Download spatial reasoning - Region 11 Math And Science Teacher Partnership

Geometric and Spatial
Reasoning
Today’s Agenda

Symbolic and Algebraic Reasoning: Review &
Sharing

Geometric Reasoning

Can a Picture Prove?

Spatial Reasoning

Baseline Assessment
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)
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.

Geometric Reasoning
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)
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”.
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.
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.
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.
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.
Using Geometry Logic

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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.
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?

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

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?
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.
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?
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 it’s clear the order
of the statements being expressed.

A Mathematical Fact

Fact:
1/4 + 1/16 + 1/64 + … = 1/3

Normally justified in a calculus class using
a
the geometric series formula 1  r , since
the series has the form a + ar + ar2 + …
where a is ¼ and r is also ¼.

How else to show this?
Its Proof?
Visual Proof that 64 = 65?
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?

Let’s use the prior discussion about
conditions for visual proofs to revisit an
old friend…
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!
 Let’s look at some possibilities for proving it
to determine their efficacy…

Pythagorean Theorem Proof #1
Is this a correct proof of the Pythagorean Theorem? Why?
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:
c 2  4  ( 12 ab)  (b  a) 2
c 2  2ab  b 2  2ab  a 2
c 2  b2  a2
a2  b2  c 2
Pythagorean Theorem Proof #2
Is this a correct proof of the Pythagorean Theorem? Why?
Verify :
32  4 2  9  16  25
5 2  25
So 32  4 2  5 2.
Try another :
5 2  12 2  25  144  169
132  169
So 5 2  12 2  132.
We can try others and they will be the same.
Therefore a 2  b 2  c 2
Pythagorean Theorem Proof #3
Is this a correct proof of the Pythagorean Theorem? Why?
Spatial Reasoning
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!

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

More with Isometric Views
Cubes can be arranged and stacked to form
3-d landscapes
 Example:

Spatial Reasoning with Isometric Views
There are four representations at work here:
1. Physical manipulatives: centimeter blocks
2. Mat plans: 2-d blueprints
3. Top, bottom and side views of the 3-d
shape
4. Isometric view
Students use spatial reasoning in shifting among
these representations.
 Today we will focus mostly on shifting from
isometric views to mat plans
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
1
1
1
Front
1
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
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

Baseline
Assessment
Baseline Assessment – Item 1
What is the shape described below?
Clue 1:
Clue 2:
Clue 3:
Clue 4:
Clue 5:
Clue 6:
Clue 7:
Clue 8:
Clue 9:
It is a closed figure with 4 straight sides.
It has 2 long sides and 2 short sides.
The 2 long sides are the same length.
The 2 short sides are the same length.
One of the angles is larger than one of the other
angles.
Two of the angles are the same size.
The other two angles are the same size.
The 2 long sides are parallel.
The 2 short sides are parallel.
Baseline Assessment - Item 2
Mat Plan
Front
Front
Baseline Assessment – Item 3
Why is this proof of the Pythagorean Theorem incorrect?
Verify :
32  4 2  9  16  25
5 2  25
So 32  4 2  5 2.
Try another :
5 2  12 2  25  144  169
132  169
So 5 2  12 2  132.
We can try others and they will be the same.
Therefore a 2  b 2  c 2
Baseline Assessment – Item 3
Why is this proof of the Pythagorean Theorem incorrect?