Page

-Place Value-Developing Understanding of

Numeration

• NCTM Number and Operation Standard• Characteristics of the Hindu- Arabic

Numeration System• Types and Examples of Base- Ten Models• Base- Ten Activities

NCTM Number and Operations Standard

http://standards.nctm.org/document/chapter3/numb.htmInstructional programs from prekindergarten

through grade 12 should enable all students to--understand numbers, ways of representing numbers, relationships among numbers, and number systems; understand meanings of operations and how they relate to one another; compute fluently and make reasonable estimates

NCTM Focal Pointshttp://www.nctm.org/standards/focalpoints.aspx?id=326

How Much Is A Million?by David Schwartz

• How Much is a Million Reading Rainbow Activities•http://pbskids.org/readingrainbow/books/episode_detail_120.html

•http://pbskids.org/readingrainbow/parents_and_teachers/activity_120.html

• How Much is a Million?•http://www.emints.org/ethemes/resources/S00001120.shtml

A Number SystemCharacterized by a infinite set of elements called numbers Basic operations can be performed on those numbers, Some generalizations or principles hold true for that particular number system.

A Numeration SystemCharacterized as consisting of a finite set of symbols for certain numbers,A set of rules governing the use of the symbols,Particular symbols used to represent numbers are known as the digits of the system.

Hindu-Arabic Numeration SystemBase of Ten

Base means a collection The base of the system is the number of objects used in the grouping process.10 is the value that determines a new collectionThere is no special symbol for tenWith zero, there are ten digits in the system, 0-9.

Page

Important CharacteristicsPositional or Place Value

The number of digits is limited Digits need to be repeated to express larger numbers.A digit takes on a value determined by the place it occupies in a number. The symbolization system is extended to the left by designating place values for successively larger groups of ten.Enables one to distinguish between the face value of a digit and its value because of its particular position in a numeral.

Important CharacteristicsMultiplicative Property

Multiplication is employed in decoding the value of each digit in a numeral.Example: 333

Each 3 has a different valueThe 3 on the left has a value of 3 times one hundred.The 3 in the middle has a value of 3 times ten.The 3 on the right has a value of 3 time one.

Important CharacteristicsAdditive Property

Numbers can be summed with respect to place valueFor example 333

(3 x 100) + (3 x 10) + (3 x 1)300 + 30 + 3 = 333

Important Characteristics

Zero as a PlaceholderAllows us to represent symbolically the absence of something

Key Ideashttp://www.linkslearning.org/Kids/1_Math/2_Illustrated_Lessons/3_Place_Value/The foundation of developing place value concepts lies in grouping activities.Explicit grouping or trading rules are defined and consistently followed.The position of a digit determines the number being represented.

Developing Place ValueRoss (1989) proposed a five stage development of place value understanding.Stage 1 – children associate two-digit numerals with the quantity they represent.

28 means the whole amount of 28.

Stage 2 – children identify the positional names but do not necessarily know what each digit represents.

Uses positional labels onlyIn 54, there are 4 ones and 5 tens

Page

Developing Place ValueStage 3 – children can identify the face value of digits in a number

In 34 the 3 means 3 tens and the 4 means 4 ones.The value of each digit may not be known.

Stage 4 – a transition stage during which true understanding of place value is constructed.

They know that the tens digit represents quantities of ten units.They can coordinate the part-whole relationship with two-digit numbers.

Developing Place ValueStage 5 – denotes a level of understanding the structure of our numeration system

They know that digits in a two-digit numeral represent a partitioning of the whole quantity into tens and onesThey know that the number represent is the sum of the parts.

Developing Place ValueA child’s first cognitive understanding of a two-digit numbers is of the whole amount.The students knows that the position on the right is the ones place and the position on the left is the tens place.The child is able to interpret each of the digits as representing the number indicated by its face value.The child knows that the leftmost digit in a two-digit numeral is a group of ten and the rightmost digit represents the single units.The child is able to construct numbers by partitioning whole groups of objects and can determine quantities even when the objects have been partitioned in nonstandard ways.

Developing Place ValueClip #18

Ms. Fisher’s 2nd grade class Solve problems using 2 different strategies

Clip #5Zenaida, a 3rd grader

Clip #7Talecia, a 3rd grader

Clip #19Grade 3-4 Classroom

Developing Place ValueKindergarten-do not formally explore place value concepts. However, they should be exposed to the place value mat and the notion of organizing large numbers of objects into groups of ten.First grade-place value concepts are formally introduced and explored at the concept and connecting levels.Second grade-place value is the main focus. Students explore all levels of place value throughout the school year.Third and Fourth grades-students are working with numbers involving three or more digits. Focus on mental computations and estimations.Fourth and Fifth grades-the ideas of whole number place value are extended to decimals.

ModelsProportional Models

The material for 10 is ten times the size of material for; 100 is ten times the size of 10 and so on.Base-ten blocks, bean sticks, Unifix cubes

Page

ModelsNonproportional Models

The materials do not maintain any size relationship.Money, counters, abacusChip Trading

Ten Frames•Lays the foundation for an understanding of Place Value •Ten frames assist children in further developing their number sense once they have acquired basic notions of counting.

•Ten frames help to bring about a more sophisticated counting technique as they favor the recognition of numbers from 1 to 10 and of the relationships among the numbers.

•Ten frames supports development of partitions of ten.•Ten frames provides a spatial organization for the dots that supports children's development of five-referenced, ten-referenced, and doubles-referenced conceptions of numbers up to ten.

•The ten frame enhances the development of mental imagery for such numbers.

Ten Frame ResearchTen Frames Rules

One rule of working with the ten-frame is that the top row of boxes must be filled before placing an object in the second row.When the top row is filled, there are five in the ten frame. When both rows are filled, there are ten.It is important to discuss the relationships students observe in a ten-frame.When the number seven is represented in the ten-frame, the top row is filled and there are two in the bottom row.This shows that seven is two more than five.

Children use counters to show ten on a ten-frame

Page

Ten frames can be used to access prior knowledge, to introduce a lesson, and as a lesson intervention.

Ten frames can be used as a lesson intervention.

Ten FramesChildren use a 10 frame as a step to support children’s movement toward more advanced procedures.

Adding it Up page 189

Ten FramesChildren place seven counters in one frame and five counters in the second frame.

7 + 5 = ?

Page

Ten Frames3 are needed to make a 10.Use 3 from 5 to complete 10. Two are left. That makes 12.

7 + 5 = ?

10 + 2 = 12

Ten Frame ResourcesNCTM Illuminations

http://illuminations.nctm.org/ActivityDetail.aspx?ID=75

SMART Technologyhttp://education.smarttech.com/ste/en-US/Ed+Resource/Lesson+activities/Notebook+activities/Browse+Notebook/United+States/Elementary/K-3/Math/Ten+Frames.htm

Base Ten BlocksBase ten blocks consist of cubes, rods, flats, and blocks.Cubes represent the ones place and look exactly like their name suggests - a small cube usually one centimeter by one centimeter by one centimeter. Rods represent the tens place and look like ten cubes placed in a row and fused together. Flats, as you might have guessed, represent hundreds, and blocks represent thousands. A flat looks like one hundred cubes place in a 10 x 10 square and attached together. A block looks like ten flats piled one on top of the other and bonded together.

Base Ten Blocks can be used to access prior knowledge, to introduce a lesson and for lesson intervention.

1st grade Chapter 10 pages 279-281

2nd grade Chapter 5 page 125

Page

1st grade Chapter 21 page 605 1st grade Chapter 22 page 627 2nd grade Chapter 10 page 267 2nd grade Chapter 10 page 273

2nd grade Chapter 12 page 329 2nd grade Chapter 12 page 333

Introducing Base-Ten Blocks

SMART Technologyhttp://education.smarttech.com/ste/en-US/Ed+Resource/Software+Resources/essentials/Mathematics/Number+concepts+and+operations/Place+value.htm

Math Stepshttp://www.eduplace.com/math/mathsteps/2/a/2.placeval.ideas.html

How to Use Base Ten Blockshttp://ulm.edu/~esmith/nctmregional/blocks.htmhttp://www.arcytech.org/java/b10blocks/instructions.htmlhttp://www.arcytech.org/java/b10blocks/b10blocks.htmlhttp://nlvm.usu.edu/en/nav/category_g_2_t_1.html

Place Value Games

Who Has…? I have…Index cards

Number RiddlesNumber tiles or cards

Folding NumbersSentence strips

Race to the FlatBase- Ten Blocks

Using the Hundreds Charts

Hundreds Chart Activitieshttp://www.primaryresources.co.uk/online/numbersquare.swfSMART Technology

http://education.smarttech.com/ste/en-US/Ed+Resource/Software+Resources/essentials/Mathematics/Number+concepts+and+operations/Counting+-+Hundred+Square+-+interactivity.htm

Page

Understanding and Interpreting Large Numbers

Number Names and Reading NumbersNever read whole numbers with the word AND between the hundreds and tens.

Expanded NotationRounding Numbers

Introducing rules for rounding often inhibits developing a sense of number relationships.

EstimationStudents need to explore estimation strategies rather than to be told rules and procedures.

ResourcesUsing Base Ten Blocks

http://www.susancanthony.com/Resources/base10ideas.html

Place Value Gameshttp://www.gamequarium.com/placevalue.html

Understanding the Use of Place Valuehttp://www.iit.edu/~smile/ma9202.html

Numbers as the Romans Dohttp://www.oliverlawrence.com/romans101/

Summary

An understanding of numbers should proceed computational procedures.Place-value tasks underlie algorithmic procedures for computation. A good understanding of numeration is a prerequisite for mental computation and estimation with whole numbers.