Chapter Fourteen

Learning Mathematics In Elementary and Middle Schools, 5e Cathcart, Pothier, Vance, and Bezuk

©2011 Pearson Education, Inc. All Rights Reserved

Chapter Fourteen

Developing Geometric Thinking and Spatial Sense

DodecagonNonagonPentagon


©2011 Pearson Education, Inc. All Rights Reserved 14-2

The van Hiele Levels of Geometric Thought

Level Description

0 – Visualization Children recognize shapes by their global, holistic appearance.

1 – Analysis Children observe the component parts of figures but are unable to explain the relationships between properties within a shape or among shapes.

2 – Informal Deduction Children deduce properties of figures and express interrelationships both within and between figures.

3 – Formal Deduction Children create formal deductive proofs.

4 – Rigor Children rigorously compare different axiomatic systems.



Connecting van Hiele Levels to Elementary School Children Most children at the elementary level are at the

visualization or analysis level of thought. Some middle school children are at the informal

deduction. Students who successfully complete a typical

high school geometry course reach the formal deduction level.

The goal is to have children at the informal deduction level or above by the end of middle school.



Comments on the Levels of Thought Levels are not age dependent, but are related to

the experiences a child has had. Levels are sequential. Experience is key in helping children move from

one level to the next. For learning to take place, language must match

the child’s level of understanding.

It is difficult for two people who are at different levels to communicate effectively.



Learning about Topology Topology is the study of the properties of figures

that stay the same even under distortions, except tearing or cutting. Place and Order – Describing where something is located

in the environment or in pictures. Focus on the following types of words

Is the pillow Inside or Outside the box?

http://uk.ixl.com/math/year-1

Where is the picture? Above, Below, Under, Between, Behind or After



Learning about Topology Topology is the study of the properties of figures that stay the same even under distortions, except tearing or

cutting.

Maze

http://www.newton.ac.uk/

Network – decision points and paths

One route to the centre is A -> B -> D -> K -> I -> M.



Learning about Topology Topology is the study of the properties of figures

that stay the same even under distortions, except tearing or cutting. Distortion of Figures

http://britton.disted.camosun.bc.ca/mug_torus_morph.gif



Learning about Euclidean Geometry

Three-Dimensional Shapes Polyhedra – three-dimensional shapes with faces

consisting of polygons, that is, plane figures with three, four, five, or more straight sides.Edge Vertices Face

curriculumsupport.education.nsw.gov.au



Learning about Euclidean Geometry Three-Dimensional

Shapes Regular polyhedra – a

regular polyhedron is one whose faces consist of the same kind of regular congruent polygons with the same number of edges meeting at each vertex of the figure.



Three-Dimensional Shapes

Shape Type and Number of Faces

Tetrahedron 4 equilateral triangles

Octahedron 8 equilateral triangles

Icosahedron 20 equilateral triangles

Hexahedron 6 squares

Dodecahedron 12 regular pentagons

oThere are five regular polyhedra



Three-Dimensional Shapes Semiregular polyhedraTruncated and stellated polyhedra

Semi-regular Polyhedron of 62 Faces.

http://www.literka.addr.com/hexsqr14.htm

stellated polyhedra



Other Three-Dimensional Shapes Discovering Euler’s Rule: Examining relationships

between faces, vertices, and edges

What relationship do you notice among the shapes?

Polyhedron Faces Vertices Edges

Tetrahedron 4 4 6

Cube 6 8 12

Octahedron 8 6 12

wikipedia.org



Other Three-Dimensional Shapes

Discovering Euler’s Rule: Examining relationships between faces, vertices, and edges

What relationship do you notice among the shapes?

Polyhedron Faces Vertices Edges

Triangular Prism 5 6 9

Square Pyramid 5 5 8



Learning about Two-Dimensional Figures

Polygons – two-dimensional figures with straight line segments

Convex – interior angles are all less than 180°; any two points in a figure can be connected by a line segment that will be completely within the figure and all diagonals will remain inside the figure.



Learning about Two-Dimensional Figures

Polygons – two-dimensional figures with straight line segments Concave – a geometric shape is concave if it has

any line segment that joins two interior points outside the figure.

wikipedia.org



Convex Polygons

Number of Sides Name of Polygon

3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon



Convex Polygons

Number of Sides Name of Polygon

8 Octagon

9 Nonagon

10 Decagon

11 Hendecagon

12 Dodecagon



TrianglesTriangles can be classified by angles and sides

SidesEquilateral – all sides equal

Isosceles – two sides equal

Scalene – no sides equal

http://www.mathsisfun.com/triangle.html



TrianglesTriangles can be classified by angles and sides

Angles Right – one angle is equal to 90°

Acute - all angles are less than 90°

Obtuse – one angle is greater than 90°

http://http://www.mathsisfun.com/triangle.html



Quadrilaterals

Quadrilateral Definition

Parallelogram A quadrilateral with two pairs of parallel sides

Rectangle A parallelogram with 90-degree angles

Square A rectangle with equal sides



Quadrilaterals Quadrilateral Definition

Trapezoid A quadrilateral with at least one pair of parallel sides

Isosceles Trapezoid A trapezoid with two nonparallel sides

Kite A quadrilateral with two nonparallel sides equal

Rhombus A parallelogram with all sides equalTrapezoid



Learning about Symmetry Symmetry – when a figure

is bisected into two congruent parts, every point on one side of the bisection line will have a reflective point on the other side of the bisection line.

http://britton.disted.camosun.bc.ca/sun.jpg



Learning about Symmetry Plane symmetry – a three-dimensional shape has

plane symmetry if a plane passing through the figure bisects it such that every point of the figure on one side of the plane has a reflection image on the other side of the plane.

Magic-squares.net Feko.info



Learning about Symmetry Rotational symmetry – when a figure is rotated

about a point for an amount less than 360°, and the rotated shape matches the original shape.

math.kendallhunt.com mathexpression.com



Rotational Symmetry of a Square

Some three- and two-dimensional shapes and figures have rotational symmetry.



Transformation Geometry Translation – a movement along a straight line

Slides Flips

Turns

learningideasgradesk-8.blogspot.com



Transformation Geometry Reflections – the movement of a figure about a line

outside the figure, on a side of the figure, or intersecting with a vertex

Rotation – the movement of a figure around a point.

intmath.comart.unt.edu

mathsisfun.com



Developing Spatial Sense

Spatial sense involves both visualization and orientation factors Spatial visualization – the ability to mentally

picture how objects appear under some rigid motion or other transformation

Orientation – the ability to note positions of objects under different orientations.



1. Using the three small pieces (two small triangles and the medium size triangle) create these five basic geometric shapes.• Square• Trapezoid• Parallelogram• Rectangle• Triangle

Task 1



Triangle



Rectangle



Trapezoid



Parallelogram



Square



1. Linear Relationships• The hypotenuse of the small triangle is congruent

to the leg of the medium size triangle.• The hypotenuse of the medium sized triangle is

congruent to twice the length of the leg of the small triangle.

• The two small triangles are congruent because:1. The legs of both triangles are congruent.2. The hypotenuse of both triangles are congruent.3. The angles of both triangles are congruent.

Explanation Task 1



1. Using the five small pieces (two small triangles, medium size triangle, rhombus, parallelogram) create these five basic geometric shapes.• Square• Trapezoid• Parallelogram• Rectangle• Triangle

Task 2



Rectangle



Trapazoid



Paralellogram



Triangle



Square



1. Using all seven tan pieces create these five basic geometric shapes.• Square• Trapezoid• Parallelogram• Rectangle• Triangle

Task 3



Rectangle



Parallelogram



Trapezoid



Triangle



Square



1. Working with Three Small Pieces• Identifying Linear Relationships• Examining Transformations

2. Working with Five Small Pieces• Application of Linear Relationship Identification• Strengthening Language Descriptions of

Transformations3. Working with Seven Pieces

• Similar Task to Three Small Pieces• Introduce concept of Ratio and Proportion

4. Examining Area is Another Lesson

Connecting the Tasks

Chapter Fourteen

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