1. for Mathematics Learning
PARTNERS
for Mathematics Learning
Grade Four
Module 2
Partners
for Mathematics Learning

2. Mathematics Learning…2
Mathematics Learning…
 Mathematics learning is about making
sense of mathematics
 Mathematics learning is about acquiring
skills and insights to solve problems
NCTM Standards 2000
Philosophy
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for Mathematics Learning

3. Problem Solving  Solving problems is not only a goal of3
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 Standards 2000
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for Mathematics Learning

4. Children with Good Number Sense4
Children with Good 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

5. Developing Number Sense . . .5
Developing Number Sense . . .
 Number Lines (Relative Magnitude)
 What numbers do points C and D represent?
 How far apart are A and B?
 Why do you think so? A B
C D E F G 75
200
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for Mathematics Learning

6. +5 Number Lines n 128 153 What number is n? Label the number line 1236
Number Lines
123 +5
128 n
153
What number is n?
Label the number line
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for Mathematics Learning

7. Number Relationships  Greater than/ less than/ equal7
Number Relationships
 Greater than/ less than/ equal
 Relative magnitude
 Different ways to name the same number
 Decomposed into a combination of other
numbers
 Composed with other numbers to name a
new number
 Multiples and factors
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for Mathematics Learning

8. Number Relationships  1026 916 1176  Which two numbers are closest?8
Number Relationships
 1026
916
1176
 Which two numbers are closest?
 Which number is closest to 950? to 1100?
 Name a number between 1026 and 1179
 Name a multiple of 25 between 1026
and 1179
 About how far apart are 1026 and 1179?
Partners
for Mathematics Learning

9. Number Relationships  1026 916 1176
 1026
916
1176
 If these are “big” numbers, what are
some small numbers?
 …numbers that make these seem small?
Partners
for Mathematics Learning

10. Games Provide Fun Practice10
Games Provide Fun Practice
 Order Up!
• Need dice or number cards
 Go Digit!
• Children create number cards to use
 Largest or Smallest
• Use dice with 0-9
 What’s My Number?
 Target 100 (or choose other target)
Partners
for Mathematics Learning

11.  Name two odd numbers whose sum is 16 11
Mystery Number Pairs
 Name two odd numbers whose sum is 16
 Name two numbers whose sum is 20 and
whose difference is 4
 Name two numbers whose product is 24
and whose sum is 11
 List 3 pairs of numbers with a difference
of 54
Partners
for Mathematics Learning

12. ? Mystery Numbers  I am not an even number  I am between 20 and 5012 ?
Mystery Numbers
 I am not an even number
 I am between 20 and 50
 The sum of my digits is 7
Partners
for Mathematics Learning

13. ? ? Mystery Numbers  I am not an even number  I am between 20 and 5013 ?
Mystery Numbers
 I am not an even number
 I am between 20 and 50
 The sum of my digits is 7
 I am a multiple of 5
 What number am I? ?
Partners
for Mathematics Learning

14. Operation Sense  A complete and flexible understanding of14
Operation Sense
 A complete and flexible understanding of
the operations
 Understand various meanings of multiplication
and division [and addition and subtraction]
 Understand the effects of multiplying and
dividing whole numbers [and of adding and
subtracting]
 Identify and use relationships between
operations to solve problems
 Understand and use properties of operations
Partners
for Mathematics Learning

15. Mastering Basic Facts  Components of Fact Mastery Development
 Development of a strong understanding of
number relationships related to the operations
 Development of efficient strategies for fact
retrieval
 Then (and only then) drill in use and selection
of these strategies
Partners
for Mathematics Learning

16. Mastering Multiplication Facts16
Mastering Multiplication Facts
 Relate new facts to existing facts
 Understand and use commutative property
 Children should use strategies for
remembering that make sense to them
 Drill activities should be individualized to
promote different strategies and address
different collections of facts
Partners
for Mathematics Learning

17.  Groups of…  4 groups of 3 4 x 3 = 12  3 groups of 4 3 x 4 = 1217
Multiplication Models
 Groups of…
 4 groups of 3
4 x 3 = 12
 3 groups of 4
3 x 4 = 12
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for Mathematics Learning

18. Multiplication Models18
Multiplication Models
 Arrays
3 x 4 = 12
6 x 2 = 12
 Are any other arrays possible for factors
of 12?
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for Mathematics Learning

19. Multiplication and Division19
Multiplication and Division
 Equivalent sentences fit this array:
xxxxxxxxxxxx
 Multiplication:
3 x 12 = 36
36 = 3 x 12
 Division:
36 ÷ 3 = 12
12 = 36 ÷ 3
12 x 3 = 36
36 = 12 x 3
36 ÷ 12 = 3
3 = 36 ÷ 12
Partners
for Mathematics Learning

20. Practice Can Be Fun  Four's a Winner  Practices facts20
Practice Can Be Fun
 Four's a Winner
 Practices facts
 Uses strategy
 Promotes mathematical
conversation
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for Mathematics Learning

21. Multiplication  A thirsty camel drinks about 20 gallons of21
Multiplication
 A thirsty camel drinks about 20 gallons of
water
 How many gallons will 80 thirsty camels
drink?
20 x 80 =
Partners
for Mathematics Learning

22. Multiplication: The Array Model22
Multiplication: The Array Model 10
20 × 80
(2 × 10) × (8 × 10)
(2 × 8) × (10 × 10)
16 × 100
1600
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for Mathematics Learning

23. 23
Do These Work?
Partners
for Mathematics Learning

24. 24
Do These Work? How?

25. Thinking About Multiplication25
Thinking About Multiplication
 Beth and Sahil finished working a giant
jigsaw puzzle. They saw that the pieces fit
into rows of 45 pieces and that there were
29 rows in all. How many pieces did the
puzzle have?
45 x 29 =
Partners
for Mathematics Learning

26. Multiplication: The Array Model26
Multiplication: The Array Model
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for Mathematics Learning

27. Multiplication: The Array Model27
Multiplication: The Array Model
Connecting to the compact algorithm:
Partners
for Mathematics Learning

28. “Deep Mathematics”  “Depth” means …that students know a lot28
“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

29. Thinking About Computation29
Thinking About Computation
 "The depressing thing about arithmetic badly
taught is that it destroys a child’s intellect and,
to some extent, his integrity. Before they are
taught arithmetic, children will not give their
assent to utter nonsense; afterwards they will.
Instead of looking at things and thinking about
them, they will make wild guesses in the
hopes of pleasing a teacher….
Partners
for Mathematics Learning

30. Thinking about Computation30
Thinking about Computation
 The essential quality for a mathematician
is the habit of thinking things out for
oneself. That habit is usually acquired in
childhood. It is hard to acquire it later."
From Mathematician’s Delight, W. W. Sawyer, University of Toronto, 1943
Partners
for Mathematics Learning

31. Learning Through Problems31
Learning Through Problems
Jungle Problem # 4
 Some kings found nine melons
 They shared them equally
 Then they found six more melons and shared
them equally
 How many kings were there and how many
melons did each king get?
 Show your work and explain your thinking
Maharajah’s Tasks
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for Mathematics Learning

32. Assessment: Scoring Rubric32
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

33. Division  Freda has a booth at the state fair33
Division 
Freda has a booth at the state fair
Her tiger’s-eye marbles are a big seller
She has 92 marbles to put equally into 4 bags
How many marbles can she put into each
bag?
92 ÷ 4 =
Partners
for Mathematics Learning

34. Division  First estimate the solution to this equation: 92 ÷ 4 =34
Division
 First estimate the solution to this
equation:
92 ÷ 4 =
 How did you make your estimate?
 Without using the traditional algorithm,
how many ways can you solve this
problem?
Partners
for Mathematics Learning

35. Division – Equivalent Sentences35
Division – Equivalent Sentences
92 ÷ 4 = 4x
= 92
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for Mathematics Learning

36. Division – Equivalent Sentences36
Division – Equivalent Sentences
92 ÷ 4 = 4x
= 92
4 x 20 = 80
(92-80 =12)
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for Mathematics Learning

37. Division – Equivalent Sentences37
Division – Equivalent Sentences
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for Mathematics Learning

38. Division – Equivalent Sentences38
Division – Equivalent Sentences
 Try using this method to solve these:
 376 ÷ 8 =
 2520 ÷ 40 =
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for Mathematics Learning

39. Division - Chunking - 720 36 x 20 Answer: 27 36 )972 252 36 x 7 -25239
Division - Chunking
36 )972
- 720
252
-252
36 x 20
36 x 7
Answer: 27
Partners
for Mathematics Learning

40. Division – Mental Strategies40
Division – Mental Strategies
 How would you solve this problem
without the traditional algorithm?
168 ÷ 21
Partners
for Mathematics Learning

41. Division – Mental Strategies41
Division – Mental Strategies
168 ÷ 21
 Fourth graders who did not know the
algorithm solved it with mental strategies
 Sixth graders "couldn't do it" because they
hadn't yet learned to "do it with 2 digits"
From research by Julia Anghileri, University of Cambridge,
shared at ICME-10, July, 2004
Partners
for Mathematics Learning

42. Division – Mental Strategies42
Division – Mental Strategies
 Try this division problem without using the
traditional algorithm
648 ÷ 3
 How many different ways can you find to
solve this problem?
Partners
for Mathematics Learning

43. Calculating with Number Sense43
Calculating with Number Sense
 "Calculating with number sense means that
one should look at the numbers first and then
decide on a strategy that is fitting – and
efficient. Developing number sense takes
time; algorithms taught too early work against
the development of good number sense.
Children who learn to think, rather than to
apply the same procedures by rote regardless
of the numbers, will be empowered. "
From Fosnot and Dolk, Young Mathematicians at Work:
Constructing Multiplication and Division, Heinemann, 2001
Partners
for Mathematics Learning

44. What’s the Story?  Think about this equation: 47 ÷ 5 =44
What’s the Story?
 Think about this equation:
47 ÷ 5 =
 Give a context in which each one of
these answers would be the correct
solution for the division of 47÷5:
9 , 9 r 2 , 10 , 9 2/5 , 9 or 10
Partners
for Mathematics Learning

45. Problem-Based Lessons45
Problem-Based Lessons
 Provide problems that require thinking, not
just following procedures and rules
For example:
 How many stamps are not on the border of
a 10 by 10 sheet of stamps?
Partners
for Mathematics Learning

46. Avoiding Meaningless Math46
Avoiding Meaningless Math
 When students reflect on their own
strategies, they are developing an
understanding of concepts out of empirical
experiences
 “You can’t tell students what a concept is,
but you can give them experiences that
help them make abstractions for
themselves.”
-Michael Mitchelmore, Sydney, Australia, quoted at ICME-10, Copenhagen, Denmark, July, 2004
Partners
for Mathematics Learning

47. Discovering Order of Operations47
Discovering Order of Operations
 Using the calculators
provided by the leader,
solve these equations
 How did we get different
answers to the same
problems?
 What answers does your
personal calculator give?
Partners
for Mathematics Learning 
4x7+3=
4+8x3=
10 – 6 ÷ 2 =
12 – =
7+2x5–3=
8 x 27 ÷ 9 =
40 ÷ 8 – 2 =

48. “Good Manners of Mathematics”48
“Good Manners of Mathematics”
What is happening here?
 4 x (7 + 3) = 40
 (4 + 8) x 3 = 36
 (10 – 6) ÷ 2 = 2
 x (5 – 3) = 11
How do the parentheses impact the solution?
Partners
for Mathematics Learning

49. R emembering O rder of O perations
107
R emembering O rder of O perations
How is the triangle a
better model than the
traditional P E M D A P E M A D S S
for remembering the
correct order?
Partners
for Mathematics Learning

50. Connections  Where in this module did you see algebra50
Connections
 Where in this module did you see algebra
embedded in the number strand?
 Which of the process standards did we
use?
 Problem Solving
 Reasoning and Proof
 Communication
 Connections
 Representation
Partners
for Mathematics Learning

51. 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

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

53. 2009 Writers Jeane Joyner, Co-PI and Project Director Partners Staff
Kathy Harris
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

54. for Mathematics Learning
PARTNERS
for Mathematics Learning
Grade Four
Module 2
Partners
for Mathematics Learning