Place Value:
The Foundation
of Number Sense
North Dakota Council of Teachers of Mathematics
presented by Kathleen Cotter Lawler
© Activities for Learning, Inc. 2015
Number Sense
• Expected to be learned through counting
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Number Sense
• Expected to be learned through counting
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Counting Model
• Adding by counting-on is a difficult skill.
• Provides poor concept of quantity.
• Ignores place value.
• Is not natural; it takes years of practice.
• Is very error prone.
• Is tedious and time-consuming.
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Problems Learning to Count
• Children with dyslexia or dyscalculia
• Children with SLI (specific language impairment)
• Children with memory challenges
• Children from low SES backgrounds
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Arithmetic Traditionally Taught
Counting
Memorizing
390 Facts
Learning
Procedures
Place
Value
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Solving
Problems
Place Value in Its Proper Place
In Treviso Arithmetic, (1478), the world’s first
printed arithmetic book, the author states there
are five fundamental operations in arithmetic:
• Numeration (place value up to millions)
• Addition
• Subtraction
• Multiplication
• Division
© Activities for Learning, Inc. 2015
Place Value in Its Proper Place
Place Value
Naming
Quantities
Visualizing
390 Facts
Learning
Procedures
Solving
Problems
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Place Value
• Place value is the foundation of modern
arithmetic.
• It must be taught, not left for discovery. • It is critical for understanding algorithms.
• Children need the big picture, not tiny snapshots.
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Place Value
Two aspects
Static (Written) • Value of a digit is determined by position.
• No position may have more than nine.
• As you progress to the left, value of each position
is ten times greater than previous position.
• Shown with place-value cards.
Dynamic (Trading) • 10 ones = 1 ten; 10 tens = 1 hundred;
10 hundreds = 1 thousand, …. • Represented on an abacus and other materials.
© Activities for Learning, Inc. 2015
Teaching Place Value
• Place value, not counting, is the key to
understanding numbers beyond ten. • Place value is best learned by:
• Subitizing quantities 1–10,
• Initially using transparent number naming,
• Composing numbers with place-value cards,
• Mastering facts with base-ten strategies,
• Trading with four-digit numbers.
© Activities for Learning, Inc. 2015
Teaching Place Value
• Place value, not counting, is the key to
understanding numbers beyond ten. • Place value is best learned by:
• Subitizing quantities 1–10,
• Initially using transparent number naming,
• Composing numbers with place-value cards,
• Mastering facts with base-ten strategies,
• Trading with four-digit numbers.
© Activities for Learning, Inc. 2015
Research on Subitizing
Finger gnosia
• Finger gnosia is the ability to know which fingers
have been lightly touched without looking.
• Part of the brain controlling fingers is adjacent to
math part of the brain.
• Children who use their fingers as
representational tools perform
better in mathematics.
—Butterworth
© Activities for Learning, Inc. 2015
Learning 1–10
Using fingers
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Learning 1–10
Using fingers
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Learning 1–10
Using fingers
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Subitizing
• Japanese children subitize numbers up to 10
by grouping in fives.
• They subitize the quantities, then use mental
strategies for learning addition and
subtraction.
• They are discouraged from using counting for
adding and subtracting. They are not taught to
count on or count back.
© Activities for Learning, Inc. 2015
Teaching Place Value
• Place value, not counting, is the key to
understanding numbers beyond ten. • Place value is best learned by:
• Subitizing quantities 1–10,
• Initially using transparent number naming,
• Composing numbers with place-value cards,
• Mastering facts with base-ten strategies,
• Trading with four-digit numbers.
© Activities for Learning, Inc. 2015
Transparent Number Naming
11 = ten 1
12 = ten 2
13 = ten 3
14 = ten 4
. . . .
19 = ten 9
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20 = 2-ten 21 = 2-ten 1
22 = 2-ten 2
23 = 2-ten 3
. . . .
. . . .
99 = 9-ten 9
Transparent Number Naming
137 = 1 hundred 3-ten 7
or
137 = 1 hundred and 3-ten 7
© Activities for Learning, Inc. 2015
Transparent Number Naming
Compared to reading
• Just as reciting the alphabet doesn’t teach reading,
counting doesn’t teach arithmetic.
• Just as we first teach the sound of the letters, we must
first teach the transparent name of the quantity.
© Activities for Learning, Inc. 2015
Transparent Number Naming
• Only 11 words are needed to count to 100 using
transparent number naming, 28 in English. (All
Indo-European languages are non-standard.)
• Asian children learn mathematics using
transparent number naming.
• They understand place value in first grade; only
half of U.S. children understand place value at
the end of fourth grade.
• Mathematics is the science of patterns. The
patterned transparent number naming greatly
helps children learn number sense.
© Activities for Learning, Inc. 2015
Transparent Number Naming
• Asian children do not struggle with the teens.
• Their languages are completely “ten-based.”
• Asian countries use the ten-based metric
system.
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Transparent Number Naming
Transition to regular names
4-ten = forty
The “ty”
means tens.
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Transparent Number Naming
Transition to regular names
6-ten = sixty
The “ty”
means tens.
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Transparent Number Naming
Transition to regular names
3-ten = thirty
“Thir” also
used in 1/3,
13 and 30.
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Transparent Number Naming
Transition to regular names
2-ten = twenty
Two used to be
pronounced
“twoo.”
© Activities for Learning, Inc. 2015
Transparent Number Naming
Transition to regular names
ten 4
Prefix -teen
means ten.
© Activities for Learning, Inc. 2015
teen 4
fourteen
Transparent Number Naming
Transition to regular names
a one left
© Activities for Learning, Inc. 2015
a left-one
eleven
Transparent Number Naming
Transition to regular names
two left
Two said
as “twoo.”
© Activities for Learning, Inc. 2015
twelve
Teaching Place Value
• Place value, not counting, is the key to
understanding numbers beyond ten. • Place value is best learned by:
• Subitizing quantities 1–10,
• Initially using transparent number naming,
• Composing numbers with place-value cards,
• Mastering facts with base-ten strategies,
• Trading with four-digit numbers.
© Activities for Learning, Inc. 2015
Composing Numbers
3-ten
30
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Composing Numbers
3-ten 7
30
7
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Composing Numbers
3-ten 7
37
0
Note the congruence in the way we say the number,
represent the number, and write the number.
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Composing Numbers
1-ten
10
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Composing Numbers
1-ten 8
10
8
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Composing Numbers
1-ten 8
18
0
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Composing Numbers
How much?
43
8
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Composing Numbers
How much?
76
8
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Composing Numbers
10-ten
100
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Composing Numbers
10-ten
100
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Composing Numbers
1 hundred
100
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Composing Numbers
109
100
9
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Composing Numbers
110
10
10
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Can You Find Activity
Objective: To practice identifying
hundreds, ten, and ones.
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Can You Find Activity
“Can you find
5
500
six hundred?”
30
7
0
8
0
2
600
20
400
7
60
800 700 1 9
3
90 8 100
3
0
0
9
0
0
40
50
10
6 200
4
© Activities for Learning, Inc. 2015
Can You Find Activity
“Can you find
5
500
3-ten nine?”
7
0
8
0
2
600
2
0
400
30
7
60
9
800 700 1
3
90 8 100
3
0
0
9
0
0
40
50
10
6 200
4
© Activities for Learning, Inc. 2015
Can You Find Activity
500
“Can you find
2 hundred 5?”
600
39
0
200
5
© Activities for Learning, Inc. 2015
20
2
70 80
400
7
60
800 700 1
3
90 8 100
3
0
0
9
0
0
40
50
10
6
4
Can You Find Activity
500
“Can you find
8 hundred 46?”
600
39
0
200
5
800
20
40
6
© Activities for Learning, Inc. 2015
2
70 80
400
7
60
1
700
3
90 8 100
300 50 900
10
4
Can You Find Activity
500
“Can you find
8 hundred 46?”
600
39
0
200
5
80
40
20
6
© Activities for Learning, Inc. 2015
2
70 80
400
7
60
1
700
3
90 8 100
300 50 900
10
4
Can You Find Activity
500
“Can you find
8 hundred 46?”
600
39
0
200
5
80
40
6
© Activities for Learning, Inc. 2015
20
2
70 80
400
7
60
1
700
3
90 8 100
300 50 900
10
4
Can You Find Activity
600
39
0
200
5
80
40
6
80
40
70
90
3
© Activities for Learning, Inc. 2015
30
20
8
72
0
10
10
7
90
50
61
0
500
4
Composing Numbers
Reading numbers backward
To read a number, students are often
instructed to start at the right (ones
column), contrary to normal reading of
numbers and text:
4258
Place-value cards encourage reading numbers in
the normal order.
© Activities for Learning, Inc. 2015
Teaching Place Value
• Place value, not counting, is the key to
understanding numbers beyond ten. • Place value is best learned by:
• Subitizing quantities 1–10,
• Initially using transparent number naming,
• Composing numbers with place-value cards,
• Mastering facts with base-ten strategies,
• Trading with four-digit numbers.
© Activities for Learning, Inc. 2015
Fact Strategies
Complete the Ten
9+5=
Take 1 from the 5
and give it to the 9.
© Activities for Learning, Inc. 2015
Fact Strategies
Complete the Ten
9 + 5 = 14
Take 1 from the 5
and give it to the 9.
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Fact Strategies
Complete the Ten
9+7=
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Fact Strategies
Complete the Ten
9 + 7 = 16
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Fact Strategies
Two Fives
8+6=
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Fact Strategies
Two Fives
8+6=
10 + 4 = 14
© Activities for Learning, Inc. 2015
Fact Strategies
Two Fives
7+5=
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Fact Strategies
Two Fives
7 + 5 =12
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Fact Strategies
Subtracting Part from Ten
15 – 9 =
Subtract 5 from 5
and 4 from 10.
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Fact Strategies
Subtracting Part from Ten
15 – 9 = 6
Subtract 5 from 5
and 4 from 10.
© Activities for Learning, Inc. 2015
Fact Strategies
Subtracting All from 10
15 – 9 =
© Activities for Learning, Inc. 2015
Fact Strategies
Subtracting All from 10
15 – 9 =
Subtract 9 from 10.
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Fact Strategies
Subtracting All from 10
15 – 9 = 6
Subtract 9 from 10.
© Activities for Learning, Inc. 2015
Fact Strategies
Going Up
15 – 9 =
Start with 9; go
up to 15.
© Activities for Learning, Inc. 2015
Fact Strategies
Going Up
15 – 9 =
1+5=6
Start with 9; go
up to 15.
© Activities for Learning, Inc. 2015
Teaching Place Value
• Place value, not counting, is the key to
understanding numbers beyond ten. • Place value is best learned by:
• Subitizing quantities 1–10,
• Initially using transparent number naming,
• Composing numbers with place-value cards,
• Mastering facts with base-ten strategies,
• Trading with four-digit numbers.
© Activities for Learning, Inc. 2015
Trading
Thousands
1000
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100
10
1
Trading
Hundreds
1000
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100
10
1
Trading
Tens
1000
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100
10
1
Trading
Ones
1000
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100
10
1
Trading
Adding
1000
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100
10
1
8
+6
Trading
Adding
1000
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100
10
1
8
+6
Trading
Adding
1000
© Activities for Learning, Inc. 2015
100
10
1
8
+6
14
Trading
Adding
1000
100
10
1
8
+6
14
Too many ones; trade
10 ones for 1 ten.
© Activities for Learning, Inc. 2015
Trading
Adding
1000
100
10
1
8
+6
14
Same answer before
and after trading.
© Activities for Learning, Inc. 2015
Trading
Bead Trading activity
1000
100
10
1
Object: To reach 1000
by adding numbers on
the cards.
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Trading
Bead Trading activity
1000
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100
10
1
7
Trading
Bead Trading activity
1000
© Activities for Learning, Inc. 2015
100
10
1
6
Trading
Bead Trading activity
1000
© Activities for Learning, Inc. 2015
100
10
1
6
Trading
Bead Trading activity
1000
© Activities for Learning, Inc. 2015
100
10
1
6
Trading
Bead Trading activity
1000
© Activities for Learning, Inc. 2015
100
10
1
6
Trading
Bead Trading activity
1000
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100
10
1
9
Trading
Bead Trading activity
1000
100
10
1
9
Another trade.
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Trading
Bead Trading activity
1000
100
10
1
9
Another trade.
© Activities for Learning, Inc. 2015
Trading
Bead Trading activity
• In the Bead Trading activity, trading
• 10 ones for 1 ten occurs frequently;
• 10 tens for 1 hundred, less often;
• 10 hundreds for 1 thousand, rarely.
• Bead trading helps the child experience the
greater value of each column from left to right.
• To detect a pattern, there must be at least three
examples in the sequence. To experience place
value as a pattern, the thousands are needed.
© Activities for Learning, Inc. 2015
Trading
Adding 4-digit numbers
1000
100
10
1
3658
+ 2738
Enter numbers from
left to right.
© Activities for Learning, Inc. 2015
Trading
Adding 4-digit numbers
1000
100
10
1
3658
+ 2738
Enter numbers from
left to right.
© Activities for Learning, Inc. 2015
Trading
Adding 4-digit numbers
1000
100
10
1
3658
+ 2738
Enter numbers from
left to right.
© Activities for Learning, Inc. 2015
Trading
Adding 4-digit numbers
1000
100
10
1
3658
+ 2738
Enter numbers from
left to right.
© Activities for Learning, Inc. 2015
Trading
Adding 4-digit numbers
1000
100
10
1
3658
+ 2738
Enter numbers from
left to right.
© Activities for Learning, Inc. 2015
Trading
Adding 4-digit numbers
1000
100
10
1
3658
+ 2738
Add starting at the
right. Write results after
each step.
© Activities for Learning, Inc. 2015
Trading
Adding 4-digit numbers
1000
100
10
1
3658
+ 2738
Trade 10 ones for 1 ten.
© Activities for Learning, Inc. 2015
Trading
Adding 4-digit numbers
1000
100
10
1
3658
+ 2738
6
Write 6.
© Activities for Learning, Inc. 2015
Trading
Adding 4-digit numbers
1000
100
10
1
1
3658
+ 2738
6
Write 1 for the extra
ten.
© Activities for Learning, Inc. 2015
Trading
Adding 4-digit numbers
1000
100
10
1
1
3658
+ 2738
6
Add the tens.
© Activities for Learning, Inc. 2015
Trading
Adding 4-digit numbers
1000
100
10
1
1
3658
+ 2738
96
Write the tens.
© Activities for Learning, Inc. 2015
Trading
Adding 4-digit numbers
1000
100
10
1
1
3658
+ 2738
96
Add the hundreds.
© Activities for Learning, Inc. 2015
Trading
Adding 4-digit numbers
1000
100
10
1
1
3658
+ 2738
96
Trade 10 hundreds for
1 thousand.
© Activities for Learning, Inc. 2015
Trading
Adding 4-digit numbers
1000
100
10
1
1
3658
+ 2738
396
Write the hundreds.
© Activities for Learning, Inc. 2015
Trading
Adding 4-digit numbers
1000
100
10
1
1
1
3658
+ 2738
396
Write the 1 for the extra
thousand.
© Activities for Learning, Inc. 2015
Trading
Adding 4-digit numbers
1000
100
10
1
1
1
3658
+ 2738
396
Add the thousands.
© Activities for Learning, Inc. 2015
Trading
Adding 4-digit numbers
1000
100
10
1
1
1
3658
+ 2738
6396
Write the thousands.
© Activities for Learning, Inc. 2015
Trading
Adding 4-digit numbers
1000
© Activities for Learning, Inc. 2015
100
10
1
1
1
3658
+ 2738
6396
Meeting the Standards
Page 5
“These Standards do not dictate curriculum or
teaching methods. For example, just because topic
A appears before topic B in the standards for a given
grade, it does not necessarily mean that topic A must
be taught before topic B. . . . Or, a teacher might
prefer to teach a topic of his or her own choosing
that leads, as a byproduct, to students reaching the
standards for topics A and B.” —CCSS
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Meeting the Standards
Page 5 summary
• Standards do not dictate curriculum or
teaching methods.
• Within a grade, topics may be taught in any
order or taught indirectly.
© Activities for Learning, Inc. 2015
Meeting the Standards
Kindergarten (K.CC)
1. Count to 100 by ones and by tens.
2. Count forward beginning from a given number.
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Meeting the Standards
Kindergarten (K.CC)
1. Count to 100 by ones and by tens.
2. Count forward beginning from a given number.
© Activities for Learning, Inc. 2015
Meeting the Standards
Kindergarten (K.CC)
1. Count to 100 by ones and by tens.
2. Count forward beginning from a given number.
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Meeting the Standards
Kindergarten (K.CC)
3. Write numbers from 0 to 20. Number Chart
© Activities for Learning, Inc. 2015
1!
6"
2!
7"
3!
8"
4!
9"
5"
10"
Meeting the Standards
Kindergarten (K.NBT)
Work with numbers 11–19.
1. Compose and partition numbers from 11 to 19
into ten ones and some further ones.
10
6
16
8
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Meeting the Standards
Grade 1 (1.OA)
1. Apply properties of operations as strategies to
add and subtract, commutative property and
associative property of addition.
6+3=9
3+6=9
© Activities for Learning, Inc. 2015
Meeting the Standards
Grade 1 (1.OA)
Extend the counting sequence.
1. Count to 120, starting at any number less than
120.
100
10
9
10
10
9
© Activities for Learning, Inc. 2015
Meeting the Standards
Grade 1 (1.NBT)
3. Compare two two-digit numbers, recording the
results of comparisons with symbols >, =, <.
46
0
© Activities for Learning, Inc. 2015
64
0
Meeting the Standards
Grade 1 (1.NBT)
3. Compare two two-digit numbers, recording the
results of comparisons with symbols >, =, <.
46
0
64
0
.
.
46 . 64"
Put two dots by greater number.
Put one dot by lesser number.
© Activities for Learning, Inc. 2015
Meeting the Standards
Grade 1 (1.NBT)
3. Compare two two-digit numbers, recording the
results of comparisons with symbols >, =, <.
46
0
64
0
.
.
46 . 64"
Put two dots by greater number.
Put one dot by lesser number.
Connect the dots.
© Activities for Learning, Inc. 2015
Meeting the Standards
Grade 1 (1.NBT)
4. Add a two-digit number and a multiple of 10.
5. Mentally find 10 more or 10 less than the
number, without having to count.
24 + 10 = __
© Activities for Learning, Inc. 2015
Meeting the Standards
Grade 1 (1.NBT)
4. Add a two-digit number and a multiple of 10.
5. Mentally find 10 more or 10 less than the
number, without having to count.
24 + 10 = 34
© Activities for Learning, Inc. 2015
Meeting the Standards
Grade 1 (1.NBT)
4. Add a two-digit number and a multiple of 10.
5. Mentally find 10 more or 10 less than the
number, without having to count.
24 – 10 = __
© Activities for Learning, Inc. 2015
Meeting the Standards
Grade 1 (1.NBT)
4. Add a two-digit number and a multiple of 10.
5. Mentally find 10 more or 10 less than the
number, without having to count.
24 – 10 = 14
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Meeting the Standards
Grade 1 (1.NBT)
6. Subtract multiples of 10 in the range 10-90
from multiples of 10.
90 – 30 = 60
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Meeting the Standards
Grade 2 (2.OA)
3. Determine whether a group of objects (up to
20) has an odd or even number of members.
Is 17 even
or odd?
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Meeting the Standards
Grade 2 (2.OA)
3. Determine whether a group of objects (up to
20) has an odd or even number of members.
Is 17 even
or odd?
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Meeting the Standards
Grade 2 (2.OA)
4. Use addition to find the total number of objects
arranged in rectangular arrays.
5 + 5 + 5 + 5 = 20
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Meeting the Standards
Grade 2 (2.NBT)
2. Skip-count by 2s, 5s, 10s, and 100s.
5,
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Meeting the Standards
Grade 2 (2.NBT)
2. Skip-count by 2s, 5s, 10s, and 100s.
5, 10,
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Meeting the Standards
Grade 2 (2.NBT)
2. Skip-count by 2s, 5s, 10s, and 100s.
5, 10, 15, . . .
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Meeting the Standards
Grade 2 (2.NBT)
2. Skip-count by 2s, 5s, 10s, and 100s.
1000
100, 200, 300, . . .
© Activities for Learning, Inc. 2015
100
10
1
Meeting the Standards
Grade 2 (2.NBT)
2. Count within 1000.
3. Read and write numbers to 1000 using base-ten
numerals, number names, and expanded form.
300
78
0
300
79
0
378, 379, 380
© Activities for Learning, Inc. 2015
300
80
Meeting the Standards
Grade 2 (2.NBT)
4. Compare two three-digit numbers based on
meanings of the hundreds, tens, and ones
digits, using >, =, and <.
700
6
60
70
706 > 670
© Activities for Learning, Inc. 2015
Meeting the Standards
Grade 3 (3.NBT)
1. Use place value understanding to round whole
numbers to the nearest 10 or 100.
10
20
30
Which is the
40
nearest ten?
50
Move the
60
fewest beads.
70
80
90
© Activities for Learning, Inc. 2015
Meeting the Standards
Grade 3 (3.NBT)
1. Use place value understanding to round whole
numbers to the nearest 10 or 100.
10
20
30
Which is the
40
nearest ten?
50
Move the
60
fewest beads.
70
80
90
© Activities for Learning, Inc. 2015
Meeting the Standards
Grade 3 (3.NBT)
1. Use place value understanding to round whole
numbers to the nearest 10 or 100.
10
20
30
Which is the
40
nearest ten?
50
Move the
60
fewest beads.
70
80
90
© Activities for Learning, Inc. 2015
Meeting the Standards
Grade 3 (3.NBT)
1. Use place value understanding to round whole
numbers to the nearest 10 or 100.
10
20
30
Which is the
40
nearest ten?
50
Move the
60
fewest beads.
70
80
90
© Activities for Learning, Inc. 2015
Meeting the Standards
Grade 3 (3.NBT)
3. Multiply one-digit whole numbers by
multiples of 10 in the range 10–90 (e.g., 9 ×
80, 5 × 60) using strategies based on place
value and properties of operations.
80 × 9 = 8 tens × 9 = 72 tens = 720
70
20
© Activities for Learning, Inc. 2015
Teaching Place Value
• Place value, not counting, is the key to
understanding numbers beyond ten. • Place value is best learned by:
• Subitizing quantities 1–10,
• Initially using transparent number naming,
• Composing numbers with place-value cards,
• Mastering facts with base-ten strategies,
• Trading with four-digit numbers.
© Activities for Learning, Inc. 2015
Place Value in Its Proper Place
Place Value
Naming
Quantities
Visualizing
390 Facts
Learning
Procedures
Solving
Problems
© Activities for Learning, Inc. 2015
Place Value:
The Foundation
of Number Sense
North Dakota Council of Teachers of Mathematics
presented by Kathleen Cotter Lawler
© Activities for Learning, Inc. 2015