## Series

# Learning Math: Number and Operations

## This video- and web-based course for K-8 teachers examines three main categories in the Number and Operations strand of the math standards.

*A video- and web-based course for K-8 teachers; 12 half-hour video programs (10 per grade band), course guide, and website.*

*Learning Math: Number and Operations*, a video- and Web-based course for elementary and middle school teachers examines the three main categories in the Number and Operations strand of *Principles and Standards of School Mathematics* (NCTM) — understanding numbers, representations, relationships, and number systems; the meanings of operations and relationships among those operations; and reasonable estimation and fluent computation. The course covers the real number system, place value, the behavior of zero and infinity, meanings and models of basic operations, percentages, and modeling operations with fractions, often with the aid of concrete, physical models that enhance understanding. It also examines Basic Number Theory topics, such as factors and multiples, as well as divisibility tests, at both practical and abstract levels. Accordingly, parts of the *Number and Operations *course may be more challenging than other *Learning Math *courses.

The course consists of 10 approximately two-and-a-half-hour sessions, each with a half-hour of video programming, problem-solving activities provided online and in a print guide, and interactive activities and demonstrations on the Web. The 10th session (choose video program 10, 11, or 12, depending on your grade level) explores ways to apply the concepts of number and operations you’ve learned in your own classroom.

### Overview of the Course

**Session 1: What Is a Number System?**

Understand the nature of the real number system, the elements and operations that make up the system, and some of the rules that govern the operations. Examine a finite number system that follows some (but not all) of the same rules, and then compare this system to the real number system. Use a number line to classify the numbers we use, and examine how the numbers and operations relate to one another.

**Session 2: Number Sets, Infinity, and Zero**

Continue examining the number line and the relationships among sets of numbers that make up the real number system. Explore which operations and properties hold true for each of the sets. Consider the magnitude of these infinite sets and discover that infinity comes in more than one size. Examine place value and the significance of zero in a place value system.

**Session 3: Place Value**

Look at place value systems based on numbers other than 10. Examine the base two numbers and learn uses for base two numbers in computers. Explore exponents and relate them to logarithms. Examine the use of scientific notation to represent numbers with very large or very small magnitude. Interpret whole numbers, common fractions, and decimals in base four.

**Session 4: Meanings and Models for Operations**

Examine the operations of addition, subtraction, multiplication, and division and their relationships to whole numbers and integers. Work with area models for multiplication and division. Explore the use of two-color chips to model operations with positive and negative numbers.

**Session 5: Divisibility Tests and Factors**

Explore number theory topics. Analyze Alpha math problems and discuss how they help with the conceptual understanding of operations. Examine various divisibility tests to see how and why they work. Begin examining factors and multiples.

**Session 6: Number Theory**

Examine visual methods for finding least common multiples and greatest common factors, including Venn diagram models and area models. Explore prime numbers. Learn to locate prime numbers on a number grid and to determine whether very large numbers are prime.

**Session 7: Fractions and Decimals**

Extend your understanding of fractions and decimals. Examine terminating and non-terminating decimals. Explore ways to predict the number of decimal places in a terminating decimal and the period of a non-terminating decimal. Examine which fractions terminate and which repeat as decimals, and why all rational numbers must fall into one of these categories. Explore methods to convert decimals to fractions and vice versa. Use benchmarks and intuitive methods to order fractions.

**Session 8: Rational Numbers and Proportional Reasoning**

Begin examining rational numbers. Explore a model for computations with fractions. Analyze proportional reasoning and the difference between absolute and relative thinking. Explore ways to represent proportional relationships and the resulting operations with ratios. Examine how ratios can represent either part-part or part-whole comparisons, depending on how you define the unit, and explore how different representations affect a ratio’s behavior in computations.

**Session 9: Fractions, Percents, and Ratios**

Continue exploring rational numbers, working with an area model for multiplication and division with fractions, and examining operations with decimals. Explore percents and the relationships among representations using fractions, decimals, and percents. Examine benchmarks for understanding percents, including percents less than 10 and greater than 100. Consider ways to use an elastic model, an area model, and other models to discuss percents. Explore some ratios that occur in nature.

**Session 10: Classroom Case Studies**

Explore how the concepts developed in this course can be applied at different grade levels through case studies of K-2, 3-5, and 6-8 teachers (former course participants), all of whom have adapted their new knowledge to their classrooms. Select Video 10 for K-2 teachers, Video 11 for 3-5 teachers, and Video 12 for 6-8 teachers.

### Who's Who

**Content Developer/Facilitator
Carol R. Findell, Ed.D.
**For more than 30 years, Dr. Findell has been a mathematics educator at the preschool through college levels. She served as editor of the National Council of Teachers of Mathematics publication

*Student Math Notes*and on the editorial board of the

*New England Mathematics Journal,*and was the author of two

*World’s Largest Math Events — Math Olympics*and

*Mathematics: The Language of the Universe.*She was a member and chairperson of the Question Writing Team for the Mathcounts Competition and was head of the writing team for the

*Figure This!*national campaign to promote mathematics education reform. Dr. Findell is co-author of many books and a frequent speaker at national conferences. She has participated in several funded projects in mathematics education and has worked with elementary, middle, and high school teachers across the country, primarily in New Hampshire, Massachusetts, and Connecticut.

**Onscreen Participants**

Monique BrinsonTeacher, Grade 3 |
Kristen MannTeacher, Grade 4 |
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Ben CarterTeacher, Grade 6 |
Doug McGlatheryCurriculum Advisor, Grades 9-10 |
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Rhonda CherrySpecial Education Teacher, Grade 7 |
Victoria MilesMath Teacher, Grade 7 |
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Donna DonnellMath Coordinator, Grades K-5 |
Maria RoystonTeacher, Grade 2 |
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L.J. DuttonTeacher, Grade 4 |
Thomas SzekelyMath Teacher, Grade 8 |
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Nancy GaffMath Teacher, Grade 8 |
Rick TaborMath Teacher, Grade 8 |
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Andrea GladkowskiMath Teacher, Grade 8 |
Jeanne WattsMath Teacher, Grade 7 |
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Elizabeth HughesTeacher, Grade 5 |
Susan WeissMath Specialist, Grades K-5 |

### Credits

**Website Production Credits**

*Learning Math: Number and Operations* is a production of WGBH Interactive and WGBH Educational Programming and Outreach for Annenberg Media.

© 2003 WGBH Educational Foundation. All rights reserved.

**Senior Producer**

Ted Sicker

**Curriculum Director**

Denise Blumenthal

**Content Developer**

Carol R. Findell, Ed.D., Boston University, Massachusetts

**Coordinating Producer**

Sanda Zdjelar

**Curriculum Developer**

Anna Brooks

**Special Projects Assistant**

Nina Farouk

**Core Advisors**

Suzanne Chapin, Boston University, Massachusetts

Bowen Kerins, Mathematics Consultant

Michelle Manes, Mathematics Teacher and Education Consultant

**Designers**

Plum Crane

Lisa Rosenthal

Christian Wise

**Web Developers**

Joe Brandt

Kit Buckley

Rishi K. Connelly

**Online Video Segment Coordinator**

Mary Susan Blout

**Business Managers**

Walter Gadecki

Joe Karaman

**Unit Managers**

Maria Constantinides

Adriana Sacchi

**Special Assistance of**

Jennifer Davis-Kay

Rebecca Evans

Julie Wolf

### Video Series Production Credits

*Learning Math: Number and Operations *is a production of WGBH Educational Foundation for Annenberg Media.

**Executive Producer**

Michele Korf

**Senior Project Director**

Amy Tonkonogy

**Producer**

Christine Dietlin

**Content Developer/Facilitator**

Carol R. Findell, Ed.D., Boston University, Massachusetts

**Advisors**

Hollee Freeman, TERC, Massachusetts

DeAnn Huinker, University of Wisconsin, Milwaukee

Miriam Leiva, University of North Carolina, Charlotte

Sid Rachlin, East Carolina University, North Carolina

**Content Editor**

Srdjan Divac, Harvard University, MA

**On Location Consultants**

Kenton G. Findell

Bowen Kerins

**Editor**

Glenn Hunsberger

**Director**

Bob Roche

**Associate Producers**

Irena Fayngold

Pamela Lipton

**Project Manager**

Sanda Zdjelar

**Additonal Editing**

Dickran H. Manoogian

**Production Manager**

Mary Ellen Gardiner

**Post Production Associate Producer**

Peter Villa

**Location Coordinator**

Mary Susan Blout

**Teacher Liaison**

Lisa Eure

**Set Design**

Irena Fayngold

**Set Assistant**

Elena Graceffa

**Camera**

Kevin Burke

Bill Charette

Lance Douglas

Larry LeCain

Steve McCarthy

Dillard Morrison

David Rabinovitz

**Audio**

Steve Bores

Chris Bresnahan

Charlie Collias

Jose Leon

Dennis McCarthy

Gilles Morin

**Interns**

Timothy Barney

Ravi Blatt

Nina Farouk

Joe Gudema

Justin Hartery

Aimee Jones

**Design**

Gaye Korbet

Daryl Myers

Alisa Placas

**On-line Editors**

Mark Geffen

John O’Brien

**Sound Mix**

John Jenkins

Dan Lesiw

**Content Graphics**

Glenn Hunsberger

**Original Music**

Tom Martin

**Narrator**

Jeff Loeb

**Business Manager**

Joe Karaman

**Unit Manager**

Maria Constantinides

**Office Coordinators**

Justin Brown

Laurie Wolf

### Special Thanks

**Session 3
Number Systems for Computers**

Deborah G. Douglas, Curator of Science and Technology, Massachusetts Institute of Technology Museum

Jason Glasgow, Principle Design Engineer, EMC Corporation

Rodney C. Marable, Principle Design Engineer, EMC Corporation

Photographs courtesy of MIT Museum, Cambridge, Massachusetts

**Session 4
How Do Computers Divide?**

Professor Charles E. Leiserson, Lab for Computer Science, Massachusetts Institute of Technology

Archival footage courtesy of Analog Devices Inc., High Speed Converter Lab and Wafer Fabrication Facilities

**Session 5
X-Ray Astronomy and Divisibility**

Jennifer Lauer, Data Systems Operations, Chandra X-Ray Observatory

Madhu Sudan, Laboratory for Computer Science, Massachusetts Institute of Technology

Images courtesy of Chandra X-Ray Observatory:

• |
Spiral galaxy NGC 4631: X-ray: NASA/UMass/D. Wang, et al. |

• |
Optical: NASA/HST/D. Wang, et al. |

• |
Antennae Galaxy: NASA/CXC/SAO/PSU/CMU |

• |
Mosaic of Galactic Center: NASA/UMass/D. Wang, et al. |

• |
Crab Nebula: NASA/CXC/SAO |

• |
Vela Pulsar: NASA/PSU/G. Pavlov, et al. |

• |
Supernova remnant E0102-72: X-ray: NASA/CXC/SAO, Optical: NASA/HST |

• |
Radio: CSIRO/ATNF/ATCA |

**Session 6
Internet Security**

Michael Szydlo, Ph.D., Research Scientist, RSA Security, Inc.

**Session 7
Babylonian Decimals**

Dr. Kim Plofker, Postdoctoral Fellow, Dibner Institute

Dr. James Armstrong, Semitic Museum, Harvard University

Archival footage courtesy of Harvard University Art Museums. Illustrations taken from I. Mary Hussey,

*Sumerian Tablets in the Harvard Semitic Museum, Part II.*© 1915 by Harvard University Press.

**Session 8
Relative Reasoning in Finance**

Geeta Aiyer, President, Walden Asset Management, A Division of US Trust Co. of Boston

Archival footage courtesy of Boston Stock Exchange, Costco Wholesale Corp., Walgreens Co., Merck and Co., Inc., and EMC, Inc.

**Session 9
The Golden Rectangle in Architecture**

Ed Tsoi, Architect

Archival footage courtesy of:

Bettmann/CORBIS, Harvard University Planning and Real Estate

Jonathan Hale, *The Old Way of Seeing.* © 1994 by Houghton Mifflin Company. Used with permission.

A. J. Bicknell, *Bicknell’s Victorian Buildings.* © 1979 by Dover Publications. Used with permission.

Matila Ghyka, *The Geometry of Art and Life.* © 1946, 1977 by Dover Publications. Used with permission.

**Session 10
Classroom Case Studies**

Donna Donnell, Swasey Central School, Brentwood, NH

Victoria Miles, Abigail Adams Intermediate School, Weymouth, MA

Susan Weiss, Solomon Schechter Day School, Newton, MA

**Site Location**

Ferryway School, Malden, MA

### Video Index

**Session 1: What Is a Number System?**

Noticing Patterns

Identity and Inverse Elements

Density

Constructing Irrational Numbers

**Session 2: Number Sets, Infinity, and Zero**

Number Sets

Comparing the Size of Number Sets

The Size of Infinity

Exploring the Graph

**Session 3: Place Value**

Converting to Base Two

Computers and Base Two

Base Four

Computers Today

**Session 4: Meanings and Models for Operations**

Quotative and Partitive Division

Using Manipulatives

Another Model

**Session 5: Divisibility Tests and Factors**

Divisibility by Six

Why Does It Work?

Divisibility by Four

**Session 6: Number Theory**

Modeling GCF

Modeling LCM

Locating Prime Numbers

Large Primes

**Session 7: Fractions and Decimals**

Unit Fractions and Decimals

Predicting Remainders

Converting Decimals to Fractions

Numbers in Ancient Bablyonia

**Session 8: Rational Numbers and Proportional Reasoning**

Modeling Operations With Fractions

Absolute and Relative Reasoning

**Session 9: Fractions, Percents, and Ratios**

Modeling Multiplication of Fractions

Golden Rectangle in Architecture

**Session 10: Classroom Case Studies**

Grades K-2: |
Susan Weiss: 2nd Grade, Solomon Schechter Day School |

Grades 3-5: |
Donna Donnell: 5th Grade, Swasey Central School |

Grades 6-8: |
Victoria Miles: 7th Grade, Abigail Adams Intermediate School |

### Reading List

**Seife, Charles (2000). Zero: The Biography of a Dangerous Idea
(pp. 6, 12-21). (Session 1)**

Reproduced with permission from Viking Penguin. © 2000 by Charles Seife. All rights reserved.

Download PDF File:

**Zero: The Biography of a Dangerous Idea**

** History and Transfinite Numbers: Counting Infinite Sets. (Session 2)
**Download PDF File:

History and Transfinite Numbers: Counting Infinite Sets.

**Chapin, Suzanne and Johnson, Art (2000). Chapters 3 and 4 in Math Matters: Understanding the Math You Teach, Grades K-6 (pp. 40-72). Sausalito: CA: Math Solutions Publications. (Session 4)**

Reproduced with permission from the publisher. © 2001 by Math Solutions Publications. All rights reserved.

Download PDF File:

Math Matters: Understanding the Math You Teach, Grades K-6

**Kilpatrick, J.; Swafford, J.; and Findell, B., ed. (2001). Adding It Up: Helping Children Learn Mathematics. A Report of the National Research Council. Washington, D.C.: National Academy Press. (Session 9) **Reproduced with permission from the publisher. © 2001 by National Academy Press. All rights reserved.

Download PDF File:

Adding It Up: Helping Children Learn Mathematics

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### Support Materials

**Session 1: What is a Number System?**

**Session 2: Number Sets, Infinity, and Zero**

**Session 3: Place Value**

**Session 4: Meanings and Models for Operations**

**Session 5: Divisibility Tests and Factors**

**Session 6: Number Theory**

**Session 7: Fractions and Decimals**

**Session 8: Rational Numbers and Proportional Reasoning**

**Session 9: Fractions, Percents, and Ratios**

**Session 10, Grades K-2: Classroom Case Studies**

**Session 10, Grades 3-5: Classroom Case Studies**

**Session 10, Grades 6-8: Classroom Case Studies**

**Appendix,** including **Glossary**

### Progress Chart

Print out the chart to keep track of your progress through the Number and Operations online course, available in PDF format.