Connie Locklear March 4, 2013 COMMON CORE MATH UPDATES PUBLIC SCHOOLS OF ROBESON COUNTY

The Three Shifts in Mathematics Focus strongly where the standards focus Focus deeply only on what is emphasized in the standards, so that students gain strong foundations Major Work for each grade level Vocabulary

Focusing attention within Number and Operations Operations and Algebraic Thinking Expressions and Equations Algebra  Number and Operations—Base Ten  The Number System  Number and Operations— Fractions  K High School

“The coherence and sequential nature of mathematics dictate the foundational skills that are necessary for the learning of algebra. The most important foundational skill not presently developed appears to be proficiency with fractions (including decimals, percents, and negative fractions). The teaching of fractions must be acknowledged as critically important and improved before an increase in student achievement in algebra can be expected.” Final Report of the National Mathematics Advisory Panel (2008, p. 18)

Coherence: Think across grades and link to major topics within grades Vertical Alignment and Horizontal Alignment Each standard is not a new event, but an extension of previous learning. The Three Shifts in Mathematics

Coherence Example: Grade 3 The standards make explicit connections at a single grade Properties of Operations Area Multiplication and Division 3.OA.5 3.MD.7c 3.MD.7a

Major Work for Third Grade Third Grade Major ClustersSupporting/Additional Clusters Operations and Algebraic Thinking  Represent and solve problems involving multiplication and division.  Understand properties of multiplication and the relationship between multiplication and division.  Multiply and divide within 100.  Solve problems involving the four operations, and identify and explain patterns in arithmetic. Number and Operations—Fractions  Develop understanding of fractions as numbers. Measurement and Data  Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.  Geometric measurement: understand concepts of area and relate area to multiplication and to addition. Number and Operations in Base Ten  Use place value understanding and properties of operations to perform multi-digit arithmetic. Measurement and Data  Represent and interpret data.  Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures. Geometry  Reason with shapes and their attributes.

Coherence means presenting mathematics so that when we put all the pieces together we have a beautiful work of art.

Rigor: Requires conceptual understanding, fluency, and application Requires equal intensity in time, activities, and resources in pursuit of all three The Three Shifts in Mathematics

Solid Conceptual Understanding Teaches more than “how to get the answer” and supports students’ ability to access concepts from a number of perspectives Students are able to see math as more than a set of mnemonics or discrete procedures Conceptual understanding supports the other aspects of rigor (fluency and application)

Fractions

Activity—Let’s Do the Math What portion of the rectangle is shaded yellow? Explain your thinking.

Mathematical Practices The CCSSM cannot be taught without embedding the eight mathematical practices 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

Standards for Mathematical Practices Teacher(s): Mathematical Topic(s): Date: 1. Makes sense of problems and perseveres in solving them ☐ Understands the meaning of the problem and looks for entry points to its solution ☐ Monitors and evaluates the progress and changes course as necessary ☐ Analyzes information (givens, constrains, relationships, goals) ☐ Checks their answers to problems and ask, “Does this make sense?” ☐ Designs a plan _________________________________________________________ Comments: 2. Reason abstractly and quantitatively4. Model with mathematics.8. Look for and express regularity in repeated reasoning ☐ Makes sense of quantities and relationships ☐ Represents a problem symbolically ☐ Considers the units involved ☐ Understands and uses properties of operations ___________________________________________ Comments: ☐ Apply reasoning to create a plan or analyze a real world problem ☐ Applies formulas/equations ☐ Makes assumptions and approximations to make a problem simpler ☐ Checks to see if an answer makes sense and changes a model when necessary ___________________________________________ Comments: ☐ Notices repeated calculations and looks for general methods and shortcuts ☐ Continually evaluates the reasonableness of their results while attending to details and makes generalizations based on findings ☐ Solves problems arising in everyday life ___________________________________________ Comments: 3. Construct viable arguments and critique the reasoning of others 5. Use appropriate tools strategically. 7. Look for and make use of structure. ☐ Uses definitions and previously established causes/effects (results) in constructing arguments ☐ Makes conjectures and attempts to prove or disprove through examples and counterexamples ☐ Communicates and defends their mathematical reasoning using objects, drawings, diagrams, actions ☐ Listens or reads the arguments of others ☐ Decide if the arguments of others make sense ☐ Ask useful questions to clarify or improve the arguments ___________________________________________ Comments: ☐ Identifies relevant external math resources (digital content on a website) and uses them to pose or solve problems ☐ Makes sound decisions about the use of specific tools. Examples may include: ☐ Calculator ☐ Concrete models ☐ Digital Technology ☐ Pencil/paper ☐ Ruler, compass, protractor ☐ Uses technological tools to explore and deepen understanding of concepts ______________________________________ Comments: ☐ Looks for patterns or structure ☐ Recognize the significance in concepts and models and can apply strategies for solving related problems ☐ Looks for the big picture or overview ___________________________________________ Comments: 6. Attend to precision. ☐ Communicates precisely using clear definitions ☐ Provides carefully formulated explanations ☐ States the meaning of symbols, calculates accurately and efficiently ☐ Labels accurately when measuring and graphing __________________________________________________________________ Comments:

CCSSM Mathematical Practice Questions for Teachers to Ask 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 Teachers ask:  What is this problem asking?  How could you start this problem?  How could you make this problem easier to solve?  How is ___’s way of solving the problem like/different from yours?  Does your plan make sense? Why or why not?  What tools/manipulatives might help you?  What are you having trouble with?  How can you check this? Teachers ask:  What does the number ____ represent in the problem?  How can you represent the problem with symbols and numbers?  Create a representation of the problem. Teachers ask:  How is your answer different than _____’s?  How can you prove that your answer is correct?  What math language will help you prove your answer?  What examples could prove or disprove your argument?  What do you think about _____’s argument  What is wrong with ____’s thinking?  What questions do you have for ____? *it is important that the teacher poses tasks that involve arguments or critiques Teachers ask:  Write a number sentence to describe this situation  What do you already know about solving this problem?  What connections do you see?  Why do the results make sense?  Is this working or do you need to change your model? *It is important that the teacher poses tasks that involve real world situations Use appropriate tools strategicallyAttend to precisionLook for and make use of structure Look for and express regularity in repeated reasoning Teachers ask:  How could you use manipulatives or a drawing to show your thinking?  Which tool/manipulative would be best for this problem?  What other resources could help you solve this problem? Teachers ask:  What does the word ____ mean?  Explain what you did to solve the problem.  Compare your answer to _____’s answer  What labels could you use?  How do you know your answer is accurate?  Did you use the most efficient way to solve the problem? Teachers ask:  Why does this happen?  How is ____ related to ____?  Why is this important to the problem?  What do you know about ____ that you can apply to this situation?  How can you use what you know to explain why this works?  What patterns do you see? *deductive reasoning (moving from general to specific) Teachers ask:  What generalizations can you make?  Can you find a shortcut to solve the problem? How would your shortcut make the problem easier?  How could this problem help you solve another problem? *inductive reasoning (moving from specific to general)

Foundation for Item and Task Development Common Core State Standards Smarter Balanced Content Specifications Smarter Balanced Item and Task Specifications Items and Performance Tasks

Cognitive Rigor and Depth of Knowledge The level of complexity of the cognitive demand Level 1: Recall and Reproduction Requires eliciting information such as a fact, definition, term, or a simple procedure, as well as performing a simple algorithm or applying a formula. Level 2: Basic Skills and Concepts Requires the engagement of some mental processing beyond a recall of information. Level 3: Strategic Thinking and Reasoning Requires reasoning, planning, using evidence, and explanations of thinking. Level 4: Extended Thinking Requires complex reasoning, planning, developing, and thinking most likely over an extended period of time.

Cognitive Rigor Matrix This matrix from the Smarter Balanced Content Specifications for Mathematics draws Educational Objectives and Webb’s Depth-of-Knowledge Levels below.

Level 1 Example Grade 8 Select all of the expressions that have a value between 0 and ∙ 8 – – ∙ (–5) 6 (–5) 10

Level 2 Example Grade 8 A cylindrical tank has a height of 10 feet and a radius of 4 feet. Jane fills this tank with water at a rate of 8 cubic feet per minute. How many minutes will it take Jane to completely fill the tank without overflowing at this rate? Round your answer to the nearest minute. A cylindrical tank has a height of 10 feet and a radius of 4 feet. Jane fills this tank with water at a rate of 8 cubic feet per minute. How many minutes will it take Jane to completely fill the tank without overflowing at this rate? Round your answer to the nearest minute.

Level 3 Example Grade 8 The total cost for an order of shirts from a company consists of the cost for each shirt plus a one-time design fee. The cost for each shirt is the same no matter how many shirts are ordered. The company provides the following examples to customers to help them estimate the total cost for an order of shirts. 50 shirts cost $ shirts cost $2370 Part A: Using the examples provided, what is the cost for each shirt, not including the one-time design fee? Explain how you found your answer. Part B: What is the cost of the one-time design fee? Explain how you found your answer. The total cost for an order of shirts from a company consists of the cost for each shirt plus a one-time design fee. The cost for each shirt is the same no matter how many shirts are ordered. The company provides the following examples to customers to help them estimate the total cost for an order of shirts. 50 shirts cost $ shirts cost $2370 Part A: Using the examples provided, what is the cost for each shirt, not including the one-time design fee? Explain how you found your answer. Part B: What is the cost of the one-time design fee? Explain how you found your answer.

Level 4 Example Grade 8 During the task, the student assumes the role of an architect who is responsible for designing the best plan for a park with area and financial restraints. The student completes tasks in which he/she compares the costs of different bids, determines what facilities should be given priority in the park, and then develops a scale drawing of the best design for the park and an explanation of the choices made. This investigation is done in class using a calculator, an applet to construct the scale drawing, and a spreadsheet.

Thank you!