Transitioning to the Common Core State Standards – Mathematics Pam Hutchison

AGENDA Party Flags Overview of CCSS-M  Standards for Mathematical Practice  Standards for Mathematical Content Word Problems and Model Drawing Math Facts Quick review – Multiplication and Division Facts Area Models, Multiplication, and Division

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 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)

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. focus coherence 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 mathematical modeling 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 know The Standards for Mathematical Content are what students should know. do The Standards for Mathematical Practice are what students should do.  Mathematical “Habits of Mind”

Standards for Mathematical Practice

Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.

CCSS Mathematical Practices OVERARCHING HABITS OF MIND 1.Make sense of problems and persevere in solving them 6.Attend to precision REASONING AND EXPLAINING 2.Reason abstractly and quantitatively 3.Construct viable arguments and critique the reasoning of others MODELING AND USING TOOLS 4.Model with mathematics 5.Use appropriate tools strategically SEEING STRUCTURE AND GENERALIZING 7.Look for and make use of structure 8.Look for and express regularity in repeated reasoning

CCSS Mathematical Practices Cut apart the Eight Standards for Mathematical Practice (SMPs) Look over each Taxedo 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 flipbooks 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 3 rd Grade) Measurement and Data Geometry

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

5 th Grade – CCSS-M Look through the CCSS-M What has changed?  What’s missing?  What’s still there but what they are asking for is different?  What’s the same?

Word Problems and Model Drawing

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 1)Read the entire problem, “visualizing” the problem conceptually 2)Decide and write down (label) who and/or what the problem is about 3)Rewrite the question in sentence form leaving a space for the answer. 4)Draw the unit bars that you’ll eventually adjust as you construct the visual image of the problem H

Steps to Model Drawing 5)Chunk the problem, adjust the unit bars to reflect the information in the problem, and fill in the question mark. 6)Correctly compute and solve the problem. 7)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 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

Multiplication

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

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 or

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) 7 units 4 units 28 squares factors product

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 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 = = 56 +

Teaching Multiplication Facts 1 st 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

Multiplication 3 x 2  3 groups of

Multiplying by 2 Doubles Facts x x 5

Multiplying by 4 Doubling 2 x 3 (2 groups of 3) 4 x 3 (4 groups of 3) 2 x 5 (2 groups of 5) 4 x 5 (4 groups of 5)

Multiplying by 3 Doubles, then add on 2 x 3 (2 groups of 3) 3 x 3 (3 groups of 3) 2 x 5 (2 groups of 5) 3 x 5 (3 groups of 5)

Teaching Multiplication Facts Group 1Group 2

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

Breaking Apart 4 7

Teaching Multiplication Facts Group 1Group 2 Group 3

Commutative property Build to automaticity

Teaching Multiplication Facts Group 1Group 2 Group 3 Group 4

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

Connecting Multiplication and Division

Division What does 6  2 mean?  Repeated subtraction group 2 groups 3 groups

Measurement Division I have 21¢ to buy candies with. If each gumdrop costs 3¢, how many gumdrops can I buy?

Fair Share Division Mr. Gomez has 12 cupcakes. He wants to put the cupcakes into 4 boxes so that there’s the same number in each box. How many cupcakes can go in each box?

Difference in counting? Measurement 4 for you, 4 for you, 4 for you And so on Like measuring out an amount Fair Share 1 for you, 1 for you, 1 for you, 1 for you 2 for you, 2 for you, 2 for you, 2 for you And so on Like dealing cards

Measurement Division What does 6  2 mean?  6 split into groups of 2

Fair Share Division What does 6  2 mean?  6 split evenly into 2 groups

Models for Division Repeated subtraction Groups  Finding the number in each group  Finding the number of groups Arrays – finding the missing side

Repeated Subtraction 21 ÷ 3  1 group for you 21 – 3 = 18 left  1 group for you18 – 3 = 15 left  1 group for you15 – 3 = 12 left  1 group for you12 – 3 = 9 left  1 group for you9 – 3 = 6 left  1 group for you6 – 3 = 3 left  1 group for you3 – 3 = 0 left 7 groups of 3

Skip Counting 30 ÷

Using Arrays 5 ? ? 3 6

Using Arrays ? )

Connection to Multiplication 42 ÷ 7 = think “7 x _?_ = 42” Fact Families

Practicing Facts

Triangle Flash Cards

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 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

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

Area Models and Multiplication

Using Arrays to Multiply 23 x 4 4 rows of 20 4 rows of 3 = 80 =

Patterns in Multiplication Groups of 4 1’s 10’s100’s groups

Using Arrays to Multiply Use Base 10 blocks and an area model to solve the following: 21 x 13

Multiplying and Arrays 21 x

Partial Products 21 x (10  20) (10  1) (3  20) (3  1)

31 14

Partial Products 3 1 x (4  1) (4  30) (10  1) (10  30)

Pictorial Representation 84 x  80 4,  4 7  80 7 

Pictorial Representation  30 2,  7 4  30 4  x 94

Pictorial Representation 347 x ,000 2, ,

Decimals 0.4 x

Try It! 0.3 x x 3 2 x x 3.1

Patterns in Multiplication x 6 Start with 3 x

Fractions 