Transitioning to the Common Core State Standards – Mathematics Pam Hutchison pam.ucdmp@gmail.com

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

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.

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.

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?”

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

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)

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

Common Core State Standards

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)

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

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”

Standards for Mathematical Practice

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.

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

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?

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?

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 http://katm.org/wp/wp-content/uploads/ flipbooks

Standards for Mathematical Content

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

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

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

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

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

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

Word Problems and Model Drawing

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.

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

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.

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

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?

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?

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?

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

Computation

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.

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

Learning Progression Concrete  Representational  Abstract

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

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

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.

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

Quick Review of Primary Number Sense

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

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

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

How Many Dots?

How Many? 

How Many?   

How Many?  

How Many?   

Base 10 Blocks

Base 10 Blocks

Base 10 Shorthand

Base 10 Shorthand

Basic Facts

Basic Facts

Basic Facts

Tens Facts  7 + 3 = 10

Tens Facts  6 + 4 = 10

Tens Facts  8 + 2 = 10

Composing and Decomposing Numbers

Part-Whole Relations

Number Bonds 4 4 4 4 4

Number Bonds – 17 17 17 17 17 17 17 17 17

3 Phases for Teaching Math Facts

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

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

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

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

Review of Addition Fact Strategies

Doubles        

Doubles            

Doubles Plus 1             

Doubles Plus 1             

Doubles Minus 1        

Making Fives       

Making Fives 4 + 3 1 2 7

Making Fives            

Making Fives            

Making Fives 6 + 6 1 1 12

Making Fives              

8 + 6             

Making Fives 8 + 6 3 1 14

Making Tens              

Making Tens 8 + 6 2 4 14

Making Tens            

Making Tens 7 + 5 3 2 12

Addition

Addition – 28 + 6

Addition – 28 + 6 Make tens 28 + 6 2 4 30 + 4 34

Addition – 28 + 6

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

Adding 2-digit numbers Miguel – 1st Grade 30 + 16 Connor – 1st Grade 39 + 25 How is the way these students solved the problems different from the way we typically teach addition?

Addition: 28 + 34

Addition – 28 + 34 Plan to make tens 28 + 34 2 32 30 + 32 62

Addition – 46 + 38 Plan to make tens 46 + 38 4 34 50 + 34 84

Addition: 28 + 34

…adds tens and tens, ones and ones… Addition: 28 + 34 …adds tens and tens, ones and ones…

… and sometimes it is necessary to compose a ten Addition: 28 + 34 … and sometimes it is necessary to compose a ten

Addition: 28 + 34

Addition: 28 + 34

Addition – 28 + 34 2 8 + 3 4 1 6 2

Addition – 546 + 278 546 + 278 1 1 8 2 4

Be careful about run on equal signs! Addition – 46 + 38 Add On Tens, Then Ones 46 + 38 Add on tens 46 + 30 = 76 Add on ones 76 + 8 = 84 Be careful about run on equal signs!

Addition – 546 + 278 Add On Hundreds, Tens, and Ones 546 + 278 = 546 + 200 = 746 + 70 = 816 + 8 = 746 816 824

Addition – 28 + 34 28 + 34 20 + 8 + 30 + 4 50 12 10 2 = 62

Addition – 46 + 38 + 30 + 8 70 + 14 84 (70 + 10 + 4) Expanded Form 40 + 6 + 30 + 8 70 + 14 84 (70 + 10 + 4)

Addition – 546 + 278 Expanded Form 500 + 40 + 6 + 200 + 70 + 8 700 + 110 + 14 810 + 14 824

Addition – 46 + 38 Add Tens, Add Ones, and Combine 46 + 38 40 + 30 = 70 6 + 8 = 14 70 + 14 = 84 This can also be done as add ones, add tens, and combine. 70 14 84

Addition – 546 + 278 546 + 278 500 + 200 40 + 70 6 + 8 700 110 14 824

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

Addition Try at least 2 different strategies on each problem 1. 57 + 6 2. 48 + 37 3. 63 + 29 4. 254 + 378 5. 538 + 296

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

Review of Subtraction Fact Strategies

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?

5 – 2     

5 – 2          

5 – 2 0 1 2 3 4 5 6 7 8 9 10 11 12

5 – 2 0 1 2 3 4 5 6 7 8 9 10 11 12

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

Making Fives       

13 – 6             

Making Tens 13 – 6 = 3 3

Making Tens 15 – 7 = 5 2

13 – 6             

Using Tens 13 – 6 = 10 3 4 7

Using Tens 15 – 7 = 10 5 3 8

Commutative Property 4 + 9 9 + 4 5 + 7 7 + 5

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

Subtraction

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

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

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

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

1. Subtraction: 53 – 38

1. Subtraction: 53 – 38

2. Subtraction: 53 – 38

2. Subtraction: 53 – 38

2. Subtraction: 53 – 38

3. Subtraction: 53 – 38

3. Subtraction: 53 – 38

4. Subtraction: 53 – 38

4. Subtraction: 53 – 38

Practice First tens, then ones 54 – 38 First ones, then tens 71 - 45

Subtraction: 53 – 38

Subtraction: 53 – 38

Subtraction: 53 – 38

Subtraction: 53 – 38

Subtraction: 53 – 38

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

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

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

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

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

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

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

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!

Subtraction Try at least 2 different strategies on each problem 1. 53 – 7 2. 58 – 36 3. 73 – 29 4. 554 – 327 5. 538 – 298

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

1. 475 2. 527 3. 792 – 231 – 375 – 299   4. 843 5. 683 6. 792 – 598 – 496 – 371 7. 475 8. 504 9. 702 – 197 – 368 – 695

Practicing Facts

Triangle Flash Cards 15 7 8

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

Assessing Facts

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 60-90 seconds If students finish before time to stop, continue to write and solve your own fact problems

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

Multiplication

Reasoning about Multiplication and Division http://fw.to/sQh6P7I

Multiplication 3 x 2 3 groups of 2 Repeated Addition 2 + 2 + 2

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

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

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

Advantages of Arrays Commutative Property of Multiplication 4 x 6 = 6 x 4

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

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

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

Teaching Multiplication Facts 1st group

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

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

Multiplication 3 x 2 3 groups of 2 2 4 6

Multiplication 3 x 2 3 groups of 2 2 + 2 + 2

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

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

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

Teaching Multiplication Facts Group 1 Group 2

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

Breaking Apart 7 4

Teaching Multiplication Facts Group 1 Group 2 Group 3

Group 3 Commutative property Build to automaticity

Teaching Multiplication Facts Group 1 Group 2 Group 3 Group 4

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