Connie LocklearMarch 4, 2013

COMMON CORE MATH UPDATESPUBLIC 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 1 2 3 4 5 6 7 8 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.7c3.MD.7a

Major Work for Third Grade

Third GradeMajor Clusters Supporting/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 MathWhat 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 PracticesTeacher(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 quantitatively 4. 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 AskMake sense of problems and

persevere in solving themReason abstractly and

quantitativelyConstruct 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 strategically

Attend to precisionLook for and make use of

structureLook for and express regularity

in repeated reasoningTeachers 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 KnowledgeThe 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 ExampleGrade 8

Select all of the expressions that have a value between 0 and 1. 87 ∙ 8–12

74

7–3

1

3

2

∙1

3

9

(–5)6

(–5)10

Level 2 ExampleGrade 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.

Level 3 ExampleGrade 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 $349.50

• 500 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 ExampleGrade 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!