.

Cone A three-dimensional shape with a base (usually a circle)joined to a vertex by a curved surface.

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Congruent Relating to geometric figures that have the same size and

shape.

Coordinate System A reference system for locating and graphing points. In

two dimensions, a coordinate system usually consists of ahorizontal axis and a vertical axis, which intersect to givethe origin. Each point in the plane is located by itshorizontal distance and vertical distance from the origin.These distances, or coordinates, form an ordered pair ofnumbers.

Cube A solid shape in which every face is a square and everyedge is the same length.

Cubic Foot The volume of a cube that is one foot wide, one foot high,and one foot deep.

Cubic Unit A unit with length, width, and height that is used tomeasure volume.

Cylinder A can shape. A solid shape with congruent parallelcircles (or other shapes) joined by a curved surface.

Decagon A polygon with 10 sides. A regular decagon has10 equal sides.

Diagonal A line segment that joins two non-adjacent vertices of apolygon or polyhedron.

Diameter A line segment passing through the center of a circle orsphere and connecting two points on the circumference.

Diamond (See Rhombus)

Dimension The number of coordinates used to express a position.

Dodecagon A polygon with twelve sides. A regular dodecagon hastwelve equal sides.

Dodecahedron A polyhedron with twelve faces. All faces of a regulardodecahedron are congruent regular pentagons.

Edge A line segment where two faces of a three-dimensionalshape meet.

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Endpoint The point(s) at the end of a ray or line segment.

Equilateral Triangle A triangle with equal sides. Each angle measures 60 degrees.

Face A flat side of a polyhedron.

Flip (See Reflection)

Geometry The branch of mathematics that deals with the position,size, and shape of figures.

Grid A network of horizontal and vertical lines that intersectto form squares or rectangles.

Hemisphere Half of a sphere, formed by making a plane cut throughthe center of a sphere.

Heptagon A polygon with seven sides. A regular heptagon hasseven equal sides.

Hexagon A polygon with six sides. A regular hexagon has sixequal sides.

Hexahedron A polyhedron with six faces. A regular hexahedron is a cube.

Hypotenuse The side opposite the right angle of a right triangle. Thehypotenuse is the longest side of a right triangle.

Icosahedron A polyhedron with 20 faces. All faces of a regularicosahedron are congruent equilateral triangles.

Isosceles Triangle A triangle with two equal sides and two equal angles.(An equilateral triangle is a special case of an isoscelestriangle.)

Kite Shape A quadrilateral with two pairs of equal adjacent sides.

Line A set of points that form a straight path extendinginfinitely in two directions. Lines are often calledstraight lines to distinguish them from curves, whichare often called curved lines. Part of a line is called aline segment.

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Line of Symmetry A line dividing a two-dimensional figure into two parts

that are mirror images of each other.

Line Segment A part of a line. A line segment has two endpoints and a

finite length.

Network A diagram consisting of arcs (branches) connectingpoints or nodes (junctions). A network may represent areal-world situation, such as road system or electroniccircuit. Sometimes the nodes are called vertices.

Node A point in a network at the end of an arc or at thejunction of two or more arcs.

Nonagon A polygon with nine sides. A regular nonagon has nineequal sides.

Obtuse Angle An angle that is greater than 90 degrees but less than 180degrees; that is, between a right angle and a straight line.

Obtuse Triangle A triangle that has one obtuse angle.

Octagon A polygon with eight sides. A regular octagon has eightequal sides.

Octahedron A polyhedron with eight faces. All faces of a regularoctahedron are congruent equilateral triangles.

Opposite Angles In a quadrilateral, angles that do not have a common linesegment.

Parallel Lines Two or more lines that are always the same distance apart.

Parallelogram A quadrilateral with opposite sides parallel and equal inlength. Opposite angles are equal.

Pentagon A polygon with five sides. A regular pentagon has fiveequal sides.

Perimeter The boundary of a plane shape or the length of a boundary.

Perpendicular At right angles.

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pi () The ratio of the circumference of a circle to its diameter.

This ratio is the same for every circle. Its value, which isfound by dividing the circumference by the diameter, is alittle more than 3.14.

Pie graph A circle marked into sectors. Each sector shows thefraction represented by one category of data. Pie graphsare also called circle graphs.

Plane A flat surface extending infinitely in all directions.

Plane Shape In geometry, a closed two-dimensional figure that liesentirely in one plane. (Polygons and circles are examplesof plane figures. An arc is not a plane figure because it is

not closed.)

Point The smallest geometric unit. A position in space, oftenrepresented by a dot.

Polygon A plane shape bounded by straight sides.

Polyhedron A solid shape bounded by flat faces,

Prism A polyhedron with at least one pair of opposite faces thatare parallel and congruent. Corresponding edges of

these faces are joined by rectangles or parallelograms.

Pyramid A polyhedron with any polygon for its base. The otherfaces are triangles that meet at a point or vertex.

Quadrilateral A polygon with four sides.

Ray One-half of a line. A set of points that form a straightpath extending infinitely in one direction. A ray has one

endpoint.

Rectangle A quadrilateral with four right angles. Opposite sidesare equal and parallel.

Rectangular Prism A box shape. A prism with three pairs of parallelopposite faces.

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Reflection A transformation of a point, line, or geometric figure that

results in a mirror image of the original.

Regular Polygon A polygon that has equal sides and equal angles.

Regular Polyhedron A polyhedron with congruent faces that are regularpolygons.

Rhombus A parallelogram with four equal sides. Opposite anglesare equal.

Right Angle An angel that has one-fourth of a full turn. A right anglemeasures 90 degrees.

Right Triangle A triangle that has one right angle.

Scalene Triangle A triangle with sides of unequal length.

Semicircle One-half of a circle, bounded by a diameter and one-halfof the circumference. Sometimes, one-half of thecircumference is called a semi-circle.

Similar Figures that have the same shape but are different sizes.

Slide (See Translation)

Solid Shape A closed, three-dimensional figure.

Sphere A ball shape. A three-dimensional shape formed by a setof points that are all the same distance from a fixed pointcalled the center. Also, the solid shape enclosed by thatset of points.

Square A rectangle with equal sides.

Square Unit A unit that has length and width used to measure area.

Examples are square inches, square centimeters, acres,etc.

Surface Part or all of the boundary of a solid. A surface may beflat or curved. (For example, a cone has one flat surfaceand one curved surface.) The surfaces of a polyhedronare called faces.

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Symmetry a. When one side of a shape is a mirror image of theother side, the shape is said to have line of symmetry.

b. When a shape can be turned through a fraction of afull rotation and still look the same, it is said to haveturning symmetry or rotational symmetry.

Tessellation An arrangement of plane shapes (usually congruentshapes) to cover a surface without overlapping or leavingany gaps.

Tetrahedron A polyhedron with four triangular faces. A tetrahedronis a triangular pyramid.

Three-Dimensional Relating to objects that have length, width, and depth.Solid figures such as polyhedra, cones, and spheres arethree-dimensional.

Transformation The changing of a geometric figure from one position toanother, according to some rule. Examples of transformations

are reflection, rotation, and translation.

Translation A transformation in which a geometric figure is moved ina line. Each point of the figure moves the same distance.

Trapezoid A quadrilateral with one pair of parallel sides of unequallength.

Triangle A polygon with three sides.

Triangular Prism A prism in which the parallel opposite faces is triangles.

Two-Dimensional Relating to objects that have length and width but notdepth. Plane shapes such as polygons and circles that aretwo-dimensional.

Vertex A point, or corner, where the sides of shape, the rays ofan angle, or the edges of a solid meet. The plural ofvertex is vertices.

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Table of Contents

PAGE

I. van Hiele Theory of Geometric Thought 2

Van Hiele Levels and Triangle Sorts. 3

Quadrilaterals and their Properties 27

Quadrilateral Sort 29

Whats My Rule 37

Quadrilateral Properties Laboratory 39

Quadrilateral Sorting Laboratory 44

Quadrilaterals and their Properties through LOGO 59

Meet the Turtle 60

Writing Geo-LOGO Procedures 67

Squares and Rectangles 70

II. Classification 73

Tibby 77

Whats In the Box 78

Missing Pieces 80Whats My Rule 81

Twenty Questions Game 83

Who Am I? Game 85

Differences-Trains and Games 95

Hidden Number Patterns 99

Attribute Networks 103

Identifying Shapes 106

Human Circle 108

Geoboard Triangles and Quadrilaterals 109Shape Hunt 113

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III. Spatial Relationships 115

Square It 117

Pick up the Sticks 120Partition the Square 122

Cutting Square Puzzles 127

Tangrams 129

Make your Own Tangrams 131

Area & Perimeter Problems/Tangrams 133

Spatial Problem Solving Tangrams 136

Symmetry 139

Butterfly Symmetry 142

Copy Cat 144Recover the Symmetry 146

Folded Shapes 150

Symmetry and Right Angles in Quadrilaterals 155

Origami: Making a Square 163

Origami: Making a Heart 165

IV. Transformational Geometry: Tessellations 171

Sums of Angles of a Triangle 174

Do Congruent Triangles Tessellate? 177

Do Congruent Quadrilaterals Tessellate? 187Tessellations by Translation 192

Tessellations by Rotation 195

Solid Geometry 197

Polyhedron Sort 198

Whats My Shape? Ask About It 199

Whats My Shape? Touch Me. 200

Take It Apart 201

Building Polyhedra 202

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V. Perimeter and Area 204

Dominoes and Triominoes 207

Tetrominoes 210

Pentominoes 215

Areas with Pentominoes 219

Hexominoes 229

Perimeters with Hexominoes 234

Perimeter and Area Part II 236

The Perimeter Is 24 Inches. Whats the Area? 238

The Area Is 24 Inches. What is the Perimeter? 242

Change the Area 243

Coordinate Geometry 246

Hurkle 247

Battleship 254

Two-Dimensional Hurkle 257

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Elementary Geometry

Module 1

Topic Activity Name Related SOL Transparencies

Handouts

Materials

Van HieleTheory of

Geometric

Thought

Lecture vanHiele levels

Triangle Sorts

K.11, K.12,K.13, 1.16

T: 1.1, 1.2, 1.3H: 1.1, 1.2, 1.3

Paper triangles

Quadrilateral

Sort

K.11, K.12,

K.13, 1.16,

2.22, 3.18, 4.15,4.17, 5.14,

5.15

T: 1.4

H: 1.4, 1.5

Paper

Quadrilaterals

Whats My

Rule

K.11, K.12,

K.13, 1.16,2.22, 3.18, 4.15,

4.17, 5.14,

5.15

T: 1.5

H: 1.4

Paper

Quadrilaterals

Quadrilateral

PropertiesLaboratory

K.11, K.12,

K.13, 1.16,2.22, 3.18, 4.15,

4.17, 5.14,

5.15

T: 1.6

H: 1.6

Geo-strips,

D-stix, orminiature

marshmallows,

& toothpicks;square corner

Quadrilaterals

& Their

Properties

QuadrilateralSorting

Laboratory

K.11, K.12,K.13, 1.16,

2.22, 3.18, 4.15,

4.17, 5.14,5.15

T: 1.6, 1.7H: 1.6, 1.7

PaperQuadrilaterals

Meet the Turtle 1.16, 3.18, 5.14 T: 1.8, 1.9,

1.10

H: 1.8

Writing Geo-

Logo

Procedures

1.16, 3.18, 5.14 H: 1.9

Quadrilaterals

& Their

Properties

Through LOGO

Squares &Rectangles

1.16, 3.18, 5.14 H: 1.10

Geo-Logo on 1

computer per

pair, overhead

view screenhooked to

computer,

(if available)



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Key Idea: The van Hiele Levels of Geometry Understanding

Description: The van Hiele theory of geometric understandingdescribes how students learn geometry andprovides a framework for structuring studentexperiences that should lead to conceptual growthand understanding. In this first session, theparticipants will explore the van Hiele levels ofgeometric understanding doing triangle sorts andcomparing their sorts to those performed byelementary students. The sorting task is

appropriate for all ages and levels of students. Itcan serve as an activity to help students advancetheir level of understanding as well as anassessment tool that can inform the teacher whatvan Hiele level the student is thinking at withregard to triangles.

Related SOL:Kindergarten

K.11 The student will identify, describe, and draw two-dimensional (plane)geometric figures (circle, triangle, square, and rectangle).

K.12 The student will describe the location of one object relative to another(above, below next to) and identify representations of plane geometricfigures (circle, triangle, square, and rectangle), regardless of their positionand orientation in space.

K.13 The student will compare the size (larger/smaller) and shape of planegeometric figures (circle, triangle, square, and rectangle).

First Grade

1.16 The student will draw, describe, and sort plane geometric figures(triangle, square, rectangle, and circle) according to number of sides,corners, and square corners.

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Activity : The van Hiele Levels of Geometry Understanding

Format: Large Group Lecture and Small Group Activity

Description: Participants will explore the van Hiele levels of geometricunderstanding by doing triangle sorts and comparing their sortsto those performed by elementary students and by consideringstudent activities appropriate for each level.

Objectives: Participants will be able to describe the developmental sequenceof geometric thinking according to the van Hiele theory of

geometric understanding and activities suitable for each level.In addition, participants will be able to assess the van Hielelevels of their students.

Related SOL: K.11, K.12, K.13, and 1.16

Vocabulary: van Hiele levels of geometric understanding, visualization,analysis, abstraction, deduction, rigor, adjacency, distinction,separation, attainment, phases of learning, information,guided orientation, explicitation, free orientation,

integration, gravity-based figures, triangle, shape orientation

Materials: Paper triangles, cut out and placed in a plastic baggy ormanila envelope (see Handout 1.1 for Triangle SortingPieces.) You will need at least one set of triangles for everythree participants. Handout 1.1, Handout 1.2, and Handout1.3 (the Crowley article), Transparency 1.1, Transparency 1.2,and Transparency 1.3.

Time Required: Approximately 1 hour


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Background: To Trainer (for lecture):

After observing their own students, Dutch teachers P.M.van Hiele and Dina van Hiele-Geldof described learning as adiscontinuous process with jumps that suggest "levels."They identified five sequential levels of geometricunderstanding or thought:1) Visualization2) Analysis3) Abstraction4) Deduction5) RigorClements and Battista (1992) proposed the existence of aLevel 0 that they call Pre-recognition.

In Grades K, 1, and 2 most students will be at Level 1. Bygrade 3, students should be transitioning to Level 2. If theSOL are mastered, students should attain Level 3 by the endof sixth grade. Students who can prove theorems usingdeductive techniques usually attain level 4. One problem isthat most current textbooks provide activities requiring onlyLevel 1 thinking up through sixth grade and teachers mustprovide different types of tasks to facilitate the developmentof the high levels of thought.

Directions: 1. Show Transparency 1.1 on the overhead unmasking theinformation about Level 1 only. Describe Level 1,visualization. Make particular note of the student's wayof identifying the rectangle. "I know it's a rectanglebecause it looks like a door and I know that the door is arectangle." Identification is based on a visual model.


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2. Unmask Level 2 on Transparency 1.1. Compare Level 1and Level 2. Properties are perceived at Level 2, but theyare isolated and unrelated. A Level 2 student would say

"I know it's a rectangle because it is closed; it has 4 sidesand 4 right angles; opposite sides are parallel; oppositesides are congruent; diagonals bisect each other; adjacentsides are perpendicular; etc...." All the properties knownare listed since the student doesn't perceive anyrelationship between the properties, e.g., one implies theother. There is no knowledge of necessary and sufficientconditions.

3. Unmask Level 3 on Transparency 1.1. Relationships betweenproperties and between figures are perceived at Level 3. Such a

student would say "I know its a rectangle because it's aparallelogram with right angles." The student will give theminimum number of properties, eliminating redundancies. ALevel 3 student can formulate meaningful definitions andunderstand inclusion relationships such as every square is aspecial type of rectangle, but not every rectangle is a square.Note that if a student is require to "know a definition" beforeattaining Level 3, it will be a memorized definition with littlemeaning to the student. His concept definition will likely notmatch his concept image.

4. Unmask Level 4 on Transparency 1.1. Generally speaking, mosthigh school geometry courses are taught at this level. Masteringthe Virginia Standards of Learning for geometry require thistype of thinking.

5. Unmask Level 5 on Transparency 1.1. This level is sometimesreferred to as the mathematicians' level; although gifted middleschools have been known to think at this level. The ability to dosymbolic logic and non-Euclidean geometry are characteristic ofthis level of thought.


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6. Turn to Additional Points on Transparency 1.1. Note for Point 1that the levels are hierarchical. Students cannot be expected towrite a geometric proof successfully unless they have

progressed through each level of thought in turn. At Point 2,mention that college students, and even some teachers, havebeen found at Level 1 while there are middle school students atLevel 3 and above. (If the SOL are mastered, students shouldattain Level 3 by the end of sixth grade.) As an example of anexperience that can impede progress (Point 3), give theillustration of the teacher who knew that the relationshipbetween squares and rectangles was a difficult one for herfourth graders. Therefore, the teacher had them memorize"Every square is a rectangle, but not every rectangle is a square."

When tested a couple of weeks later, half the studentsremembered that a square is a type of rectangle, while the otherhalf thought that a rectangle was a type of square. It was almostimpossible for these students to learn the true relationshipbetween squares and rectangles now because every time theyheard the words square and rectangle together, they insisted onrelying on their memorized sentence rather than on theproperties of the two types of figures.

7. Turn to Properties of Levels on Transparency 1.1. As anexample of separation, you can use the meaning of the word

"square." When someone such as a teacher who is thinking atLevel 3 or above says "square," the word conveys the propertiesand relationships of a square such as having four congruentsides; having four congruent angles; having perpendiculardiagonals; and being a type of polygon, quadrilateral,parallelogram, and rectangle. To a student thinking at Level 1,the word "square" will only evoke an image of something thatlooks like a square such as a CD case or first base. The sameword is being used, but it has an entirely different meaning tothe teacher and the student. The teacher must keep in mindwhat the meaning of the word or symbol is to the student andhow the student thinks about it.


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8. For Attainment, mention that there are five phases of learningthat lead to understanding at the next higher level. Then turn tothe Phases of Learning on Transparency 1.1. Tell the

participants that the activities that they are going to be doingduring the course of the workshops are structured to providedthese experiences for students.

9. Divide the participants into small groups. Pass out the sets ofcutout triangles, at least one set per three participants. Instructthe participants to lay out the pieces with the letters up. Don'tcall them triangles. Tell the participants that objects can begrouped together in many different ways. For example, if wesorted the shapes that make up the American flag (the redstripes, the white stripes, the blue field, the white stars), we

might sort by color. In this case we would put the white stripesand the stars together because they are white, the red strips inanother group because they are red, and the blue field by itselfbecause it is the only blue object. Another way to sort the flagparts would be to put all the strips and the blue field togetherbecause they are all rectangles and all the stars together becausethey are not rectangles. If needed, you can demonstrate atriangle sort using pieces cut from Transparency 1.2. Haveparticipants sort the shapes into groups that belong together,recording the letters of the pieces they put together and the

criteria they used to sort. Have them sort two or three times,recording each sort.10. Ask the participants for some of their ways of sorting. Expect

answers like "acute, right, and obtuse triangles" or "scalene,isosceles, and equilateral". Have them compare their ways withthose of other groups.

11. Ask them how they think their students would sort thesefigures. Show sample sorts in Transparency 1.3 and askthe participants to conjecture the criteria used for sortingand the van Hiele level of the sorter. Sample 1 is a lowLevel 1 sort where the student is sorting strictly by sizeand may not even know that the figures are triangles.Sample 2 is another Level 1 sort. Here the student thinksthat triangles must have at least two sides the samelength or possibly that triangles must be symmetric.Sample 3 is another Level 1 sort. This student alsobelieves that


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triangles must have at least two sides the same length orpossibly that triangles must be symmetric. Additionally,this student recognized the figures with right angles or"corners" as a separate category. The Sample 4 sort is at

least a Level 2 or 3 sort in which the sorter focuses on thelengths of the sides, criteria which separates the figuresinto categories which overlap. The student has actuallysorted into groups where no sides have the same length,where two sides have the same length, and where allsides have the same length. It is unclear whether thestudent knows that equilateral triangles are a type ofisosceles triangle. The Sample 5 sort focuses on parts ofthe figures and so is a Level 2 sort, but the student doeshave the vocabulary to adequately describe the figures.

The Sample 6 sort is similar to the Sample 4 sort, but theword "Perfect" is incorrect and indicates that the studentmay be thinking more of the shape as a whole rather thanof the individual parts. This sort is probably Level 2.


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Virginia Department of Education Transparency 1.1 Page 9

The van Hiele Levels

Level 1: Visualization. Geometric figures

are recognized as entities, without any

awareness of parts of figures or relationships

between components of the figure. A student

should recognize and name figures, anddistinguish a given figure from others that

look somewhat the same.

"I know it's a rectangle because it looks like a

door and I know that the door is a rectangle."

Level 2: Analysis. Properties are perceived,but are isolated and unrelated. A student

should recognize and name properties of

geometric figures.

"I know it's a rectangle because it is closed, it

has 4 sides and 4 right angles, opposite sidesare parallel, opposite sides are congruent,

diagonals bisect each other, adjacent sides are

perpendicular,...


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Level 3: Abstraction. Definitions are

meaningful, with relationships being perceived

between properties and between figures. Logicaimplications and class inclusions are understood

but the role and significance of deduction is not

understood. I know its a rectangle because it's

a parallelogram with right angles."

Level 4: Deduction. The student can construct

proofs, understand the role of axioms and

definitions, and know the meaning of necessary a

sufficient conditions. A student should be able t

supply reasons for steps in a proof.

Level 5: Rigor. The standards of rigor &

abstraction represented by modern geometries

characterize level 5. Symbols without referents

be manipulated according to the laws of formal

logic. A student should understand the role andnecessity of indirect proof and proof by

contrapositive.


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Additional Points

1. The learner cannot achieve one level

without passing through the previous levels.

2. Progress from one level to another is more

dependent on educational experience than onage or maturation.

3. Certain types of experiences can facilitate

or impede progress within a level or to a

higher level.


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Properties of Levels

Adjacency: what was intrinsic in the

preceding level is extrinsic in the current level

Distinction: each level has its own linguistic

symbols and its own network of relationshipsconnecting those symbols

Separation: two individuals reasoning at

different levels cannot understand one another

Attainment: the learning process leading tocomplete understanding at the next higher

level has five phases: inquiry, directed

orientation, explanation, free orientation and

integration


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Phases of Learning

Information: Gets acquainted with the

working domain (e.g., examines examples and

non-examples)

Guided orientation: Does tasks involving

different relations of the network that is to beformed (e.g., folding, measuring, looking for

symmetry)

Explicitation: Becomes conscious of the

relations, tries to express them in words, and

learns technical language whichaccompanies the subject matter (e.g.,

expresses ideas about properties of figures)

Free orientation: learns, by doing more

complex tasks, to find his/her own way in

the network of relations (e.g., knowingproperties of one kind of shape, investigates

these properties for a new shape, such as

kites)


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Integration: Summarizes all that he/she has

learned about the subject, then reflects on

his/her actions and obtains an overview of the

newly formed network of relations now

available (e.g., properties of a figure aresummarized)


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Triangle Sorting Pieces

AB C D E

F G

H

JK L

M

N

O P

Q

RS

TU

V

W

XY

Z


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Sample Student Sort 1

H

C

J

K

O

U

R W

X

B

Q

S

Z

T

E

G

N

DL

M

V

A

Medium

Large

P

F

Small


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T

NOT Triangles

HC

J

K

O U

R

W X

B

Q S

Z

E

G N

DLM

VA

Triangles

P


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HC

J

K

O

U R

WX

B

Q

S

Z T EG N

D

LM

V A

Look alike...

N is smaller...NOT triangles

Triangles

look like ramps... NOT triangles


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H

C

J

K

O

U

WX

B

Q

S

ZT E

G

N

D L

MV

A

SCALENE

ISOSCELES

P

F

EQUILATERAL

Y


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KR

W

X

S

same shape &small size

H CB

one longest side

T E G P

irregular & very narrow

J

O U

2 sides are similar,one is shorter

Q

F

Y

2 sides are similar,one is longer

ZL

V

A

3 uneven sides

N

DM

irregular sides


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H

C

J

K

OU

R WX

B

Q

S

Z

T

E

G

N

DLM

V

A

P

F Y

Every side has a different size

Perfect

Triangles

2 sides =, 3rd side smaller or larger



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Triangle Sorting Pieces

AB C D E

F G

H

J

KL

M

N

O P

Q

RS

TU

V

W

XY

Z

Virginia Department of Education Handout 1.1 Page 22


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Student Triangle Sorting Pieces



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page 2



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A B

C

DE

F

G



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page 2

H

I

H

JK



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Key Idea: Quadrilaterals and their Properties

Description: Participants will explore quadrilaterals and their propertiesthrough the use of various manipulatives such as sorting piecesand geo-strips. The sequence of activities is designed to facilitatean increase in a learner's van Hiele level of thinking aboutquadrilaterals from Level 1 to Level 3. First, the participants learnhow to determine the van Hiele levels of their own students byanalyzing how they sort a set of quadrilateral pieces. Then theyplay the game "What's My Rule?" to develop the ability to classifyquadrilaterals by various attributes and to focus on more than oneattribute at a time. The participants also construct parallelograms,rectangles, rhombi, and squares using D-stix, geo-strips, toothpicks,

or other manipulatives and make observations while the figures areflexed (Level 2). Finally, the participants identify relationshipsbetween parallelograms, rectangles, rhombi, squares, trapezoids,kites, and darts through a lab that culminates in the creation of aquadrilateral family tree (Level 3).While these activities are presented with quadrilaterals, most ofthem are easily adapted to triangles and other polygons.

Related Geometry SOL:Kindergarten

K.11 The student will identify, describe, and draw two-dimensional (plane)geometric figures (circle, triangle, square, and rectangle).

K.12 The student will describe the location of one object relative to another (above,below, next to) and identify representations of plane geometric figures (circle,triangle, square, and rectangle), regardless of their position and orientation inspace.


Grade One1.16 The student will draw, describe, and sort plane geometric figures (triangle,

square, rectangle, and circle) according to number of sides, corners, and squarecorners.

Grade Two2.22 The student will compare and contrast plane and solid geometric shapes

(circle/sphere, square/cube, and rectangle/rectangular solid).

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Grade Three3.18 The student will analyze two-dimensional (plane) and three-dimensional

(solid) geometric figures (circle, square, rectangle, triangle, cube, rectangularsolid [prism], square pyramid, sphere, cone, and cylinder) and identifyrelevant properties, including the number of corners, square corners, edgesand the number and shape of faces, using concrete models.

Grade Four4.15 The student will

a) identify and draw representations of points, lines, line segments, rays, andangles, using a straightedge or ruler, andb) describe the path of shortest distance between two points on a flat surface.

4.17 The student willa) analyze and compare the properties of two-dimensional (plane) geometricfigures (circle, square, rectangle, triangle, parallelogram, and rhombus) andthree-dimensional (solid) geometric figures (sphere, cube,, and rectangularsolid [prism]);b) identify congruent and noncongruent shapes, andc) investigate congruence of plane figures after geometric transformations suchas reflection (flip), translation (slide) and rotation (turn), using mirrors, paperfolding, and tracing.

Grade Five5.14 The student will classify angles and triangles as right, acute, or obtuse.

5.15 The student, using two-dimensional (plane) figures (square, rectangle,triangle, parallelogram, rhombus, kite, and trapezoid) willa) recognize, identify, describe, and analyze their properties in order todevelop definitions of their figures;b) identify and explore congruent, noncongruent, and similar figures;c) investigate and describe the results of combining and subdividing shapes;d) identify and describe a line of symmetry; ande) recognize the images of figures resulting from geometric transformationssuch as translation (slide), reflection (flip), or rotation (turn).

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Activity: Quadrilateral Sort

Format: Small Group/Large Group

Description: Participants sort quadrilaterals in three different ways and comparetheir sort to what they predict their students would do. They thenrelate the sorts to the van Hiele levels of geometric understandingof quadrilaterals.

Objectives: After performing their own sorts, participants will be able todistinguish the way students at various van Hiele levels ofgeometric understanding may sort quadrilaterals.

Related SOL: K.11, K.12, K.13, 1.16, 2.22, 3.18, 4.15, 4.17, 5.14, 5.15

Vocabulary: depends on participants. Terms such as parallel, congruent, rightangle, square, rectangle, trapezoid, parallelogram, rhombus, kite,and dart are likely.

Materials: Paper Quadrilaterals from Handout 1.4, cut out and placed in aplastic baggy or manila envelope. You will need at least one set ofquadrilaterals for every three participants. Handouts 1.4, 1.5,

Transparency 1.4.

Time Required: Approximately 20 minutes

Directions: 1) Divide the participants into small groups. Pass out the sets ofcutout quadrilaterals, at least one set per three participants.Instruct the participants to lay out the pieces with the letters up.Don't call them quadrilaterals. Tell the participants that objectscan be grouped together in many different ways. For example,if we sorted the shapes that make up the American flag (the red

stripes, the white stripes, the blue field, the white stars), wemight sort by color and put the white stripes and the starstogether because they are white, the red strips in another groupbecause they are red, and the blue field by itself because it is theonly blue object. Another way the flag parts could be groupedwould be all the strips and the blue field together because theyare all rectangles and all the stars

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together because they are not rectangles. Have them sort theshapes into groups that belong together, recording the letters ofthe pieces they put together and the criteria they used to sort.Have them sort two or three times, recording each sort.

2) Ask the participants for some of their ways of sorting. Havethem compare their ways with those of other groups.

3) Ask them how they think their students would sort thesefigures.

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Quadrilateral Sorting Pieces



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Quadrilateral Sorting Pieces



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Student Quadrilateral Sorting Pieces



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Student Quadrilateral Sorting Piecespage 2



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A

B

C

D

E F



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page 2

G

H

I

J

J

L



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Activity: What's My Rule?


Description: A sorter separates the quadrilaterals according to a hidden rulewhile other players try to figure out what the sorting rule is.

Objectives: After playing the game, participants will be able to classifyquadrilaterals by various attributes. In children, this gamedevelops the ability to attend to more than one characteristic of afigure at the same time.

Related SOL: K.11, K.12, K.13, 1.16, 2.22, 3.18, 4.15, 4.17, 5.14, 5.15

Vocabulary: depends on participants. Terms such as parallel, congruent, rightangle, square, rectangle, trapezoid, parallelogram, rhombus, kite,and dart are likely.

Materials: Paper Quadrilaterals, cut out and placed in a plastic baggy ormanila envelope (see Handout 1.4 for Quadrilateral Sorting Pieces.)You will need at least one set of quadrilaterals for every three orfour participants. Transparency 1.5.


Directions: 1) Divide the participants into groups or 3 or 4. Pass out the setsof cutout quadrilaterals, one set per group.

2) Display Transparency 1.5 and go over the rules of the game.One participant in each group is the sorter. The sorter writesdown a "secret rule" to classify the set of quadrilaterals into twoor more piles and uses that rule to slowly sort the pieces as theother players observe.

3) At any time, the players can call "stop" and guess the rule. Afterthe correct rule identification, the player who figured out therule becomes the sorter. The correct identification is worth 5points. A correct answer, but not the written one, is worth 1point. Each incorrect guess results in a 2-point penalty. Thewinner is the first one to accumulate 10 points.

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WHAT'S MY RULE?

Rules

1. Choose one player to be the sorter. Thesorter writes down a "secret rule" toclassify the set of quadrilaterals intotwo or more piles and uses that rule toslowly sort the pieces as the otherplayers observe.

2. At any time, the players can call"stop" and guess the rule. Thecorrect identification is worth 5points. A correct answer, but notthe written one, is worth 1 point.Each incorrect guess results in a 2-point penalty.

3. After the correct rule identification,the player who figured out the rule

becomes the sorter.4. The winner is the first one to

accumulate 10 points.



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Activity: Quadrilateral Properties Laboratory


Description: Participants construct parallelograms, rectangles, rhombi,and squares using D-stix, geo-strips, or toothpicks andmarshmallows and make observations as the figures areflexed.

Objectives: The participant will be able to identify the main properties ofparallelograms, rectangles, rhombi, and squares.

Related SOL: K.11, K.12, K.13, 1.16, 2.22, 3.18, 4.15, 4.17, 5.14, 5.15

Vocabulary: quadrilateral, parallelogram, rectangle, rhombus, square,opposite sides, opposite angles, parallel

Materials: D-stix, geo-strips, or miniature marshmallows and toothpicks cutinto two different lengths; square corner (the corner of an indexcard or book); Handout 1.6; and Transparency 1.6


Directions: 1) Divide the students into groups of 3 or 4 and direct each group toexperiment as you ask questions. Be sure to model constructingthe polygons and flexing them.

2) Ask the participants to pick two pairs of congruent segments andconnect them as shown below. Have them flex the figure todifferent positions.

. .

. .

Virginia Department of Education Quadrilateral Properties Lab Page 39


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Ask what stays the same? (lengths of the sides, the opposite sidesare parallel, opposite angles are the same, sum of angles,perimeter)What changes? (size of angles, area, lengths of diagonals)

What do you notice about the opposite sides of this quadrilateral?(They remain parallel and congruent.)______________________________________A parallelogram is a quadrilateral with opposite sides parallel.______________________________________

What is the sum of the interior angles of this quadrilateral? (360)What do you notice about the opposite angles? (congruent)

Note to Trainer: Some participant will likely turn the strips so that

they cross forming two triangles. If no one does, you should. Askif this figure is a polygon. Elicit from the group what the essentialelements of a polygon are, i.e.,a) composed of straight line segments connected end to endb) simple (the segments don't cross)c) closedd) lies in a plane (e.g. if you take a wire square and twist it so

that it isn't flat, it is no longer a polygon)

3) Make one of the angles a right angle (You can use the squarecorner to check your accuracy.) What happens to the other angles?(They become right angles.) Will this always be true when youmake one angle of a parallelogram a right angle? (Yes) How doyou know? (The sum of the angles in a parallelogram is 360. Oneangle is given as 90. Its opposite angle must be the same or 90.Subtracting these two angles from 360, the remaining two angles,which are congruent since they are opposite angles in aparallelogram, must total 180. Therefore, each is 90. Note: Thisis Level 3 thinking.) Is it still a parallelogram? (Yes) Is it still aquadrilateral? (Yes) Is it still a polygon? (Yes) What other name,besides polygon, quadrilateral, and parallelogram, can be given to

it now? (rectangle)______________________________________A rectangle is a parallelogram with four right angles.______________________________________

4) Make a parallelogram that has all four sides equal in length. Whatis another name for this parallelogram? (rhombus)



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______________________________________A rhombus is a parallelogram with four congruent sides.______________________________________

5) Flex the figure to different positions.

..

..

.What stays the same? (lengths of the sides, the opposite sides areparallel, opposite angles are the same, sum of angles, perimeter)What changes? (size of angles, area, lengths of diagonals)What is the sum of the interior angles of this quadrilateral? (360)What do you notice about the opposite angles? (congruent)Is it still a quadrilateral? (Yes) Is it still a polygon? (Yes)

6) Make one of the angles of this rhombus a right angle, checkingwith your square corner. What happens to the other angles? (Allright angles) Is it still a parallelogram? (Yes) What other name,besides polygon, quadrilateral, parallelogram, and rhombus, canbe given to this new figure? (square)______________________________________A square is a parallelogram with four congruent sides and fourright angles.______________________________________

Is it a rectangle? (Yes) How do you know? (It has four rightangles.)

7) Pass out Handout 1.6 (Transparency 1.6) and discuss thedefinitions for quadrilateral, parallelogram, rectangle, rhombus,and square. Discuss the examples of each, noticing theirorientations and how each example fits the definition even thoughthey aren't necessarily the stereotype figure usually seen. Askwhat implications the fact that a Level 1 student recognizes shapes

by comparing them to a known shape has for teaching.



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TYPES OF QUADRILATERALS

A parallelogram is a

quadrilateral with opposite sides parallel .

These sides

are parallel.

A quadrilateral is a

four sided polygon.

A rectangle is a quadrilateral

with all right angles.

A rhombus is a quadrilateral

with all sides congruent.

A trapezoid is a

quadrilateral with exactly

one pair of parallel sides.These sides

are parallel.

A square is a quadrilateral

with 4 right angles and

4 congruent sides .

A dart is a concave

quadrilateral.

A kite is a quadrilateral with 2

pair of consecutive

congruent sides but not all

congruent sides.



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TYPES OF QUADRILATERALS

A parallelogram is a

quadrilateral with opposite sides parallel .

These sides

are parallel.

A quadrilateral is a

four sided polygon.

A rectangle is a quadrilateral

with all right angles.

A rhombus is a quadrilateral

with all sides congruent.

A trapezoid is a

quadrilateral with exactly

one pair of parallel sides.These sides

are parallel.

A square is a quadrilateral

with 4 right angles and

4 congruent sides .

A dart is a concave

quadrilateral.

A kite is a quadrilateral with 2

pair of consecutive

congruent sides but not all

congruent sides.



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Activity: Quadrilateral Sorting Laboratory


Description: Participants record which quadrilaterals meet the variousdescriptions listed in the properties table, determine which sets areidentical and are subsets of one another, attach labels to eachcategory, and create a quadrilateral family tree.

Objectives: The participant will be able to identify the relationships betweenparallelograms, rectangles, rhombi, squares, trapezoids, kites, anddarts.

Related SOL: K.11, K.12, K.13, 1.16, 2.22, 3.18, 4.15, 4.17, 5.14, 5.15

Vocabulary: quadrilateral, parallelogram, rectangle, rhombus, square,trapezoid, kite, dart, and proper subset

Materials: Paper Quadrilaterals (Handout 1), cut out and placed in a plasticbaggy or manila envelope. You will need at least one set ofquadrilaterals for every three or four participants. Transparencies1.6 and 1.7 and Handouts 1.6 and 1.7.


Directions: 1) Pass out Handout 1.6. Divide the participants into groups of 3or 4 and direct each group to experiment and answer thequestions on the handout.

2) After the participants have filled out the table, have pairs ofgroups compare their answers, and reconcile any discrepancies.

3) Have the students continue with Steps 5-10. Refer toTransparency 1.6 as needed while discussing the results of #10.Participants may refer to Handout 1.6 for Step 10.

4) For Step 11 the participants can construct the family tree assmall groups or as a large group. Discuss various possibilitiesfor the entries using Transparency 1.7.

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Quadrilateral Table

___________________________________________________

1. has 4 right angles

___________________________________________________

2. has exactly one pair of parallel sides

___________________________________________________

3. has two pair of opposite sides congruent

___________________________________________________

4. has 4 congruent sides

___________________________________________________

5. has two pair of opposite sides parallel

___________________________________________________6. has no sides congruent

___________________________________________________

7. has two pair of adjacent sides congruent, but not all sides congruent

___________________________________________________

8. has perpendicular diagonals

___________________________________________________

9. has opposite angles congruent

___________________________________________________

10. is concave

___________________________________________________

11. is convex

___________________________________________________

12. its diagonals bisect one another

___________________________________________________

13. has four sides

___________________________________________________

14. has four congruent angles___________________________________________________

15. has four congruent sides and four congruent angles___________________________________________________



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OTHERS

QUADRILATERALS

PARALLELOG

RAMSR

HOMBI


OTHERS

QUADRILATERALS

PARALLELOG

RAMSR

HOMBI


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Quadrilateral Sorting Laboratory

Objective: The participant will be able to identify the relationships between

parallelograms, rectangles, rhombi, squares, trapezoids, kites, anddarts.

Materials: Paper Quadrilaterals, cut out and placed in a plastic baggy or manilaenvelope (See P-2.). One set per 3 or 4 people is desirable.

Directions: 1) Spread out your Quadrilateral Set with the letters facing up so youcan see them.

2) Find all of the quadrilaterals having 4 right angles. List them byletter alphabetically in the corresponding row of the Table.

3) Consider all of the quadrilaterals again. Find all of thequadrilaterals having exactly one pair of parallel sides. List them

by letter alphabetically in the corresponding row of the Table.4) Continue in this manner until the Table is complete.5) Which category is the largest? What name can be used to describe

this category?6) Which lists which are the same? What name can be used to

describe quadrilaterals with these properties?7) Are there any lists that are proper subsets of another list? If so,

which ones?8) Are there any lists that aren't subsets of one another that have some

but not all members in common? If so, which ones?

9) Which lists have no members in common?10) Label each of the categories in the Table with the most specific namepossible using the labels kite, quadrilateral, parallelogram,rectangle, rhombus, square, and trapezoid. For example, #1 - aquadrilateral which has 4 right angles is a rectangle. (Having 4right angles isn't enough to make it a square; it would need 4congruent sides as well.)

11) Compare your results to that of the other Lab Groups. Then fill outthe family tree by inserting the names kites, rectangles, squares,and trapezoids into the appropriate places on the diagram.

Virginia Department of Education Handout 1.7Page 56


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Quadrilateral Table

___________________________________________________

1. has 4 right angles

___________________________________________________2. has exactly one pair of parallel sides

___________________________________________________

3. has two pair of opposite sides congruent

___________________________________________________

4. has 4 congruent sides

___________________________________________________

5. has two pair of opposite sides parallel

___________________________________________________

6. has no sides congruent

___________________________________________________

7. has two pair of adjacent sides congruent, but not all sides congruent

___________________________________________________

8. has perpendicular diagonals

___________________________________________________

9. has opposite angles congruent

___________________________________________________

10. is concave___________________________________________________

11. is convex

___________________________________________________

12. its diagonals bisect one another

___________________________________________________

13. has four sides

___________________________________________________

14. has four congruent angles

___________________________________________________

15. has four congruent sides and four congruent angles___________________________________________________



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OTHERS

Q

UADRILATERALS

PARALLELO

GRAMSR

HOMBI



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Key Idea: Quadrilaterals and their Properties Through LOGO

Description: Participants will explore quadrilaterals and their propertiesthrough the use of Geo-Logo, an inexpensive type of LOGO createdunder the sponsorship of the National Science Foundation. Afterthe basic commands of the language are introduced, the sequenceof activities is designed to facilitate an increase in a learner's vanHiele level of thinking about quadrilaterals from Level 1 to Level 3.While these activities are presented with quadrilaterals, most ofthem are easily adapted to triangles and other polygons.

Related Geometry SOL:KindergartenK.11 The student will identify, describe, and draw two-dimensional (plane)

geometric figures (circle, triangle, square, and rectangle).K.12 The student will describe the location of one object relative to another (above,

below, next to) and identify representations of plane geometric figures (circle,triangle, square, and rectangle), regardless of their position and orientation inspace.


Grade One1.16 The student will draw, describe, and sort plane geometric figures (triangle,

square, rectangle, and circle) according to number of sides, corners, and squarecorners.

Grade Three3.18 The student will analyze two-dimensional (plane) and three-dimensional

(solid) geometric figures (circle, square, rectangle, triangle, cube, rectangularsolid [prism], square pyramid, sphere, cone, and cylinder) and identifyrelevant properties, including the number of corners, square corners, edgesand the number and shape of faces, using concrete models.

Grade Five5.14 The student will classify angles and triangles as right, acute, or obtuse.

Virginia Department of Education Quadrilaterals through LOGOPage 59


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Activity: Meet the Turtle

Format: Individual / Small Group

Description: Participants become familiar with the basic commands of Geo-Logoby drawing their initials; making a square, a rectangle, and anequilateral triangle; and drawing a house.

Objectives: After drawing their initials; making a square, a rectangle, and anequilateral triangle; and drawing a house; participants will befamiliar with the commands in Geo-Logo and be able to describethe geometric properties of the figures they have made.

Related SOL: 1.16, 3.18, and 5.14

Vocabulary: See handout of Logo commands. Additionally, right angle,congruent, interior angles, and opposite sides

Materials: Geo-Logo on one computer per pair of participants or onecomputer per person, overhead viewscreen hooked to computer (if

available), Handout 1.8, Transparencies 1.9, 1.10, and 1.11.


Directions: 1) Divide the participants into pairs, sharing one computer, orhave one participant per computer. Pass out Handout 1.8.Demonstrate how to start Geo-Logo.

2) Display Transparency 1.9 and discuss the basic commands.

3) Have participants complete Handout 1.8, recording theirprocedures and answers. Monitor their progress, helping whenneeded.

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Geo-Logo Commands

Shorthand Means Meaning

pd Pen down The turtle's pen will now leave a track.

fd x Forward x The turtle moves forward x turtle steps.

bk x Back x The turtle moves backward x turtle steps.

rt x Right x The turtle turns right x degrees.

lt x Left x The turtle turns left x degrees.

home Home The turtle returns to his initial position

in the middle of the screen.

pu Pen up The turtle's pen no longer leaves a track.

fill Fill Fills a closed shape or the entiredrawing window with the current turtle's

color, starting at the current turtle's

position.

ht Hide turtle The turtle becomes invisible.

st Show turtle The turtle becomes visible.

setc Set color The turtle and its track changes color.

The color names are: white, black, grey,

grey2, yellow, orange, red, pink, violet,blue, blue2, green, green2, brown,

brown2, grey3.

print colors A list of available colors is printed.

make-points {A. B} Makes points and shows them on the

drawing window - you can use tool.

sp Show point The points and labels are shown.

hp Hide point Points become invisible.

jt [_, _] Jump to The turtle moves to the point whosecoordinates are given without

leaving a path.



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Making a Square



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Making an Equilateral Triangle



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Meet the Turtle

Click twice on the program titled "Geo-Logo Turtle Paths." After the Geo-Logo title page

appears on the screen, click on "Free Explore" to begin the program. Now you will see a

screen labeled "Untitled (Drawing)" at the center top. This center part is the turtle'sdomain. The left part is the Command Center. The turtle should be in his "Home position"on the center part of the screen. Try moving him around using the following commands:

Shorthand Means Meaningpd Pen down The turtle's pen will now leave a track.

fd x Forward x The turtle moves forward x turtle steps.

bk x Back x The turtle moves backward x turtle steps.rt x Right x The turtle turns right x degrees.

lt x Left x The turtle turns left x degrees.

home Home The turtle returns to his initial position in

the middle of the screen.pu Pen up The turtle's pen no longer leaves a track.

fill Fill Fills a closed shape or the entire drawing window

with the current turtle's color, starting at the currentturtle's position.

ht Hide turtle The turtle becomes invisible.

st Show turtle The turtle becomes visible.setc color Set color The turtle and its track changes color. The color

names are: white, black, grey, grey2, yellow, orange,

red, pink, violet, blue, blue2, green, green2, brown,brown2, grey3

print colors A list of available colors is printed.make-points {A. B} Makes points and shows them on the drawing

window - you can use toolsp Show point The points and labels are shown.

hp Hide point Points become invisible.

jt [_, _] Jump to The turtle moves to the point whose coordinates aregiven without leaving a path.



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Record your procedures and answers to the following questions.

1. Have the turtle draw your first initial. Record the commands you used.

2. Move the turtle around to find the length and width of the turtle's screen in turtle steps.

It is approximately _____ turtle steps wide and _____ turtle steps tall. Erase the

screen.

3. Draw a square that is 50 turtle steps on a side. Record the commands you used.

4. Draw a rectangle that is 60 turtle steps by 40 turtle steps. Record the commands you

used.

5. Draw an equilateral triangle that is 80 turtle steps on a side. Record the commandsyou used.

6. You can use a short cut to writing all the commands out. For example, if I wanted tomake a rectangle 80 turtle steps by 30 turtle steps, I could type in the commands fd 80

rt 90 fd 30 rt 90 fd 80 rt 90 fd 30 rt 90 or I could use the repeat command to repeat the

parts that are done over.

It would look like this: repeat 2[fd 80 rt 90 fd 30 rt 90]

Draw a square that is 100 turtle steps on a side using the REPEAT command.

7. Draw a house using squares, rectangles, and equilateral triangles. Record the

commands.



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Activity: Writing Geo-Logo Procedures


Description: By making a procedure for a 60-step square, participants learn howto use the repeat command and write logo procedures and usevariables, using the Teach Turtle function.

Objectives: Participants will write procedures using the repeat command withvariables and use the Teach Turtle function.

Related SOL: 1.16, 3.18, and 5.14

Vocabulary: See handout of Logo commands. repeat, procedure, and TeachTurtle

Materials: Geo-Logo on one computer per pair of participants or onecomputer per person, overhead viewscreen hooked to computer (ifavailable), Handout 1.9


Directions: 1) Divide the participants into pairs, sharing one computer, orhave one participant per computer. Pass out Handout 1.9.Have participants start Geo-Logo.

2) Introduce the concept of a procedure as teaching theturtle a new vocabulary word. Have students completeHandout 1.9, helping individuals as needed.

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Writing Geo-Logo Procedures

1. You can teach the turtle vocabulary so that you don't have to type in the set ofcommands every time. First, type in the commands you want to teach the turtle in the

command center. This is called writing a procedure. Press the "Teach Turtle" buttonin the upper left part of the screen and name the procedure when asked. The computerwill add the word "end" to the procedure and move it to the Teach screen.

Let's define a procedure for making a square 50 steps on a side.

Type repeat 4[fd 50 rt 90] and press return.Now press the Teach-Turtle button. Type the word Square and press return. What

happened? Now type Square in the command center and press return. Did the turtle

follow your directions?

2. Now we want to make a square that is 60 steps on a side. Go to the Teach section.

What do you have to change in your procedure to make the new square? Go ahead andchange it using the arrow key and the delete key to move around the screen just as in a

word processor. Try out your new procedure.

3. You can have more than one procedure written in the Teach section at any time. Just

be sure that each one ends with end.

4. It can be an awful nuisance to have to keep editing to get different squares. Wouldn't

it be nice if we could tell the turtle how long we wanted a side of the square to be

when we ran the procedure? Well, guess what! We can do just that using variables for

the things we want to change. For example, go to the Teach section and let's edit thatprocedure you have for a 60 step square.

to square

repeat 4[fd 60 rt 90]end

to look like this:

to square: side

repeat 4[fd :side rt 90]end



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The way Logo designates a variable is to put a colon in front of the variable name,

which can be a single letter or a series of letters. I could have called the variable :side,:s, :x or even :George if I wanted to. It usually helps to call it something that will help

you remember what it stands for. When you run the procedure, you type in the nameof the procedure (square in this case) and what number you want used for the variable.The computer will take that value and plug it in wherever it finds the variable. Now

flip back to your graphic page and type square 100. Try square 50. Try some other

values.

If you need more than one variable in your procedure, just type their names in one at a

time separated by a space after the name of the procedure. When you run it you just

type the name of the procedure followed by the values you want used, separated byspaces. The computer will automatically take the first value and plug it into the first

variable, etc.



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Activity: Squares and Rectangles


Description: The participants write a procedure for a square using a variable forthe length of side and a procedure for a rectangle using variablesfor the length and width. After trying to make a square with therectangle procedure and a non-square rectangle with the squareprocedure, participants derive the inclusion relationship betweensquares and rectangles, which requires van Hiele level 3 thinking.

Objectives: Participants will derive the inclusion relationship between squares

and rectangles after writing procedures for each.

Related SOL: 1.16, 3.18, and 5.14

Vocabulary: See handout of Logo commands. inclusion relationship

Materials: Geo-Logo on one computer per pair of participants or onecomputer per person, overhead viewscreen hooked to computer (ifavailable), Handout 1.10


Directions: 1) Divide the participants into pairs, sharing one computer, orhave one participant per computer. Pass out Logo Handout1.10. Have participants start Geo-Logo.

2) Have students complete Handout 1.10, helpingindividuals as needed.

3) Discuss the relationship between squares and rectangles

and ask participants how this lesson could be used toteach this relationship.

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Squares and Rectangles

1. Make a procedure using variables for a square. Make a procedure for a rectangle using

variables. Record them.

2. Now, draw a square 50 turtle steps on a side using the Rectangle Procedure. How did youdo it? Record what you did.

3. Now, draw a rectangle 50 turtle steps by 100 turtle steps using the Square Procedure.

Record what happened.

4. What mathematical principle did you just illustrate?



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Elementary Geometry

Module 2

Topic Activity Name Related

SOL

Transparencies

Handouts

Materials

Tibby 1.20 32 or 60 pieceset of

attribute pieces

Whats In the

Box?

K.11, K.12,

K.13, K.17,

1.20

32 or 60 piece

set of

attribute pieces

Missing Pieces K.17, 1.20 32 or 60 pieceset of

attribute pieces

Whats My Rule? K.17, 1.20 T: 2.1 32 or 60 pieceset of

attribute pieces

Twenty

QuestionsGame

K.17, 1.20 32 or 60 piece

set ofattribute pieces

Who Am I?Game

K.17, 1.20 T: 2.2H: 2.1

32 or 60 pieceset of

attribute pieces

Differences

Trains & Games

K.17, 1.20 T: 2.3

H: 2.2

32 or 60 piece

set ofattribute pieces

Hidden NumberPatterns

K.17, 1.20 T: 2.4H: 2.3

32 or 60 pieceset of

attribute pieces

Classification

Attribute

Networks

K.17, 1.20 T: 2.5

H: 2.4

32 or 60 piece

set of

attribute pieces

Human Circle K.11, K.12,

1.16, 1.17,2.20, 2.22,

3.18, 4.17

Enough pieces of string,

each of the same length(5-10 ft) for each

participant butone; chalk

Geoboard

Triangles andQuadrilaterals

K.11, K.12,

1.16, 1.17,2.20, 2.22,

3.18

T: 2.6

H: 2.5

Identifying

Shapes

Shape Hunt

Virginia Department of Education Page72


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Key Idea: Classification Using Attribute Block Materials

Description: Participants will explore the concept of classifying; a basic process ofmathematical thinking that is essential to many concepts that aredeveloped in the grades K-5 mathematics curriculum. Classificationinvolves the understanding of relationships. Classification activities(observing likenesses and differences) can be presented throughproblem solving situations and provide students with the opportunityto develop logical reasoning abilities. Logical reasoning skills andespecially the meaningful use of the language of logic (if-then, and, or,not, all, some) are valuable across all areas of mathematics. Anunderstanding of classification, or the recognition of the variousattributes of items, is also an essential skill to patterning (extending,exploring, and creating patterns or sequences). These classification

skills can be taught through a variety of materials; attribute blocks willbe the manipulative used for this session.

Attribute

Materials: Attribute materials are sets of objects that lend themselves to beingsorted and classified in different ways. Natural or unstructured attributmaterials include such things as sea shells, leaves, the childrenthemselves, or the set of the children's shoes. The attributes are theways that the materials can be sorted. For example, hair color, height,and gender are attributes of children. Each attribute has a number of

different values: for example, blond, brown, or red (for the attribute ofhair color), tall or short (for height), male or female (for gender).

A structured set of attribute pieces has exactly one piece for everypossible combination of values for each attribute. For example, severalcommercial sets of plastic attribute materials have four attributes: color(red, yellow, blue), shape (circle, triangle, rectangle, square, hexagon),size (big, little), and thickness (thick, thin). In the set just describedthere is exactly one large, red, thin triangle as well as each of all othercombinations. The specific values, number of values, or number of

attributes that a set may have is not important.

The value of using structured attribute materials (instead of unstructuredmaterials) is that the attributes and values are very clearly identified and easilyarticulated to students. There is no confusion or argument concerning what valuea particular piece possesses. In this way we can focus our on the reasoning skillsthat the materials and

Virginia Department of Education Introduction to Classification Page 73


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activities are meant to serve. Even though a nice set of attribute pieces maycontain geometric shapes or different colors and sizes, they are not very goodmaterials for teaching shape, color, or size. A set of attribute shapes does notprovide enough variability in any of the shapes to help students develop anything

but very limited geometric ideas. In fact, simple shapes, primary colors, and twosizes are usually chosen because they are most easily discriminated and identifiedby even the youngest of students.

Related Geometry SOL:KindergartenK.11 The student will identify, describe, and draw two-dimensional (plane) geometric

figures (circle, triangle, square, and rectangle).K.12 The student will describe the location of one object relative to another (above,

below, next to) and identify representations of plane geometric figures (circle,

triangle, square, and rectangle) regardless of their position and orientation inspace.

K.13 The student will compare the size (larger/smaller) and shape of plane geometricfigures (circle, triangle, square, and rectangle).

K.17 The student will sort and classify objects according to similar attributes (size,shape, and color).

K.18 The student will identify, describe, and extend a repeating relationship (pattern)found in common objects, sounds, and movements.

Grade One1.16 The student will draw, describe, and sort plane geometric figures (triangle, square

rectangle, and circle) according to number of sides, corners, and square corners.1.20 The student will sort and classify concrete objects according to one or more

attributes, including color, size, shape, and thicknesses.1.21 The student will recognize, describe, extend, and create a wide variety of patterns,

including rhythmic, color, shape, and numeric. Patterns will include both growinand repeating patterns. Concrete materials and calculators will be used bystudents.

Grade Two2.25 The student will identify, create, and extend a wide variety of patterns, using

numbers, concrete objects, and pictures.

Grade Three3.18 The student will analyze two-dimensional (plane) and three-dimensional (solid)

geometric figures (circle, square, rectangle, triangle, cube, rectangular solid[prism], square pyramid, sphere, cone, and cylinder) and identify relevantproperties, including the number of corners, square corners, edges and the numbeand shape of faces, using concrete models.



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3.24 The student will recognize and describe a variety of patterns formed usingconcrete objects, numbers, tables, and pictures and extend the patterns, using samor different forms (concrete objects, numbers, tables, and pictures).

Grade Four4.21 The student will recognize, create, and extend numerical and geometric patterns,

using concrete materials, number lines, symbols, tables, and words.

Grade Five5.20 The student will analyze the structure of numerical and geometric patterns (how

they change or grow) and express the relationship, using words, tables, graphs, ora mathematical sentence. Concrete materials and calculators will be used.

Note:On the following page you will find a black-line master of the 32-Piece Attribute BlockShapes. Copy the page on red, blue, green and yellow cover stock or construction paperand laminate the pages, if possible. Finally, cut out the shapes and place them in baggiesbefore using them for instruction.


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Activity: Tibby

Format: Large Group

Description: The teacher selects a particular student attribute such as shirt colorand sorts the children accordingly. The participants guess thesorting rule.

Objectives: Participants will use logical reasoning to identify specificattributes used to sort them into groups.

Related SOL: 1.20

Vocabulary: Tibby

Materials: Participants themselves

Time Required:Approximately 5 minutes

Directions: 1) Select an attribute, such as the color of a participant's shirt, hair,or some other attribute. Don't tell the participants what hasbeen selected. Call a participant's name and have him/herstand up and say, "You're a Tibby" if the participant has on the

color of the shirt you're thinking of (or other attribute);otherwise, say "You're not a Tibby". Continue choosingparticipants that are Tibbys and not Tibbys. Have participantstry to guess what makes a participant a Tibby or not a Tibby.

2) Let participants take the lead and select a characteri

Documents

Geometry Elementary