Models, language, and symbols are all used to describe or communicate a mathematical concept. By supplying students with various models, students are able to see different representations of a concept. While students are used to hearing the concept verbally, and seeing it with mathematical symbols, models provide a real-world, hands-on connection. This, in turn, enables teachers and students to communicate a concept using alternate methods.

By the end of fourth grade students should have a mastery of the models, language, and symbols of whole numbers. Students should also be adept at applying the operations and properties of whole numbers. A student in grades 5, 6, or 7 who does not have mastery of these concepts will find mathematics challenging and often frustrating. In fifth grade, students expand their understanding of our number system to the rational numbers. Without a solid foundation in whole numbers students will often fall further behind. It is important to assess a student’s competence in the whole number system and remediate these concepts before expanding to the rational number system.

The models represented below will be utilized in the additional teaching strategies portion of the whole numbers lessons.
One of the most important models of whole numbers is the subset of whole numbers, the counting numbers (1, 2, 3, …). We use these numbers to count objects. For example, given a bag of oranges a student can count that there are 10 oranges in a bag. Given 14 bags of oranges, a student knows that the number 14 represents the number of bags of oranges and 14 bags of 10 oranges represents 140 oranges.

An array model can help a student count the oranges. Later we’ll look at the operation of multiplication, a model of repeated addition, which makes counting objects more efficient.
Students enjoy sorting and counting objects. The activity of sorting is a very important part of the development of algebraic reasoning. Students should be encouraged to sort and count objects in many different contexts. Sorting and counting different arrays of Unifix cubes is an interesting and valuable activity.

For example, the whole number two hundred forty-three, 243, can be modeled using unifix cubes. If each cube represents 1 unit then a tower of ten cubes would represent 10. Thus, a bundle of ten towers (each of 10 cubes) would represent 100. A model of the number 243 using unifix cubes would be:
Using base ten blocks as a model, 243 could be represented by 2 flat 10 by 10 arrays of 100, 4 long towers of 10, and 3 individual units.
Money is another model that can be introduced. While we concentrate on whole numbers, 243 cents can be represented by showing 243 pennies in various arrangements.
Multiple representations of different models of numbers are valuable concepts that will provide necessary background knowledge for a student’s future study of the rational number system.
The language of mathematics is a complex system. While we use the Arabic number system to represent whole numbers, we also use word representations to express numbers. For example, on a check we might write the amount $236.00 using the Arabic system. However, we also write “two hundred thirty-six” in words on another line.

In language, we represent the place value of a number by the value of the place the number holds. In the example $236.00, the 2 represents the number two hundred (200). In order to communicate clearly, it is important for children to express whole numbers in Arabic as well as word forms. 
   
Arabic Form Word Form
1,203 One thousand two hundred three
136 One hundred thirty-six
 
Notice that “and” has not been used in the word form. Including the word “and” in the word form of a whole number is a common mistake for children and adults. We should not use “and” in these representations because “and” represents a decimal point. It is not necessary to identify the decimal point when expressing whole numbers. The word “and” will be very important later when we discuss rational numbers.
 
Arabic numbers are the standard system for representing whole number values. The symbols for whole numbers are the ten digits 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. Each digit can represent a different amount depending on its place value. For example, 423 represents 400 + 20 + 3 in expanded form. Our system of ten digits and place value allow the number to be written in the condensed form of 423. Zero should be recognized for its importance in the Arabic System. Without zeros acting as place-holders, we would not have clear representations for numbers such as one-thousand (1,000).

The Roman Numeral System is another way to represent whole numbers. While this system is not as common, we do see whole number representations in our world. For example, Super Bowl XXIX was in San Francisco. In a conversation we would say, "Super Bowl twenty-nine." The year a film was made is often shown using Roman numerals.

Another important form for representing whole numbers is expanded form. In second grade students are introduced to representing whole numbers in expanded form. Representing numbers in this form is necessary for children to have a complete understanding of whole numbers.
Arabic Form Expanded Form Roman Numeral Form
3,452 3,000+400+50+2 MMMCDLII
103 100+00+3 CIII