Building Geometric Thinking with Hands-On Tasks & in Virtual Environments

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Building Geometric Thinking with Hands-On Tasks
in Virtual Environments

• Jean J. McGehee
• jeanm_at_uca.edu
• University of Central Arkansas

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Today

• Geometric Habits of Mindfrom Paper Folding to
  Using Sketchpadin the context of rich problems
• Transformations a Connecting big idea
• The role of good definitions Quadrilaterals
• Connecting Sketchpad to the Number and Algebra
  strands

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Goals of FGT

• Strengthen understanding of geometry
• Enhance capacity to recognize and describe
  geometric thinking
• Increase attention to students thinking
• Enhance understanding of students geometric
  thinking
• Prepare to advance students geometric thinking

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The Geometry Curriculum in Arkansas

• Lets take a quick look at the frameworks..\..\..\
  Desktop\frameworks\geometry_06.doc
• Even my student interns at UCA notice how much
  repetition there is in the curriculume.g. The
  Triangle Sum
• We do need to revisit ideasbut we need to do it
  with value added.

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The FGT Project in NE Arkansas

• Two school districts in grades 5-11
• I wanted the teachers at all grade levels to
  share their strengths and understand the
  curriculum vertically.
• I wanted them to share their students work and
  their ideas so that they gained an appreciation
  for each other.

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All levels benefited.

• The 5th and 6th grade teachers enjoyed the hands
  on activities, and these very same activities are
  useful even after the high school geometry
  course.
• With Algebra II and beyond we really downplay
  geometryyet these kids have to take the ACT or
  SAT. With these problems they will also have to
  explain their thinking.

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Its hard to get teachers to focus on content for
fun when they deal with state tests.

• Cathy (6th grade) and Cindy (Geometry) are
  teachers who trust rich problems, inquiry
  investigations, and projects BEFORE the State
  Tests as a means to prepare for criteria
  tests---Their scores show it!!

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Another teacher from Arkansas reported

• One of my 6th grade FGT participants told me that
  her students' scores on the Arkansas assessment
  increased from 46 proficient and advanced in
  2005 to 68 in 2006. That was great news, but it
  gets better.
• She credits the increase to their better
  understanding of geometry and measurement than in
  years past. She says that is a direct result of
  the problems we did (and she did with her
  students) in FGT. In fact, after the testing her
  students told her that the problems on the test
  were
• like what they had done in class "except they
  didn't give us any paper to fold".

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Structure of FGT

• The Structured Exploration Process guides the
  activities in each part of FGT sessions. There
  is a cycle of doing math and exploring student
  thinking.
• The Geometry Habits of Mind framework provides a
  lens to analyze geometric thinking.

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Three content strands

• Focus the work on different important areas of
  geometry measurement.
• They are
• Properties
• Transformations
• Measurement

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The Structured Exploration Process

• Stage 1 Doing mathematics
• Stage 2 Reflecting on the mathematics
• Stage 3 Collecting student work
• Stage 4 Analyzing student work
• Stage 5 Reflecting on students thinking

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FGT G-HOMs

• Reasoning with Relationships
• Generalizing Geometric Ideas
• Looking for Invariants
• Balancing Exploration Reflection

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More about FGT and G-HOMs later

• First, lets do exercise our own geometric
  thinking
• Folding, Making Squares, Congruent Halves
• Paper-Folding Constructions
• Tangrams
• Dissecting Shapes
• Comparing Triangles
• We will start in detailbut I may have to
  summarize the latter problems.

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Do Math--Ideally

• Work problem individually 5-10 minutes
• Work problem in groups 25 minutes
• Last 10 minutes groups prepare report either on
  transparency or chart paper
• Reflect on the problem Identify G-HOMs 25-30
  minutes

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Lets do more with paper folding-Start at b

• Construct a triangle with exactly ¼ the area of
  the original square. Explain how you know it has
  ¼ the area
• Construct another triangle that also has ¼ the
  area, which is not congruent to the first one you
  constructed. Explain how you know is has ¼ the
  area
• Construct a square with exactly /12 the area of
  the original square. Explain how you know it has
  ½ the area
• Construct another square, also with ½ the area
  which is oriented differently than the one you
  constructed in (d). Explain how you know it has
  ½ the area

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Review Investigating Area by Folding

• Some comments on the challenge problem.
• Recall that it was relatively easy to find a
  square that is ¼ of the original.
• We all found one square that is ½ of the
  original.
• I want to show you a quilters approach
• Also I want to show you this problem in a fun
  book.

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A Quilters Solutiondoes it work?
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The Number Devil

• This little devil beguilesRobert into dreams
  togive him a glimpse of the beauty
  powernumbers.
• In this case, the squareroot of 2.

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Student intern gave students two squares and asked
How many black squares fit into the red
square? Show how you knew this.
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Hands-On Sketchpad

• I have learned both in PD and classes to start
  with Hands-On
• Making a gallery of chart paper reports and
  walking through the gallery is a wonderful way to
  summarize the problem.
• Sketchpad provides a way to solidify conjectures
  and make a bridge to proof.

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Basic Paper Folding

• The perpendicular bisector is the most basic
  fold. Who can describe this for me? How do you
  know?
• Construct a line that is parallel to your
  original segment. Describe your method. How do
  you know your new line is a parallel line to the
  original segment?
• Now start with a fresh segment each time and
  constructan isosceles trianglean equilateral
  trianglea square

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Analyzing Student Work

• What are the important mathematical ideas in the
  problem?
• What strategies do you want to foster and why?
• What is the evidence that a student used a
  strategy? Is it related to a G-HOM?

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Student work on Paper Folding

• What do you think students typically do?
• How do you think students use geometric language?
• Go Back to the Demand of the Task.
• Are we actually requiring students to write and
  speak the language of geometry?
• Or do we practice Multiple Choice items and work
  problems in which the language task is low?

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Paper Folding related toTangrams

• You are familiar with the square, but can you
  make a rectangle that are not squares2 ways?

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Lets explore the area problems

• Tangrams on Sketchpad
• More shapes with the same areaan understanding
  based on properties rather than memorized
  formulas.

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Dissecting Shapes

• The ability to dissect and transform shapes is
  important.
• Students are also exploring invariance and
  properties.

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Dissecting Shapes--Conclusions
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Comparing Triangles
Start with a piece of paper (you can also use
different size rectangular paper). Fold your
paper so that point A is directly on top of point
C. Some triangles appear. In the picture
belowyou should see 3 triangles.
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Comparing Triangles
Start with another piece of paper. This time
fold A onto any point between D and C. Again
there are 3 triangles which are all right
triangles. What else do you notice about the
triangles?
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In your report, consider

• Describe your construction method in pictures and
  words.
• Before you tried your method, why did you think
  it would work?
• Were there methods you tried that didnt work?
  What were they?
• What are the properties of the constructed
  shapes? How do you know your shape has these
  properties.

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Sorting by Symmetry more Advanced Properties
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Transformations
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Miraa transition to the Computer

• Rotation
• Translation
• Finding Centers of Rotation

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Coordinate work

• Wumps

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Dilations
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The Role of Definitions

• To me it appears a radically vicious method,
  certainly in geometry, . . .to supply a child
  with ready made definitions, to be memorized
  after being more or less carefully explained.. .
  .The evolving of a workable definition by the
  childs own activity stimulated by appropriate
  questions, is both interesting and highly
  educational.
• Bechara, Blandford, 1908

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Development of Definitions

• Descriptive Defining
• Constructive Defining
• Hierarchical vs. Partition Defining
• The Role of Construction Measurement

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Quadrilateral Activities

• Geometric Structures If we had time, we would
  go through these activitiesyou may think they
  are repetitive, but students need all of these
  experiences to deal more flexibly with properties
  and definitions.
• Lets do Sketchpad activity from Restructuring
  Proof think about this activity from High and
  Low levels.

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Geometric Thinking Task Demand Categories

• Memorization
• What is the formula for the area of a
  triangle?State the SAS congruence postulate
• Procedures without Connections
• Given this drawing, find the area of the
  triangle?
• Given these marked triangles, are they congruent?
• Procedures with Connections
• Draw a rectangle around the triangle and find
  the area.
• Fold the paper and identify the relationship
  between the triangles.
• Doing mathematics
• If we dont want to count the squares that cover
  the triangle, how can we find the area?
• Verify by measurement Reason through your
  conjecture about the triangles.

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Reasoning with relationships

• Actively looking for and applying geometric
  relationships, within and between geometric
  figures. Internal questions include
• How are these figures alike?
• In How many ways are they alike?
• How are these figures different?
• What would I have to do to this object to make
  it like that object?

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Generalizing geometric ideas

• Wanting to understand and describe the "always"
  and the "every" related to geometric phenomena.
  Internal questions include
• Does this happen in every case?
• Why would this happen in every case?
• Can I think of examples when this is not true?
• Would this apply in other dimensions?

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Investigating invariants

• An invariant is something about a situation that
  stays the same, even as parts of the situation
  vary. This habit shows up, e.g., in analyzing
  which attributes of a figure remain the same when
  the figure is transformed in some way. Internal
  questions include
• How did that get from here to there?
• What changes? Why?
• What stays the same? Why?

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Sustaining reasoned exploration

• Trying various ways to approach a problem and
  regularly stepping back to take stock. Internal
  questions include
• "What happens if I (draw a picture, add to/take
  apart this figure, work backwards from the ending
  place, etc..)?"
• "What did that action tell me?"

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Sketchpad is not limited to Geometry

• Making figures for any handout-the Pentagon
• The capabilities of the hide/show buttons and
  easy text abilities make it ideal for puzzles
• It gives a visual representation of
  algebra-graphs and algebra tiles.

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CONCRETE/ PICTORIAL
VERBAL
GRAPH
ALGEBRA
NUMERICAL/ TABLE
SYMBOLIC
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Teaching with the Pentagram
CONCRETE/ PICTORIAL
GRAPH
VERBAL
NUMERICAL/ TABLE
SYMBOLIC
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Lizs Pattern
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Factoring

• I have a PowerPoint for you and Sketches that are
  interactive with the tiles.

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Formulas Graphing

• Making sense of geometric formulas
• A sketch that could be done on NSpirebut it
  works on the computer, too

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The Possibilities are Endless!

• I have a CD for you with many sketchesstart
  playing with themimagine how you can use
  themeven change them.
• When you think of a concept for a
  lessonvisualize the geometric representation of
  it, then either play with GSP or e-mail me.
• Any questions? Comments?