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

2. Equipartitioning  Equipartitioning means What experiences2
Equipartitioning
 Equipartitioning means
“Sharing Fairly”
What experiences
have students had
with sharing fairly?
Partners
for Mathematics Learning

3. What Happens in K and 1 st Grades?3
What Happens in K and 1 st Grades?
 Share collections of objects fairly and reassemble
 Kindergarten: share between 2 people
 1 st grade: share between 2-6 people
 Share rectangles and circles among two or four
people and reassemble
 1 st grade
Partners
for Mathematics Learning

4. What Happens in 2nd Grade?4
What Happens in 2nd Grade?
 Look at the Essential Standards
 Fairly share collections and reassemble them
 Fairly share a rectangle or circle among two,
four, eight; three and six people and
reassemble it
 Name the share as “1/nth of the whole”
 Predict that the size of a fair share decreases
as the number of shares increases, and vice
versa
Partners
for Mathematics Learning

5. Equipartitioning  Parts of a whole  Continuous or area models5
Equipartitioning
 Parts of a whole
Poland
France
 Continuous or area models
 Equal shares by sub-dividing a whole
Partners
for Mathematics Learning

6. Equipartitioning  Parts of a group  Discrete or set models6
Equipartitioning
 Parts of a group
 Discrete or set models
 Dividing into subsets of equal size (equal
numbers of elements)

7. Equipartitioning, Division, Fractions7
Equipartitioning, Division, Fractions 
All focus on equal portions
Understanding the “whole” is important
Often referred to as “fair shares”
Models may look different depending on
the context
Partners
for Mathematics Learning

8. Equipartitioning, Division, Fractions8
Equipartitioning, Division, Fractions
 How could 2 people fairly share the cake?
The gumballs?
 How could you divide the cake and the
gumballs into 2 equal parts?
 Show 1/2 of the cake and 1/2 of the
gumballs
Partners
for Mathematics Learning

9. Division as Equipartitioning9
Division as Equipartitioning
 Get at least 15 two-color counters
 Use them to solve these problems:
 Larry and 3 friends are sharing 12 cupcakes.
How many cupcakes will each child get if each
gets a fair share?
 Larry, Mary, Devin and Martez each have 3
brownies. How many brownies did Larry’s
mom bake?
Partners
for Mathematics Learning

10. Student Thinking  Tom, Renee, and Sara collected some10
Student Thinking
 Tom, Renee, and Sara collected some
apples. Each person got five apples. What
was the total number of apples collected?
 What conceptual understanding related to
equipartitioning do students have?
 How do we start to build understanding of the
relationships between the whole and the equal
parts?
Partners
for Mathematics Learning

11. Relationships : Wholes and Parts11
Relationships : Wholes and Parts
 Needs to be made explicit
 Whole was 15 apples
 5 apples represented 1/3 of the whole
 5 represented a fair share for 3 people
 Bringing together each of the three thirds
makes the whole
 If 5 people shared the same whole,
the fair share would be smaller
Partners
for Mathematics Learning

12. Classroom Problems  Write several problems for 2 nd graders that12
Classroom Problems
 Write several problems for 2 nd graders that
address division
as equipartitioning
 Ideas to include
 Equal sized groups
 Determine initial size of a group given
the equal parts
 Describe what happens when collections
are divided fairly and what happens
when reassembled
Partners
for Mathematics Learning

13. Considering Set Models13
Considering Set Models
 The whole is the total set of objects and
subsets of the whole are fractional parts
 Number of objects not size is important
 Examples: counters, people, M & Ms,
any discrete objects
Partners
for Mathematics Learning

14. Fair Shares: Rectangles14
Fair Shares: Rectangles
 Fold a square in half
 Cut your square
 How do you know each half is equal? Talk
with a neighbor
 Is your half equivalent to your neighbor’s?
 What key ideas do you want 2 nd graders to
understand in this activity?
Partners
for Mathematics Learning

15. Key Ideas  Each part is one-half of the whole square15
Key Ideas 
Each part is one-half of the whole square
The two parts make the square
Why are these not halves (non-examples)?
Why may some students think they are?
 If we started with different size squares,
would the one-halves be equivalent?
Partners
for Mathematics Learning

16. 16
One Half? How Many Ways?
Partners
for Mathematics Learning

17. 17
One Half? How Many Ways?
Partners
for Mathematics Learning

18. Rectangles and Circles18
Rectangles and Circles
 2nd graders share
rectangles and circles
among 2, 4, 8, 3, and 6 people
 What experiences do we need to provide
students?
 Context
 Models
Partners
for Mathematics Learning

19. 19
People Fractions
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for Mathematics Learning

20. 20
Goofy
Fractions
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for Mathematics Learning

21. Context  Maria’s mom baked  How can Maria share the brownies with 322
Context
 Maria’s mom baked
a pan of brownies
 How can Maria share the brownies with 3
friends and herself?
Partners
for Mathematics Learning

22. Get to 30 Play the game What mathematics is addressed by the game?
Partners
for Mathematics Learning

23. Contextual Problems  With a partner brainstorm problems that23
Contextual Problems
 With a partner brainstorm problems that
match the Essential Standard:
“Explain the division of rectangles and
circles to accommodate different numbers
of people”
 Share your ideas
Partners
for Mathematics Learning

24. Making Sense of Fractions24
Making Sense of Fractions
 Students need:
 Multiple models
 Experience with
multiple contexts
 Time
 Language
Partners
for Mathematics Learning

25.  Content is new (proportional vs. additive 25
Fractions: Reasons for Difficulties
 Content is new (proportional vs. additive
reasoning) and notation is different
 Conceptual-development experiences are limited;
symbols are often emphasized
 Instruction is
 Too abstract
 Too procedural
 Without connections
 With limited models
 Without meaningful contexts

26. Fraction Research Students’ difficulties in understanding26
Fraction Research
Students’ difficulties in understanding
fractions stem from learning fractions by
using rote memorization of procedures
without a connection with informal ways
of solving problems involving fractions.
Steffe and Olive, 2002
Partners
for Mathematics Learning

27. No Symbols for Fractions27
In 2nd Grade:
No Symbols for Fractions 1 2
Partners
for Mathematics Learning

28. Math & Literature  What happens to each child’s share28
Math & Literature
 What happens to each child’s share
as the doorbell rings?
 What happens to each
child’s share when
Grandma arrives?
Partners
for Mathematics Learning

29. Compensatory Principle29
Compensatory Principle
 The principle states that the bigger the
unit, the smaller the number of that unit
needed
 1/8 is fewer than 1/4; 1/4 is few cookies
than 1/2 of the cookies
 What experiences might lead students in
developing an understanding of this idea?
Partners
for Mathematics Learning

30. When Concepts Are Not Well-Developed…  NAEP Question:30
When Concepts Are Not
Well-Developed…
 NAEP Question:
Estimate the sum of 12/13 and 7/8
a. 1
b. 2
c. 19
d. 21
 How did you decide your estimate?
 Which answer did most 13 year old
students choose?
Partners
for Mathematics Learning

31. Well-Developed…NAEP Question31
When Concepts Are Not
Well-Developed…NAEP Question
 Estimate the sum of 12/13 and 7/8
a b. 2
c d. 21
 Students look at fractions as though they
represent two separate whole numbers
 This leads to misinterpretation and an
inability to access the reasonableness of
results
Partners
for Mathematics Learning

32. Chinese Proverb “ Tell me and I'll forget; show me and I may remember;32
Chinese Proverb
“ Tell me and I'll forget;
show me and I may remember;
involve me and I'll
understand. ”
Partners
for Mathematics Learning

33. What is Measurement?  A count of how many units are needed to33
What is Measurement?
 A count of how many units are needed to
fill, cover, or match the attribute of the
object being measured
 A fundamental mathematical process
interwoven throughout all strands of
mathematics
Partners
for Mathematics Learning

34. Why Measurement?  An essential link between34
Why Measurement?
 An essential link between
mathematics and other disciplines such as
science, art, music, and social studies
 It makes mathematics real and tangible for
students giving them a handle on their
world
 What are the implications for teaching
measurement?

35. Measurement: Big Ideas35
Measurement: Big Ideas
 An object can be described and
categorized in multiple ways (attributes)
 The measurement of a specific numerical
attribute tells the number of units
 The process of measurement is similar for
all attributes, but the measurement system
and tool vary according to the attribute
 Measurements are accurate to the extent
that the appropriate unit/tool is used
properly

36. Measurement Standards36
Measurement Standards
 Look at the Essential
Standards for
Measurement
 Look at the Clarifying
Objectives
Partners
for Mathematics Learning

37. Attributes  What are the attributes of the present?37
Attributes
 What are the attributes of the present?
 Which of these are measurable?
Partners
for Mathematics Learning

38. Process of Measurement38
Process of Measurement
 Determine the attribute
to be measured
 Choose an appropriate unit
that has the same attribute
 Choose the tool
 Determine how many of that unit is needed
by filling, covering, or matching the object
Partners
for Mathematics Learning

39. A Cloak for a Dreamer Covering space with nonstandard units39
A Cloak for a Dreamer
Covering space with
nonstandard units
leaving no gaps
or overlaps
Partners
for Mathematics Learning

40. Measuring a Square  Use the pattern blocks to cover the square40
Measuring a Square
 Use the pattern blocks to cover the square
 Only use one type of pattern block
Partners
for Mathematics Learning

41. Computer Activities www.arcytech.org/java/patterns/patterns_j.shtml41
Computer Activities
Partners
for Mathematics Learning

42. Measurement Components42
Measurement Components 
Conservation
Transitivity
Partitioning
Unit Iteration
Compensatory Principle
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for Mathematics Learning

43. Components of Measurement43
Components of Measurement
 Conservation: an object maintains the
same size if it is rearranged, transformed,
or divided in various ways
Partners
for Mathematics Learning

44. Conservation Examples44
Conservation Examples
 Students explore conservation
 Take a clay ball, cut it in half. Then roll one
half into a ball and other into a snake. Do they
have the same mass?
 Pour the same amount of liquid from one
container to another. How tall is the liquid in
the container? Does the
amount stay the same?

45. Components of Measurement45
Components of Measurement
 Transitivity : two objects can be compared
in terms of a measurable attribute using a
third object
If A = B and B = C, then A = C
If A < B and B < C, then A < C
If A > B and B > C, then A > C

46. Partitioning  Larger units can be subdivided into46
Partitioning
 Larger units can be subdivided into
equivalent units…mentally subdividing
 Choose a place in the room (table, wall)
 Mentally divide it in half
 Choose a unit of measure (marker, handspan)
 Determine the measurement and double it
Partners
for Mathematics Learning

47. Components of Measurement47
Components of Measurement
 Units : the attribute being measured
dictates the type of unit used; the unit must
have the same attribute as the attribute to
be measured
Partners
for Mathematics Learning

48. Components of Measurement48
Components of Measurement
 Unit iteration : units must be repeated in
order to determine the measure
Partners
for Mathematics Learning

49. Iteration Experiences49
Iteration Experiences
 What questions might
you ask to encourage
these students to focus
on the placement of
units?
 How can you provide opportunities for
students to use iteration?
Partners
for Mathematics Learning

50. Compensatory Principle50
Compensatory Principle
 The compensatory principle states that
the bigger the unit, the smaller the number
of that unit is needed
 Turn to your partner and give an example
of what this means
 What happens when the unit used to
measure is smaller? Do you need more or
fewer units?
Partners
for Mathematics Learning

51. Compensatory Principle in Action51
Compensatory Principle in Action
 Students measure the top of the cabinet
and it is 60 cubes long
 Then they measure it in craft sticks and
find out that it is 5 sticks long
 It takes more of the smaller units (cubes) to
measure the cabinet than
the larger unit (sticks)
Partners
for Mathematics Learning

52. Measurement Estimate  Choose two spots on a surface  Mark each spot52
Measurement Estimate
 Choose two spots on a surface
 Mark each spot
 Estimate the halfway point between the
two spots and mark it
 Cut a piece of string to represent the
distance between the two spots
 Fold it in half and use it to measure the
halfway point
 How close was your estimate?
Partners
for Mathematics Learning

53. Connecting Data & Measurement53
Connecting Data & Measurement
 What took place in the Giant Step activity?
 How could iteration have been highlighted
during this activity?
 How could the compensatory principle have
been more explicit during this activity?
Partners
for Mathematics Learning

54. Capacity and Volume Capacity is the focus for primary children54
Capacity and Volume
 Capacity refers to the amount a container
will hold
 Pitchers and bottles
 Scoops, cups, quarts measure liquids
 Volume is often though of as the amount of
space an object takes up
 Non-Standard units such wooden cubes,
marbles, tennis balls teddy bear counters
Capacity is the focus for primary children
Partners
for Mathematics Learning

55. Nonstandard Units of Capacity55
Nonstandard Units of Capacity
 Containers or scoops: small containers
that are filled and poured repeatedly into
the container being measured; the number
of scoops provides the measure
 Solid units: small congruent objects that fill
of small objects provides the measure
Partners
for Mathematics Learning

56. C apacity: C ompensatory P rinciple56
C apacity: C ompensatory P rinciple
 How can the compensatory principle be
developed through the concept of capacity
in second grade?
Partners
for Mathematics Learning

57. Mass  Mass is the amount of57
Mass
 Mass is the amount of
matter in an object
 Weight is a measure of the pull or force of
gravity on an object
 On the moon there is less gravity so an
object weighs less
 The mass of an object is identical on the
moon and on Earth
Partners
for Mathematics Learning

58.  Children can use their hands to estimate58
Exploring Mass
 Children can use their hands to estimate
which of two objects is heavier
 Children place marbles on one side of a
balance scale to find the mass of a book
using marbles as a non-standard measure
Partners
for Mathematics Learning

59. Classroom Experiences59
Classroom Experiences
 Look at the measurement handouts
 Each group will work through the
Scavenger Hunt and the Measuring
Rectangles
 As you work discuss how the activity
addresses the Essential Standards
Partners
for Mathematics Learning

60. Measuring Rectangles  How would students60
Measuring Rectangles
 How would students
order the rectangles?
 What attributes do they use?
 Do they realize that
when they consider different attributes, the
rectangles may be ordered differently?
 Can they cover it with no gaps or overlaps?
 Other ideas?
Partners
for Mathematics Learning

61. Scavenger Hunt  How could you extend the activity? Partners 61
for Mathematics Learning

62. Measurement Jigsaw  Roll This Weigh Side by Side Guess and Count62
Measurement Jigsaw 
Roll This Weigh
Side by Side
Guess and Count
Paper Clip Measurement
Which Takes Longer?
Partners
for Mathematics Learning

63.  2nd Grade Essential Understanding63
Marker board
Paths
chair
bookcase
 2nd Grade Essential Understanding
 Describe relationships of objects using
proximity, position, direction, and turns
 Look at the handout, “Paths Lesson”
Partners
for Mathematics Learning

64. NCTM Applets http://illuminations.nctm.org/64
NCTM Applets
 Hiding Ladybug Adventures
 Ladybug Mazes
 Turtle Pond
Partners
for Mathematics Learning

65. Measurement  Is one of the most widely used applications65
Measurement
 Is one of the most widely used applications
of mathematics
 Connects to geometry, data, number, and
problem solving
 Experience provide opportunities to learn
and understand
Partners
for Mathematics Learning

66. Personal Reflection  Opportunities to learn and understand in an66
Personal Reflection
 Opportunities to learn
and understand in an
environment that is
interesting, challenging, and enjoyable
provide students with strong foundations
upon which to build
 What strengths does your instructional
program currently have?
 What three challenges will you take from this
module for your classroom?
Partners
for Mathematics Learning

67. Renee Cunningham Kitty Rutherford Robin Barbour Mary H. Russell 67
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.
Partners
for Mathematics Learning

68. PML Dissemination Consultants68
PML Dissemination Consultants
Susan Allman
Julia Cazin
Ruafika Cobb
Anna Corbett
Gail Cotton
Jeanette Cox
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Lisa Davis
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Donna Godley
Cara Gordon
Tery Gunter
Barbara Hardy
Kathy Harris
Julie Kolb
Renee Matney
Tina McSwain
Marilyn Michue
Amanda Northrup
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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

69. 2009 Writers Partners Staff Kathy Harris Rendy King Tery Gunter69
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
Partners
for Mathematics Learning

70. for Mathematics Learning
PARTNERS
for Mathematics Learning
Grade Two
Module 3
Partners
for Mathematics Learning