# 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

### Website Production Credits

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

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

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

Joe Karaman

Unit Managers
Maria Constantinides

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

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

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

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

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

Zero: The Biography of a Dangerous Idea

History and Transfinite Numbers: Counting Infinite Sets. (Session 2)
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)
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.
Adding It Up: Helping Children Learn Mathematics
Continued…
Continued…
Continued…
Continued…
Continued…
Continued…
Continued…

### Support Materials

Introduction

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.