Download Developing Geometric Reasoning Part 1

Class 4:
Part 1
Common Core State Standards (CCSS) Class
April 4, 2011
Reading – Chapter 8
Implementing Talk in the Classroom: Getting Started
Five Principals of Productive Talk
Principle 1: Establishing and Maintaining a Respectful,
Supportive Environment
Principle 2: Focusing Talk on the Mathematics
Principle 3: Providing for Equitable Participation in Classroom
Talk
Principle 4: Explaining Your Expectations About New Forms of
Talk
Principle 5: Trying Only One Challenging New Thing at a Time
Journal Review by Your Peers
 Read and react to each other’s journal entries about
the five principals of productive talk
 Use Post-it-Notes to record your comments in each
journal
 Pass the journals to the right until you get your own
journal back
 Discuss as a group – What stands out after reading all
the journals?
Learning Intentions
We Are Learning To …
 Understand the Common Core State Standards
Alignment Task
 Reflect on how the mathematical tasks we select
address the Standards for Mathematical Practice and
Content Standards
Success Criteria
We will know we are successful when we can
explain to others the Common Core State Standards
Alignment Task and when we can identify the
Standards for Mathematical Practice and Content
Standards in the mathematical tasks we teach.
Common Core State
Standards Alignment Task 3
 Present your completed Task 3 to the teachers at your
table, briefly reviewing the task and student work.
 Why did you select the task you did?
 What did the students’ work reveal?
 What did you see and hear when your students worked
on this task?
Common Core Alignment
Task 3
Discuss differences/ similarities between the three tasks
in relation to the Standards for Mathematical Practices.
Final Project
 Review CCSS Project
 Questions about expectations?
Developing Geometric
Reasoning: Grades 3-5
Common Core State Standards (CCSS)
Learning Intentions
We Are Learning To:
 Examine how students’ understanding of geometry
develops through defined stages, the van Hiele Model.
 Consider implications for instruction to move students
along in developing their geometric reasoning.
 Compare, contrast, and connect properties of
quadrilaterals.
Success Criteria
 We will know we are successful when we can articulate
how Mathematical Practice Standards 1, 2 and 5—
sense making, reasoning, and tools—are infused in
mathematical tasks or lessons for a standards’ content
progression.
Wisconsin
Common
Core
Standards
Domain
Content strand across grades:
Geometry
Cluster
“Big Idea” that groups together
a set of related standards.
Standards
Statements that define what
students should understand and
be able to do at a grade level.
A Content Standards Progression
Domain: Geometry
Clusters:
3: Reason with shapes and their attributes
4: Draw and identify lines and angles, and classify shapes
by properties of their lines and angles
5: Classify 2-dimensional figures into categories based on
their properties
Standards: 3.G.1, 3.G.2, 4.G.1, 4.G.2, 4.G.3, 5.G.3,
5.G.4
van Hiele Levels of Geometric
Reasoning
 Level 0: Visualization Recognize figures as total
entities, but do not recognize properties.
 Level 1: Analysis (Description) Identify properties of
figures and see figures as a class of shapes.
 Level 2: Informal Deduction Formulate
generalizations about relationships among properties of
shapes; Develop informal explanations.
van Hiele Levels of Geometric
Reasoning
 Level 3: Deduction Understand the significance of
deduction as a way of establishing geometric theory
within an axiom system. See interrelationship and role
of undefined terms, axioms, definitions, theorems and
formal proof. See possibility of developing a proof in
more than one way.
 Level 4: Rigor Compare different axiom systems (e.g.,
non-Euclidean geometry). Geometry is seen in the
abstract with a high degree of rigor, even without
concrete examples.
How do students progress in developing
geometric reasoning?
 How would you recognize each of these levels of
thinking in your students’ work?
 Considering the first three levels, where would you
place the majority of the lessons that you teach?
“I believe that development is more dependent on
instruction than on age or biological maturation and
that types of instructional experiences can foster, or
impede, development.”
Pierre M. van Hiele
Quadrilateral Sorting Activity
Facilitator: Give all a voice.
Recorder: Take notes on
recording sheet.
Directions
1. Remove a card from the envelope.
2. Sort the quadrilateral shapes by the directions on the card.
3. Record the sort and discussion on the recording sheet.
4. Push all of the shapes back into a pile.
5. Pass the envelope to the left and repeat steps 1–4 with the
next card.
Property Criteria
Property Criteria
• all right angles
• at least one right angle
• both pairs of opposite sides congruent
• both pairs of opposite sides parallel
• both pairs of opposite angles congruent
• congruent diagonals • diagonals bisect each other
• at least one pair of parallel sides
• exactly one pair of parallel sides
What is a Trapezoid?
 Some authors choose to define trapezoid as a
quadrilateral with at least one pair of parallel sides.
 That definition is more inclusive and leads to the
conclusion that all parallelograms are trapezoids.
Trapezoid definitions
Everyday Math
Expressions
Scott Foresman
A quadrilateral
that has exactly
one pair of parallel
sides
(K & 1st)
(3rd & 4th)
A quadrilateral
with only one pair
of parallel sides.
A quadrilateral with
at least one pair of
parallel sides.
(5th)
A quadrilateral with
one pair of parallel
sides.
(5th)
A quadrilateral
that has exactly
one pair of
parallel sides
Maybe by Middle School?
CMP
Glencoe
Holt
A quadrilateral
with at least one
pair of opposite
sides parallel.
(6th) A quadrilateral
with one pair of
opposite sides
parallel.
A quadrilateral
with exactly one
pair of parallel
sides.
(7th) A quadrilateral
with one pair of
parallel sides.
(8th) A quadrilateral
with exactly one
pair of parallel
opposite sides.
Parallelogram Definitions
Everyday Math
Expressions
Scott Foresman
•A quadrilateral with
two pairs of parallel
sides.
(K-2nd)
A quadrilateral in
which both pairs of
opposite sides are
parallel and
opposite angles are
congruent.
(3rd & 4th)
A quadrilateral in
which opposite sides
are parallel.
•Opposite sides of a
parallelogram are
congruent.
•Opposite angles in
a parallelogram have
the same measure.
(3rd-5th)
A quadrilateral
with both pairs of
opposite sides
parallel.
(5th)
A quadrilateral with
both pairs of opposite
sides
parallel.
True or False
A square is
 a special kind of rectangle.
 It is a rectangle in which all four sides are the same
length
A parallelogram is
 a special kind of trapezoid.
 It is a trapezoid with two pairs of parallel sides.
True or False
A rhombus
 is a special kind of kite.
 It is a kite in which all four sides are the same
length.
Reflect
 How do these tasks engage you in the content learning
infused with practices?
(Mathematical Practices Standards 1, 2, 5)
 How do these tasks help you to better understand the
mathematics?
 Standards: 3.G.1, 3.G.A, 4.G.1, 4.G.2, 4.G.3, 5.G.3, 5.G.4
Big Ideas of Geometry
 Two- and three-dimensional objects can be described,
classified and analyzed by their attributes.
 Objects can be oriented in an infinite number of ways. The
orientation of an object does not change the other attributes
of the object.
 Some attributes of objects (e.g. area, volume, perimeter,
surface area) are measurable and can be quantified using
unit amounts.
 Objects can be constructed from or decomposed into other
objects. In particular, any polygon can be decomposed into
triangles.
Development Through the van Hiele
Levels
 Level is not affected by biological age.
 Level is affected by degree of experience.
 In order to progress through the levels, instruction must
be sequential and intentional.
 When instruction (or materials or vocabulary, etc.) is at
an inappropriate level, students will not be able to
understand the instruction. They may be able to
memorize it, but with no understanding of material.
What other
practices
were
infused
in the
content
learning?
Provide
specific
examples.
Summary
We were learning to recognize three of the Standards for
Mathematical Practices—sense making, reasoning, and
tools— within a chosen Content Standards progression.
We will know we are successful when we can
articulate how both a Content Standard and a Standard
for Mathematical Practice are infused in a math lesson in
the classroom.