Teaching Models

Place Value

The Hindu-Arabic system of enumeration we use is described as a base-ten positional number 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. In future chapters the importance of the base-ten positional system will become apparent. All standard computational algorithms for whole numbers and decimals that students learn in elementary school are based on the base-ten positional numeration system.

In a whole number, 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
or
100,000
10 × 1,000
or
10,000
100 × 10
or
1,000
10 × 10
or
100
1 × 10
or
10
1

1

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 writing a number. For the number 241,375, these are as follows.

Standard Form
Expanded Form
Word Form
241,375
200,000 + 40,000 + 1,000 + 300 + 70 + 5
two hundred forty-one thousand, three hundred seventy-five

The digit 4 is in the ten thousands place. Its value is 4 ten thousands, or 40,000. The digit 7 is in the tens place. Its value is 7 tens, or 70.

The place-value chart can be extended to include greater numbers.

Millions Thousands Ones
Hundreds Tens Ones Hundreds Tens Ones Hundreds Tens Ones
100,000,000 10,000,000 1,000,000 100,000 10,000 1,000 100 10 1
5 2 6 0 4 1 3 7 5

As an aid in reading 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 three places to the left of the ones period. The millions period consists of the three places to the left of the thousands period. The numerals in each period are read as a three-digit number, and the period name is then added (except for the ones period). Thus, 526,041,375 is read as five hundred twenty-six million, forty-one thousand, three hundred seventy-five.


Teaching Model 1.3: Use Logical Reasoning


Houghton Mifflin Math Grade 4