Geometric and Spatial Reasoning

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Geometric and SpatialReasoning
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Todays Agenda

• Symbolic and Algebraic Reasoning Review
  Sharing
• Geometric Reasoning
• Can a Picture Prove?
• Spatial Reasoning
• Baseline Assessment

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Symbolic and Algebraic Reasoning Review

• Symbolic and Algebraic Reasoning is interwoven
  throughout the MN math standards, but
  specifically, students should
• Justify steps in generating equivalent
  expressions by identifying the properties
  used...
• (Minnesota Math Standard 9.2.3.7, Grade 9-11, MN
  Dept of Ed, 2007)

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Reasoning Review

• What did you learn most from the Symbolic and
  Algebraic Reasoning baseline and summative
  assessments?
• Describe at least two ways that you helped your
  students use Symbolic and Algebraic reasoning.
• Describe one classroom situation where you saw a
  student exhibit growth in Symbolic or Algebraic
  reasoning.

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Geometric Reasoning
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Geometric Reasoning

• Construct logical arguments and write proofs of
  theorems and other results in geometry, including
  proofs by contradiction. Express proofs in a form
  that clearly justifies the reasoning, such as
  two-column proofs, paragraph proofs, flow charts
  or illustrations.
• (Minnesota Math Standard 9.3.2.4, Grade 9-11, MN
  Dept of Ed, 2007)

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Van Hiele Levels of Geometric Understanding

• Levels provide a way to characterize student
  understanding
• Level 0 Visualization
• students are able to recognize and name figures
  based on visual characteristics
• students can make measurements
• groupings are made based on appearances and not
  necessarily on properties
• Example Students see squares turned on their
  corner as diamonds.

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Van Hiele levels

• Level 1 Analysis
• students can consider all shapes within a class
  rather than a specific shape
• focus on properties
• Example Students see rectangles as having right
  angles and parallel sides. But they may insist
  that a square is not a rectangle.

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Van Hiele levels

• Level 2 Informal Deduction
• students can develop relationships between and
  among properties
• proofs arise here, informally
• focus on relationships among properties of
  geometric objects
• Students can recognize relationships between
  types of shapes.
• Example They can recognize that all squares are
  rectangle, but not all rectangles are squares.

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Van Hiele levels

• Level 3 Deduction
• students can use logic to establish conjectures
  made at Level 2
• student is able to work with abstract statements
  about geometric properties and make conclusions
  based on logic rather than intuition
• focus on deductive axiomatic systems for geometry
  such as Euclidean
• Example Students can prove that the base angles
  in an isosceles triangle are congruent.

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Van Hiele levels

• Level 4 Rigor
• students appreciate the distinctions and
  relationships between different axiomatic systems
• focus on comparisons and contrasts among
  different axiomatic systems of geometry
• Example Students understand the parallel
  postulate and its meaning in the Euclidean system
  compared to its meaning in the spherical geometry
  system.

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Using Geometry Logic

• Can you determine the shape that satisfies all of
  the following clues?
• It is a closed figure with straight sides.
• It has only two diagonals.
• Its diagonals are perpendicular.
• Its diagonals are not congruent.
• It has a diagonal that lies on a line of
  symmetry.
• It has a diagonal that bisects the angles it
  joins.
• It has a diagonal that bisects the other
  diagonal.
• It has a diagonal that does not bisect the other
  diagonal.
• It has no parallel sides.
• It has two pairs of consecutive congruent sides.

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What Van Hiele Level?

• Discuss what Van Hiele level you think the
  preceding geometry logic problem was.
• What grade level for students?
• How does this problem compare to the logic puzzle
  from the Math Reasoning Session, where you were
  determining the construction sequence for city
  buildings?

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Developing Reasoning Via Open-Ended Problems

• Open-ended problems encourage reasoning from
  students (NCTM Book on Open-Ended Problems)
• Students use geometric reasoning in making
  observations about a figure
• Students then use reasoning in formulating
  arguments to support their observations
• Their observations can indicate their level of
  understanding

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Open-Ended Problem
In the figure below, BF and CD are angle
bisectors of the isosceles triangle ABC. CF is
the angle bisector of exterior angle ACH. Step
1 Find as many relations as you can. Step 2
What van Hiele level is required for each?
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A Construction Problem

• You are given two intersecting straight lines
  and a point P marked on one of them, as in the
  figure below. Show how to construct, using
  straightedge and compass, a circle that is
  tangent to both lines and that has the point P as
  its point of tangency.

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Can a Picture Prove Something?

• Discuss your answer to the question above with
  your colleagues.
• OK, why or why not?
• Any examples to support your claim?

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Does a picture prove?

• Debate exists over this in the mathematics
  community.
• M. Giaquinto (noted math philosopher) noted that
  there is a distinction between discovery and
  demonstration.
• Discovery (of new theorems, facts, etc.) is often
  very visual for the expert.
• Demonstration (proof) can only be accomplished
  visually if its clear the order of the
  statements being expressed.

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A Mathematical Fact

• Fact
• 1/4 1/16 1/64 1/3
• Normally justified in a calculus class using the
  geometric series formula , since the
  series has the form a ar ar2 where a is ¼
  and r is also ¼.
• How else to show this?

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Its Proof?
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Visual Proof that 64 65?
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Visual proof necessary conditions (Hanna
Sidoli, 2007, p. 75)

• Reliability that the underlying means of
  arriving at the proof are reliable and that the
  result is unvarying with each inspection
• Consistency That the means and end of the proof
  are consistent with other known facts, beliefs,
  and proofs.
• Repeatability That the proof may be confirmed
  by or demonstrated to others.
• Discuss the conditions above with your
  colleagues. Do you agree?

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• Lets use the prior discussion about conditions
  for visual proofs to revisit an old friend

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The Pythagorean Theorem!

• The sum of the squares of the lengths of the legs
  on a right triangle is equal to the square of the
  length of the hypotenuse. That is, if a is the
  length of one leg, and b is the length of the
  other leg, and c is the length of the hypotenuse,
  then a2 b2 c2.
• It is the ultimate marriage between geometry and
  algebra!
• Lets look at some possibilities for proving it
  to determine their efficacy

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Pythagorean Theorem Proof 1
Is this a correct proof of the Pythagorean
Theorem? Why?
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Pythagorean Theorem Proof 1
More (algebraic) detail added here.

• The area of the big square is c2. This equals
  the area of the four right triangles plus the
  area of the smaller inside square.
  Algebraically

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Pythagorean Theorem Proof 2
Is this a correct proof of the Pythagorean
Theorem? Why?
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Pythagorean Theorem Proof 3
Is this a correct proof of the Pythagorean
Theorem? Why?
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Spatial Reasoning
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Spatial Reasoning

• We have used spatial reasoning in several
  activities so far but these have been all
  geometric.
• Q What is an activity which uses spatial
  reasoning but is not so dependent upon standard
  geometry?
• A Isometric views!

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Isometric Views

• What is an isometric view?
• It is a 2-d picture which portrays a 3-d shape.
• The standard isometric view shows three faces of
  the 3-d object, top, right side and left side.
• Example Cube

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More with Isometric Views

• Cubes can be arranged and stacked to form 3-d
  landscapes
• Example

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Spatial Reasoning with Isometric Views

• There are four representations at work here
• Physical manipulatives centimeter blocks
• Mat plans 2-d blueprints
• Top, bottom and side views of the 3-d shape
• Isometric view
• Students use spatial reasoning in shifting among
  these representations.
• Today we will focus mostly on shifting from
  isometric views to mat plans

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Isometric View to Mat Plan - I

• The mat plan for the given isometric view is
  below.
• The value in each box refers to the number of
  cubes stacked on that square
• Notice that the values are arranged in an L,
  like the iso view

Front
Front
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Isometric View to Mat Plan - I

• Create the mat plan for the given isometric view
• Discuss your plan with your colleagues.
• Can there be different mat plans for one
  isometric view? Why?

Front
Front
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Extending Spatial Reasoning

• The book pictured uses Cuisenaire Rods to
  encourage spatial reasoning
• Three views are given, students then construct
  the shape with the rods
• Position, color and length are all used as clues

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BaselineAssessment
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Baseline Assessment Item 1
What is the shape described below? Clue 1 It
is a closed figure with 4 straight sides. Clue 2
It has 2 long sides and 2 short sides. Clue 3
The 2 long sides are the same length. Clue 4
The 2 short sides are the same length. Clue 5
One of the angles is larger than one of the other
angles. Clue 6 Two of the angles are the
same size. Clue 7 The other two angles are the
same size. Clue 8 The 2 long sides are
parallel. Clue 9 The 2 short sides are parallel.
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Baseline Assessment - Item 2
Mat Plan
Front
Front
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Baseline Assessment Item 3
Why is this proof of the Pythagorean Theorem
incorrect?
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Baseline Assessment Item 3
Why is this proof of the Pythagorean Theorem
incorrect?