Sarah Roach
10/26/11
Chapter 20: Geometric Thinking and Geometric Concepts
Pedagogy Leader
This chapter looks at the different types of symmetry
Geometry Goals for Students
 Spatial sense and geometric reasoning
o How students think and reason about shape and space.
 The specific geometric content found in your state or district objectives.
o Knowing about symmetry, triangles, parallel lines, etc.
Spatial Sense and Geometric Reasoning
 Spatial sense is an intuition about shapes and the relationships among shapes.
o Spatial sense includes the ability to mentally visualize objects and spatial
relationships-to turn things around in your mind.
Geometric Content
 Four content goals for geometry:
o Shapes and Properties- a study of the properties of shapes in both two and
three dimensions and the study of the relationships built on properties.
o Transformation- a study of translations, reflections, rotations (slides, flips,
and turns), the study of symmetries, and the concept of similarity.
o Location- refers to coordinate geometry or other ways of specifying how
objects are located in the plane or in space.
o Visualization- the recognition of shapes in the environment, developing
relationships between two and three-dimensional objects and the ability to
draw and recognize objects from different perspectives.
The Development of Geometric Thinking
 The van Hiele theory is the greatest influence in the American geometry
curriculum.
The van Hiele Levels of Geometric Thought
 Five-level hierarchy ways of understanding spatial ideas.
 The five levels describe the thinking processes used in geometric contexts.
o This includes how we think and what types of geometric ideas we think
about, instead of how much knowledge we have.
 The five levels are:
o Level 0: Visualization- shapes and what they look like
 Students can name and recognize the different shapes by the
shapes appearances.
 The goal is for students to see how shapes are alike and different.
 An activity that can be used to demonstrate this level is give to
give the students multiple different shapes and let them list the
similarities and differences.
o Level 1: Analysis- classes of shapes rather than an individual shape
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
Students are able to consider all shapes with in a class rather than a
single shape on their desk.
 Example: Students can figure out what makes a rectanglefour sides, opposite sides parallel , opposite sides same
length, four right angles, congruent diagonals, etc.
 Students get the thought of properties of shapes in their head such
as symmetry, perpendicular and parallel lines.
 An activity that can be used with students is to hand out different
groups of shapes but let the shapes in a group be the same color
and then let the students list the different properties of each shape.
o Level 2: Informal Deduction- the properties of shapes
 During this level students will start using logical reasoning.
 As a teacher you could allow students to work in groups and have
them form “if-then” statements about the properties of shapes.
o Level 3: Deduction- relationships between properties of geometric objects
 Students are able to examine more than just the properties of
shapes.
 During this level the students learn the “true” definition of the
geometric shapes.
 Students start thinking about deductive axiomatic systems for
geometry.
o Level 4: Rigor- deductive axiomatic systems for geometry
 Usually college students majoring in math.
 Characteristics of the van Hiele Levels:
o The levels are sequential.
o The products of thought at each level are the same as the objects of
thought at the next.
o The levels are not age dependent.
o Geometric experiences are the most influencing advancement through the
levels. Students should explore, talk about, and interact with content at the
next level while increasing experiences at their current level.
o When language is at a level higher than that of a student there will be a
lack of communication.
Implications for Instruction
 Teachers should know that the experiences they provide are the most important to
moving the students up the development chart.
 There should be growth in the students’ geometric thinking over the year.
 There are 3 instruction levels that teachers should follow to guide your interaction
with students.
o Instruction at Level 0
 Involve a lot of sorting and classifying.
 Allow the students to draw, build, make, compose, and decompose
shapes in both two and three dimensions.
 In order for students to move to the next level test their
knowledge.
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o Instruction at Level 1
 Focus on properties of figures instead of simple identification.
 Apply ideas to entire classes of figures instead of individual
models.
 Before students move on ask them questions as to “why”
that is so.
o Instruction at Level 2
 Encourage the making and testing of hypotheses or conjectures.
 Examine properties of shapes to determine conditions for different
shapes or concepts.
 Use the language of informal deduction (all, some, none, if…then,
what if).
 Encourage students to attempt informal proofs.
 Task Selection and Levels of Thought:
o Examine the descriptors for the first two levels while they perform
activities.
Learning about Shapes and Properties
 Children work with two and three-dimensional shapes and find out what makes
the shapes alike and different then they discover the properties of shapes.
Shapes and Properties for Level-0 Thinkers
 Triangles should be introduced with the vertex shown other than just the top and
not just show equilateral triangles.
 Use shapes with curved sides, straight sides, and a combination of these.
 Sorting and Classifying:
o Children classify shapes as “dented” and “looks like a tree” or “points up”.
o An activity that can be used for classifying shapes is to allow students to
try and guess a secret shape that is in a folder. They will ask questions so
that the person with the folder can only reply with a yes or no answer.
 Composing and Decomposing Shapes:
o Students should get to explore how shapes fit together to form larger
shapes and how larger shapes can be made of smaller shapes.
 A good way to demonstrate this is by using tangrams, mosaic
puzzles, geoboards, and pattern blocks.
 Have the students draw these patterns in a book so they can
keep a recording of what they come up with.
o Students should realize that triangles are used with many building of
bridges.
 Skeletal models can be a helpful demonstration.
 Tessellations are a tilting of the plane using one or more shapes in a repeated
pattern with no gaps or overlaps.
o This is very helpful in having the students figure out what forms can go
together.
Shapes and Properties for Level-1 Thinkers
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 During this level the students learn the proper names of the shapes and their
properties.
 The definition of two and three dimension is introduced.
 Shape definitions include relationships between and among shapes.
 Special Categories of Two-Dimensional Shapes:
o Simple Closed Curves
 Concave, Convex
 Symmetrical, Nonsymmetrical
 Polygons
 Concave, Convex, Symmetrical, Nonsymmetrical, and
Regular
o Triangles
 Sides- Equilateral, Isosceles, Scalene
 Angles- Acute, Right, Obtuse
o Convex Quadrilaterals
 Kite
 Trapezoids
 Isosceles Trapezoid
 Parallelograms
 Rhombi, Rectangles, Squares
 Special Categories of Three-Dimensional Shapes:
o Sorted by Edges and Vertices
 Spheres and “egglike” shapes
o Sorted by Faces and Surfaces
 Polyhedrons
o Cylinders
 Right Cylinder, Prism, Rectangular Prism, and Cube
o Cones
 Circular Cone, Pyramid
 Sorting and Classifying Activities:
o “Triangle Sort”- have students create a chart. On the left have the three
angles and on top have the three sides and have the students draw a
triangle for each.
o “Property Lists for Quadrilaterals”- students create properties of
parallelograms, rhombi, rectangles, and squares. During this activity is a
great time to introduce new vocabulary and symbols.
 Construction Activities:
o Quadrilaterals can be described in terms of its diagonals using the
conditions of length, ratio of parts, and whether it is perpendicular or not.
 Circles:
o Important for the ratio between measures of the circumference and the
diameter.
o Students also get introduced to pi.
 Dynamic Geometry Software:
o Helpful when drawing points, lines, and geometric figures.
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o Distances, lengths, areas, angles, slopes, and perimeters can be measured
by moving the shape around.
o Lines can be drawn perpendicular or parallel to other lines or segments.
o Angles and segments can be drawn congruent to other angles and
segments.
o A point can be placed at the midpoint of a segment.
o A figure can be reflected, rotated, or dilated.
o The object of this software is to prove that the shape stays the same no
matter how many times you move it.
Shapes and Properties for Level-2 Thinkers
 The focus is now on explorations that include logical reasoning.
 Teachers should encourage conjecture and exploration of informal deductive
arguments.
 Students should follow simple proofs and explore ideas that connect directly to
algebra.
 Definitions and Proofs:
o Minimal Defining Lists (MDL)
o MDL is more involved with logical thinking than examining shapes.
o MDL could be a definition of the shapes.
o Conjecture is a statement whose truth has not yet been determined.
 The Pythagorean Relationship:
o This relationship states that if a square is constructed on each side of a
right triangle, the areas of the two smaller squares will together equal the
area of the square on the longest side, the hypotenuse.
 Finding Versus Explaining Relationships:
o The focus is on reasoning or deductive thinking.
Learning about Transformations
 Transformations are changes in position or size of a shape.
 Movements that do not change the size or shape of the object moved are rigid
motions.
 Three type of rigid motions are:
o Translations (slides)
o Reflections (flips)
o Rotation (turns)
Transformations for Level-0 Thinkers
 Basic concepts of slides, flips, and turns and the development of line symmetry
and rotational symmetry.
 Slides, Flips, and Turns:
o The goal for this level is to help students recognize these transformations
and to begin to explore their effects on simple shapes.
o To introduce this, teachers can start with nonsymmetrical shapes.
o Reflections are flips over vertical or horizontal lines.
 Line and Rotational Symmetry:
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o Line Symmetry or Mirror Symmetry is when a shape can be folded on a
line so that the two halves match.
o The fold of the line is the line of reflection. The portion of the shape on
one side of the line is reflected onto the other side. This shows the
connection between line symmetry and transformations.
 To demonstrate line symmetry you can use pattern blocks on one
side of a paper that is separated in half.
o A plane of symmetry in three dimensions is analogous to a line of
symmetry in two dimensions.
o Rotational symmetry or point symmetry is when a shape can be rotated
about a point and land in a position exactly matching the one in which it
began.
 A way to test rotational symmetry is to trace the shapes “footprint”
on a piece of paper. The order of rotational symmetry will be the
number of ways that the shape can fit into its footprint without
flipping it over.
Transformations for Level-1 Thinkers
 Students begin to analyze transformations a bit more analytically and to apply
them to shapes that they see.
 Two types of activities used for this are compositions of transformations and
using transformations to create tessellations.
 Composition of Transformations:
o A figure can be reflected over a line, and then that figure can be rotated
about a point.
o Composition is a combination of two or more transformations.
 Similar Figures and Proportional Reasoning:
o Two figures are similar if all of their corresponding angles are congruent
and the corresponding sides are proportional.
o A dilation is a nonrigid transformation that produces similar figures.
o Figures that are dilated are larger and smaller figures.
 Tessellations Revisited:
o M.C. Escher is known for his tessellations.
o A regular tessellation is made of a single tile that is a regular polygon.
o A semiregular tessellation is made of two or more tiles. Each are both
regular polygons.
o A vertex can be described by the series of shapes meeting at a vertex.
Transformations for Level-2 Thinkers
 Pentominoes are shapes made from 5 squares each touching at least one other
square by sharing a full side.
Learning about Location
 Location activities involve analysis of paths from point to point as on a map and
the use of coordinate systems.
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Location for Level-0 Thinkers
 In pre-k and kindergarten the students learn everyday positional descriptions-over,
under, near, far, between, left, and right.
Location for Level-1 Thinkers
 Use of the coordinate grid is to examine transformations in a more analytic
manner.
 There are four quadrants; this is when we introduce the negative coordinates.
 When coordinates are multiplied the shape stays the same but the size changes.
 Dilation is a transformation that is not rigid because the shape changes.
Location for Level-2 Thinkers
 This is when students use logical reasoning.
 Coordinate Transformations Revisited:
o Students could be asked the following questions:
 How should the coordinates be changed to cause a reflection if the
line of reflection in not the y-axis but is parallel to it?
 Can you discover a single rule for coordinates that would cause a
reflection across one of the axes followed by a rotation of a quarter
turn? Is that rule the same for the reverse order-a quarter turn
followed by a reflection?
 If two successive slides are made with coordinates and you know
what numbers were added or subtracted, what number should be
added or subtracted to get the figure there in only one move?
 What do you think will happen if, in a dilation, different factors are
used for different coordinates?
 Applying the Pythagorean Relationship:
o The geometric version is about area.
o The students should be able to follow the rationale if shown proofs.
o Teachers should lead students through the procedure of finding the length
of one line you give them information to compute the lengths of other
lines.
o Students will see that all they need are the coordinates of the two
endpoints to compute the areas of all three squares and the length of the
hypotenuse.
 Slope:
o Show students what a slope is by drawing different slanted lines.
o The “steepness” of a line is an attribute that can be measured like other
measurable attributes.
o The coordinate grid provides a reference (x-axis) and the numbers to use
in the measurement.
o The convention for measuring the steepness of a line or the slope is based
on the ideas of the rise and run between any two points on the line.
o The rise is the vertical change from the left point to the right point.
Positive if up and negative if down.
o The run is the horizontal distance from the left point to the right point.
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o Slope=rise/run
o Vertical lines have no slope. Therefore, it is called undefined.
o Horizontal lines have a slope of 0 as a result of the definition.
Learning about Visualizations
 Visualization= geometry done with the mind’s eye.
 Students should be able to create mental images of shape and then turn them
around mentally.
 Visualization is anything that requires students to think about a shape mentally, to
manipulate or transform a shape mentally, or to represent a shape as it is seen will
contribute to the development of the students’ visualizations skills.
Visualization for Level-0 Thinkers
 Students are bound to thinking about shapes in terms of the way they look.
 Finding out how many different shapes can be made with a given number of
simple tiles demands that students mentally flip and turn shapes in their minds
and find ways to decide if they have found them all.
 Students should also be able to think about solid shapes in terms of their faces or
sides.
Visualization for Level-1 Thinkers
 Students should know the degree of attention that must be given to the particular
properties of shapes.
 Students should also be able to identify and draw two-dimensional figures to build
three-dimensional figures from two-dimensional images.
Visualization for Level-2 Thinkers
 Connecting Earlier Activities to Level-2 Visualization:
o Students can make predictions about the types of slices that are possible.
o An appropriate visualization activity using pentominoes are:
 How many hexominoes are there? A hexomino is made of six
squares following the same rule as for pentominoes. Since there
are 35 hexominoes, devising a good logical scheme for
categorizing the shapes is one of the few ways there are of
knowing they have all been found.
 Instead of putting together five squares, students can find all of the
arrangements of five cubes. These shapes are called pentominoids.
Shapes made of cubes in which adjoining cubes share a complete
face are called polyminoids.
 The Platonic Solids:
o A polyhedron is a three-dimensional shape with polygons for all faces.
o Platonic solids is the name given to the set of completely regular (each
face is a regular polygon and every vertex has exactly the same number of
faces joining at that point) polyhedrons.
o A tetrahedron is a four-sided solid.
o An octahedron is an eight-sided solid.
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o An icosahedron will have 20 sides.
o There is only one solid made of squares; a hexahedron has three at each
point and 6 all together. (cube)
o A dodecahedron has three at each point and 12 in all. It is the only solid
with pentagons.
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