1. for Mathematics Learning1
PARTNERS
for Mathematics Learning
Grade Five
Module 4
Partners
for Mathematics Learning

2. Big Ideas in Geometry  Shapes or groups of shapes can be2
Big Ideas in Geometry
 Shapes or groups of shapes can be
classified by their properties
 Two-dimensional shapes are combined
to make three-dimensional shapes
 Area, perimeter, and volumes are
examples of measurable attributes in
geometry
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for Mathematics Learning

3.  Shapes can be described in terms of their3
Big Ideas in Geometry
 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
 Coordinate systems can be used to
describe locations precisely
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for Mathematics Learning

4. Shapes & Properties Attributes Classifying Developing Refining4
Shapes & Properties
Refining
Concepts
Exploring
Attributes
of Shapes
Classifying
Shapes
Developing
Vocabulary
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for Mathematics Learning

5. V an H iele L evels of G eometric T hought5
V an H iele L evels of G eometric T hought 
Level 0
Level 1
Level 2
Level 3
Visualization (Recognition)
Description/Analysis
Abstract/Relational
Formal Deduction (high school
geometry)
 Level 4 Rigor (college level geometry)
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for Mathematics Learning

6. Shapes and Properties  Look at this shape6
Shapes and Properties
 Look at this shape
 What are some of its properties?
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for Mathematics Learning

7. Shapes and Properties  Find these properties in your environment:7
Shapes and Properties
 Find these properties in your environment:
 Parallel lines
 Right angles
 Shapes with “dents” (concave)
 Solids like a cylinder
 Solids like a pyramid
 Shapes with rotational symmetry
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for Mathematics Learning

8. What is a Polygon?  All of these are Polygons8
What is a Polygon?
 All of these are Polygons
 None of these are Polygons
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for Mathematics Learning

9. What is a Polygon?  Which of these is a Polygon?9
What is a Polygon?
 Which of these is a Polygon?
 What attributes does a polygon have that
makes it a polygon?
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for Mathematics Learning

10.  Create a polygon to fit each description10
Constructing Polygons
 Create a polygon to fit each description
 4 sides and 4 right angles
 3 sides and 1 right angle
 5 sides
 12 sides
 4 sides with exactly 2 sides parallel
 4 sides with 2 pairs of sides perpendicular
 3 sides with 2 sides perpendicular

11. Regular and Irregular Polygons11
Regular and Irregular Polygons
 These are regular polygons
 These are irregular polygons
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for Mathematics Learning

12. Triangles  Sort the triangles into three groups so that12
Triangles
 Sort the triangles into three groups so that
no two triangles belong in more than one
group
 Write a description of each group
 Now sort the triangles again into three
different groups so that no triangle belongs
in two groups
 Write a description of each of these groups
Partners
for Mathematics Learning

13. Triangles  Fill in the chart below with a sketch of a13
Triangles
 Fill in the chart below with a sketch of a
triangle that fits both labels
 Are any impossible ones?
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for Mathematics Learning

14. Quadrilaterals  Find all the quadrilaterals Sort them into groups14
Quadrilaterals 
Find all the quadrilaterals
Sort them into groups
Are there overlaps?
Draw a Venn diagram to sort the shapes
Can you find a different way to sort them?
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for Mathematics Learning

15. Diagonals in Quadrilaterals\ 15
Diagonals in Quadrilaterals
 Draw the diagonals in each quadrilateral
 What properties can you identify in the
diagonals?
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for Mathematics Learning

16. Diagonals in Quadrilaterals16
Diagonals in Quadrilaterals
Properties of
Diagonals
Parallelogram
Rectangle
Rhombus
Square
Trapezoid
Kite
form
congruent
triangles
bisecteach
other
are
Are
perpendicular
Bisect
opposite
angles
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for Mathematics Learning

17. Diagonals in Quadrilaterals17
Diagonals in Quadrilaterals
Properties of
Diagonals
Parallelogram
Rectangle
Rhombus
Square
Trapezoid
Kite
form
congruent
triangles
2opposite
pairs
yes
1pair
2pairs
(adjacent)
bisecteach
other
are
Are
perpendicular
Bisect
opposite
angles
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for Mathematics Learning

18. Diagonals in Quadrilaterals18
Diagonals in Quadrilaterals
 How many diagonals in these
quadrilaterals?
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for Mathematics Learning

19. Diagonals in Quadrilaterals19
Diagonals in Quadrilaterals
 Concave quadrilaterals do have 2
diagonals.
 What is different?
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for Mathematics Learning

20. Quadrilaterals  Look at this set of parallelograms.20
Quadrilaterals
 Look at this set of parallelograms.
 What are its properties of sides? angles?
symmetry? diagonals?
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for Mathematics Learning

21. Quadrilaterals  Classify the quadrilaterals by labeling the21
Quadrilaterals
 Classify the quadrilaterals by labeling the
parts of this Venn diagram:
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for Mathematics Learning

22. 22
Quadrilaterals
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for Mathematics Learning

23. True or False?  If it is a square, it is also a rhombus23
True or False? 
If it is a square, it is also a rhombus
Some parallelograms are rectangles
All rectangles are squares
If it has exactly two lines of symmetry, then
it must be a quadrilateral
 No triangles have diagonals
 All triangles have 3 congruent sides
 All trapezoids have exactly 2 parallel sides
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for Mathematics Learning

24. True or False?  If it is a square, it is also a rhombus. T24
True or False? 
If it is a square, it is also a rhombus. T
Some parallelograms are rectangles. T
All rectangles are squares. F
If it has exactly two lines of symmetry, then it
must be a quadrilateral. F
 All triangles have no diagonals. T
 All triangles have 3 congruent sides. F
 All trapezoids have exactly 2 parallel sides. T
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for Mathematics Learning

25. Algebra Connection: Diagonals25
Algebra Connection: Diagonals
 How many diagonals in a triangle?
 a quadrilateral?
 A pentagon?
 Any polygon?
 How many diagonals in a polygon with n
sides?
 What’s the rule?
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for Mathematics Learning

26. Angles  What are angles?  What makes conceptualizing the size of an26
Angles D E B A F
 What are angles?
 What makes conceptualizing the size of an
angle challenging for students?
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for Mathematics Learning

27. Angles  “The protractor is one of the most poorly school.”27
Angles
 “The protractor is one of the most poorly
understood measuring instruments in
school.”
John Van de Walle
 Why is measuring the size of an angle so
difficult for students?
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for Mathematics Learning

28. Challenges of a Protractor28
Challenges of a Protractor
 Units (degrees) are very small
 No angles are shown on protractor; only
little marks around the edge
 Numbers go both clockwise and
counterclockwise on a typical protractor
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for Mathematics Learning

29. Angle Size  Students need to practice telling the29
Angle Size
 Students need to practice telling the
difference between a small and a large
angle prior to measuring angles
 What activities might provide this?
Partners
for Mathematics Learning

30. Reading a Protractor  How do you help students understand how30
Reading a Protractor
 How do you help students understand how
to use a typical protractor?
 What experiences need to come before
students try to use a protractor?
Partners
for Mathematics Learning

31. Angle Units  Use a straightedge to draw a narrow angle  Cut it out31
Angle Units
 Use a straightedge to draw a narrow angle
on your card
 Cut it out
 Use the wedge as a unit to measure
angles, counting the number of wedges
that fit into a particular angle
Partners
for Mathematics Learning

32. Making a Protractor  Fold the piece of waxed paper in half, creasing32
Making a Protractor
 Fold the piece of waxed paper in half, creasing
the fold tightly
 Fold in half again so that the folded edges
match
 Fold along the diagonal from the folded corner
 Fold again from the folded corner to bring
together the two sides that form the folded
corner
 Cut or tear off the edge about 4-5 inches from
the vertex and unfold
Partners
for Mathematics Learning

33. Making a Tool for Measuring Angles33
Making a Tool for Measuring Angles
 Compare your waxed paper tool to a
traditional protractor
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for Mathematics Learning

34. Measuring Angles  Make your answer card34
Measuring Angles
 Make your answer card
 Decide on the angle you want to use
 Draw an angle on the index card,
measuring very carefully
 Plan good “incorrect” answers
 Place the answer choices in appropriate
locations
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for Mathematics Learning

35. Measurement and Geometry35
Measurement and Geometry
 How does geometry overlap with
measuring angles?
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for Mathematics Learning

36.  “Perhaps the biggest error in measurement36
Measurement
 “Perhaps the biggest error in measurement
instruction is the failure to recognize and
separate two types of objectives: first,
understanding the meaning and technique
of measuring a particular attribute unit and,
second, learning about the standard units
commonly used to measure that attribute.”
John Van de Walle, Teaching Student-Centered Mathematics
Partners
for Mathematics Learning

37. Triangles  Use a straight edge to make a large37
Triangles
 Use a straight edge to make a large
triangle
 Place a visible dot on each vertex
 Rip off each of the angles
 Carefully join the angles at the dots and
tape or glue them down
 What do you notice?
Partners
for Mathematics Learning

38. Angle Sums  Measure each angle of each polygon38
Angle Sums
 Measure each angle of each polygon
 Record your findings in the appropriate
chart
 Find the sum of the angles for each figure
 Compare your results to those of your
classmates
 What conclusion(s) about the sum of the
angles do you draw from the results?
Partners
for Mathematics Learning

39. Angle Sums  The sum of the angle measures of a triangle is 180 °39
Angle Sums
 The sum of the angle measures of a
triangle is 180 °
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for Mathematics Learning

40. Angle Sums  The sum of the angle measures of any40
Angle Sums
 The sum of the angle measures of any
quadrilateral is 360 °
 Any quadrilateral can be divided into two
triangles, each with an angle sum or 180 °
Partners
for Mathematics Learning

41. Angle Sums  Divide a pentagon into triangles41
Angle Sums
 Divide a pentagon into triangles
 Use only diagonal lines and do not cross any
lines
 How many triangles did you make?
 How does that impact the sum of the angles of
the pentagon?
 Do the same with a hexagon
 What is the pattern?
Partners
for Mathematics Learning

42. Describe the Polygon  Create a “Wanted” poster for one of the42
Describe the Polygon
 Create a “Wanted” poster for one of the
“culprits” below:
 A right scalene triangle
 A rhombus
 A trapezoid
 An obtuse isosceles triangle
 A regular pentagon
Partners
for Mathematics Learning

43. Creating Nets for 3-D Shapes43
Creating Nets for 3-D Shapes
 A pentomino is made from 5 congruent
squares, with squares touching only by a
whole side (“edge to edge construction”)
 Make as many different pentomino shapes
as you can. How many can you make?
OK Not OK
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for Mathematics Learning

44. Creating Nets for 3-D Shapes44
Creating Nets for 3-D Shapes
 Are these pentominoes different or
congruent?
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for Mathematics Learning

45. Creating Nets for 3-D Shapes45
Creating Nets for 3-D Shapes
 Which of the pentomino pieces can be
folded into a topless box?
 What rectangles can be created from the
pentomino pieces that can be folded into a
topless box?
Partners
for Mathematics Learning

46. Area and Perimeter  What is the relationship46
Area and Perimeter
 What is the relationship
between the area of the
T-shaped net and the surface area of the
topless box it creates?
 Are the perimeter of the net and the sum of
the lengths of the edges of the 3-D topless
box also equivalent?
Partners
for Mathematics Learning

47. Volume: Rectangular Prisms47
Volume: Rectangular Prisms
 Use centimeter cubes to create rectangular
prisms
 How many different prisms can you make
with these cubes: 24 32 36
 Record the dimensions of each prism that
you make and the total number of cubes
used
Partners
for Mathematics Learning

48. Connections  Which of the process standards did we48
Connections
 Which of the process standards did we
use?
 Problem Solving
 Reasoning and Proof
 Communication
 Connections
 Representation
 In Grade 5 what are strong connections
between geometry and measurement?
Partners
for Mathematics Learning

49. 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

50. 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

51. 2009 Writers Partners Staff Kathy Harris Freda Ballard, Webmaster
Rendy King
Tery Gunter
Judy Rucker
Penny Shockley
Nancy Teague
Jan Wessell
Stacy Wozny
Amanda Baucom
Julie Kolb
Partners Staff
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
Partners
for Mathematics Learning

52. for Mathematics Learning52
PARTNERS
for Mathematics Learning
Grade Five
Module 4
Geometry
Partners
for Mathematics Learning