Teaching Models

Place Value

A numeration system consists of a set of symbols and rules for using these symbols. Such a system makes it possible to communicate number ideas. The symbols used are called numerals. Early numeration systems used tally marks, such as I or four tally marks crossed plus one tally mark.

Later, tally marks were grouped by fives so the numeral 6 became 4tally marks crossed plus 1 tally mark and |||||||||| became 2 groups of 4 tally marks crossed. As greater numbers were needed, other symbols were invented. These systems used the idea of putting objects in groups of 5 or 10 to aid in counting.

Roman numerals: XVI
(10 + 5 + 1)
Egyptian numerals: ∩ | | | | | |
(10 + 1 + 1 + 1 + 1 + 1 + 1)

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. 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 seen in a place-value chart.

Hundreds Tens Ones
10 groups of 10
1 group of 10


Since grouping is essential in the use of such a numeration system, children need many experiences putting objects into groups of 10. When writing numbers in word form, notice that hyphens are used for numbers between 21 and 99, except for those ending in 0. There are three basic ways of showing a number. For the number 89 these are:

Standard Form
Expanded Form
Word Form
80 + 9

The expanded form of a number makes it clear that the value of a digit depends on its place in the numeral.

89 → 80 + 9 → (8 × 10) + 9

Teaching Model 5.3: Identify Place Value

Houghton Mifflin Math Grade 2