1. Grade Three PARTNERS for Mathematics Learning Module 3 Partners 1

2. What is Geometry? “geo” - means earth “metry” – means measure2
What is Geometry?
“geo” - means earth
“metry” – means measure
“Measurement of the earth”
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for Mathematics Learning

3. NCTM Reminds Us…  Geometry is an opportunity for3
NCTM Reminds Us…
 Geometry is an opportunity for
students to explore relationships among
geometric objects and their component parts
 Geometry is more than definitions, it is about
describing relationships and reasoning
Principles and Standards for School Mathematics, 2000
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for Mathematics Learning

4. Why Study Geometry?  Geometry can provide a more complete4
Why Study Geometry?
 Geometry can provide a more complete
appreciation of the world
 Geometry can be found in the structure of
the solar system, in rocks and crystals, in
plants and flower, even in animals
 Virtually everything that humans create
have geometric forms
Partners
for Mathematics Learning

5. Different Perspectives5
Different Perspectives
 Beginning experiences with
geometric concepts should focus
on helping students perceive and
communicate about the world around
them from a spatial perspective
Partners
for Mathematics Learning

6.  Fold the large sheet of paper into 4 sections; 6
Different Perspectives
 Fold the large sheet of paper into 4 sections;
fold accordion style
 Fold the 2 pictures into 4 sections
 Number and cut apart sections
 Alternating picture strips, glue on large paper
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for Mathematics Learning

7.  An intuition about shapes and the 7
Spatial Reasoning
 An intuition about shapes and the
relationships among shapes
 The ability to visualize mentally objects
and spatial relationships
 The ability to use geometric ideas to
describe and analyze the world
 Leads to an appreciation of geometric form
in art, nature, and architecture
Partners
for Mathematics Learning

8. Spatial Visualization8
Spatial Visualization
 “Some students' capabilities with
geometric and spatial concepts exceed
their number skills. Building on these
strengths fosters enthusiasm for
mathematics and provides a context in
which to develop number and other
mathematics concepts.”
(Razel and Eylon 1991)

9. Geometric Shapes  The area of mathematics that deals with9
Geometric Shapes
 The area of mathematics that deals with
lines, shapes and space.
 Plane Geometry is about (2-dimensional)
flat shapes like lines, circles and triangles
 Solid Geometry is about solid
(3-dimensional) shapes like spheres
and cubes
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for Mathematics Learning

10.  Shapes or groups of shapes can be 10
Big Ideas
 Shapes or groups of shapes can be
classified by their properties (attributes)
 Shapes can be described in terms of their
location and viewed from different
perspectives; geometric figures can be
moved in a plane without changing their
size or shape
Partners
for Mathematics Learning

11. What Are Attributes?  What other attributes in the real world do11
What Are Attributes?
 Attributes are characteristics of an object
such as color, shape, size, and texture
 What other attributes in the real world do
students need to recognize?
Partners
for Mathematics Learning

12. Plane Geometry  Plane geometry is about shapes12
Plane Geometry
 Plane geometry is about shapes
like lines, circles and triangles that can be
drawn on a flat surface called a plane
(think of an endless piece of paper)
 A two dimensional shape has
 Width
 Breadth
 But no thickness
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for Mathematics Learning

13. Two-Dimensional Shapes13
Two-Dimensional Shapes
 Shapes or forms drawn on flat surfaces
such as paper have different attributes
 Most of the shapes have sides and corners
 The circle and oval do not have any
corners and only one curved side
 Different shapes have different numbers of
sides and corners and form different
patterns
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for Mathematics Learning

14.  Polygons  Not polygons C Polygons, Not Polygons A B Partners 14
for Mathematics Learning

15. Polygons: 2-Dimensional Shapes15
Polygons: 2-Dimensional Shapes
 A polygon is a closed, plane figure
formed by three or more line segments
called sides
 Polygons are many-sided figures and
named by the number of sides they have
 If all sides of a polygon are congruent and
all angles are congruent, then the polygon
is referred to as a regular polygon
Partners
for Mathematics Learning

16. Quick Images in the Classroom16
Quick Images in the Classroom 
Flash an image for 3 seconds
Remove or cover the image
Draw a sketch of what you saw
Flash the image again, for revision and cover
Show the image again and leave it visible
Participants revise or complete drawings
Discuss the mental images using attributes,
properties and spatial relationships
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for Mathematics Learning

17. 17
Quick Images
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for Mathematics Learning

18. What Did You See?
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for Mathematics Learning

19. 19
Quick Image
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for Mathematics Learning

20. What Did You See?
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for Mathematics Learning

21. Triangles or Not Triangles21
Triangles or Not Triangles
 Identify triangles below
 Use properties of triangles to explain
your reasoning
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for Mathematics Learning

22. Triangles or Not Triangles22
Triangles or Not Triangles
 Select a triangle and a shape that
is a non-example
 How are the shapes alike? Different?
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for Mathematics Learning

23. Building Triangles  Using straws and pipe cleaners, work with23
Building Triangles
 Using straws and pipe cleaners, work with
your partner to construct different triangles
 Triangle with a 3”, a 4” and a 5 “ straw
 Triangle with two sides equal
 Triangle with all sides equal
 What would you want your students to
notice about all of the triangles? How are
they alike? How are they different?
Partners
for Mathematics Learning

24.  Although a vertex and an angle are related 24
Vertices and Sides
 Although a vertex and an angle are related
they are not the same thing
 A vertex (corner) is the actual point where
two rays intersect (meet)
 Two sides of polygons meet at a vertex;
the measure of angles that are formed
varies for different polygons
 The plural or vertex is vertices
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for Mathematics Learning

25. What Is an Angle?  The arms of angles are called25
What Is an Angle?
 The arms of angles are called
rays - two rays come from one
endpoint (vertex) to form an angle
 Rays of an angle can extend indefinitely
 The measure of the angle is the amount
of turning between the two rays (sides) as
measured in degrees
Partners
for Mathematics Learning

26. Angle Measures  A right angle is an angle which measures 90°26
Angle Measures
 A right angle is an angle
which measures 90°
 An obtuse angle is one that is
more than 90°but less than 180°
 An acute angle is less than 90°
 A straight angle is 180°
 A complete circle is 360°
Partners
for Mathematics Learning

27. Angles  Measures of angles are related to the27
Angles
 Measures of angles are related to the
number of degrees around the point that
is the center of a circle
 Angles can be named by the rays and the
vertex, i.e., angle abc a c b
Partners
for Mathematics Learning

28. Modeling Angles (Turns)28
Modeling Angles (Turns)
 Stand and face straight ahead
 Demonstrate a quarter turn to the right (a
90˚ turn - called a right angle)
 If you continue turning in the same
direction, how many 90˚ turns to the
starting point?
 Is there such a thing as a left angle?
 Model right, acute, and obtuse angles with
your arms
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for Mathematics Learning

29.  An angle can be viewed through rotation 29
Turns and Rotations
 An angle can be viewed through rotation
 The amount of rotation is the measure of
the angle
 A circle represents a full rotation (360
degrees; a line represents half of a
rotation (180 degrees)
 How can you know when you have
turned a sufficient amount to model an
obtuse angle (greater than 90˚ and less
than 180˚)?
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for Mathematics Learning

30.  Use arms and hands to form angles 30
Classroom Angle Model
 Use arms and hands to form angles
designated by a teacher or student
 Use popsicle sticks, coffee stirrers,
toothpicks, elastic strips to construct angles
and model angles
 Ask students to draw a picture/design using
a designated number of each type of angle
 On a given picture, trace acute angles in blue,
obtuse angles in green, and right angles in red
Partners
for Mathematics Learning

31. Exploring Angles  Trace the angles in your triangles with your31
Exploring Angles
 Trace the angles in your triangles with your
finger; identify angles
 How might you test to determine if an angle is
a right angle?
 Identify angles that are greater or less than a
right angle?
 The interior angles of a triangle always add
up to 180°
Partners
for Mathematics Learning

32. Angle Search  Use an index card as a right32
Angle Search
 Use an index card as a right
angle measuring tool
 Complete the Angle Hunt chart
 Find 3 or more objects that fit
in each category
Judy’s right angle
measuring tool
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for Mathematics Learning

33. Classifying Triangles33
Classifying Triangles
 Triangles are classified by their angles and
by their sides
 Angles: acute – less than 90°
obtuse – greater than 90°
 Sides: scalene – no sides equal
isosceles – at least two sides equal
equilateral – all sides equal
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for Mathematics Learning

34. Parallel Lines  Parallel lines go in the same34
Parallel Lines
 Parallel lines go in the same
direction in the same plane
and never intersect
 With partners, list other examples of parallel
lines students might find in the environment
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for Mathematics Learning

35. F H Perpendicular Lines  Two lines are perpendicular if they meet at35
Perpendicular Lines
 Two lines are perpendicular if they meet at
right angles
 Find lines that are perpendicular
 Where do you find right angles?
T L F H
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for Mathematics Learning

36. Constructing Line Segments36
Constructing Line Segments
 Use geoboards to construct examples
of parallel, intersecting and
perpendicular line segments
 Record and label each example
on geodot paper
Partners
for Mathematics Learning

37.  Identify the properties of a quadrilateral 37
Quadrilaterals
 Identify the properties of a quadrilateral
 Record the likenesses and differences of
two shapes on the slide
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for Mathematics Learning

38. Constructing Quadrilaterals38
Constructing Quadrilaterals
 Construct three different four-sided
polygons (cut straws as needed)
 Construct one polygon with more than
four sides
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for Mathematics Learning

39. Comparing Quadrilaterals39
Comparing Quadrilaterals
Trace two shapes on white paper
 Using a T-chart, record common attributes as
well as differences in the two shapes
Same
Different
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for Mathematics Learning

40. True or False  All squares are rectangles but all40
True or False
 All squares are rectangles but all
rectangles are not squares
 Write three more true/false statements
about quadrilaterals
 Share with your partner and pick one to
share with the group
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for Mathematics Learning

41. Properties Define Shapes41
Properties Define Shapes
 Shapes can be grouped by properties
 Properties that identify classes of
geometric shapes remain the same even
when a shape is oriented differently in
space, enlarged, or shrunk proportionally
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for Mathematics Learning

42.  Geometric definitions are based on42
Two-Dimensional Shapes
 Geometric definitions are based on
identifying a set of shapes that have
common properties; properties determine
how shapes are classified
 Are all two-dimensional
shapes, polygons?
Partners
for Mathematics Learning

43. Reasoning by Properties43
Reasoning by Properties
 Students are reasoning by properties when
they identify features shared by different
shapes
 Properties refer to specific geometric
features of shapes
 Number of sides
 Length of sides
 Size of angles of a polygon
Partners
for Mathematics Learning

44. Growing Polygons  The Greedy Triangle  Students can model the44
Growing Polygons
 The Greedy Triangle
 Students can model the
story of the triangle’s
adventure and can see
the relationship of one
polygon to others
Partners
for Mathematics Learning

45. Patterns and Relationships45
Patterns and Relationships
 What do you
know about
polygons based
on the patterns
found on the
table?
know about a
decagon?
Partners
for Mathematics Learning

46. Constructing Polygons46
Constructing Polygons
 Record all constructions on geodot
paper, labeling figure and angles
 On a geoboard construct a right triangle
with one vertex at 1,0 and another at 2,3
 Where might be the third vertex be?
 Construct a figure with no right 3 1
Partners
for Mathematics Learning
angles. Name the points
 Construct a figure with three
right angles and 7 sides

47. 2 - Dimensional Shapes  Consider shapes A, B, C, and D47
2 - Dimensional Shapes
 Consider shapes A, B, C, and D
 How are the figures alike and how are
they different?
 Number of sides
 Number of angles
 Size of angles C A
 Length of sides D B
Partners
for Mathematics Learning

48. Polygon to Polygons? not okay okay48
Polygon to Polygons?
 Cut one square in half on the diagonal to
make two congruent triangles
 What are ways to put two triangles
together following the rule: all sides must
be adjacent to another side
not okay
okay
Partners
for Mathematics Learning

49. Combining Four Triangles49
Combining Four Triangles
 As a table group, cut 30 squares
on the diagonal
 Find all possible arrangements when
combining 4 triangles ( 2 of each color)
 Each triangle must touch one side of
another triangle – sides must be the same
length and match exactly (congruent)
 Tape together each 4 triangle shape
Partners
for Mathematics Learning

50. Graphic Representation50
Graphic Representation
 Sort the four-triangle shapes
into columns on a class graph
 Discuss properties of shapes by identifying
number of sides
 Length of sides
 Number of angles
 Size of angles
 Select two shapes to compare and contrast
Partners
for Mathematics Learning

51.  Distribute pattern blocks to each group  Explore ways to make51
Pattern Blocks Triangles
 Distribute pattern blocks to each group
 Explore ways to make
similar triangles with
the pattern blocks
 Identify the number
of green triangles
it would take to make
each of these figures?
Partners
for Mathematics Learning

52. Quadrilateral Sort  Select one way to sort quadrilaterals52
Quadrilateral Sort
 Select one way to sort quadrilaterals
 With all sides the same length
 With all angles the same measure
 With all sides the same length; angles not the
same measure
 with all angles the same measure; not all sides
the same length
Is there a different way you might describe or
name the figures in your sort of quadrilaterals?
Partners
for Mathematics Learning

53. Triangle Sort  Select one way to sort triangles53
Triangle Sort
 Select one way to sort triangles
 With all sides the same length
 With all angles the same measure
 With all sides the same length; angles not the
same measure
Is there a different way you might describe
the figures in your sort of triangles?
Partners
for Mathematics Learning

54. Sorting Squares  What are different classifications54
Sorting Squares
 What are different classifications
in which to include a square?
 For each different group that includes a
square, find one or more different shape
cards that also match the rule
Partners
for Mathematics Learning

55. Reasoning by Properties55
Reasoning by Properties
 As geometric reasoning develops,
students begin to identify relationships
among attributes; that is they begin to
“reason by properties”
(Fox, 2000)
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for Mathematics Learning

56. Big Idea  Shapes can be described in56
Big Idea
 Shapes can be described in
terms of their location and viewed
from different perspectives
 Coordinate systems can be used to
describe locations precisely
Partners
for Mathematics Learning

57. Coordinate Grids  Perpendicular lines; axes,57
Ordered pairs
(3,4) and (4,3)
refer to two
different
coordinates
Coordinate Grids
 Perpendicular lines; axes,
labels like number lines
 Numbers are used to locate
points, ordered pairs of
numbers on a plane
 The point where the x -axis
and y -axis intersect is the
origin (0,0) The x -coordinate
comes first, followed
by the y -coordinate
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for Mathematics Learning

58. Using a Coordinate Grid58
Using a Coordinate Grid
 Add one more
point on the grid
and connect to
create a triangle
 Name the points
to locate the
triangle 1 2 3 4 5 6 7 8
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for Mathematics Learning

59. Tic-Tac-Toe  Use ordered pairs to  Label the y axis red59
Tic-Tac-Toe
o o x
 Use ordered pairs to x x x
locate 4 points in a
row diagonally,
vertically or
horizontally
 Label the y axis red
and the x axis green
 Record coordinates
that create 4 in a row 1 2 3 4 5 6 7 8
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for Mathematics Learning

60. Vocabulary Opportunities60
Vocabulary Opportunities
 Students become familiar with and develop
understanding of vocabulary through many
opportunities to hear and use words their
meanings in context, discussions, and
writing
 Class chart
 Frayer model
 Math journal
 Other ideas?
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for Mathematics Learning

61. 61
Frayer Model
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for Mathematics Learning

62. Ways to Build Vocabulary62
Ways to Build Vocabulary
 Because “sense-making” often precedes
vocabulary, provide repeated experiences
with the same terms and concepts
 Teachers use the correct mathematical
terms in “context”
 Explain meanings of vocabulary to students
informally, not with formal definitions
 Through conversations encourage everyone
to agree on “classroom” vocabulary
Partners
for Mathematics Learning

63. Communication and Reasoning63
Process Standards:
Communication and Reasoning
 Why provide opportunities for students to
write and talk about mathematics?
 ...”If mathematics were treated as a native
language rather than a foreign language,
using repeated exposure and immersion,
everyone could learn a significant amount
of mathematics”
Zalman Usiskin, 1996
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for Mathematics Learning

64. Polygons - A Closer Look64
Polygons - A Closer Look
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for Mathematics Learning

65. 2-Dimensional to 3-Dimensional65
2-Dimensional to 3-Dimensional
 Polyhedron , means "many faces”
 Trace faces of a variety of boxes
 Draw pictures of 3-D shapes
Face – 3-D term
A flat surface on a
polyhedron
Faces are polygons
Edge – 3-D term
formed where two faces
coincide
Partners
for Mathematics Learning

66.  Interpreting 2-D representations of 3-D66
Why Study Geometry?
 Interpreting 2-D representations of 3-D
models becomes increasingly important as
tools (such as computer-aided design)
become more visual
 Visual skills are central to many
occupations including scientists, architects,
artists, engineers, land developers, etc.
 Home: building a fence, planning a garden,
arranging furniture, decorating with art
Partners
for Mathematics Learning

67.  According to the theories of two Dutch67
Van Hiele Theory
 According to the theories of two Dutch
educators, the development of geometric
thinking progresses through 5 levels,
sequentially
 Students must pass through all prior
levels to arrive at any specific level.
 Understanding the levels at which
students are functioning impacts planning
of appropriate tasks and investigations
Partners
for Mathematics Learning

68. Van Hiele Theory  By third grade, students typically shift from68
Van Hiele Theory
 By third grade, students typically shift from
identifying attributes to developing
understanding of properties, to recognizing
interrelationships among properties
 This anticipates informal
deductive reasoning
Geometric experience is
the most important factor
that contributes to
growth in geometric
thinking
Van Hiele Theory
Partners
for Mathematics Learning

69. Essential Standards  Look at the grade 3 Essential69
Essential Standards
 Look at the grade 3 Essential
Standards and Objectives
 Talk with your neighbor about how
your instructional plans may change
because of the emphasis
in geometry
Partners
for Mathematics Learning

70.  Spatial reasoning and geometric70
Why Study Geometry?
 Spatial reasoning and geometric
explorations are important forms of problem
solving, a major reason for studying math
 The ability to look at situations visually,
geometrically, and analytically makes
students better problem solvers
 Geometry plays a major role in
measurement, ratio and proportion,
and fraction concepts
Partners
for Mathematics Learning

71. Geometry … “is grasping space…. that space in which71
Geometry …
“is grasping space…. that space in which
the child lives, breathes and moves. The
space that the child must learn
to know, explore, conquer, in
order to live breathe, and move
better in it”
Freudenthal, Mathematics as an Educational Task
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for Mathematics Learning

72. Renee Cunningham Kitty Rutherford Robin Barbour Mary H. Russell 72
DPI Mathematics Staff
Everly Broadway, Chief Consultant
Renee Cunningham Kitty Rutherford
Robin Barbour Mary H. Russell
Carmella Fair Johannah Maynor
Amy Smith
Partners for Mathematics Learning is a Mathematics-Science
Partnership Project funded by the NC Department of Public Instruction.
Permission is granted for the use of these materials in professional
development in North Carolina Partners school districts.
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for Mathematics Learning

73. PML Dissemination Consultants73
PML Dissemination Consultants
Susan Allman
Julia Cazin
Ruafika Cobb
Anna Corbett
Gail Cotton
Jeanette Cox
Leanne Daughtry
Lisa Davis
Ryan Dougherty
Shakila Faqih
Patricia Essick
Donna Godley
Cara Gordon
Tery Gunter
Barbara Hardy
Kathy Harris
Julie Kolb
Renee Matney
Tina McSwain
Marilyn Michue
Amanda Northrup
Kayonna Pitchford
Ron Powell
Susan Riddle
Judith Rucker
Shana Runge
Yolanda Sawyer
Penny Shockley
Pat Sickles
Nancy Teague
Michelle Tucker
Kaneka Turner
Bob Vorbroker
Jan Wessell
Daniel Wicks
Carol Williams
Stacy Wozny
Partners
for Mathematics Learning

74. 2009 Writers Partners Staff Kathy Harris Rendy King Tery Gunter74
2009 Writers
Partners Staff
Kathy Harris
Rendy King
Tery Gunter
Judy Rucker
Penny Shockley
Nancy Teague
Jan Wessell
Stacy Wozny
Amanda Baucom
Julie Kolb
Freda Ballard, Webmaster
Anita Bowman, Outside Evaluator
Ana Floyd, Reviewer
Meghan Griffith, Administrative Assistant
Tim Hendrix, Co-PI and Higher Ed
Ben Klein , Higher Education
Katie Mawhinney, Co-PI and Higher Ed
Wendy Rich, Reviewer
Catherine Stein, Higher Education
Please give appropriate credit to the Partners
for Mathematics Learning project when using the
materials.
Jeane Joyner, Co-PI and Project Director
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for Mathematics Learning

75. Grade Three PARTNERS for Mathematics Learning Module 3 Partners 75