Creating an Investigative Geometry Lesson

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Creating an Investigative Geometry Lesson

• Jim Rahn
• LL Teach, Inc.
• www.llteach.com
• www.jamesrahn.com
• James.rahn_at_verizon.net

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• Recent research on how students learn mathematics
  indicates students should
• Be doing hands-on activities with emphasis on
  exploring relationships and seeing patterns
  rather than on calculating answers
• Working in cooperative groups settings
• to learn to communicate about mathematics,
• to see various approaches to problems and
• to learn to support each other in a learning
  atmosphere
• Be engaged in problem-solving lessons
• Be asked to justify their answers
• Engage with constructive assessments.

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• Geometry lessons that are investigative-based can
  
• Engage students in problems before showing how to
  solve them.
• Engage students in encountering mathematical
  ideas and terminology as needed for their
  investigation or to summarize and communicate
  their findings.
• Engage students in problem solving in every
  lesson.

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Collinear Points
A, B, and C are collinear points
M, N, and H are not collinear points
D, E, F and G are collinear points
I, J, and K are not collinear points
What are collinear points?
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Median of a Triangle
Segments LO and PS are medians of a triangle
Segments ED and JG are not medians of a triangle
What is a median of a triangle?
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• The psychology behind the investigation approach
  is straightforward
• Students engaged in an interesting investigation
  will be motivated to learn ideas, terminology, or
  techniques that help solve their problem.
• Most students dont easily make sense of answers
  to questions they havent asked
• If we tell them how to solve problems theyve
  never tried to solve, they wont appreciate the
  need for, nor the usefulness of, the methods
• Ideas that arise in many different contexts and
  in answer to students own questions can be
  understood more deeply and thus can be applied to
  solve other problems more readily
• Students experience right away the applications
  and problem solving so necessary for retention

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• Investigation-based courses are predicated on a
  belief that our goal is to help students learn to
  solve significant problems, not just problems
  requiring an already-known method. Therefore,
  lessons should be designed so
• Students are given opportunities to solve
  problems without a pre-made algorithm
• Students will be prepared to solve problems that
  will arise in their futures
• Students will think about a problem and look at
  mathematics as a tool that could model the
  situation and lead to a solution
• Students are not just taught algorithms but
  rather thinking, problem solving, and
  communication.

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• The investigative philosophy acknowledges that
  most mathematics problems have a variety of valid
  approaches.
• Acknowledging the validity of a variety of
  approaches helps include students who otherwise
  might feel they cant do math.
• Students who understand that there are multiple
  valid approaches to solving problems score better
  on standardized tests
• Students are not immobilized if they cant
  remember the right method for solving problems

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van Hiele Levels

• Developed by Pierre M. van Hiele and his wife
  Dina van Hiele-Geldof

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• The Van Hiele levels of geometric reasoning are
  sequential. Students must pass through all prior
  levels to arrive at any specific level.
• These levels are not age-dependent in the way
  Piaget described development.
• Geometric experiences have the greatest influence
  on advancement through the levels.
• Instruction and language at a level higher than
  the level of the student may inhibit learning.

John A. Van de Walle, Elementary and Middle
School Mathematics Teaching Developmentally, 4th
ed. (New York Addison Wesley Longman, 2001), pp.
310-11.
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Level 0--Basic Visualization

• The student identifies, names, compares and
  operates on geometric figures (e.g., triangles,
  angles, intersecting or parallel lines) according
  to their appearance as a whole.

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Level 1--Analysis

• The student analyzes figures in terms of their
  components and relationships among components and
  discovers properties/rules of a class of shapes
  empirically (e.g., by folding, measuring, using a
  grid or diagram).

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Level 2--Informal Deduction

• The student logically interrelates previously
  discovered properties/rules by giving or
  following informal arguments.

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Level 3--Deduction

• The student proves theorems deductively and
  establishes interrelationships among networks of
  theorems.

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Level 4--Rigor

• The student establishes theorems in different
  postulational systems and analyzes/compares these
  systems.

Well, if we consider the implications of this
figure drawn in a non-Euclidean plane . . .
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What shape is this?
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Match the response to the van Hiele level

• It is a rectangle because it is a quadrilateral
  with four right angles, the opposites are
  parallel, and consecutive sides are
  perpendicular.
• I can prove its a rectangle if its a
  parallelogram with one right angle.
• I know its a rectangle if its a quadrilateral
  with four right angles.
• It is a rectangle because it looks like a door.
• Well, if I draw that rectangle on a sphere . . .

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Item 15, NAEP test 2003
Alan says that if a figure has four sides, it
must be a rectangle. Gina does not agree. Which
of the following figures shows that Gina is
correct?
A) B) 
C) D) 
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Item 10, NAEP test 2003
In the figure above, WXYZ is a parallelogram.
Which of the following is NOT necessarily true?  
A) Side WX is parallel to side ZY. B) Side XY
is parallel to side WZ. C) The measures of
angles W and Y are equal. D) The lengths of
sides WX and ZY are equal. E) The lengths of
sides WX and XY are equal.
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Investigating with Patty Paper
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Patty Paper Constructions
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Patty Paper Constructions

• Draw a line segment on one sheet of patty paper.
  Discover a way to make a duplicate segment
  without using the marks on a ruler. Describe
  your method.
• Draw any angle on one sheet of patty paper.
  Discover a way to make a duplicate angle without
  using a protractor. Describe your method.

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Patty Paper Constructions

• Use one your angles from the last activity.
  Discover a way to create an angle bisector of the
  angle.
• Describe your method.
• How do you know you have an angle bisector?

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Patty Paper Constructions

• Draw a line segment on a sheet of patty paper.
  Discover a way to create a perpendicular bisector
  of the line segment without using a protractor or
  ruler.
• Describe your method.
• How do you know you have created a perpendicular
  bisector?

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Patty Paper Constructions

• Draw a line on a sheet of patty paper. Select
  and label any point on the line. Discover a way
  to create a perpendicular to the line through the
  indicated point.
• Describe your method.
• How do you know you have created a perpendicular
  line?

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Patty Paper Constructions

• Draw a line on a sheet of patty paper. Select
  and label a point off the line. Discover a way
  to create a perpendicular through the indicated
  points that is perpendicular to the line.
• Describe your method.
• How do your line is perpendicular to the original
  line?

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An Investigation with Patty Paper

• On a sheet of patty paper draw a line segment.
  Label the two endpoints A and B. Construct the
  perpendicular bisector of the segment.
• Select several points on the perpendicular
  bisector and label them.
• Using a ruler compare the distance between each
  of these points and the two endpoints. Label
  these measurements on the patty paper. Describe
  what you find to be true.
• Compare your results with others around you.
  What conclusion can you make about every point
  that is equidistant (the same distance) from the
  endpoints of a line segment?

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Think about this statement

• When is this statement true?
• When is this statement false?

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Creating Points of Concurrency
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• Work in groups
• Each person will draw a different triangle
• scalene acute triangle
• scalene obtuse triangle
• scalene right triangle
• Make three copies of your triangle

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• On one of the triangles fold (or construct) the
  three angle bisectors.

Write a description on the patty paper that
describes the definition of an incenter.
Incenter
Compare your triangle to other triangles. What
do you notice?
What is special about this point of concurrency?
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• Compare the incenter with others in your group.
  What do you notice?

Incenter
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• On the second copy of the triangle fold the three
  perpendicular bisectors of the sides

Compare your triangle to other triangles. What
do you notice?
Circumcenter
Write a description on the patty paper that
describes the definition of an circumcenter.
What is special about this point of concurrency?
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• Compare the circumcenter with others in your
  group. What do you notice?

Circumcenter
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• Place the two triangles on top of each other.
  What do you notice?
• Is the incenter the same as the circumcenter?

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• On a third copy of your triangle construct the
  three medians.

Compare your triangle to other triangles. What
do you notice?
Centroid
Write a description on the patty paper that
describes the definition of an centroid.
What is special about this point of concurrency?
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• Compare the centroid with others in your group.
  What do you notice?

Centroid
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• Compare your three triangles. What do you notice
  about the incenter, circumcenter, and centroid?

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• On the fourth copy of your triangle construct the
  three altitudes.

Compare your triangle to other triangles. What
do you notice?
Orthocenter
Write a description on the patty paper that
describes the definition of an orthocenter.
What is special about this point of concurrency?
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• Compare orthocenter with others in your group.
  What do you notice?

Orthocenter
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• Compare your four triangles. What do you notice
  about the incenter, circumcenter, centroid, and
  orthocenter?

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Making the investigation more dynamic
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Creating an Investigative Geometry Lesson

• Jim Rahn
• LL Teach, Inc.
• www.llteach.com
• www.jamesrahn.com
• James.rahn_at_verizon.net