1. Module 8 PARTNERS for Mathematics Learning Grade 3 Partners

2. Algebra As A Way of Thinking2
Algebra As A Way of Thinking
“It is essential for students to learn
algebra as a style of thinking involving
the formalization of patterns,
functions, and generalizations; and as
a set of competencies involving the
representations of quantitative
relationships.”
Silver, E. A “Algebra for All.”
Mathematics Teaching in the Middle School
Partners
for Mathematics Learning

3. 3
Which Operation?
Determine which operation (+, –, × , ÷) will
make the equation true:
 10 o 6 = 20 – 4
 27 o 3 = 6 + 3
 24 ÷ 3 = 2 o 4
o 3+8
 6x4+8=8
o 1
 1+0=1
Partners
for Mathematics Learning

4. Exploring Algebraic Thinking4
Exploring Algebraic Thinking
Equality
Big Ideas 2,3,4
Patterns
Big Idea 1
Repeating
&
Growing
Patterns
Properties
&
Order of
Operations
Equations
&
Variables
Functional
Relationships
Partners
for Mathematics Learning

5.  Equations and inequalities5
Equivalence: A Big Idea of Algebra
 Equations and inequalities
are used to express relationships
between quantities
 Unknown quantities can be represented
using variables that are letters or symbols
(4 x n) + 2 = 14
Partners
for Mathematics Learning

6. When comparing real numbers x and y,  x is greater than y (x > y) 6
Inequalities
When comparing real numbers x and y,
there are three possibilities:
 x is greater than y (x > y)
 x is less than y (x < y)
 x is equal to y (x = y)
 When asked to solve an inequality like x
-3 > 0, find all possible values of x that
make the inequality true
 x = 10 makes the inequality true: 10 – 3 > 0
Partners
for Mathematics Learning

7.  Operation properties are generalized7
Big Idea in Algebra
 Operation properties are generalized
guidelines for operating with numbers
Partners
for Mathematics Learning

8.  Properties are central to the study8
Properties for Multiplication
 Properties are central to the study
of arithmetic and algebra
 Explain operation properties for
multiplication
 Share examples of commutative,
associative, distributive properties
Partners
for Mathematics Learning

9. Properties of Multiplication9
Properties of Multiplication
Partners
for Mathematics Learning

10. Goals of Number Talks  To let reason, not the10
Goals of Number Talks
9x8 “I subtracted because
I know 10x8 = 80.
So 9x8 =72.
 To let reason, not the
teacher, be the authority
in determining whether
or not a strategy works
 To use mental arithmetic in
developing numerical reasoning
 To learn basic facts through reasoning
and discussion instead of isolated drill
Partners
for Mathematics Learning

11. Number Talks To ……..  Increase fluency and proficiency11
Number Talks To ……..
 Increase fluency and proficiency
in operations with small
and large numbers
 To address ideas of equality
 To utilize properties of number
 Solve problem in your head
 When you have solved, put your thumb up in
front of your chest to show one solution
 Try to solve in a different way; add fingers
Partners
for Mathematics Learning

12. = Equality  Equations are mathematical sentences12 =
Equality
 Equations are mathematical sentences
with equal signs
 The expressions on either side of the
equals sign have the same value
 Algebra involves a language of symbols
used to describe expressions

13. 13
Balancing Equations

14. Balancing Equations x 4) x 3 = (4 x 3) x 18 = ( x 18) + 1814
Balancing Equations
( 9
x 4) x 3 = (4 x 3) x
18 = (
x 18) + 18
4 x 4 = 16 x
7x6x5=7x(
x 5)
8 x 78 = (8 x ) + (8 x 8)
Partners
for Mathematics Learning

15. Using What You Know  Use what you know15
Using What You Know
 Use what you know
 Decompose 42 into
 What do you know?
 30 ÷ ÷ 3
 What mathematics are you using?
 If a number is a divisor of two numbers,
then it will also be a divisor of their sum
 Mental calculation - Go from what you
know to what you do not know
 Decompose and solve: 98 ÷ 7
Partners
for Mathematics Learning

16. Venn Diagrams – Relationships16
Venn Diagrams – Relationships
Partners
for Mathematics Learning

17. Venns: Using What You Know17
Venns: Using What You Know 33
Partners
for Mathematics Learning

18. Venns: Using What You Know18
Venns: Using What You Know
Partners
for Mathematics Learning

19. True/False Number Sentences19
True/False Number Sentences
“True/false and open number sentences
have proven particularly productive as a
context for discussing equality. These
number sentences can be manipulated in
a variety of ways to create situations that
may challenge students’ conceptions and
provide a context for discussion.”
Thomas Carpenter, Megan Franke, & Linda Levi
Thinking Mathematically , p.15
Partners
for Mathematics Learning

20. True or False? 3x7=7+7+7 6 x 7 = 35 + 7 49 = (5 x 7) + (2 x 7)20
True or False?
3x7=7+7+7
6 x 7 =
49 = (5 x 7) + (2 x 7)
(8 x 30) + (8 x 8) = 8 x 38
2 x 8 x 4 = 20
9 x 7 = 10 x 7 – 7
Partners
for Mathematics Learning

21. True or False? 4 x 9 = (4 x 3) x (9 ÷ 3)21
True or False?
4 x 9 = (4 x 3) x (9 ÷ 3)
(54 ÷ 6) = (30 ÷ 6) + (24 ÷ 6)
48 x 1= 49
28 x 0 = 28
42 ÷ 3 = (30 ÷ 3) + (12 ÷ 3)
7 = 35 ÷ 5
35 ÷ 5 = 5 ÷35
Partners
for Mathematics Learning

22. Properties of Multiplication22
Properties of Multiplication
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for Mathematics Learning

23. Leading to Conjectures23
Leading to Conjectures
 Write a series of equations based
on an arithmetic property third grade
students should think about
 Commutative Property of Multiplication
 Associative Property of Multiplication
 Identity Property of Multiplication
 Multiplication involving zero
 Distributive Property of Multiplication
 Other
Partners
for Mathematics Learning

24. Why Conjectures?  Properties of Number Operations24
Why Conjectures?
 Properties of Number Operations
 Learning of arithmetic becomes easier
 Critical for learning algebra
 Focus on relationships
 Classes of Numbers
 Ideas related to odd and even numbers
 Operations of even and odd numbers
If you add an even number
and an odd number, is the
sum even or odd?
If you multiply an even number
times an odd number, is the
product odd or even?
Partners
for Mathematics Learning

25. Reciprocal Relationship25
Reciprocal Relationship
 A Reciprocal relationship between multiplication
and division depends on understanding the
relationship of parts to whole
 If we know the parts ( number of groups and the
number in each group ) we can figure out the
whole
 If we know the whole and one part ( number of
objects in each group ) we can figure out the
other part ( how many groups )
 If we know the whole and ( number of groups ),
we can figure out the other part ( number in each
group )
Partners
for Mathematics Learning

26. 26
Circus Hunt Game
Partners
for Mathematics Learning

27. Circus Hunt Game  How is getting into three groups different27
Circus Hunt Game
 How is getting into three groups different
from getting into groups of three? Discuss
with a partner
Partners
for Mathematics Learning

28. Grouping and Partitioning28
Grouping and Partitioning
Example:
 Megan has 6 bags of cookies. There are 3
cookies in each bag. How many cookies
does Megan have altogether?
Multiplication:
 When the total number of cookies
Megan has altogether is unknown, the
problem is a Multiplication problem
Partners
for Mathematics Learning

29. Division: Two Types Equal Group Problems  Partitive Division29
Equal Group Problems
Division: Two Types
 Partitive Division
 When the number of cookies in
each bag is unknown (sharing)
 Measurement Division
 When the number of groups or bags is
unknown
Partners
for Mathematics Learning

30. Equal Group Problems  Partitive Division30
Equal Group Problems
 Partitive Division
 Rick has 24 apples. He wants to share them
equally with 4 of his friends. How many
apples will each friend receive?
 Measurement Division
 Rick has 24 apples. He put them into bags
containing 6 apples each. How many bags
will Mark use?
How might children solve these differently?
Partners
for Mathematics Learning

31. 31
Factors of x12, 2x6, 3x4
Partners
for Mathematics Learning

32. Building Square Tables32
Building Square Tables
 The third grade class is responsible for the
table arrangements for the school math fair
 Students decided all table tops will be in
the shape of a square .
 Table tops will be constructed from square
tiles
 The smallest table will be constructed from
exactly one square tile equal
equal
Partners
for Mathematics Learning

33. Building Square Tables33
Building Square Tables
 Each additional table will need enough
square tiles to make the next largest
square table
 Use color tiles to build the 4 th table top
 Record the number of square tiles needed
for each table. Look for patterns
Partners
for Mathematics Learning

34. Building Square Tables34
Building Square Tables 
Determine how each step in the pattern differs
from the preceding step
What patterns might third graders find? 
 What conjectures can you make related to this pattern?
Partners
for Mathematics Learning

35. Building Square Tables35
Building Square Tables
 Describe patterns in the table
 What relationship do you see between the
table number and the number of squares
needed to build the table?
Partners
for Mathematics Learning

36. Square Numbers 3x3  Square numbers are sometimes36
* * *
Square Numbers
3x3
 Square numbers are sometimes
presented in exponent form ²
 The explicit rule for generalizing ²
the square number pattern is n ²
where n stands for the term
6 ² or 36
 The sixth square number is * * * *
Partners
for Mathematics Learning

37. Problem Based Lessons Research has indicated that37
Problem Based Lessons
Research has indicated that
beginning with problem
situations yields greater
problem-solving competence
and equal or better
computational competence

38. Why Focus on Problems?  The single most important principle for38
Why Focus on Problems?
 The single most important principle for
improving the teaching of mathematics is
to allow the subject of mathematics to be
problematic for students (Hiebert et al.,
1996)
 Developing students’ abilities to
solve word problems is critical for
algebra readiness
Partners
for Mathematics Learning

39. Max and the Peanuts  One clown’s pet monkey, Max, has some39
Max and the Peanuts
 One clown’s pet monkey, Max, has some
peanuts.
 Max divides the peanuts evenly among
himself and 4 other monkeys.
 After the peanuts are all given out, there
are 3 peanuts left over
 How many peanuts did Max have to start
with?
Partners
for Mathematics Learning

40. Multiplicative Comparison40
Multiplicative Comparison
 “ Compare” - multiplication or division
 “Megan baked four times as many cookies as
Alex. Alex baked 3 cookies. How many
cookies did Megan bake?”
 “Kaneka baked 15 chocolate chip cookies.
She baked three times as many cookies as
Alex. How many cookies did Alex bake?”
Partners
for Mathematics Learning

41. Multiplicative Comparison41
Multiplicative Comparison
 “ Compare” - multiplication or division
 “Alex baked 36 cookies this month. Last
month he baked 12 cookies. How many times
as many cookies did he bake this month as
last month?”
Alex (this month) 36
Alex (last month) 12
Partners
for Mathematics Learning

42. Sharing Bananas  Some monkeys found 9 bananas42
Sharing Bananas
 Some monkeys found 9 bananas
They shared them equally
Then they found 6 more bananas and
shared them equally
 How many monkeys were there?
How many bananas did each monkey get?
Partners
for Mathematics Learning

43. Making Rectangles  In groups, trace, cut and label factors for43
Making Rectangles
 In groups, trace, cut and label factors for
 all possible arrays for several square numbers
 all possible arrays for several non square
numbers
 Search for patterns in the number of factors for
square numbers versus non square numbers
 What conjectures can you make
related to the number of factors?
10 x 10 = 100
100 is a square number

44. Doubling and Halving  Marty wants to cut a strip of6 44
Doubling and Halving 9
 Marty wants to cut a strip of
paper to cover a shelf. The back
of her paper has a 9 x 6 grid
 If Marty doubles the 9 and then
partitions the 6 in half, what will
be the new measurement
(factors) for her paper
 What mathematics is used when
solving the problem?
Partners
for Mathematics Learning

45. How Many in a Bag?  Tyler has 36 video games and he45
How Many in a Bag?
 Tyler has 36 video games and he
wants to put the games in boxes.
If Tyler puts the same number
of video games in each box,
how many boxes could Tyler pack?
 What questions might you ask
to determine students’
understanding?
Partners
for Mathematics Learning

46. What are the Factors?  The school art room has some stools with46
What are the Factors?
 The school art room has some stools with
3 legs and some chairs with 4 legs
 If there is a total of 26 legs on the stools and
chairs, how many stools and chairs are in the
art room. (Remember there are some of each)
 Share your solution strategies
 What mathematics is used in solving
the problem?
Partners
for Mathematics Learning

47. Sharing Brownies  Renee is baking  Create a table to47
Sharing Brownies
 Renee is baking
brownies for a class
of students. Renee
plans to give each
student 3 brownies
 Create a table to
show the number of
brownies needed for
a given number of
students
Renee just learned there are 24 students.
How many brownies does she need to
bake?
Partners
for Mathematics Learning

48. Value of a Penny  Ramona is saving pennies in a48
Value of a Penny
 Ramona is saving pennies in a
bucket
 On the first day, Ramona
puts in 1 penny; on the second day,
she put in 2 pennies; on the third day,
3 pennies
 If she continues this pattern, how much
money will Ramona save in 10 days?
In 20 days?
 Organize the data in a function table
Partners
for Mathematics Learning

49. Interpreting Remainders49
Interpreting Remainders
 A clown has 30 balloons to share fairly
with 7 children. How many balloons
will each child receive?
 Cathy has 36 cookies. She wants to
share them equally with 8 friends.
What will be each friend’s share?
 A carpenter needs to cut a 6 meter
board of wood into 4 equal pieces.
How long will each piece of wood be?
 A rope is 25 meters long. How
many 8 meter ropes can be made?
Partners
for Mathematics Learning

50. Saving for a Wii Game  Mario’s Dad gives him $3 each week for50
Saving for a Wii Game
 Mario’s Dad gives him $3 each week for
emptying the trash and helping with
chores. Mark is saving his money to buy a
Wii game. How much money will Mario
save after three weeks? Four weeks?
 How many weeks will Mario need to save
money if the Wii game he wants costs $48
dollars, including tax.
Partners
for Mathematics Learning

51. Factors and Multiples  Mark is planning a picnic for the51
Factors and Multiples
 Mark is planning a picnic for the
24 swimmers (including Mark) on his team.
Mark can buy hot dogs in packages of 12
and hot dog buns in packages of 6. Mark plans to
give each swimmer the same number of hot dogs
and buns and there will be no leftovers.
 What would be the least number of hot dog packages
and the least number of bun packages Mark can buy?
 Mark decides to give each swimmer more than one
hot dog and more than one bun, what is the least
number of packages he can buy with no leftovers?
Partners
for Mathematics Learning

52. Part of a Set  Three fourths of the crayons in Mr.52
Part of a Set
 Three fourths of the crayons in Mr.
Hester’s box of two dozen crayons are
broken.
 How many unbroken crayons are in the
box?
Partners
for Mathematics Learning

53. Find Area - Using Multiplication53
Find Area - Using Multiplication
 Denny has 4 strips of wood to
make a flower garden. His wood
is already cut in strips of 3 meters
and 4 meters. If Denny builds a rectangular
garden, how much space will be inside
Denny’s garden?
 Work with a partner to create one square
meter
 Group with other participants to model the
area of Denny’s garden
Partners
for Mathematics Learning

54. Connections Through Context54
Connections Through Context
 Mario is carrying sandy soil to a flower
garden. He can only carry 200 grams of
soil at a time. How many trips will Mario need to
make in order to carry 1 kilogram of soil to the
garden? How many trips for 5 kilograms?
 Jessie is 130 centimeter tall. The tallest tree in
his yard is 10 times as tall as Jesse. How many
centimeters tall is the tree?
 There are 12 computers in the classroom. ¾ of
the computers are being used by students. How
many computers are not being used?
Partners
for Mathematics Learning

55.  Mathematical situations can be analyzed55
Big Ideas in Algebra
 Mathematical situations can be analyzed
through patterns, functions, and other
relationships; those mathematical
situations can be represented in multiple
ways (words, pictures, numbers, symbols)
 Equations and inequalities are used to
express relationships between quantities

56. Algebraic Thinking in Classrooms56
Algebraic Thinking in
Classrooms
 Identify relationships among numbers and
number operations
 Consider properties of operations rather
than calculating by rote
 Use contexts to help make sense of
symbolic manipulation
 Recognize patterns and relationships to
make conjectures

57. Thinking Mathematically57
Thinking Mathematically
 Learning mathematics involves learning
ways of thinking. It involves learning
powerful mathematical ideas rather than a
collection of disconnected procedures for
carrying out calculations.
Thinking Mathematically by Carpenter, Franke, Levi
Partners
for Mathematics Learning

58. Algebraic Thinking  Three processes that support algebraic thinking58
Algebraic Thinking
 Three processes that support
algebraic thinking
 Generalizing
 Formalizing
 Justifying
Partners
for Mathematics Learning

59. “Think of all that you know about mathematics. Algebra is about making 59
“Think of all that you know about
mathematics. Algebra is about making
it richer, more connected, more general,
and more explicit.”
- Ricardo Nemirovsky, as quoted in Erick Smith
Partners
for Mathematics Learning

60.  With a partners, discuss the Essential60
Essential Standards
 With a partners, discuss the Essential
Standards for Algebra
 How might ideas of
algebraic thinking
impact your instruction
in mathematics?
Partners
for Mathematics Learning

61. Renee Cunningham Kitty Rutherford Robin Barbour Mary H. Russell 61
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

62. PML Dissemination Consultants62
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

63. 2009 Writers Partners Staff Kathy Harris Rendy King Tery Gunter63
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

64. Module 8 PARTNERS for Mathematics Learning Grade 3 Partners