This document outlines an agenda for a professional development session focused on implementing the Standards for Mathematical Practice. It includes discussions and activities to help teachers understand each standard and strategies for bringing the standards into their classrooms. The session aims to help teachers explore how to make students active problem solvers, encourage mathematical reasoning and modeling, and use tools strategically when solving problems. Sample problems and templates are provided to demonstrate ways to incorporate the standards into instruction.

1. Fall 2013
Putting the
Mathematical
Practices Into Action

2. Welcome
“Who’s in the Room”

3. Norms
• Listen as an Ally
• Value Differences
• Maintain Professionalism
• Participate Actively

4. http://maccss.ncdpi.wikispaces.net/Home
Materials Found on Mathematics Wiki

5. Session Outcome
• Understand the Standards for Mathematical
Practice
• Explore strategies for implementing the
Standards effectively

6. What does it mean to be
Mathematically Proficient?

7. Turn and Talk
• Are students who can remember formulas
or memorize algorithms truly mathematically
proficient, or are there other skills that are
necessary?
• Is the correct answer the ultimate goal of
mathematics, or do we expect a greater
level of competence?

8. For the first time, mathematical processes are
elevated to essential expectations, changing
our view of math to encompass more than just
content.
The goal now is to apply, communicate, make
connections, and reason about math content
rather than simply compute.

9. What are
The Standards for
Mathematical Practice?

12. What has been your biggest
challenge with the
implementation of
The Standards for
Mathematical Practice?

13. Overarching habits of
mind of a productive
Mathematical Thinker

14. Exploring Standard 1
Make Sense of Problems and
Persevere in Solving them.

15. Understanding the Standard
• What do we do each day in our classroom
to build mathematical thinkers?
• What do we do to keep our students actively
engaged in solving problems?
• How do we help our students develop
positive attitudes and demonstrate
perseverance during problem solving?

16. How Do We Get There?
Brainstorming Strategies…
1. Choose an Operation
2. Draw a Picture
3. Find a Pattern
4. Make a Table
5. Guess and Check
6. Make an Organized List
7. Use Logical Reasoning
8. Work Backward Page 11

17. The Holiday Tree
The Partin family counted the different types of
ornaments on the town’s holiday tree. Here is
the list of what they saw.
stars – 24
gingerbread men – 14
snowflakes – 12
reindeer – 18
candy canes – 6
Six of the reindeer had red noses. What fraction
of the reindeer had red noses? Tell how you
would get the answer.
Page 14

18. The Holiday Tree
The Partin family counted the different types of ornaments
on the town’s holiday tree. Here is the list of what they
saw.
stars – 24
gingerbread men – 14
snowflakes – 12
reindeer – 18
candy canes – 6
What fraction of the ornaments were snowflakes?
What fraction of the ornaments were edible?
If 6 of the stars were silver, what fractions of the stars were
not silver?
Page 14

19. What questions could you ask?
Look on page 19 for some additional suggestions.

20. What questions could you ask?
Shipley Aquarium
Admission Cost
Adults - $8.00
Children (ages 3 and over) - $6.50
Children (ages 2 and under) – Free
Look on page 19 for some additional suggestions.

21. Traditional Problems vs.
Rich Problems
• We can ask questions that stifle learning by
prompting a quick number response.
– What is the answer to number 3 on your
worksheet?
– What is 5 x 4?
• We can ask questions that promote
discussion, thinking, and perseverance.

22. Sort the math questions.
Check your arrangement on page 22

23. page 26

24. Reflecting on strengthening student
problem solving experience…
1. Do I routinely provide opportunity for my students
to share their solutions and processes with
partners, groups, and the whole class?
2. Do I show my students that I value process (how
they did it) rather than simply the correct answer?
3. Do I pose problems that require perseverance?
Do I use thoughtful questions to guide and
encourage students as they struggle with
problems?

26. Exploring Standard 2
Reason abstractly and
quantitatively.

27. Understanding the Standard
• What can we do in our classrooms each day
to help students build a strong
understanding of numbers (quantities)?
• How do we help students convert problems
to abstract representations?
• What can we do to help students
understand what numbers stand for in a
given situation?

28. How Do We Get There?
• Number Webs
• Headline Stories
– Post It
– Book It
– Reverse It
– Match It
– Question It
• Pinch Cards

29. Let’s Create a Number Web!
Select a number.

30. Number Web
Number Webs encourage
flexibility with numbers. page 34

31. Headline Stories
• Headlines sum up a story.
• Equations are like newspaper headlines—short
and to the point.
• Equations are connected to word problems the
same way a headline is connected to a news
story.
• Using headlines can help you see students
understandings and misunderstandings.

32. Headline Stories
page 36

33. Headline Stories
page 36

34. Headline Stories
Headline: 52÷4 =
Headline stories can be as easy or as difficult
as you make them!
Students might be asked to write problems
about equations that include fractions,
decimals, percents, or variables.
Let’s look at some variations found on page 39.

35. Pinch Cards
Pinch cards are an all-pupil response technique.
There were 6 soccer teams in the league and 12 players on each
team. How many players were in the league?
The 4 members of the High Rollers Bowling Team scored 120, 136,
128, and 162. What was the team’s mean score?
page 41

36. Avoiding Key Words
• Key words are misleading.
• Many problems have no key words.
• The key word strategy sends a terribly
wrong message about doing mathematics.
A sense making strategy will always work.
Van de Walle & Lovin (2006)

37. Read and Discuss page 33
• What is contextualization and
decontextualization?
• Why is it important?
• Discuss at your table.

38. Contextualize and
Decontextualize
120 students and 5 chaperones went on the
field trip. Each bus held 35 people. How many
buses were needed?
Decontextualize: consider the data, the action of the
problem, and create an equation to represent the problem in
an abstract way
Contextualize: refer back to the context of the problem to
determine if the answer makes sense

39. • Understanding the units and quantities
within a problem is an important factor in
making sense of the numbers within the
problem.
• Labeling answers forces students to refer
back to the context of the problem.

40. page 40

41. Exploring Standard 3
Construct viable arguments
and critique the reasoning
of others.

42. Understanding the Standard
• What do we do in the classroom to get students to
justify their answer and defend their process for
finding the answer?
• How do we help students understand math skills
and concepts so they can construct viable
arguments?
• How do we help students consider and judge the
reasonableness of other answers and strategies?

43. How Do We Get There?
• Eliminate It
• Agree or Disagree?
• My 2 Cents

44. Eliminate It!
• As a group, decide on the concept that
should be eliminated with reasoning or math
data to back up your decision.
• There may be more than one way to
eliminate an item!
• Create your own.

45. Eliminate It
page 50

46. Agree or Disagree?
• 75% is more than 2/3.
• Tell why you agree or disagree.

47. Agree or Disagree?

48. Agree or Disagree?

49. Agree or Disagree?
• Jim has 12 pencils and Annie has 8. Jim has more
than Annie.
• 7 + 3 and 4 + 6 are the only ways to make 10.
• 9 is an even number.
• 6 tens and 3 ones is the same as 5 tens and 13
ones.
• 3 jars of peanut butter for $7.50 is a better deal
than 4 jars of peanut butter for $10.20.
page 53

50. My 2 Cents

51. Constructing Arguments
• Read page 44-46
• What is the difference between an assertion
and an argument?
• Be prepared to share your thinking.

52. Assertion vs. Argument
• Assertion: a statement of what students want us to
believe without support or reasoning.
– The answer is correct “because it is,” “because I know
it,” or “because I followed the steps.”
• Argument: a statement that is backed up with
facts, data, or mathematical reasons
• Constructing viable arguments is not possible for
students who lack an understanding of math skills
and concepts.

53. Page 58

54. Exploring Standard 4
Model with mathematics.

55. Understanding the Standard
• As teachers, we model with mathematics
routinely in our classrooms. Should students
be able to model? Why?
• How do students modeling mathematics
look?
• How does student modeling of mathematics
affect instruction?

56. How Do We Get There?
• Model It
• Part-Part-Whole mats (addition & subtraction)
• Bar Diagrams (multiplication & division)
• Bar Diagrams (solving equations)

57. How would you model…
• 123 + 57
• 1 – 1
3
• 3.4 + 5.07

58. Part-Part-Whole Mat
• There were 2 yellow lollipops and 3 red
lollipops. How many lollipops were there?

59. Part-Part-Whole Mat
• There were 6 children. 3 were boys. How
many were girls?

60. Part-Part-Whole Mat
• There were 5 cupcakes. Jan ate some.
There were 2 left. How many did she eat?

61. Bar Diagram
• There are 3 boxes with 6 toys in each box.
How many toys are there?
6 6 6

62. Bar Diagram
• 18 toys are packed equally into 3 boxes.
How many toys are in each box?
18

63. Bar Diagram
• 18 toys are packed 6 to a box. How many
boxes are needed?
18

64. Bar Diagram
• 18 toys are packed 6 to a box. How many
boxes are needed?
18
6 6 6

65. Page 74

66. Assessment Tips
• Tell me what your model represents.
• Why did you choose this model?
• Did creating a model help you any way? If so,
how?
• Did you get any insights by looking at your
model?
• Is there another way you might model this
problem or idea? How?
Page 75

67. Exploring Standard 5
Use appropriate tools
strategically.

68. Understanding the Standard
• What are tools used by our students?
• Why is it important to use tools?
• Tools enhance our students’ mathematical
power by assisting them as they perform
tasks.
• The ability to select appropriate tools is an
important reasoning skill.

69. Which tool is more efficient?
There is often more than one tool that
will work for a task, but some tools are
more efficient than others.
Paper & Pencil
Mental Math
Calculator

70. Solve using your assigned tool!
1. 5 x 6
2. 23 x 15
3. Estimate the cost of
2 pies @ $3.75 each Cereal @ $3.20 each
Milk @ $1.79 gal Bananas @ 59 cents/lb
1. 236 x 0 x 341
2. What comes next 3, 7, 15, 31, ___
3. A local TV store had a sale on TV’s. They sold 7 for
$1,699.95 each. They made a profit of $169.00 on
each TV. What did the store pay for the 7 TVs?
A. $1,183.00 C. $13,082.65
B. $10,716.65 D. $11,899.65

71. How Do We Get There?
• Use tools appropriately
• Number Lines (It’s Close to…)
• Rulers (broken ruler, magnified inch)
• Mental Math
– Number Partners
– In My Head?

72. • Students benefit from opportunity to select a
tool that makes sense for the math task and
to evaluate which tool is most efficient for
that task.
• Not only do our students need to be able to
select appropriate tools, they must be able
to effectively use those tools. (page 80-81)

73. Number Lines

74. Rulers
Page 84

75. Folding Paper
• Fold the strip in half. Open it, and mark ½ at
the center fold.
• Refold the strip in half and fold it in half
again. Label 1/4, 2/4, 3/4 on the three folds.

76. Questions
• Are the sections equal in size?
• Do the fraction labels make sense? Why?
• Where is 0? Why?
• Where is 1? Why?
• Why are 1/2 and 2/4 on the same fold?

77. Folding Paper
• Refold the paper and then fold it in half one
more time.
• Open the paper, place a mark on each fold
and indicate what each of the new marks
represent.

78. Folding Paper
• Why is there more than one fraction on
some folds?
• Does it make sense that those fractions are
on the same fold? Why?
• Which of those fractions is easiest to
understand? Would you say 1/2 or 2/4 or
4/8? Why?

79. Number Partners (Mental Math)
• Find a Number Partner that Makes 10
5 4 9 3 6 1 7 2 5 8
Page 86
What are some modifications for this task?

80. In My Head? (Mental Math)
Do I use paper & pencil or do it in my head?
–734 x 82
–63 x 4
–1/4 + 2/8
–930 ÷ 3
Page 86-87
Students need to identify tools that increase their efficiency with math tasks.

81. Page 88-89

82. Exploring Standard 6
Attend to precision.

83. Understanding the Standard
• Why is precision important in mathematics?
• What does it mean to be precise?
• What can we do in the classroom to
promote precise communication in
mathematics?

84. How Do We Get There?
• Estimate and Exact
• Vocabulary
– Word Webs
– Word Walls
– Sort and Label
– Mystery Words
– Translate the Symbol
– Word Boxes

85. Estimate and Exact
• Buying bags of candy to put in party treat
bags
• Measuring the dimensions of the doorway to
install a screen door
• Buying pizzas for a class party
• Buying carpeting for a living room floor

86. Estimation Skills
• Will the sum of 8 + 7 be greater than or less than
20? Why?
• Is the difference of 81 and 29 closer to 40, 50, or
60? Why?
• Is the sum of 1/3 + 4/8 greater than or less than 2?
Why?
• How would you estimate the product of 2.4 and
63? Will the product be between 2 x 60 and 3 x
60? Why or why not?

87. Word Webs
Select a word or
phrase.

88. Sort and Label

89. Sort and Label
• sum, minus, join, compare, subtract, add, take
apart, plus
• pint, foot, measuring cup, ounce, inch, scale, yard,
pound, ruler
• square, trapezoid, hexagon, rectangle, rhombus,
triangle, pentagon
• expression, equation, addition, operation,
inequality, comparison, variable, division

90. Mystery Words
area perimeter
volume length

91. Translate the Symbol
• 4 dollars and 10 cents is greater than 4
dollars and 5 cents
• One-fourth of 16 is 4
• Doubling a number then adding six more

92. Translate the Symbol
• 12 = 7 + 5
• 4 + x = 6
• 3 x 4 > 2 x 5

93. Which Is More Challenging?
• 4 dollars and 10 cents
is greater than 4
dollars and 5 cents
• One-fourth of 16 is 4
• Doubling a number
then adding six more
• 12 = 7 + 5
• 4 + x = 6
• 3 x 4 > 2 x 5

94. Word Boxes
word
Definition
Real life example
Picture
Other words

97. Page 102-103

98. Exploring Standard 7
Look for and make use of
structure.

99. Understanding the Standard
• How do we help students discover patterns
in the number system?
• What can we do to help students make
sense of mathematics through the use of
structure?

100. Can you see the pattern?
1/2 = .50 1/3 = .33 1/5 = .20
1/4 = .25 1/6 = .167 1/10 = .10
1/8 = .125 1/12 = .083 1/20 = .05
1/16 = .0625 1/24 = .0467 1/40 = .025
page 107

101. Properties, Patterns, and Functions
There were 10 children at the party. How
many were boys and how many were girls?
Boys Girls
0 10
1 9
2 8
3 7
4 6
5 5
6 4
7 3
8 2
9 1
10 0

102. Perimeter Patterns
If there was a row of 50 connected
equilateral triangles, what would the
perimeter measure?

103. Number Flexibility
There was 1 ½ cupcakes left on the plate and
Liam and Molly decided they would eat them.
How much might each person have eaten?
Be ready to justify your answers.

104. How Do We Get There?
• Pattern Cover Up
• Pattern in the Hundreds Chart or
Multiplication Chart
• Ratio Tables
• Number Lines

105. Pattern Cover Up
2 5 9 14 20

106. Hundreds Chart
On a hundred chart, can students explain the vertical and
horizontal pattern? Do they see diagonal patterns?

107. Multiplication Chart
Can students explain vertical and horizontal patterns? Do
they see diagonal patterns? Can they find patterns that
explore equivalent fractions or proportions?

110. Ratio Table
Margo was painting flowers on the classroom
mural. Every flower had 3 leaves. How many
leaves were on the mural after 7 flowers had
been painted.
Number of flowers 1 2 3 4 5 6 7
Number of leaves 3 6 9 12 15 18 21

111. Ratio Table
• Each chicken has two legs. How many legs
are on 4 chickens?
• 2 jars of peanut butter cost $4.50. How
much do 8 jars cost?
• To make one dozen ice-cream sandwiches,
Katie used ¾ gallon of ice cream. How
much ice cream did she need for 60 ice
cream sandwiches?

112. Number Line

113. page 117-118

114. Exploring Standard 8
Look for and express
regularity in repeated
reasoning.

115. Understanding the Standard
• Why is it important for students to recognize
repetition and reason why it is happening?
• When you think about repeated reasoning,
what do you think about?

116. How Do We Get There?
• Exploring Repetition
• Finding Shortcuts
• Organizing and Displaying Data to Discover
Rules
• Classroom Investigations

117. Exploring Repetition
2 + 1 = 3
5 + 1 = 6
3 + 1 = 4
6 + 1 = 7
4 + 1 = 5
What do you notice?
What conclusion can you draw?

118. Exploring Repetition

119. Investigations to Find Shortcuts
• Read first paragraph under heading
Investigations to Find Shortcuts - page 124
• Turn and Talk – about your experience with
“Memorization”

120. Investigations to Find Shortcuts
• What do you notice?
• Do you see any pattern?
• Did you notice anything
interesting about the
solution?

121. Organizing and Displaying Data
to Discover Rules
Alice jumps rope faster then anyone in her class.
She can jump 8 times in 4 seconds. How long will it
take her to jump 40 times? Justify your answer.
Page 127

122. Orchestrating Classroom Investigations
to Discover Shortcuts
1. Opportunities for all students to gather data
with partners or teams.
2. Creating compilations of class data.
3. Observing compiled data and discussing
insights.
Page 128

123. Page 130

124. Our “To Do” List
• Introduce all teachers to the practice standards.
• Provide examples to illustrate the standards.
• Encourage ongoing reflection about the standards.
– Professional development
– Faculty meetings
– Book study or PLCs
– Grade level teams
– Math coaches

125. Let’s Do Some
Math!

126. Practice the Mathematical Practices
Three different veterinarians each help a total
of 63 dogs and cats in a week, but each
veterinarian helps a different number of dogs
and cats. How many dogs and cats could
each veterinarian have helped?

127. Use tangrams to
create the Cat…

128. What is the
fractional value
of your cat’s
tail?

129. How can you find
the fractional value
of each piece?

130. What questions do
you have?

131. Updates
• NCCTM Conference: Oct 31-Nov 1
• K-2 Mid-Year Assessments
• 3-5 Formative Tasks
• Vocabulary Document

132. DPI Mathematics Section
Kitty Rutherford
Elementary Mathematics Consultant
919-807-3841
kitty.rutherford@dpi.nc.gov
Denise Schulz
Elementary Mathematics Consultant
919-807-3839
denise.schulz@dpi.nc.gov
Johannah Maynor
Secondary Mathematics Consultant
919-807-3842
johannah.maynor@dpi.nc.gov
Vacant
Secondary Mathematics Consultant
919-807-3934
Jennifer Curtis
K – 12 Mathematics Section Chief
919-807-3838
jennifer.curtis@dpi.nc.gov
Susan Hart
Mathematics Program Assistant
919-807-3846
susan.hart@dpi.nc.gov

133. For all you do for our students!