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

2. Video Overview Welcome to the first of six modules of2
Video Overview
Welcome to the first of six modules of
Partners professional development
for teachers of fifth grade
Partners
for Mathematics Learning

3. NC Essential Standards3
NC Essential Standards
 Take a brief look at the new NC Essential
Standards:
 What’s new in Number and Operations?
 What’s new in Algebra?
 What looks familiar?
Partners
for Mathematics Learning

4. Partners 2009: 5 th Grade Modules4
Partners 2009: 5 th Grade Modules
 1 Number and Operations: Number Sense
and Multiplication
 2 Number and Operations: Division and
Fractions
 3 Measurement
 4 Geometry
 5 Data; Number and Operations: Decimals
 6 Statistics (Data); Process Standards
Partners
for Mathematics Learning

5. Mathematics Learning…5
Mathematics Learning…
 Mathematics learning is about making
sense of mathematics
 Mathematics learning is about acquiring
skills and insights to solve problems
NCTM Principles and Standards 2000
Philosophy
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for Mathematics Learning

6.  What should children be learning?6
In Elementary Classrooms…
 What should children be learning?
 Big ideas of mathematics
 Content of the essential standards
 Concepts and procedures
 How do children learn?
 Through processes of reasoning,
communicating, representing ideas, making
connections, solving routine and non-routine
problems
Partners
for Mathematics Learning

7. Developing Mathematical Power7
Developing Mathematical Power
What do teachers need to do to…
 Empower students to…
 Understand mathematics
 Use mathematics
 Enjoy mathematics
 Have confidence in themselves as
mathematics students
 Design instruction around problem solving,
reasoning, and sense-making
Partners
for Mathematics Learning

8. Problem Solving  Problem solving means engaging in a task8
Problem Solving
 Problem solving means engaging in a task
for which the solution or solution path are
not known in advance
 The process of problem solving should
permeate the entire program and provide
the context in which skills and concepts
can be learned
NCTM Standards 1989, 2000
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for Mathematics Learning

9.  Solving problems is not only a goal of9
Problem Solving
 Solving problems is not only a goal of
learning mathematics but also a major
means of learning mathematics
 Choose problems that engage students
 Create environment that encourages
exploration, risk-taking, sharing, and
questioning – developing confidence in
students engaged in problem-solving activities
NCTM Principles and Standards 2000
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for Mathematics Learning

10. Foundational Ideas 10  Number sense develops…  Over time
 Through many
experiences
 Along side
operation sense
 Number “glues” all
strands together
 Number sense is
foundational to
successful problem
solving
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for Mathematics Learning
Foundational Ideas

11. Challenges Using Number Sense11
Challenges Using Number Sense
 Find ways to write the numbers 1 – 10
using exactly four 3’s
 Find ways to write the numbers 11 – 20
using exactly five 2’s
 Write the numbers 0 – 50 using exactly
five 4’s and any symbols:
+, ! × , ÷ , ( ), !
Partners
for Mathematics Learning

12. Number Sense Is. . .  A person’s understanding of number concepts,12
Number Sense Is. . .
 A person’s understanding of number concepts,
operations, applications of numbers & operations
 The ability to use this understanding in flexible
ways to make decisions and to develop useful
strategies for using numbers and operations
 The expectation that numbers are useful and that
mathematics is logical and makes sense
 The ability to use numbers and applications of
numbers to communicate, process, and interpret
information
-from McIntosh, Reys, Reys, & Hope , Number SENSE, Grades 4-6
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for Mathematics Learning

13. Students with Number Sense…13
Students with Number Sense…
 Have well-understood number meanings
 Understand multiple interpretations and
representations of numbers
 Recognize the relative and absolute
magnitude of numbers
 Appreciate the effect of operations on
numbers
 Have developed a system of personal
benchmarks
NCTM Curriculum and Evaluation Standards , 1989
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for Mathematics Learning

14. Number Relationships Can Be…14
Number Relationships Can Be…
 Compared as greater than,
less than, or equal
 Decomposed into a
combination of other numbers
 Composed with other numbers
to name a new number
 Named in different ways
 Categorized as multiples,
factors, powers, roots
Partners
for Mathematics Learning

15. Numbers in the Real World15
Numbers in the Real World
 Look in newspapers, magazines,
billboards, and other sources for numbers
in these categories:
 A number in the millions
 A prime number
 A fraction greater than ½
 A decimal greater than .75
 The factors of 48
 A mixed number
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for Mathematics Learning

16. Number Lines What are the missing values? What number is n?16
Number Lines
987
992 n
1017
What are the missing values?
What number is n?
How might you label the number line?
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for Mathematics Learning

17. Patterns in Multiples  Notice all the multiples of a given17
Patterns in Multiples
 Notice all the multiples of a given
number on the “Patterns in Multiples” Chart
 Compare the pattern each number makes
with other number patterns
 What are the similarities? What are the
differences?
 What do these patterns tell you about the
relationships of these numbers?

18. Discovering Primes and Squares18
Discovering Primes and Squares
 As a table group, cut out all rectangular
arrays that can be made with each of the
numbers 2 through 20 (a 3x4 array is the
same as a 4x3 array)
 Organize arrays and list the dimensions of
the arrays you made for each number
 What do you notice about the number of
arrays that can be made for each number?
Partners
for Mathematics Learning

19.  Are there numbers for which only one19
Discovering Primes and Squares
 Are there numbers for which only one
array can be made?
 What are these numbers called?
 What kinds of numbers have more
than two factors?
Partners
for Mathematics Learning

20. Discovering Primes and Squares20
Discovering Primes and Squares
 For which of the numbers did you have an
odd number of factors?
 How is the set of arrays for these numbers
different from other numbers?
Partners
for Mathematics Learning

21. Exploring Factors  What is the smallest possible21
Sample
Classroom
Questions
Exploring Factors
 What is the smallest possible
number that has exactly 9 factors?
 What is the smallest composite
square number?
 Find a number with more than 9 factors
 Roll come in packs of 6; hot dogs come in
packs of 8. How many packs of each
should we buy to use everything?
Partners
for Mathematics Learning

22. What Then Is Prime Factorization?22
What Then Is Prime Factorization?
 Factoring a number using only prime factors
 180 = 15 x 12
3 x 5 x 3 x 2 x 2 = 180
 Identifying all of the prime numbers that,
when multiplied together, equal the value
 30 = 3 x 2 x 5
 18 = 3 x 3 x 2 or
18 = 3 2 x 2
Partners
for Mathematics Learning

23. What Then Is Prime Factorization?23
What Then Is Prime Factorization?
 Create 2 different “strings” of factors for
each product below
 24 x 15 = 360
 21 x 12 = 252
 48 x 60 = 2,880
 What is the longest “string” you can find?
 Why may the number 1 not be used as a
factor?
Partners
for Mathematics Learning

24. What Then Is Prime Factorization?24
What Then Is Prime Factorization?
 Did you find these “strings”?
 24 x 15 = 360
• 2 x 2 x 2 x 3 x 3 x 5 = 360
 21 x 12 = 252
• 3 x 7 x 2 x 2 x 3 = 252
 48 x 60 = 2,880
• 2x2x2x2x3x2x3x2x5
 What was your strategy?
Partners
for Mathematics Learning

25. What Then Is Prime Factorization?25
What Then Is Prime Factorization?
 Factoring with “trees” builds on students’
knowledge of number facts and divisibility 24 24 3 x 8
2x4 6 4 x
2x3 x 2x2 3 x 3 x
2 x 2x2
 Determine: A fifth grade exploration or
essential standard?
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for Mathematics Learning

26. Numerical Guess My Rule26
Numerical Guess My Rule
What are the labels?
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for Mathematics Learning

27. Numerical Guess My Rule27
Numerical Guess My Rule
Why are these labels reasonable?
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for Mathematics Learning

28. Numerical Guess My Rule28
Numerical Guess My Rule
Find the Rules
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for Mathematics Learning

29. Could  I am thinking of a number Mystery Numbers29
Mystery Numbers
 I am thinking of a number
 It is greater than 5 x 10
 It is less than 100
 It is even
Could
it be
60?
Partners
for Mathematics Learning

30. It could  I am thinking of a number Mystery Numbers30
Mystery Numbers
 I am thinking of a number
 It is greater than 5 x 10
 It is less than 100
 It is even
 It is not 70 or less
 It is not a multiple of 4
 It is not a multiple of 3
Nope.
It could
not be
60!
Partners
for Mathematics Learning

31. think  I am thinking of a number Mystery Numbers31
Mystery Numbers
 I am thinking of a number
 It is greater than 5 x 10
 It is less than 100
 It is even
 It is not 70 or less
 It is not a multiple of 4
 It is not a multiple of 3
 It is less than 80
Partners
for Mathematics Learning
Let me
think

32.  I am thinking of a number32
Mystery Numbers
 I am thinking of a number
 It is greater than 5 x 10
 It is less than 100
 It is even
 It is not 70 or less
 It is not a multiple of 4
 It is not a multiple of 3
 It is less than 80
Partners
for Mathematics Learning

33. - = ? ? + Mystery Numbers  If you add 5 to my mystery number you will33
Mystery Numbers
 If you add 5 to my mystery number you will
get the same result as when you subtract
my mystery number from 89
 What is my number? = + ? - ?
Partners
for Mathematics Learning

34. Mystery Numbers  Write the mystery number clue symbolically:34
Mystery Numbers
 Write the mystery number clue symbolically:
n + 5 = 89 – n
 Try these:
23 + a = 39 – a
8 x b = 54 – b
40 – c = c + 14
Partners
for Mathematics Learning

35. Computational Fluency35
Computational Fluency
 “By the end of [grade 5] students should be
computing fluently with whole numbers….
Students exhibit computational fluency when
they demonstrate flexibility in the
computational methods they choose,
understand and can explain these methods,
and produce accurate answers efficiently.
The computational methods that a student uses
should be based on mathematical ideas that the
student understands well.”
From Principles and Standards for School Mathematics, NCTM, 2000, page 152, emphasis added
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for Mathematics Learning

36. Computational Fluency36
Computational Fluency
 “Flexibility with a variety of computational
strategies is an important tool for successful daily
living. It is time to broaden our perspective of
what it means to compute.”
 “No student should be permitted to use any
strategy without understanding it.”
John Van de Walle, Teaching Student-Centered Mathematics, Grades 3-5, pages
Partners
for Mathematics Learning

37. Computational Fluency37
Computational Fluency
 What are we trying to achieve when
we teach arithmetic today?
 Rote Calculators or
 Problem Solvers who exhibit arithmetic
fluency*
* Fluency means computing
accurately, efficiently, and flexibly
Partners
for Mathematics Learning

38. Research Is Showing…  Teaching traditional algorithms too early38
Research Is Showing…
 Teaching traditional algorithms too early
impedes the development of number sense
 Students not taught traditional algorithms
during the first 5 years of school (ages 5-11)
 Acquired and developed good number sense,
and
 Even after they were taught standard
algorithms, they preferred their own methods
From research of Alistair McIntosh, University of Tasmania, and others, reported at ICME-10, Copenagen, 2004
Partners
for Mathematics Learning

39. Research Is Showing…  Rather than algorithms, the students…39
Research Is Showing…
 Rather than algorithms, the students…
 Performed mental calculations
 Explained their own strategies for
computations, and
 In the process, developed number sense
From research of Alistair McIntosh, University of Tasmania, Australia, and others, reported at ICME-10, 2004
Partners
for Mathematics Learning

40. Research Is Showing…  At least short term, there is evidence that a40
Research Is Showing…
 At least short term, there is evidence that a
strategies approach to mental computation
 Has a positive effect on students’ competence,
confidence and enjoyment
 Is a viable alternative classroom approach to
teaching mental computation
 Is consonant with a constructivist approach to
mathematics teaching
 Improves students’ ability to discuss, explain, and
justify orally
From research of Alistair McIntosh, University of Tasmania, and others, reported at ICME-10, 2004
Partners
for Mathematics Learning

41. Research Is Showing…  An emphasis on traditional algorithms and on41
Research Is Showing…
 An emphasis on traditional algorithms and on
mental methods of speed and accuracy
(rather than strategies)
 Does not lead to number sense
 Provides an inefficient method of improving
the mental computation skills especially of less
confident/competent students
 Inhibits flexible thinking
From research of Alistair McIntosh, University of Tasmania,
and others, reported at ICME-10, 2004
Partners
for Mathematics Learning

42. What Are Our Goals and Challenges?42
What Are Our Goals and Challenges?
 “The ultimate purpose of arithmetic
instruction is the development of the ability
to THINK in quantitative situations”
William Brownell, 1934
Partners
for Mathematics Learning

43. Alternative Strategies43
Alternative Strategies
 Student-developed strategies have
advantages for children over traditional
algorithms
 They are number oriented rather than digit
oriented
 They are left-handed rather than right-handed
 They are flexible rather than rigid
 They often employ operation properties
adapted from John Van De Walle, Teaching Student-Centered Mathematics, Grades 3-5, 1997
Partners
for Mathematics Learning

44.  Traditional approaches emphasize place44
Digit vs. Number Orientation
 Traditional approaches emphasize place
value often modeled on Base 10 materials
with standard vertical algorithms
 Rather than being digit oriented, a number
sense approach or invented strategies
approach is number oriented – not
separating a digit from its value within
the number
Partners
for Mathematics Learning

45. Research Says…  Children need informal , reliable , but not45
Research Says…
 Children need informal , reliable , but not
necessarily standard, methods of computing
 Children don't spontaneously develop the
standard algorithms
 Children need the choice not to use the algorithms
-Alistair McIntosh, University of Tanzania, emeritus, at ICME-10, 2004
 If you use it, you must understand why it works
and be able to explain it
-John Van de Walle, Teaching Student-Centered Mathematics, Grades 3-5, 2006
Partners
for Mathematics Learning

46. Higher Expectations - Not Lower46
Higher Expectations - Not Lower
 This position does not mean lower
expectations for computational expertise
 Rather, we have higher expectations for
accurate computing with understanding
 Time invested in understanding results in
long-term learning
Partners
for Mathematics Learning

47. Multiplication: Mental Strategies47
Multiplication: Mental Strategies
 Without using the traditional algorithm, how
would you solve these?
14 x 15
325 x 4
333 x 20
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for Mathematics Learning

48. What Are the Misunderstandings ?48
What Are the Misunderstandings ?
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for Mathematics Learning

49. Estimating Solutions  What is a reasonable estimate for49
Estimating Solutions
 What is a reasonable estimate for
35 x 48?
 How did you make your estimate?
 Why would an estimate be helpful when
computing?
Partners
for Mathematics Learning

50. 50
How Do These Work?

51. Thinking about Multiplication51
Thinking about Multiplication
 Beth and Sahil finished working a giant
jigsaw puzzle
 They saw that the pieces fit into rows of 49
pieces and that there were 35 rows in all
 How many pieces did the puzzle have?
35 x 49 =
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for Mathematics Learning

52. Multiplication: The Array Model52
Multiplication: The Array Model
1200
270
200 45
1715
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for Mathematics Learning

53. Multiplication: The Array Model53
Multiplication: The Array Model
 Connecting the traditional algorithm
35 x 49 = 49
x35
245
1470
1715
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for Mathematics Learning

54. Distributive Property54
Distributive Property 
35 x (30 + 5) x (40 + 9)
30 x (40 + 9) + 5 x (40 + 9)
(30 x 40) + (30 x 9) + (5 x 40) + (5 x9)
Partners
for Mathematics Learning

55. Hard Arithmetic is Not Deep Mathematics55
Hard Arithmetic is Not Deep Mathematics
 “Depth” means …that students know a lot about
multiplication before they deal with an algorithm
for performing multiplication
 “Depth” does not mean making all students
master arithmetic procedures earlier or with
more digits…. Focusing on more arithmetic
procedures or more digits at the expense of
deeper explorations and problem solving is not
the same as raising our expectations for all
students
Cathy Seeley, President, NCTM, 2004, in Teaching children Mathematics , October 2004
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for Mathematics Learning

56. Learning Through Problems56
Learning Through Problems
Jungle Problem #5
 There are 83 crocodiles
 Each maharaja (king) takes the same
amount of crocodiles for his fish pond
 There are 5 crocodiles left over
 How many crocodiles did each maharaja
get?
 Show your work and explain your thinking
Partners
for Mathematics Learning

57. Assessment: Scoring Rubric57
Assessment: Scoring Rubric
Score
Indicator
Noanswer,or
Wronganswerbasedoninappropriateplan 1
Incorrectanswerbutusesanappropriatestrategy,or
Onlycompletesonestepoftheproblem,or
Correctanswerwithinaccurateexplanation,
notrelatedtotheproblem 2
Correctanswerbutincompleteorunclearexplanation,or
Correctanswer,butminorerrorsinwork 3
Correctanswerwithappropriatestrategyused,and
Correctsolutionclearlystated,and
Clearandcorrectwrittenexplanationofprocess
Partners
for Mathematics Learning

58. Assessment – Scoring Rubric58
Assessment – Scoring Rubric
 Rubric for Jungle Problem #5 (83 Crocodiles)
 Number of Correct Solutions (0 to 5 points)
 No incorrect solutions (0 or 1 point)
 Work shown, appropriate strategy (0 or 1 point)
 Explanation fits problem (0 or 1 point)
 Explanation and/or work especially clear
(0 or 1 point)
 Solution clearly stated (0 or 1 point)
Total Score (0 to 10 points)
Partners
for Mathematics Learning

59. Problem-Based Lessons59
Problem-Based Lessons
 “If students are told how to solve each
problem, their mental energy is
squandered on “remembering” instead of
more worthwhile thinking. …problem
solving leads to understanding.”
-from Maharaja’s Tasks, Beyond Activities Project, California Department of Education
 Students with good number sense are
primed to become good problem solvers
Partners
for Mathematics Learning

60. 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

61. 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

62. 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

63. Grade Five PARTNERS for Mathematics Learning Module 1 Partners 63