Teaching Models

Place Value

The Hindu-Arabic system of enumeration we use is described as a base-ten positional number system or a decimal system. It begins with special symbols called digits to represent the first nine counting numbers—1, 2, 3, 4, 5, 6, 7, 8, 9. There is also a very important tenth digit, 0, that is used to represent an empty column. Hindus are credited with the development of this system, and Arabs with introducing it to western Europe.

The importance and power of a positional numeration system cannot be understated. It is truly one of the most significant inventions of civilization.

For whole numbers, the digit farthest to the right is in the ones place. Moving to the left, each digit has a place value equal to 10 times that of the digit to its right. This can be understood by looking at a place-value chart.

Hundred Thousands Ten Thousands Thousands Hundreds Tens Ones
10 × 10,000
10 × 1,000
100 × 10
10 × 10
1 × 10


Numerals are written symbols for numbers. When writing numbers in word form, hyphens are used for numbers between 21 and 99 (except those ending in 0).There are three basic ways of showing a number. For the number 41,375, these are as follows:

Standard Form
Expanded Form
Word Form
   40,000 + 1,000 + 300 + 70 + 5
   forty-one thousand, three hundred seventy-five

For numbers greater than 999, the digits are grouped into periods of three that are set off by commas. The ones period consists of the digits in the hundreds, tens, and ones places. The thousands period consists of the digits in the three places to the left of the ones period. The millions period consists of the three places to the left of the thousands. Thus 34,507,843 is read thirty-four million, five hundred seven thousand, eight hundred forty-three (with no mention of the ones period).

Teaching Model 1.2: Place Value: Ones, Tens, and Hundreds

Houghton Mifflin Math Grade 3