1. Transitioning to the Common Core State Standards – Mathematics
Pam Hutchison

2. AGENDA Party Flags Overview of CCSS-M Word Problems and Model Drawing
Standards for Mathematical Practice
Standards for Mathematical Content
Word Problems and Model Drawing
Strategies – Addition and Subtraction

3. Expectations
We are each responsible for our own learning and for the learning of the group.
We respect each others learning styles and work together to make this time successful for everyone.
We value the opinions and knowledge of all participants.

4. Erica is putting up lines of colored flags for a party.
The flags are all the same size and are spaced equally along the line.
1. Calculate the length of the sides of each flag, and the space between flags.
Show all your work clearly.
2. How long will a line of n flags be?
Write down a formula to show how long a line of n flags would be.

5. CaCCSS-M Find a partner Decide who is “A” and who is “B”
At the signal, “A” takes 30 seconds to talk
Then at the signal, switch, “B” takes 30 seconds to talk.
“What do you know about the CaCCSS-M?”

6. CaCCSS-M “What do you know about the CaCCSS-M?”
Using the fingers on one hand, please show me how much you know about the CaCCSS-M

7. National Math Advisory Panel Final Report
“This Panel, diverse in experience, expertise, and philosophy, agrees broadly that the delivery system in mathematics education—the system that translates mathematical knowledge into value and ability for the next generation — is broken and must be fixed.” (2008, p. xiii)

8. Common Core State Standards
Developed through Council of Chief State School Officers and National Governors Association

9. Common Core State Standards

10. How are the CCSS different?
The CCSS are reverse engineered from an analysis of what students need to be college and career ready.
The design principals were focus and coherence. (No more mile-wide inch deep laundry lists of standards)

11. How are the CCSS different?
Real life applications and mathematical modeling are essential.

12. How are the CCSS different?
The CCSS in Mathematics have two sections:
Standards for Mathematical CONTENT
and
Standards for Mathematical PRACTICE
The Standards for Mathematical Content are what students should know.
The Standards for Mathematical Practice are what students should do.
Mathematical “Habits of Mind”

13. Standards for Mathematical Practice

14. Mathematical Practice
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.

15. CCSS Mathematical Practices
REASONING AND EXPLAINING
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning of others
Make sense of problems and persevere in solving them
OVERARCHING HABITS OF MIND
Attend to precision
MODELING AND USING TOOLS
Model with mathematics
Use appropriate tools strategically
SEEING STRUCTURE AND GENERALIZING
Look for and make use of structure
Look for and express regularity in repeated reasoning

16. CCSS Mathematical Practices
Cut apart the Eight Standards for Mathematical Practice (SMPs)
Look over each Tagxedo image and decide which image goes with which practice
The more frequently a word is used, the larger the image
Using the Standards for Mathematical Practice handout…did you get them right?
Glue the Practice title to the appropriate image.
What did you notice about the SMPs?

27. Reflection
How are these practices similar to what you are already doing when you teach?
How are they different?
What do you need to do to make these a daily part of your classroom practice?

28. Supporting the SMP’s Summary
Questions to Develop Mathematical Thinking
Common Core State Standards Flip Book
Compiled from a variety of resources, including CCSS, Arizona DOE, Ohio DOE and North Carolina DOE
flipbooks

29. Standards for Mathematical Content

30. Content Standards
Are a balanced combination of procedure and understanding.
Stressing conceptual understanding of key concepts and ideas

31. Content Standards
Continually returning to organizing structures to structure ideas
place value
properties of operations
These supply the basis for procedures and algorithms for base 10 and lead into procedures for fractions and algebra

32. “Understand” means that students can…
Explain the concept with mathematical reasoning, including
Concrete illustrations
Mathematical representations
Example applications

33. Organization K-8 Domains
Larger groups of related standards. Standards from different domains may be closely related.

34. Domains K-5 Counting and Cardinality (Kindergarten only)
Operations and Algebraic Thinking
Number and Operations in Base Ten
Number and Operations-Fractions (Starts in 3rd Grade)
Measurement and Data
Geometry

35. Organization K-8 Clusters Standards
Groups of related standards. Standards from different clusters may be closely related.
Standards
Defines what students should understand and be able to do.
Numbered

37. Word Problems and Model Drawing

38. concrete – pictorial – abstract
Model Drawing
A strategy used to help students understand and solve word problems
Pictorial stage in the learning sequence of
concrete – pictorial – abstract
Develops visual-thinking capabilities and algebraic thinking.

39. Steps to Model Drawing
Read the entire problem, “visualizing” the problem conceptually
Decide and write down (label) who and/or what the problem is about
Rewrite the question in sentence form leaving a space for the answer.
Draw the unit bars that you’ll eventually adjust as you construct the visual image of the problem H

40. Steps to Model Drawing
Chunk the problem, adjust the unit bars to reflect the information in the problem, and fill in the question mark.
Correctly compute and solve the problem.
Write the answer in the sentence and make sure the answer makes sense.

41. Missing Numbers 1
Mutt and Jeff both have money. Mutt has $34 more than Jeff. If Jeff has $72, how much money do they have altogether? H

42. Missing Numbers 2
Mary has 94 crayons. Ernie has 28 crayons less than Mary but 16 crayons more than Shauna. How many crayons does Shauna have?

43. Missing Numbers 3
Bill has 12 more than three times the number of baseball cards Chris has. Bill has 42 more cards than Chris. How many baseball cards does Chris have? How many baseball cards does Bill have?

44. Missing Numbers 4
Amy, Betty, and Carla have a total of 67 marbles. Amy has 4 more than Betty. Betty has three times as many as Carla. How many marbles does each person have?

45. Representation
Getting students to focus on the relationships and NOT the numbers!

46. Computation

47. Teaching for Understanding
Telling students a procedure for solving computation problems and having them practice repeatedly
rarely results in fluency
Because we rarely talk about how and why the procedure works.

48. Teaching for Understanding
Students do need to learn procedures for solving computation problems
But emphasis (at earliest possible age) should be on why they are performing certain procedure

49. Learning Progression
Concrete
 Representational
 Abstract

50. Research
Students who learn rules before they learn concepts tend to score significantly lower than do students who learn concepts first
Initial rote learning of a concept can create interference to later meaningful learning

51. Fact Fluency
Institute of Educational Sciences Practice Guide “Assisting Students Struggling with Mathematics: Response to Intervention for Elementary and Middle Schools”
Recommends approximately 10 minutes per day building fact fluency

52. Fact Fluency The intent IS NOT to administer basic fact tests!
Teachers need to build basic fact strategy lessons for conceptual development, which builds fluency.

53. Fact Fluency
Fact fluency must be based on an understanding of operations and thinking strategies.
Students must
Construct visual representations to develop conceptual understanding.
Connect facts to those they know
Use mathematics properties and relationships to make associations

54. Quick Review of Primary Number Sense

55. SUBITIZING
Ability to recognize the number of objects in a collection, without counting
When the number exceeds this ability, counting becomes necessary

56. Perceptual Subitizing
Maximum of 5 objects
Helps children
Separate collections of objects into single units
Connect each unit with only one number word
Develops the process of counting

57. Conceptual Subitizing
Allows children to know the number of a collection by recognizing a familiar pattern or arrangement
Helps young children develop skills needed for counting, addition, and subtraction
Helps develop sense of number and quantity

58. How Many Dots?

59. How Many? 

60. How Many? 
 

61. How Many?  

62. How Many?   

63. Base 10 Blocks

64. Base 10 Blocks

65. Base 10 Shorthand

66. Base 10 Shorthand

67. Basic Facts

68. Basic Facts

69. Basic Facts

70. Tens Facts 
7 + 3 = 10

71. Tens Facts 
6 + 4 = 10

72. Tens Facts 
8 + 2 = 10

73. Composing and Decomposing Numbers

74. Part-Whole Relations

75. Number Bonds 4 4 4 4 4

76. Number Bonds – 17 17 17 17 17 17 17 17 17

77. 3 Phases for Teaching Math Facts

78. Concept Learning Goal Techniques
Understand the meaning of addition, subtraction, multiplication, and division
Techniques
Use concrete objects, pictures, and symbols to develop

79. Fact Strategies Goal Techniques Recognize clusters of facts
Understanding relationships between facts
Techniques
Addition and Subtraction – counting on, counting back, making 5’s, making 10’s, doubles, doubles plus 1, compensation, derived facts
Multiplication and Division – skip counting, repeated addition, repeated subtraction, derived facts, distributive property

80. Automaticity Goal Techniques
Know facts so that they can be recalled quickly and accurately, and be retained over time
Techniques
Schedule short frequent practices
Reinforce facts already known
Use concepts and strategies to develop missing facts

81. Math Fact Strategies Direct modeling / Counting all
Counting on / Counting back / Skip Counting
Derived Fact Strategies
Composing / Decomposing
Mental strategies
Automaticity

82. Review of Addition Fact Strategies

83. Doubles        

84. Doubles            

85. Doubles Plus 1             

86. Doubles Plus 1             

87. Doubles Minus 1        

88. Making Fives       

89. Making Fives 1 2 7

90. Making Fives            

91. Making Fives            

92. Making Fives 1 1 12

93. Making Fives              

94. 8 + 6             

95. Making Fives 3 1 14

96. Making Tens              

97. Making Tens 2 4 14

98. Making Tens            

99. Making Tens 3 2 12

100. Addition

101. Addition –

102. Addition –
Make tens 2 4
30 + 4 34

103. Addition –

104. 28 + 6 3 4 Addition – 28 + 6 1 8 ones + 6 ones = 14 ones
+ 6 3 4
8 ones + 6 ones = 14 ones
14 ones = 1 ten + 4 ones
2 tens + 1 ten = 3 tens

105. Adding 2-digit numbers Miguel – 1st Grade 30 + 16 Connor – 1st Grade
How is the way these students solved the problems different from the way we typically teach addition?

106. Addition:

107. Addition –
Plan to make tens 2 32
30 + 32 62

108. Addition –
Plan to make tens 4 34
50 + 34 84

109. Addition:

110. …adds tens and tens, ones and ones…
Addition:
…adds tens and tens, ones and ones…

111. … and sometimes it is necessary to compose a ten
Addition:
… and sometimes it is necessary to compose a ten

112. Addition:

113. Addition:

114. Addition –
2 8
+ 3 4 1 6 2

115. Addition –
546
+ 278 1 1 8 2 4

116. Be careful about run on equal signs!
Addition –
Add On Tens, Then Ones
Add on tens = 76
Add on ones = 84
Be careful about run on equal signs!

117. Addition –
Add On Hundreds, Tens, and Ones = = = =
746
816
824

118. Addition – 50 12 10 2
= 62

119. Addition – 46 + 38 + 30 + 8 70 + 14 84 (70 + 10 + 4) Expanded Form 84
( )

120. Addition –
Expanded Form
824

121. Addition –
Add Tens, Add Ones, and Combine 46
+ 38
= 70
6 + 8 = 14
= 84
This can also be done as add ones, add tens, and combine. 70 14 84

122. Addition –
546
+ 278
6 + 8
700
110 14
824

123. Addition – 46 + 38 46 + 38 Add a nice number 46 + 40 = 86
Compensate
Add a nice number = 86
(Think: 40 is 2 too many)
Compensate 86 – 2 = 84

124. Addition Try at least 2 different strategies on each problem

125. Vertical vs Horizontal
Why do students need to be given addition (and subtraction) problems both of these ways?
= 279
+ 54

126. Review of Subtraction Fact Strategies

127. Understanding Subtraction
Katie had 5 candy hearts. She gave 2 of them to Nick. How many hearts does Kate have left for herself?
Katie has 5 candy hearts. Nick has 2 candy hearts. How many more does Katie have?

128. 5 – 2     

129. 5 – 2          

130. 5 – 2

131. 5 – 2

132. Subtraction How do you currently teach subtraction?
“Take-away”
“The distance from one number to the other”
Additional Strategies

133. Making Fives       

134. 13 – 6             

135. Making Tens
13 – 6 = 3 3

136. Making Tens
15 – 7 = 5 2

137. 13 – 6             

138. Using Tens
13 – 6 = 10 3 4 7

139. Using Tens
15 – 7 = 10 5 3 8

140. Commutative Property
4 + 9
9 + 4
5 + 7
7 + 5

141. Fact Families
7 + 5 = 12
5 + 7 = 12
12 – 5 = 7
12 – 7 = 5

142. Subtraction

143. Developing Subtraction
Connor – 1st Grade
25 – 8
Connor – 1st Grade
70 – 23

144. Subtraction: 43 – 6
Take Away Tens, Then Ones
43 – 6 = 37 3 3

145. Subtraction: 73 – 46
Take Away Tens, Then Ones
73 – 46 = 27 40 6

146. Subtraction: 73 – 46
Take Away Tens, Then Ones
73 – 46 = 27 40 6 3 3

147. 1. Subtraction: 53 – 38

148. 1. Subtraction: 53 – 38

149. 2. Subtraction: 53 – 38

150. 2. Subtraction: 53 – 38

151. 2. Subtraction: 53 – 38

152. 3. Subtraction: 53 – 38

153. 3. Subtraction: 53 – 38

154. 4. Subtraction: 53 – 38

155. 4. Subtraction: 53 – 38

156. Practice
First tens, then ones
54 – 38
First ones, then tens

157. Subtraction: 53 – 38

158. Subtraction: 53 – 38

159. Subtraction: 53 – 38

160. Subtraction: 53 – 38

161. Subtraction: 53 – 38

162. Subtraction: 73 – 46 – 46 2 7 Regrouping and Ten Facts 73 6
– 46 6
60 – 40 = 20 2 7

163. Subtraction: 42 – 29 – 29 1 3 Regrouping and Ten Facts 42 10 + 2 3 - 9
– 29
10 + 2 3
- 9 1
30 – 20 = 10 1 3

164. Subtraction: 57 – 34 57  34 (50 + 7)  (30 + 4) 20 + 3 = 23
(50 + 7)  (30 + 4)
Do I have enough to be able to subtract? 20 + 3
= 23

165. Subtraction: 52 – 34 52  34 (50 + 2)  (30 + 4) (40 + 12)  (30 + 4)
(50 + 2)  (30 + 4)
( )  (30 + 4)
Do I have enough to be able to subtract? 10 + 8
= 18

166. Suppose I slide the line down 1 space?
Subtraction – 87
Constant Differences 86
299 87
300
Suppose I slide the line down 1 space?
299 – 86 =

167. Subtraction: 73 – 46
Constant Differences 73
– 46
+ 4
= 77
+ 4
= 50 27

168. Subtraction: 73 – 46
Regrouping by Adding Ten 73
– 46 13 5 27

169. Subtraction – Adding On
471 – 285
Start at 285 Add 5
Now at 290 Add 10 (15)
Now at 300 Add 100 (115)
Now at 400 Add 70 (185)
Now at 470 Add 1 (186)
Now at 471 – DONE!

170. Subtraction Try at least 2 different strategies on each problem
– – 36
– – 327
– 298

171. Subtraction Planning your strategy Not all problems are created equal!
What strategy would be the most effective.
NOT “one size fits all”

172.
– – – 299
– – – 371
– – – 695

173. Practicing Facts

174. Triangle Flash Cards 15 7 8

175. Flash Card Practice Facts I Know Quickly
Facts I Can Figure Out Quickly
Facts I Am Still Learning
Create 1 representation for each fact
Create 2 representations for each fact

176. Assessing Facts

177. Fluency Assessments 20-25 facts 2 colors of pencils (or pens)
After 60 seconds, call switch. Students change the color of the pencil they are using.
Give students another seconds
If students finish before time to stop, continue to write and solve your own fact problems

178. Advantages All students get to finish!
Let’s you assess both fluency and accuracy.

179. Multiplication

180. Reasoning about Multiplication and Division

181. Multiplication
3 x 2
3 groups of 2
Repeated Addition

182. Multiplication 3 rows of 2
This is called an “array” or an “area model”

183. Advantages of Arrays as a Model
Models the language of multiplication
4 groups of 6 or
4 rows of 6

184. Advantages of Arrays as a Model
Students can clearly see the difference between (the sides of the array) and the (the area of the array)
factors
product
7 units
4 units
28 squares

185. Advantages of Arrays Commutative Property of Multiplication
4 x = x 4

186. Advantages of Arrays Associative Property of Multiplication
(4 x 3) x = x (3 x 2)

187. Advantages of Arrays
Distributive Property
3(5 + 2) = 3 x x 2

188. Advantages of Arrays as a Model
They can be used to support students in learning facts by breaking problem into smaller, known problems
For example, 7 x 8 8 8 5 3 4 4 7 7 35 + 21
= 56 28 + 28
= 56

189. Teaching Multiplication Facts
1st group

190. Group 1 Repeated addition Skip counting Drawing arrays and counting
Connect to prior knowledge
Build to automaticity

191. Multiplication
3 x 2
3 groups of 2 1 3 5 2 4 6

192. Multiplication
3 x 2
3 groups of 2 2 4 6

193. Multiplication
3 x 2
3 groups of 2

194. Multiplying by 2
Doubles Facts
3 + 3
2 x 3
5 + 5
2 x 5

195. Multiplying by 4 Doubling 2 x 3 (2 groups of 3) 4 x 3 (4 groups of 3)

196. Multiplying by 3 Doubles, then add on 2 x 3 (2 groups of 3)

197. Teaching Multiplication Facts
Group 1
Group 2

198. Group 2 Building on what they already know Distributive property
Breaking apart areas into smaller known areas
Distributive property
Build to automaticity

199. Breaking Apart 7 4

200. Teaching Multiplication Facts
Group 1
Group 2
Group 3

201. Group 3
Commutative property
Build to automaticity

202. Teaching Multiplication Facts
Group 1
Group 2
Group 3
Group 4

203. Group 4 Building on what they already know Distributive property
Breaking apart areas into smaller known areas
Distributive property
Build to automaticity