Series
Learning Math: Number and Operations
This video and webbased course for K8 teachers examines three main categories in the Number and Operations strand of the math standards.
Learning Math: Number and Operations is available at archive.learner.org/resources/series171.html.
A video and webbased course for K8 teachers; 12 halfhour video programs (10 per grade band), course guide, and website.
Learning Math: Number and Operations, a video and Webbased 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 Operationscourse may be more challenging than other Learning Mathcourses.
The course consists of 10 approximately twoandahalfhour sessions, each with a half hour of video programming, problemsolving 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 twocolor 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 nonterminating decimals. Explore ways to predict the number of decimal places in a terminating decimal and the period of a nonterminating 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 partpart or partwhole 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 K2, 35, and 68 teachers (former course participants), all of whom have adapted their new knowledge to their classrooms. Select Video 10 for K2 teachers, Video 11 for 35 teachers, and Video 12 for 68 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 coauthor 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 Brinson Teacher, Grade 3 
Kristen Mann Teacher, Grade 4 

Ben Carter Teacher, Grade 6 
Doug McGlathery Curriculum Advisor, Grades 910 

Rhonda Cherry Special Education Teacher, Grade 7 
Victoria Miles Math Teacher, Grade 7 

Donna Donnell Math Coordinator, Grades K5 
Maria Royston Teacher, Grade 2 

L.J. Dutton Teacher, Grade 4 
Thomas Szekely Math Teacher, Grade 8 

Nancy Gaff Math Teacher, Grade 8 
Rick Tabor Math Teacher, Grade 8 

Andrea Gladkowski Math Teacher, Grade 8 
Jeanne Watts Math Teacher, Grade 7 

Elizabeth Hughes Teacher, Grade 5 
Susan Weiss Math Specialist, Grades K5 
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 DavisKay
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
Online 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
XRay Astronomy and Divisibility
Jennifer Lauer, Data Systems Operations, Chandra XRay Observatory
Madhu Sudan, Laboratory for Computer Science, Massachusetts Institute of Technology
Images courtesy of Chandra XRay Observatory:
•  Spiral galaxy NGC 4631: Xray: 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 E010272: Xray: 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 K2:  Susan Weiss: 2nd Grade, Solomon Schechter Day School 
Grades 35:  Donna Donnell: 5th Grade, Swasey Central School 
Grades 68:  Victoria Miles: 7th Grade, Abigail Adams Intermediate School 
Reading List
Seife, Charles (2000). Zero: The Biography of a Dangerous Idea
(pp. 6, 1221). (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 K6 (pp. 4072). 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 K6
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 K2: Classroom Case Studies
Session 10, Grades 35: Classroom Case Studies
Session 10, Grades 68: Classroom Case Studies
Appendix, including Glossary
Progress Chart
Print out the chart below to keep track of your progress through the Number and Operations online course. It is also availiable in PDF format.  

Session 10: Classroom Case Studies
