Developing Geometric
Reasoning K-2
Common Core State Standards (CCSS) Class
March 14, 2011
Paige Richards and Dana Thome
Learning Intentions
We Are Learning To:
 Recognize how Mathematical Practices 1 and 2—
sense making and reasoning, —are connected to a
selected standards’ content progression for geometric
reasoning.
 Identify how students will develop and demonstrate
Practices 1 and 2 in their work and discussions.
Success Criteria
 We will know we are successful when we can articulate
how Mathematical Practice Standards 1 and 2 —sense
making and reasoning—are infused in mathematical
tasks or lessons for a standards’ content progression.
Wisconsin
Common
Core
Standards
Domain
Content strand across grades:
Operations & Algebraic Thinking
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:
K: Identify and describe shapes (squares, circles,
triangles, rectangles, hexagons, cubes, cones,
cylinders, and spheres).
Analyze, compare, create, and compose shapes.
1 & 2: Reason with shapes and their attributes.
Standards: K.G.1; K.G.2; K.G.3; K.G.4; K.G.5; K.G.6;
1.G.1; 1.G.2; 2.G.1
Laying the Foundation in PK-K
Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones,
cylinders, and spheres).
1. Describe objects in the environment using names of shapes, and describe the relative positions of these
objects using terms such as above, below, beside, in front of, behind, and next to.
2. Correctly name shapes regardless of their orientations or overall size.
3. Identify shapes as two-dimensional (lying in a plane, “flat”) or three-dimensional (“solid”).
Analyze, compare, create, and compose shapes.
4. Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using
informal language to describe their similarities, differences, parts (e.g., number of sides and
vertices/“corners”) and other attributes (e.g., having sides of equal length).
5. Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing
shapes.
6. Compose simple shapes to form larger shapes. For example, “Can you join these two triangles with full
sides touching to make a rectangle?”
What is a Triangle?
On the 3-by-5 card write a definition of a triangle.
Considering Triangle
Characteristics
 Turn the card over.
 Draw a triangle.
 Draw another triangle that is different from the triangle
you just drew.
 Draw a third triangle that, again, is different than the
previous triangles.
 Finally, draw a fourth triangle that is different than any
of the previous triangles. What is different about the
triangles? What is similar?
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.
“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
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?
Tricky Triangles
Envelope. . . has a selection of shapes.
Goal... Sort the shapes into 2 groups. “Triangles” and “Not Triangles”
Process...
• Pull out a card (without looking).
• Show it to the table group.
• Explain why it is or isn’t a triangle.
Pass the Envelope... to the next person and repeat the process until
all the shapes are sorted.
Group Discussion...What defines a triangle?
Reviewing Student Work
Assign Roles
Facilitator: Give all a voice.
Recorder: Take notes on record sheet.
Directions Distribute one or two work samples to each group
member.
Review what “student understands” and for “student
misconceptions” (silently).
Taking turns, present your observations.
Table Group Discussion: What are some instructional
implications you will take back to your school?
Tricky Triangles
Percent of Correct Responses for MPS Students
in Grades 3-11
Results for Figure C
Percent of Correct by Grade
Results for Figure H
Percent of Correct by Grade
Results for Figure I
Percent of Correct by Grade
Results for Figure J
Percent of Correct by Grade
Revisiting Your Triangle Definition
 Review and revise your definition of a triangle.
 Highlight the main ideas you want to emphasize with
your students.
 Share definitions of triangles.
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?
(Content Standards K.G.1; K.G.2; K.G.3; K.G.4; K.G.5;
K.G.6; 1.G.1; 1.G.2; 2.G.1)
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.
Grade 1
Reason with shapes and their attributes.
1. Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining
attributes (e.g., color, orientation, overall size) ; build and draw shapes to possess defining attributes.
2. Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quartercircles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right
circular cylinders) to create a composite shape, and compose new shapes from the composite shape.1
3. Partition circles and rectangles into two and four equal shares, describe the shares using the words
halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the
whole as two of, or four of the shares. Understand for these examples that decomposing into more
equal shares creates smaller shares.
_________________
1
Students do not need to learn formal names such as “right rectangular prism.”
Grade 2
Reason with shapes and their attributes.
1. Recognize and draw shapes having specified attributes, such as a given number of angles
or a given number of equal faces.1 Identify triangles, quadrilaterals, pentagons,
hexagons, and cubes.
2. Partition a rectangle into rows and columns of same-size squares and count to find the
total number of them.
3. Partition circles and rectangles into two, three, or four equal shares, describe the shares
using the words halves, thirds, half of, a third of, etc., and describe the whole as two
halves, three thirds, four fourths. Recognize that equal shares of identical wholes need
not have the same shape.
_________________
1
Sizes are compared directly or visually, not compared by measuring.