Mathematics Rigor with Common Core State Standards

DPI AIG Team SessionNCAGT Conference

March, 2012

Kitty Rutherford kitty.rutherford@dpi.nc.gov

Robin Barbour robin.barbour@dpi.nc.gov

www.corestandards.org

Critical Areas

Critical AreaCritical Area

Focal PointsFocal Points

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Mathematical Mathematical PracticesPractices

Grade Level

Overview

Grade Grade LevelLevelDomainDomain Standards

ClusterCluster

High School Themes

• Number and Quantity

• Algebra

• Functions

• Modeling

• Geometry

• Statistics and Probability

High School Standards Notation

Perform operations on matrices and use matrices in applications. 6. (+) Use matrices to represent and manipulate data, e.g., to represent

payoffs of incidence relationship in a network.

11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y =g(x intersect are the solutions of the equations f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.★

K-8 DomainsDomains K 1 2 3 4 5 6 7 8

Counting and Cardinality                  

Operations and Algebraic Thinking                  

Number and Operations in Base Ten                  

Measurement and Data                  

Geometry                  

Number and Operations - Fractions                  

Ratios and Proportional Relationships                  

The Number System                  

Expressions and Equations                  

Statistics and Probability                  

Functions

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.

Standards for Mathematical Practices

Mathematical practices describe the habits of mind of mathematically proficient students…

• Who is doing the talking?

• Who is doing the thinking?

• Who is doing the math?

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Make sense of problems and preserve in solving them.

When planning, ask

“What task can I give that will build student

understanding?”rather than

“How can I explain clearly so they will understand?”

Grayson Wheatley, NCCTM, 2002

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61

77

84

41

57

69

15 1613

54

24

17

0

10

20

30

40

50

60

70

80

90

100

Australia Czech Republic Hong Kong J apan Netherlands United States

Using ProceduresMaking Connections

31

1618

20 19

59

8

52

4648

37

00

10

20

30

40

50

60

70

80

90

100

Australia Czech Republic Hong Kong J apan Netherlands United States

Using Procedures

Making Connections

Types of Math Problems Presented

How Teachers ImplementedMaking Connections Math

Problems

Lesson ComparisonUnited States and Japan

The emphasis on skill acquisition is evident in the steps most common in U.S. classrooms

The emphasis on understanding is evident in the steps of a typical Japanese lesson

•Teacher instructs students in concept or skill

•Teacher solves example problems with class

•Students practice on their own while teacher assists individual students

•Teacher poses a thought provoking problem

•Students and teachers explore the problem

•Various students present ideas or solutions to the class

•Teacher summarizes the class solutions

•Students solve similar problems

17

What do you see?

40 4

10 2

30 4

18

Predict some additional data.

40 4

10 2

30 4

19

How close were you?

40 4

10 2

30 4

20 3

20

All the numbers – so?

45 4

25 3

15 2

40 4

10 2

30 4

20 3

21

Where are you?

Roller Coaster 45 4

Ferris Wheel 25 3

Bumper Cars 15 2

Rocket Ride 40 4

Merry-go-Round 10 2

Water Slide 30 4

Fun House 20 3

22

Fill in the blanks.

Ride ??? ???Roller Coaster 45 4

Ferris Wheel 25 3

Bumper Cars 15 2

Rocket Ride 40 4

Merry-go-Round 10 2

Water Slide 30 4

Fun House 20 3

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The Amusement Park

Ride Time Tickets

Roller Coaster 45 4

Ferris Wheel 25 3

Bumper Cars 15 2

Rocket Ride 40 4

Merry-go-Round 10 2

Water Slide 30 4

Fun House 20 3

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The Amusement Park

The 4th and 2nd graders in your school are going on a trip to the Amusement Park. Each 4th grader is going to be a buddy to a 2nd grader.

Your buddy for the trip has never been to an amusement park before. Your buddy want to go on as many different rides as possible. However, there may not be enough time to go on every ride and you may not have enough tickets to go on every ride.

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The bus will drop you off at 10:00 a.m. and pick you up at 1:00 p.m. Each student will get 20 tickets for rides.

Use the information in the chart to write a letter to your buddy and create a plan for a fun day at the amusement park for you and your buddy.

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The Amusement Park

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.

Standards for Mathematical Practices

Year Standards To Be TaughtStandards To Be

Assessed

2011 – 2012 2003 NCSCOS 2003 NCSCOS

2012 – 2013 CCSS CCSS (NC)

2013 – 2014 CCSS CCSS (NC)

2014 – 2015 CCSS CCSS (SBAC)

Common Core State Standards Adopted June, 2010

Mathematics Claims

The Smarter Balanced Assessment Consortium has released a document outlining four claims about what mathematically proficient students can do.

The claims are a synthesis of the Standards for Mathematical Practice, and form the guiding principles to be used in creating assessments.

Mathematics Claim 1 & 2

• Students can explain and apply mathematical concepts and carry out mathematical procedures with precision and fluency.

• Students can frame and solve a range of complex problems in pure and applied mathematics.

Mathematics Claim 3 & 4 • Students can clearly and precisely

construct viable arguments to support their own reasoning and to critique the reasoning of others.

• Students can analyze complex, real-world scenarios and can use mathematical models to interpret and solve problems.

http://www.k12.wa.us/smarter/

RIGOR

ConceptualUnderstanding

Application

Skills and Procedures

Rigor through Standards

Rigor though Standards

6th Grade Critical Area:

Students use the meaning of fractions, the meanings of multiplication and division, and the relationship between multiplication and division to understand and explain why the procedures for dividing fractions make sense. Students use these operations to solve problems.

• Skipping material will create gaps in learning which jeopardizes foundational content needed to maximize the likelihood of success in High School Mathematics

Content Acceleration

• Compacting 3 years of content into 2 is supported by research; 2 years into 1 is considered too challenging

• Considering high school courses is essential when making middle school recommendations

6th Grade 7th Grade 8th Grade

100 %100% 7th grade ??% 8th grade

??% 8th grade Math 1 Standards

Option 1:

Option 2:

6th Grade 7th Grade 8th Grade

100 % 6th grade Remaining 7th gradeRemaining 8th grade

Math 1 Standards??% 7th grade ??% 8th grade

High School Courses in Middle SchoolGetting Students Ready

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•Integer Exponents (8.EE.1)

•Multiplication and Division with Scientific Notation (8.EE.4)

•Solving Systems by Substitution (8.EE.8)

•Volume of Pyramids, Cones and Spheres (7.G.6, 8.G.9)

•Surface Area of Pyramids (6.G.4, 7.G.6)

At A GlanceInstructional Implications High School

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•Angles (7.G.5, 8.G.5)

•Using Pythagorean Theorem in 3-D Figures (8.G.7)

•Mean Absolute Deviation (6.SP.5c)

•Two-way Tables (8.SP.4)

•Qualitative Graphs (8.F.5)

•Graphing Proportional Relationships (7.RP.2a, b, c, d, 8.EE.5)

At A GlanceInstructional Implications Math One

www.ncdpi.wikispaces.net

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www.ncdpi.wikispaces.net

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Grade Band Pages

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Crosswalk Documents

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The crosswalks reflect a comparison between the Common Core State Standards and the North Carolina Standard Course of Study. They inform educators about how the current standards align with the CCSS Standards.

Mathematics Crosswalk

CAUTION!!

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CONTENT APPEARING TO BE THE SAME MAY ACTUALLY BE

DIFFERENT!!

The CCSS Requires CLOSE Reading!!!

Unpacking Documents

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The purpose of the Unpacking Documents is to increase student achievement by ensuring educators understand the new standards.

The “unpacking” of the standards done in these documents is an effort to answer a simple question “What does this standard mean that a student must know and be able to do?” and to ensure the description is helpful, specific and comprehensive for educators.

Unpacking – At a Glance

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Unpacking – Standards for Mathematical Practice

Unpacked Content

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Common Core GlossaryTable 1. Common multiplication and division situations

K-5 Units

Students learn mathematics by exploring mathematically-rich tasks and sharing strategies, ideas, and approaches with one another. (practices)

K Adding and Subtraction

1st Exploring Two-Digit Numbers

2nd Two- & Three-Digit Addition & Subtraction

3rd Unit on Area and Perimeter

4th Fractions

5th Fractions04/20/23 • page 58

Format of the Lessons

The phases of the lesson:

• Engage - Brief opening activity

• Explore - Mathematically-rich task

• Explain - Discussion of task and concepts

• Elaborate - Follow-up activity

• Evaluate - description of formative and summative assessments

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Additional Wiki Information

Year Who Can Apply Nomination

Deadline

Application

Deadline

2012Elementary Teachers

Grades K - 6April 1, 2012 May 1, 2012

2013Secondary Teachers

Grades 7 - 12April 1, 2013 May 1, 2013

www.paemst.org

Presidential Awards for Excellence in

Mathematics and Science Teaching

Elementary Mathematics Add-on Licensure

• 18-hour Graduate program (6 courses)• Participating Universities

– East Carolina University– Appalachian State University– NC State University– UNC Chapel Hill– UNC Charlotte– UNC Greensboro– UNC Wilmington

• Dr. Sid Rachlin (rachlins@ecu.edu)

Webinars

Archived Webinars:- November 17th: CCSS and Math I Standards- January 10th: K-12 Getting Started:

Organization Tools and Instructional Planning Model

- February 9th: Making Mathematics Accessible (K-12)

- March 8th: K – 2 Assessment and Calendar Time

http://illustrativemathematics.org/  http://commoncoretools.wordpress.com K, Counting and Cardinality;K–2, Operations and Algebraic ThinkingK, Counting and Cardinality; K–5, Operations and Algebraic Thinking K–3, Categorical Data; Grades 2–5, Measurement Data* (data part of the Measurement and Data Progression)3-5, Number and Operations – Fractions 6-7, Ratio and Proportional Relationships6-8, Progression for Statistics and Probability6–8, Expressions and Equations

ADDITIONAL RESOURCES:

QUESTIONS

COMMENTS

Kitty Rutherfordkitty.rutherford@dpi.nc.gov

Robin Barbourrobin.barbour@dpi.nc.gov

Contact Information

Website: www.ncdpi.wikispaces.net