Download Developing Linear Thinking: A Progression from PK-K-Grade 2

Exploring Polygons, Part 1
Core Mathematics Partnership
Building Mathematical Knowledge and
High-Leverage Instruction for Student
Success
2:45 – 4:00
July 21, 2016
Learning Intention and Success Criteria
We are learning to…
- Understand and classify polygons.
We will be successful when we can…
- Define polygons.
- Classify polygons according to attributes.
…the most important goals for elementary
geometry according to three categories.
K-6 Geometry Progressions
 Geometric shapes, their components (e.g.
sides, angles, faces), their properties, and
their categorization based on those
properties.
 Composing and decomposing geometric
shapes
 Spatial relations and spatial thinking.
What is a polygon?
Trio: Silent Sorting
Write 2 sticky notes:
Polygons
Not-A-Polygon
WITHOUT TALKING…
Select a card from the envelope.
Place the figure under the appropriate category.
Move on to the next person without talking.
**Signal to indicate you are done.
Refining Your Definition
Study the sorted cards.
Discuss cards you agree with and cards you
disagree with.
As a trio…refine your definition of a polygon
and reorganize your cards (if needed).
Be ready to share your definition and any cards
that challenged your thinking.
A Polygon is…
A polygon is a closed figure with line segments
as sides, and vertices.
- Grade 4 Shapes and Symmetry, Investigations v.2
Joe made two inventions for Science.
He claims both of them are polygons.
Is Joe correct?
or Why not?
Invention A
Invention B
Review your Frayer model
• Revise your definition. You can make a clean
copy or add information/examples to your
paper.
Frayer Model
Definition in your own words
Facts/characteristics
A polygon is a closed figure with line
segments as sides, and vertices
Examples
Sides are segments (no curves)
Figure is closed (no holes)_
Sides form vertices
Polygon
Nonexamples
Thinking a little bit further…
• Find shapes 21, 22, 23, 25
• How are they the same? How are they
different?
• What would you name this category of
shapes?
Polygon Capture: Pairs
• Take the cards out of the envelope and lay the
cards face down to make 2 piles:
Angles
Sides
•
•
•
•
Spread the polygon shape cards on your table.
Take 1 card from each pile – read directions.
Capture all polygons that meet the criteria on
both cards.
Polygon Capture: Pairs
Switch players.
Round 1: Remove Wild Card/ Steal Cards
Round 2: Add Wild Card/Steal Card
MP6 and MP7
What mathematics are students engaged in as
they play Polygon Capture?
How would Polygon Capture helping students
develop MP6 and MP7?
What adjustments might you make for students
at your grade level?
Trapezoids! Who Knew?
“A quadrilateral
sides.”
at least one pair of parallel
OR
“A quadrilateral with
sides.”
one pair of parallel
How does YOUR curriculum define trapezoids?
Both EngageNY and Investigations use the
Inclusive definition of trapezoid:
Polygon Riddles
Polygon Riddles
Work in a trio:
Scatter polygon card shapes face-up on desk.
Person 1: Select a card.
Read each clue one at a time.
Pause between reads to allow duo to eliminate cards that
do not fit the clue.
Person 2 & 3: Work together to eliminate shape cards that do
not fit the clues read.
Switch roles. Play several times.
Sample Polygon Riddles
Clue A: My polygon is a quadrilateral.
Clue B: My polygon has exactly two sides that
are parallel.
Clue C: My polygon has two right angles.
Clue D: The longer parallel side is about twice
as long as the shorter parallel side.
Clarifying van Hiele Levels
Visual Level
Recognize figures as total entities, but do not
recognize properties.
Descriptive Level/Analytic
Identify properties of figures and see figures as
a class of shapes defined by their properties.
Informal Deduction [Analytic]
Formulate generalizations about relationships
among properties of shapes; Develop informal
explanations.
Sample Polygon Riddles
Clue A: My polygon is a quadrilateral.
Clue B: My polygon has exactly two sides that
are parallel.
Clue C: My polygon has two right angles.
Clue D: The longer parallel side is about twice
as long as the shorter parallel side.
Thinking a little bit further…
• Study your Polygon Riddle cards
• Look for a clue you can “drop” from each card.
• Test it!
Clarifying van Hiele Levels
Visual Level
Recognize figures as total entities, but do not
recognize properties.
Descriptive Level/Analytic
Identify properties of figures and see figures as
a class of shapes defined by their properties.
Informal Deduction [Analytic]
Formulate generalizations about relationships
among properties of shapes; Develop informal
explanations.
Thinking about the van Hiele Levels
Consider our polygon activities through the lens
of the van Hiele levels. What do you notice?
What are you thinking?
Open to page 78 in your “Essential
Understandings” book and read the bottom half
of the page.
One perspective on evaluating the
van Hiele Levels
“… allows for the possibility that a student can
develop two consecutive levels of reasoning at the
same time, although what usually happens is that
the acquisition of the lower level is more complete
than the acquisition of the higher level. In fact, we
observed that not all students used a single level of
reasoning, but some of them used several levels at
the same time, probably depending on the difficulty
of the problem.”
- Gutierrez, Jaime & Fortuny (1991), p. 250
One perspective on evaluating the
van Hiele Levels
“… allows for the possibility that a student can
develop two consecutive levels of reasoning at the
same time, although what usually happens is that
the acquisition of the lower level is more complete
than the acquisition of the higher level. In fact, we
observed that not all students used a single level of
reasoning, but some of them used several levels at
the same time, probably depending on the difficulty
of the problem.”
- Gutierrez, Jaime & Fortuny (1991), p. 250
Shall We Play a Game?
Quadrilateral Sorting Activity
①Remove a card from the envelope.
②Sort the quadrilateral shapes by the directions
on the card.
③Record the sort and discussion on the
recording sheet.
④Push all of the shapes back into a pile.
⑤Pass the envelope to the left and repeat steps
1–4 with the next card.
CCSSM Geometry Standards
Reason with shapes and their attributes.
3.G.1. Understand that shapes in different
categories (e.g., rhombuses, rectangles, and others)
may share attributes (e.g., having four sides), and
that the shared attributes can define a larger
category (e.g., quadrilaterals). Recognize
rhombuses, rectangles, and squares as examples of
quadrilaterals, and draw examples of quadrilaterals
that do not belong to any of these subcategories.
CCSSM Geometry Standards
Draw and identify lines and angles, and classify
shapes by properties of their lines and angles.
4.G.2. Classify two-dimensional figures based on
the presence or absence of parallel or
perpendicular lines, or the presence or absence
of angles of a specified size. Recognize right
triangles as a category, and identify right
triangles.
CCSSM Geometry Standards
Classify two-dimensional figures into categories
based on their properties.
5.G.3. Understand that attributes belonging to a
category of two-dimensional figures also belong to
all subcategories of that category. For example, all
rectangles have four right angles and squares are
rectangles, so all squares have four right angles.
5.G.4. Classify two-dimensional figures in a
hierarchy based on properties.
Thinking a little bit further…
Read the section: Geometric shapes,
components and properties – in the K-6
Geometry Progressions Pages 2 - 3.
Also read page 18 and study the Venn diagram.
As you read….
How did the activity push you level of geometric
reasoning in relation to the van Hiele levels?
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.
A rhombus is a special kind of parallelogram.
It is a parallelogram in which all four sides are the same
length.
Learning Intention and Success Criteria
We are learning to…
- Understand and classify polygons.
We will be successful when we can…
- Define polygons.
- Classify polygons according to attributes.
Disclaimer
Core Mathematics Partnership Project
University of Wisconsin-Milwaukee, 2013-2016
This material was developed for the Core Mathematics Partnership project
through the University of Wisconsin-Milwaukee, Center for Mathematics and
Science Education Research (CMSER). This material may be used by schools to
support learning of teachers and staff provided appropriate attribution and
acknowledgement of its source. Other use of this work without prior written
permission is prohibited—including reproduction, modification, distribution,
or re-publication and use by non-profit organizations and commercial
vendors.
This project was supported through a grant from the Wisconsin ESEA Title II,
Part B, Mathematics and Science Partnerships.