Shaping Up in
Geometry
K-5 Hot Topic Workshop
December 7, 2012
Complete a pattern block puzzle for fun!
Why are these tasks important for young children?
Select the figure that
best fits YOU.
Broken Squares
Form 5 congruent squares with
the puzzle pieces
 Each square should be the
same size and have equal
dimensions

Broken Squares
What is the math content
involved in this task?
 Why is this an ideal
problem solving task?

Geometry
“geo” – means earth
“metry” – means measure
“Measurement of the earth”
A tool for understanding the world.
The study of the physical world in which we
live through mathematical relationships.
The study of space and spatial relationships.
Geometry
“Geometry offers students an aspect of
mathematical thinking that is different from,
but connected to, the world of numbers.”
• Fosters enthusiasm for mathematics
• Provides a context to develop number and
other mathematical concepts
• Geometry and spatial concepts can exceed
some students’ numerical skills
PSSM, 2000, p.97
Geometry in the CCSSM
“Perhaps. . .it must be the case that some
mathematical strands take priority over others.
But we must never forget geometry or miss the
opportunities to connect it with the other content
areas. Geometry opens doors and minds for some
students that other parts of mathematics leave
closed.”
Let’s Not Forget Geometry!
by NCTM President Michael Shaughnessy
NCTM Summing Up, October 2011
www.nctm.org
Geometry
Develop spatial sense
through the study of:
1. Shapes & Properties
2. Visualization
3. Transformation
4. Location
Geometry

Shapes and Properties


Visualization


The recognition of shapes in the environment, developing
relationships between two- and three-dimensional objects, and the
ability to draw and recognize objects from different perspectives.
Transformation


Study of the properties of shapes in both two and three dimensions,
as well as the relationships built on properties.
Study of translations, reflections, and rotations and the study of
symmetries.
Location

Refers primarily to coordinate geometry or other ways of specifying
how objects are located in the plane or in space.
Aligning the Standards


Read the K-5 Geometry CCSSM standards
Sort into the four Geometry Big Ideas:
 Shapes & Properties
 Visualization
 Transformation
 Location
Aligning the Standards

As you vertically review the K-5 Geometry
standards:



What patterns do you notice?
What standards are new? What standards
have remained the same?
What parts of geometry are no longer a part of
the K-5 curriculum?
Connections to Measurement
Topic
Grade
Objectives
Perimeter
& Area
3rd
grade
3.MD.5,6,7
Angle
Measures
4th
grade
4.MD.5,6,7
Volume
5th
grade
5.MD.3,4,5
Geometry

Shapes and Properties

Study of the properties of shapes in
both two and three dimensions, as well
as the relationships built on properties.
Guess My Rule


Person fits one mystery rule
Person fits second mystery rule
Where should people go that fit both rules?
What if someone does not fit either rule?



Observe each person
Think: What attribute might each one
represent?
Test conjectures by placing participants
Exploration of Attributes

Share your thinking as
you played Guess My Rule
and describe the process
that led to uncovering the
mystery rules.
1. Were there other attributes or
observable characteristics that could
be a rule?
Exploration of Attributes
2. How do non-examples
provide information
about the rule?
3. Why are there so many possibilities
before you know the exact rule?
4. What would be some attributes that
are not observable?
Attributes

Characteristics or ways
that materials can be sorted
Unstructured attribute materials
– Each attribute has a number of
different values
– For example: sea shells, leaves, people,
children’s shoes
Attributes
•girl
•dark hair
•wearing shoes
Attributes

Characteristics or ways
that materials can be sorted
Structured attribute materials
-Have exactly one solution for every
combination of values for each attribute
-For example: commercial attribute blocks
Attributes
•large
•red
•thin
•rectangle
Which of the two types of attribute
materials were used for the
“Guess My Rule” game?
Attributes

~
Yekttis Cards




Select two attributes to focus on but do not
reveal the “secret” attributes to your group
Allow group members to place cards that
match your attribute inside the Venn circles
The rules may be guessed once a card has
been placed in the center of the Venn
diagram
Repeat with three rules and a Venn diagram
Shape Sorts


Select 2 shapes and tell
how they are alike and different
Group Activity:


1. One person selects 1 shape
2. Group finds all other shapes that are like the
selected shape
What attributes did you use to sort the shapes?
Geoboard Geometry




Use a rubber band to create a figure on
the geoboard
Be sure each person in your group has
a different figure
What do the figures have in common?
How are the figures different?
Exploration of Shapes

Attribute Trains


Create a shape using a chenille stem
(geoboard or notecard). One participant will
be the engine and lead the attribute train.
Connect to the train if your shape matches
one attribute of the shape before.
Try the activity with two attributes
or with differing attributes.
Polygons

A polygon is a closed, connected shape in a
plane consisting of a finite number of line
segments that do not cross each other.
 The line segments making the polygon are
called its sides.
 The points where line segments meet are
called the vertices of the polygon.
side
vertex
Polygons

Types of Polygons
# of sides
3
polygon
triangle
4
5
6
8
quadrilateral
pentagon
hexagon
octagon
n
n-gon
Polygons
Hexagons
Pentagons
Not Polygons
Attributes


“Rather than simply learning the names of basic
shapes, they learn to recognize the attributes of
shapes, to notice how shapes are alike and different.”
“The name of a geometric figure does not just name
but actually defines that figure. The name carries
with it the particular attributes of the shapes.
However, when children learn the name of a shape
without understanding what attributes define that
shape, they can end up with misconceptions.”
Attributes


“Children are more apt to focus on the attributes if they are
asked to describe the attributes in their own words using
whatever language makes sense to them. The first step in
language development should be learning to see, to notice, to
discriminate; the next step should be determining which
shapes with similar attributes go together and why.”
“Children should learn formal labels when they are ready to
apply the label to many different versions of a particular
shape. Learning the language prematurely can only cause
confusion and misconceptions and keep children from looking
closely at important attributes.”
--Source: Understanding Geometry
by Kathy Richardson
Venn Diagrams
Show relationships among collections of
shapes and objects
 Shows how certain
sets are related

Attributes, Properties, &
Features

Attribute & Features




Used interchangeably to indicate any characteristic of
a shape
Defining characteristics (ex-”straight sides”)
Non-defining characteristics (ex-”right-side up”)
Property


Attributes that indicate a relationship between
components of shapes
Examples


“Having parallel sides”
“Having all sides of equal lengths”
The van Hiele Levels of
Geometric Thought





Level
Level
Level
Level
Level
0:
1:
2:
3:
4:
Visualization/Recognition
Descriptive/Analytic
Abstract/Relational
Formal Deduction
Rigor
*Sequential Levels—
These levels are not dependent on age; rather
they are dependent on experiences.
The van Hiele Levels of
Geometric Thought

rectangle
looks like
a door
Level 0: Visualization/Recognition


Can see a geometric figure
as a whole and describe a shape
by what it looks like
Examples:
May be able to see how shapes are alike and
different, but uses irrelevant visual properties
 Unable to think of variations for figures
 Sorts and classifies shapes based on their appearances;
however, often inconsistently classifies figures
 Refers to visual prototypes and are misled by
orientation

The van Hiele Levels
of Geometric Thought

Level 1: Descriptive/Analytic


rectangle has 4
sides & opposite
sides are
parallel
Are able to identify relations within a single
figure; may not interrelate figures or properties
of figures
Descriptors:




Appreciates that a collection of shapes goes together
because of properties
Sorts figures in terms of only one property
Unable to see that some shapes are a subgroup of
another set of shapes (rectangle as a square)
Bases classification of figures on few examples
The van Hiele Levels of
square is a
rectangle &
Geometric Thought
parallelogram

Level 2: Abstract/Relational
Can define a particular shape in a general way
and begin to tell when there is enough
information to have a definition rather than a
description
 Descriptors:

Makes sense of definitions and are aware of connections
among figures
 See that some shapes are a subgroup of another set of
shapes - classifies figures hierarchically
 Understands interrelationships between figures

(squares as rectangles and parallelograms)
The van Hiele Levels of
Geometric Thought

Level 3: Deduction


Understands the significance of definitions,
theorems, and proofs are understood
(High School Geometry)
Level 4: Rigor

Able to compare and contrast different
geometries and axiomatic systems
(College-level Geometry)
Video
Partners for Mathematics Learning

Consider the level you are on and the
geometry experiences you had in school.
Implications for Instruction

Comparing shapes—
how are the shapes
alike and how are
they different
Create four-sided shapes on the geoboards.
Determine how the shapes are alike
and how they are different.
Implications for Instruction

Involve lots of sorting and classifying;
Sort and resort and resort and resort
and resort
Create as many different, closed, four-sided figures
as possible. Sort and resort the quadrilaterals using
different attributes.
Implications for Instruction

Include a variety of examples of shapes—
ample opportunities to draw, build, make,
put together, and take apart shapes
Implications for Instruction

Have students generate and construct
models of shapes
Implications of Instruction

Focus more on properties/attributes of figures
rather than on simple identification.
“Before definitions are introduced students
need to experience activities that develop
their critical observation skills and focus on
classification of figures based on properties
or characteristics rather than ‘naming’
figures”
“Watch What You Say” by Sally Roberts
Teaching Children Mathematics
Implications of Instruction


Develop more precise ways to describe
shapes—focus on describing the properties of
shapes and learning specialized vocabulary.
*Conjectures
Apply ideas to entire classes of figures
(ex: all rectangles) rather than on individual
models.
Implications of Instruction
The study of Geometry
requires
thinking
and doing!
Shape Sorts

Sort the shapes into groups



How did you sort the shapes?
What attributes did you use to sort the shapes?
Sort the shapes again



How did you sort the shapes?
What attributes did you use to resort the shapes?
What is the fewest number of groups I can put the
shapes into? Is there a way to sort the shapes into
more than 10 groups?
Guess my Rule





Look at the shapes
Think about the attributes of the shapes
Decide what the mystery rule might be, test
it by identifying shapes that would fit the
rule you are testing.
Do not blurt out the rule!
As you have more information, refine your
guess about the rule; keep testing
Share thinking and describe the process that
led to identifying the rule
Guess My Rule
Guess My Rule
Geometry Terminology
Term
Definition
Point, line and plane are formally undefined
but can be described as:
Point
A tiny dot. A point is an idealized version of a
dot, having no size or shape.
Line
An infinitely long, stretched string that has no
beginning or end. A line is an idealized version
of such a string, having no thickness.
Plane
An infinite flat piece of paper that has no
beginning or end. A plane is an idealized
version of such a piece of paper, having no
thickness.
Line
Segment
Part of a line lying between two points on a
line. These two points are called the endpoints
of the line segment.
Ray
Part of a line lying on one side of a point on the
line. Think of a ray as having a beginning, but
no end.
Example
Angles

An angle is an amount of rotation about
a fixed point.
OR


The region between two rays with a
common endpoint.
Angles are measured in degrees.
Angles

Angles are measured in degrees.
30̊
360̊
Angles

Angles are measured with a protractor.
120̊
Misconceptions about Angles


Size of angle is related to
the length of the ray
A wide angle with short sides may seem
smaller than a narrow angle with long
sides
D
A
H
E
F
B
G
Classifying Angles
Angles can be classified as:
 Acute - measurements are less than 90°


Obtuse - measurements are
greater than 90°
Right - measurements are
equal to 90°
Classifying Angles
Angles can be classified as:
 Straight - measurement equals 180°
http://www.mathsisfun.com/angles.html
Line Configurations


Parallel: two lines in a plane that never
intersect
Perpendicular: two lines in a plane that
intersect at a
90° angle
Triangles
Polygon
Definition
TRIANGLES
scalene
A triangle with no sides of the same length
isosceles
A triangle that has at least two sides of the same length
equilateral
A triangle that has three sides of the same length
Triangles
Polygon
Definition
TRIANGLES
right
A triangle with one right angle
acute
A triangle with all acute angles
obtuse
A triangle with one obtuse angle
Quadrilaterals

How are quadrilaterals related?
Quadrilaterals
Polygon
Definition
QUADRILATERALS
square
quadrilateral with 4 right angles whose sides all have the
same length
rectangle
quadrilateral with 4 right angles
rhombus
quadrilateral whose sides all have the same length
parallelogram quadrilateral for which opposite sides are parallel
trapezoid
quadrilateral with one pair of opposite sides are parallel
Triangle Investigations

Triangle Sort



Sort the collection of triangles into three
groups so that no triangle belongs to
two groups.
Create a description for each group.
Repeat the activity & sort again.
Regular Polygons

All sides have the same length and all
angles are equal
Student Misconceptions

1. An angle must have one horizontal ray.
2. A right angle is an angle that points to the right.

3. A segment must be vertical if it is the side of a figure.


4. A square is not a square if the base is not horizontal.
5. Every shape with four sides is a square.

6. A figure can be a triangle only if it is equilateral.

These conceptual misconceptions often can be traced
to a student’s focus on a limited number of exemplars of the shape plus the
student’s tendency to consider common features as essential to the concept.
Exploration of Shapes

Geoboard Shapes

Use a rubber band to create a polygon on the
geoboard -- Sort into groups
How did you sort the shapes?
What attributes did you use to sort the shapes?

Create a type of quadrilateral on the geoboard and
sort into groups
Exploration of Shapes



What’s My Shape?—Secret Shape Folders
Secret Shapes with
pattern blocks
Wanted Posters
Exploration of Shapes

What am I?/20 Questions/Mystery Shape






Select one shape from your set and hide it
Groups take turns asking yes or no questions
to determine the mystery shape
When students think they have the correct
shape, they should hold it up
What is the first best question to ask? Why?
Is there another question that is equally good?
If you were left with a square, rhombus, and a
rectangle, what would be a good question to ask?
Exploration of Shapes
Literature Connection:
The Greedy Triangle
by Marilyn Burns

Additional Activities:
Create a Geometry Story, Polygon Pictures,
Scavenger Hunt, Clue Cards, I Spy Game
Roping In Quadrilaterals




Arrange sorting circles into a Venn
diagram
Select attribute cards and label each ring
Place the appropriate quadrilateral pieces
in each ring according to the label
Note: Some rings will overlap to form
intersections and some will not
Classifying Shapes
Create a Venn Diagram
 Select two rules (attributes) and label
with a post-it note
 Sort the shapes
into the appropriate
categories

What is the purpose of this activity?
Describe the challenges your
students might encounter.
NCTM’s Illuminations: Shape Sorter
http://illuminations.nctm.org/ActivityDetail.aspx?ID=34
Exploration of Shapes

Yarn Shapes

By working together, use the large
piece of yarn to create various
shapes
Poster Creation


Select one shape
Create a poster


Include multiple examples of the shape
Include attributes and properties of the
shape
Geometry

Visualization

The recognition of shapes in the
environment, developing relationships
between two- and three-dimensional
objects, and the ability to draw and
recognize objects from different
perspectives.
Spatial Visualization





What activities do you use in your
classroom to improve students’ spatial
visualization skills?
Shape Hunt
Shape Puzzles
Quick Images & Quick Build
What’s Missing?
What’s Missing?
What’s Missing?
Cutting and Rearranging Figures

Cutting Corners:
Cut one sheet of paper into two smaller
rectangles with one straight cut.
Are the rectangles congruent?
 Cut a new sheet of paper into two triangles.
Are the triangles are congruent?
 Cut a new sheet of paper to form
two new shapes.
 Continue with an equilateral
triangle and a trapezoid.

Building Polygons
with Pattern Blocks

Making Shapes with Triangles


Put 1 - 6 triangles together to create
various polygons.
Record each shape on triangle paper and
cut it out.
Super Source CD
ETA/Cuisenaire
Building Polygons
with Pattern Blocks

Additional Activities:

Only Two Blocks


What shapes can you make with the trapezoid
and triangle? (The entire side of one shape
should touch the entire side of the other.)
Select two blocks from a bag of pattern blocks.
Create a new shape using the two blocks.
Super Source
ETA/Cuisenaire
Okay
Not Okay
Building Polygons
with Tangrams

Additional Activities:

One Change At a Time

Make shapes with tangrams. Move one
tangram piece to create a new shape.
Super Source CD
ETA/Cuisenaire
Building Polygons
with Color Tiles

Making Rectangles with Color Tiles




Make as many different rectangles as you
can with color tiles. Use from 1 to 6 tiles
for each rectangle.
Look over your rectangles to be sure that
each rectangle is different.
Record and cut out each different
rectangle.
Compare and count rectangles.
Super Source CD
ETA/Cuisenaire
Developing Spatial Sense
with Snap Cubes

Make a Copy with Snap Cubes
Partner 1:
 Set up a barrier between you and your
partner.
 Behind the barrier, build a structure using 8
to 12 snap cubes.
 Give directions to your partner to build your
structure one step at a time.
 Remove the barrier to see if structures are
identical.
 Switch roles and try the activity again.
Technology Resource
National Library of Virtual Manipulatives
http://nlvm.usu.edu/en/nav/vlibrary.html

Virtual Manipulatives
Geometry

Transformation

Study of translations,
reflections, and rotations and
the study of symmetries.
Symmetry

Moved to
4th grade!
Line Symmetry—

Sort shapes into groups –
~Figures that have symmetry and figures that
do not.
~Number of Lines of Symmetry
What is line symmetry?
Line Symmetry (Reflection/Mirror)


A figure has line symmetry if it can be divided
in half and each half is a reflection of the other.
A line of symmetry for a two-dimensional figure
as a line across the figure such that the figure
can be folded along the line into matching
parts.
Line of Symmetry
Regular Polygons

Explore lines of symmetry on regular
polygons. What do you notice?
Developing Spatial Sense
with Snap Cubes

Mirror Images with Snap Cubes




Use 12 snap cubes to create a 1-layer
design.
Place design on a folded piece of grid paper
so the edges of some of the cubes touch the
fold. Trace around the design and cut it out
keeping the paper folded. (Do not cut along
the fold line.)
Exchange designs with partner and challenge
them to build the mirrored image of your
design.
Check with the grid paper design.
Developing Spatial Sense
with Snap Cubes

Mirror Images with Snap Cubes



How did you know how to build the mirrored image of
your partner’s design?
How would the paper design change if a different part of
the snap cube design touched the fold?
Can you find anything else in the classroom that has a
line of symmetry?
Extension:
Investigate how the paper design changes when
different sides of the snap cube design are placed along
the fold line.
Symmetry Activities



Monster Molly
Mirror Shapes
What other types of symmetry
activities would you suggest?
Geometry

Location

Refers primarily to coordinate
geometry or other ways of
specifying how objects are
located in the plane or in space.
Coordinate Grid


Add two more
points on the grid
and connect to
create a pentagon
Name the
coordinates
Moved to
5th grade!
Coordinate System

Literature Connection:
Fly on the Ceiling
by Dr. Julie Glass
Random House
Coordinate System

Walk This Way


Create a large coordinate grid on the floor
Practice locating various points on the grid
and walking paths between points
Describe the path between two points on
the plane.
How does this activity help students better
understand the coordinate grid?
Coordinate System

Tic-Tac-Toe




Select a partner and decide who will be an X and
who will be an O
Take turns placing an X or an O on the coordinate
grid—where the lines intersect, not the spaces
In order to win, you must have four in a row
(vertical, horizontal, or diagonal)
Write the ordered pair for each point on the
T-chart
What patterns do you notice from multiple games?
What strategies might students use to win?
Coordinate System

Additional Activities:




Battleship/Hit or Miss
Box Lid Treasure Hunt
Cookie Sheet Grid
X/Y Coordinate Grid Geoboard
2-D Connections to 3-D


What are the relationships of twodimensional and three-dimensional
shapes?
How does knowing one help when
exploring properties of the other?
3-Dimensional Shapes

Shape Sorts


Sort the shapes into groups
Repeat and sort the shapes
a different way
How did you sort the shapes?
What attributes did you use to sort the
shapes?
3-Dimensional Shapes
Two Out of Three



Find two 3-D shapes that are alike
Justify your reasoning
Repeat with 3 different shapes
Which properties did you use to
determine which shapes were alike?
3-Dimensional Shapes
Mystery Shapes
 Shape Sorts
 Scavenger Hunt

Standards for
Mathematical Practices
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning
of others
Model with mathematics
Use appropriate tools strategically
Attend to precision
Look for and make use of structure
Look for and express regularity in
repeated reasoning
GEOMETRY
VOCABULARY
Geometry
List as many
geometry terms
as you can.
How do we learn geometric terms?
Not from definitions
(They’re for refining meanings.)
 What’s a triangle? (Definition)
 Which of these are triangles?

How do we learn geometric terms?

What is a square?
Four equal sides and closed
Four equal sides
?
Contrast is Essential

All of these are thingos.

None of these is a thingo.

Which of these are thingos?
a.
b.
c.
d.
e.
f.
Contrast is Essential

Need extreme examples

Need fairly close non-examples
Triangles?
Shapes & Properties
Refining
Vocabulary
Exploring
Attributes
of Shapes
Classifying
Shapes
Developing
Vocabulary
5th
4th
3rd
2nd
1st
K
Vocabulary









What strategies have
you used to help
students learn math
vocabulary?
Create word walls and word lists
Use vocabulary repeatedly in context
Graphic Organizers
Word Sorts
Word Study – origin, roots, prefixes, suffixes
Compare and Contrast
Similarities and Differences
Art – illustrations, acting it out, singing
Games
Essential Characteristics
Non-essential Characteristics
trapezoid
Examples
Non-examples
http://www.mathopenref.com/trapezoid.html
Interactive Wall Displays
Geometry Review Games

Jeopardy

Triangle Trivia

Talk a Mile a Minute
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Talk a Mile a Minute
trapezoid
four
parallel
quadrilateral
pattern block
polygon
Talk a Mile a Minute
Perpendicular
lines
angle
right
intersect
cross
Games
PowerPoint Games:
 Wheel of Fortune
 Are You Smarter Than a 1st Grader?
 Password
 Jeopardy
 Who Wants to Be a Millionaire
http://jc-schools.net/tutorials/PPT-games/
http://people.uncw.edu/ertzbergerj/ppt_games.html
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Review Games

Concentration

Matching Cards

Charades/Role Play

Flash Cards
http://quizlet.com/
Review Games

Hide & Seek

Pictionary

Scavenger Hunts

Board Games
http://jcschools.net/tutorials/vocab/wordgames-vocab.html
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Review Games

I Spy

Beach Ball Review


What am I?/Twenty Questions
Shape Riddles
Spinners and Cubes
http://www.toolsforeducators.com/dice/
Vocabulary Games

Dominoes
http://www.toolsforeducators.com/dominoes/science.php

Bingo
www.teach-nology.com
http://jc-schools.net/tutorials/vocab/wordgames-vocab.html
http://www.toolsforeducators.com/bingo/
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Vocabulary Games

Vocabulary Game

Silent Outburst

I Have, Who Has
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