Three Little Pigs, Three Blind Mice, and Tangrams: Exploring Geometric Relationships

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Three Little Pigs, Three Blind Mice, and
Tangrams Exploring Geometric Relationships

• David S. Allen, Ed.D.
• Assistant Professor, KSU
• Melisa J. Hancock,
• Teacher in Residence, KSU
• Emily R. Finney
• Teacher, Somewhere in CA

2004 NCTM Annual ConventionPhiladelphia, PA
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National Standards

• National Standard Geometry
• Analyze characteristics and properties of two-
  and three-dimensional geometric shapes and
  develop mathematical arguments about geometric
  relationships.
• Specify locations and describe spatial
  relationships using coordinate geometry and other
  representational systems
• Apply transformations and use symmetry to analyze
  mathematical situations
• Use visualization, spatial reasoning, and
  geometric modeling to solve problems.

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Pierre van Hiele

• Level 0 Visualization
• The objects of thought at level 0 are shapes
  and what they look like.
• Students recognize and name figures based on the
  global, visual characteristics of
  the figure.
• Children at this level are able to make
  measurements and even talk about properties of
  shapes, but these properties are not abstracted
  from the shapes at hand.
• It is the appearance of the shape that defines it
  for the student.
• A square is a square because it looks like a
  square.
• The products of thought at level 0 are
  classes or groupings of shapes that seem to be
  alike.

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Pierre van Hiele

• Level 1 Analysis
• The objects of thought at level 1 are classes
  of shapes rather than individual shapes.
• Students at this level are able to consider all
  shapes within a class rather than a single shape.
  
• At this level, students begin to appreciate that
  a collection of shapes goes together because of
  properties.
• Students operating at level 1 may be able to list
  all the properties of squares, rectangles, and
  parallelograms but not see that these are
  subclasses of one another, that all squares are
  rectangles and all rectangles are parallelograms.
  
• The products of thought at level 1 are the
  properties of shapes.

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Pierre van Hiele

• Level 2 Informal Deduction
• The objects of thought at level 2 are the
  properties of shapes.
• As students begin to be able to think about
  properties of geometric objects without the
  constraints of a particular object, they are able
  to develop relationships between and among these
  properties.
• If all four angles are right angles, the shape
  must be a rectangle. If it is a square, all
  angles are right angles. If it is a square, it
  must be a rectangle.
• With greater ability to engage in if-then
  reasoning, shapes can be classified using only
  minimum characteristics.
• Four congruent sides and one right angle can
  define a square.
• The products of thought at level 2 are
  relationships among properties geometric objects.

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Pierre van Hiele

• Level 3 Deduction
• The objects of thought at level 3 are
  relationships among properties of geometric
  objects.
• Earlier thinking has produced in students
  conjectures concerning relationships among
  properties. Are these conjectures correct? Are
  they true?
• As this analysis takes place, a system complete
  with axioms, definitions, theorems, corollaries,
  and postulates begins to develop and can be
  appreciated as the necessary means of
  establishing truth.
• At this level, students begin to appreciate the
  need for a system of logic that rests on a
  minimum set of assumptions and from which other
  truths can be derived.
• This is the level of the traditional high school
  geometry course.
• The products of thought at level 3 are deductive
  axiomatic systems for geometry.

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Levels of Application
Van de Walle, J. A. (2003). Elementary and Middle
School Mathematics Teaching Developmentally
(Fifth Edition). New York, NY. Addison Wesley
Longman.
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Tangrams

• The Tangram Lesson
• Available on Website (Written for fourth grade)
• Uses the 5E-Inquiry Model
• Applicable to all grades (you must determine
  where your students are and what types of
  experiences they are ready for)
• 2. Goals and Objectives
• Given a set of tan pieces TLW identify linear
  relationships between the pieces using correct
  geometric vocabulary with no errors.
• Given a set of tangram puzzles TLW identify the
  correct transformations acted upon each tan piece
  as the pieces are shifted during the puzzle
  completion with no errors.
• Given a sheet of graph paper with a shape drawn
  upon it TLW draw the shape in a second location
  after the shape has been acted upon by two
  geometric transformations.

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Task 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

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Triangle
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Rectangle
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Trapezoid
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Parallelogram
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Square
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Explanation Task 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
• The legs of both triangles are congruent.
• The hypotenuse of both triangles are congruent.
• The angles of both triangles are congruent.

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Explanation Task 1

• 2. Transformations
• Flips or Reflections
• Slides or Translations
• Turns or Rotations
• Triads
• Three Little Pigs
• The Trinity
• The Three Tangram Pieces

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Task 2

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

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Rectangle
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Trapazoid
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Paralellogram
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Triangle
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Square
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Explanation Task 2

• Linear Relationships
• The leg of the small triangle is congruent to the
  side of the square.
• The leg of the small triangle is congruent to the
  small side of the parallelogram.
• Therefore the side of the square is congruent to
  the small side of the parallelogram.
• The hypotenuse of the small triangle is congruent
  to the long side of the parallelogram.
• The leg of the medium triangle is congruent to
  the long side of the parallelogram.

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Task 3

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

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Rectangle
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Parallelogram
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Trapezoid
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Triangle
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Square
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Connecting the Tasks

• Working with Three Small Pieces
• Identifying Linear Relationships
• Examining Transformations
• Working with Five Small Pieces
• Application of Linear Relationship Identification
• Strengthening Language Descriptions of
  Transformations
• Working with Seven Pieces
• Similar Task to Three Small Pieces
• Introduce concept of Ratio and Proportion
• 4. Examining Area is Another Lesson

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Levels of Application
Van de Walle, J. A. (2003). Elementary and Middle
School Mathematics Teaching Developmentally
(Fifth Edition). New York, NY. Addison Wesley
Longman.
33
Three Little Pigs, Three Blind Mice, and
Tangrams Exploring Geometric Relationships

• David S. Allen, Ed.D.
• Assistant Professor, KSU
• Melisa J. Hancock,
• Teacher in Residence, KSU
• Emily Finney
• Teacher, Somewhere in CA

2004 NCTM Annual ConventionPhiladelphia, PA
34
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