NCTM Number and
Operations Standard
-Place ValueDeveloping Understanding of
Numeration
http://standards.nctm.org/document/chapter3/numb.htm
Instructional 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 Points
„ http://www.nctm.org/standards/focalpoints.aspx?id=3
26
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• NCTM Number and Operation Standard
• Characteristics of the Hindu
- Arabic
Numeration System
• Types and Examples of Base
- Ten Models
• Base
- Ten Activities
How Much Is A Million?
A Number System
by David Schwartz
Characterized 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.
„
• 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/etheme
s/resources/S00001120.shtml
A Numeration System
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Hindu-Arabic Numeration System
Characterized 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.
ƒ Base of Ten
ƒ Base means a collection
ƒ The base of the system is the number of
objects used in the grouping process.
ƒ 10 is the value that determines a new
collection
ƒ There is no special symbol for ten
ƒ With zero, there are ten digits in the
system, 0-9.
Page
Important Characteristics
Important Characteristics
ƒ Positional or Place Value
ƒ The number of digits is limited
ƒ Digits need to be repeated to express larger numbers.
ƒ A digit takes on a value determined by the place it
occupies in a number.
ƒ The symbolization system is extended to the left by
designating place values for successively larger
groups of ten.
ƒ Enables one to distinguish between the face value of
a digit and its value because of its particular position
in a numeral.
ƒ Multiplicative Property
ƒ Multiplication is employed in decoding the
value of each digit in a numeral.
ƒ Example: 333
ƒ Each 3 has a different value
ƒ The 3 on the left has a value of 3 times one
hundred.
ƒ The 3 in the middle has a value of 3 times ten.
ƒ The 3 on the right has a value of 3 time one.
Important Characteristics
Important Characteristics
ƒ Additive Property
ƒ Numbers can be summed with
respect to place value
ƒ For example 333
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ƒ (3 x 100) + (3 x 10) + (3 x 1)
ƒ 300 + 30 + 3 = 333
Key Ideas
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Zero as a Placeholder
Allows us to represent
symbolically the absence of
something
Developing Place Value
http://www.linkslearning.org/Kids/1_Math/2_
Illustrated_Lessons/3_Place_Value/
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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.
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Ross (1989) proposed a five stage development
of place value understanding.
Stage 1 – children associate two-digit numerals
with the quantity they represent.
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Stage 2 – children identify the positional names
but do not necessarily know what each digit
represents.
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Page
28 means the whole amount of 28.
Uses positional labels only
In 54, there are 4 ones and 5 tens
Developing Place Value
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Stage 3 – children can identify the face value of
digits in a number
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Developing Place Value
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In 34 the 3 means 3 tens and the 4 means 4 ones.
The value of each digit may not be known.
They know that digits in a two-digit
numeral represent a partitioning of the
whole quantity into tens and ones
„ They know that the number represent is the
sum of the parts.
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Stage 4 – a transition stage during which true
understanding of place value is constructed.
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They know that the tens digit represents quantities
of ten units.
They can coordinate the part-whole relationship
with two-digit numbers.
Developing Place Value
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Developing Place Value
A 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.
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Clip #18
„ Ms. Fisher’s 2nd grade class
„ Solve problems using 2 different strategies
Clip #5
„ Zenaida, a 3rd grader
Clip #7
„ Talecia, a 3rd grader
Clip #19
„ Grade 3-4 Classroom
Models
Developing Place Value
ƒ
Stage 5 – denotes a level of understanding
the structure of our numeration system
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Kindergarten-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.
Page
Proportional 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
Models
„
Ten Frames
Nonproportional Models
„ The materials do not maintain any size
relationship.
„ Money, counters, abacus
„ Chip Trading
•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, tenreferenced, and doubles-referenced conceptions of numbers up to
ten.
•The ten frame enhances the development of mental imagery for
such numbers.
Ten Frame Research
Ten Frames Rules
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Children use counters
to show ten on a tenframe
Page
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.
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 Frames
Ten Frames
Children use a 10 frame as a step to support children’s
movement toward more advanced procedures.
Children place seven counters in one
frame and five counters in the second
frame.
7 + 5 = ?
Adding it Up page 189
Page
Ten Frames
Ten Frame Resources
3 are needed to make a 10.
Use 3 from 5 to complete 10.
Two are left. That makes 12.
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7 + 5 = ?
10
+
2
„
= 12
Base Ten Blocks can be used to
access prior knowledge, to introduce a
lesson and for lesson intervention.
Base Ten Blocks
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NCTM Illuminations
„ http://illuminations.nctm.org/ActivityDetail.aspx?
ID=75
SMART Technology
„ http://education.smarttech.com/ste/enUS/Ed+Resource/Lesson+activities/Notebook+act
ivities/Browse+Notebook/United+States/Element
ary/K-3/Math/Ten+Frames.htm
Base 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.
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 12 page 329
2nd grade Chapter 10 page 267
2nd grade Chapter 12 page 333
Introducing
Base-Ten Blocks
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SMART Technology
„ http://education.smarttech.com/ste/enUS/Ed+Resource/Software+Resources/essentials/Mathematics/
Number+concepts+and+operations/Place+value.htm
Math Steps
„ http://www.eduplace.com/math/mathsteps/2/a/2.placeval.ideas.h
tml
How to Use Base Ten Blocks
„ http://ulm.edu/~esmith/nctmregional/blocks.htm
„ http://www.arcytech.org/java/b10blocks/instructions.html
„ http://www.arcytech.org/java/b10blocks/b10blocks.html
„ http://nlvm.usu.edu/en/nav/category_g_2_t_1.html
Using the Hundreds Charts
Place Value Games
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Who Has…? I have…
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Number Riddles
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Folding Numbers
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Index cards
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Number tiles or cards
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Hundreds Chart Activities
http://www.primaryresources.co.uk/online/n
umbersquare.swf
SMART Technology
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Sentence strips
Race to the Flat
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2nd grade Chapter 10 page 273
Base
- Ten Blocks
Page
http://education.smarttech.com/ste/enUS/Ed+Resource/Software+Resources/essentia
ls/Mathematics/Number+concepts+and+operati
ons/Counting+-+Hundred+Square++interactivity.htm
Understanding and
Interpreting Large Numbers
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Resources
Number Names and Reading Numbers
„ Never read whole numbers with the word AND
between the hundreds and tens.
Expanded Notation
Rounding Numbers
„ Introducing rules for rounding often inhibits
developing a sense of number relationships.
Estimation
„ Students need to explore estimation strategies
rather than to be told rules and procedures.
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Summary
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Using Base Ten Blocks
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.
Page
http://www.susancanthony.com/Resources/base10ide
as.html
Place Value Games
„ http://www.gamequarium.com/placevalue.htm
l
Understanding the Use of Place Value
„ http://www.iit.edu/~smile/ma9202.html
Numbers as the Romans Do
„ http://www.oliverlawrence.com/romans101/