Working Definition of Place Value in MATH


Place value is what we call the system of assigning value to the place a numerical digit holds in numbers (such as 4 hundreds and 2 ones in 402 and 2 hundreds and 4 ones in 204. It applies to situations, like our base-10 number system, in which the place or position something occupies changes its value. The value of a dart in the bull’s eye place is different from the value of a dart on the outer edge of a dart board just as the value of a 5 in the tens place is different from the value of a 5 in the ones place. In the base-10 system, the difference between adjacent places is 10x.

From the Virginia State Department of Education Standards of Learning:
3rd grade Standard 3.1 Statements in italics are added information

  • Numerals are symbols that represent value and quantity.
  • The structure of the base-10 number system is based upon a simple pattern of tens, where each place is ten times the value of the place to its right. This is known as a ten-to-one place-value relationship. Numbers are formed in patterns with the numerals 1-9 in each set of ten (decade).
  • Place value refers to the value of each digit and depends upon the position of the digit in the number. In the number 7,864, the eight is in the hundreds place, and the value of the 8 is eight hundred. When we count larger numbers of objects, we can use base-10 groupings to help (counting by ones, tens, hundreds, etc.)
  • Flexibility in thinking about numbers — or “decomposition” of numbers (e.g., 12,345 is 123 hundreds, 4 tens, and 5 ones) — is critical and supports understandings essential to multiplication and division.
  • Whole numbers may be written in a variety of formats:
    • Standard: 123, 456
    • Written: one hundred twenty-three thousand, four hundred fifty-six; and
    • Expanded: (1 x 100,000) + (2 x 10,000) + (3 x 1,000) + (4 x 100) + (5 x 10) + (6 x 1).
  • Numbers are arranged into groups of three places called periods (ones, thousands, millions, and so on). Places within the periods repeat (hundreds, tens, ones). Commas are used to separate the periods. Knowing the place value and period of a number helps students find values of digits in any number as well as read and write numbers. Number patterns and relationships in and between periods and numbers can be used to learn and retain basic facts.(Knowing 7+2=9 helps with the understanding of 17+12=29, for example, or 700 +200=900.
  • Decimals are written as an extension of the place-value system. Each place to the left of the decimal gets ten times smaller than the previous place.

Teaching Place Value:
Place Value understanding requires an integration of new and difficult-to-construct concepts of grouping by tens, with procedural knowledge of how groups are recorded in our place value scheme , how numbers are writren and how they are spoken. (Van de Walle,2006)

Thus, giving students opportunities to count quantities in a variety of ways is crucial to students development of number sense.
The following ideas are from Teaching Student Centered Mathematics Grades K-3.
Have them
1. fill in tables:
Bag of:
Number Word
Tens
bottle caps

Singles
beans


pennies


2. Fill in ten frames for a specific number, then note how many tens and how many ones

3. Loop dots in groups of ten to show a number like 73.

4. Show 67 three different ways:
I I I I I I I I I I
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .

tens

ones_

I I I I I I I I I I
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .

tens

ones_

I I I I I I I I I I
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .

tens

ones___


5. Play Base 10 riddles such as
  • I have 23 ones and 4 tens. What's my number?
  • I have 30 ones and 3 hundreds. What's my number?
  • If you put 4 more tens woith me, I would be 115, What's my number?
  • I have 13 ones. I am between 60 and 70. What's my number?
  • My number is 65. I have 25 ones. How many tens do I have?
  • My number is 421. I have 22 tens How many hundreds do I have?

that include equivalent ways of making numbers rather just the regular number of tens and ones.

6. Begin with a blank or nearly blank hundreds chart. Students fill in one number and then find its nearest neighbors, the numbers to the left, right, above and below. The na ask the studnets to describe the relationship of the numbers (The numbers on either side are one more of less and the numbers above and below are 10 more or less.)

5 distinct levels of understanding of place value:
Page 141 in Van de Walle's book.