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 sixth grade, students should have a mastery of the models, language, and symbols of integers. Students should also be adept at applying the operations and properties of integers. A student in grade 5 studies the models, language, and symbols of negative integers. In addition, at grade 5 students should master the operations of addition and subtraction of integers. The operations of integer multiplication and division are not expected to be introduced until grade 6, since the conceptual understanding of these two operations is abstract and difficult for students to master in their initial study of integers.

It should be noted that standards identifying integer calculations appear as early as grade 3 Measurement and Geometry 1.3. However, in the study of geometry, the integers referred to here are the positive integers. The notion of an inverse relationship or opposite relationship between addition and subtraction is first introduced in grade 1 Number Sense 2.2 and continued in grade 2 Number Sense 2.1. These standards begin the development of background knowledge necessary for further study of inverse operations in grade 5 and the inverse property in grade 7.

Students who do not have a complete understanding of integers and their operations will have difficulty mastering the seventh grade standards necessary for success in Algebra 1. Students who have been identified as needing remediation in grades 7, 8, and 9 will need to focus on integer models and operations. Success in Algebra is dependent on a thorough understanding of integers.

The models represented below will be utilized in the additional teaching strategies portion of the integer lessons.

Understanding the coordinate graph model and all of its components is essential. Students are expected to have mastered this concept in grade 5 as stated in Algebra and Functions 1.4.

The temperature model uses 0° as the midpoint and uses above zero to represent positive integers and below zero to represent negative integers.

The football field model uses the line of scrimmage as 0 and yards gained as positive integers and yards lost as negative integers.

The sea level model uses the elevation of the ocean as 0 and above sea level as positive integers and below sea level as negative integers.

Another model that is easy for students to understand is the notion of owing money compared to depositing money into a savings account. If you have $5.00 and you need to pay $8.00, you now have - $3.00 or you owe $3.00.

When introducing integers it is important to allow students time to compare the whole number line and the integer number line.

In grade 6 students are introduced to integer multiplication and division. The concept of integer multiplication is a difficult concept for students to master. While they have learned that adding two negative integers results in a smaller negative integer, the multiplication of two negative integers results in a positive integer. It is important to provide students different models for practice and mastery.

The language of integers is extremely important. Students who have not mastered the language of integers struggle with integer concepts in their study of Algebra. A complete understanding of the language of integers is important for completing calculations involving integer properties and operations.

In the introduction and development of the language of integers, it is necessary to clearly differentiate the operations of addition and subtraction versus the signs of quality, positive and negative. Students struggle with the idea of subtract or minus compared to negative. One way to help students differentiate the two operations is to use “the opposite of” when interpreting a negative sign. The idea of opposite is modeled by the number line showing the positive integers to the right of zero and the negative integers to the left of zero. Students should frequently model verbal statements expressing integers. For example: 3 + -4 should be read as “three plus the opposite of four.” The opposite of 4, or the additive inverse of 4 is negative 4. 3 - –4 is read as “three decreased by the opposite of four.” Students will benefit by modeling the language of integers while also illustrating the model on a number line. For example,

Additive inverse is defined as the number that can be added to another quantity so that the result will be zero. For example, the additive inverse of 4 is -4 since 4 + -4 = 0. The additive inverse of -6 is 6 since -6 + 6 = 0.

The negative integers are a result of the need for an opposite direction. For instance, in locating points on the earth’s surface, the distance in degrees north of the equator is positive, and south of the equator negative. For example, the latitude of Toronto is +44° and the latitude of Rio de Janeiro is -23°. The distance in degrees of latitude between these two cities is represented by the expression +44° - -23° = 63° or an additive inverse.

The language of the multiplication of integers can be thought of as repeated addition (multiplication). For example: 2 x -3 would be to add two groups of negative or -3 + -3 = -6.

Multiplication becomes more confusing when we multiply two negative integers. For example:
3 x 4 = add three groups of 4: 4 + 4 + 4 = 12
3 x -4 = add three groups of the opposite of 4: -4 + -4 + -4 = -12
-3 x 4 = take away three groups of 4: -4 – 4 - 4 = -12
-3 x -4= take away three groups of the opposite of 4: --4 - -4 - -4 = 12

The language of division can be illustrated by understanding it as the inverse operation of multiplication. 15 ÷ 3 can be expressed as “how many groups of 3 are in 15?” Students will benefit from seeing the fraction form of division by also focusing instruction and practice on using the representation of . Students will benefit from modeling the language of division while also modeling division with a physical model. The number line is a good model since it can illustrate direction (positive/negative) as well as quantity.

When considering integers, the language of division can be challenging for students. -15 ÷ 3 means “how many groups of 3 are in -15.” It seems that there are no groups of positive 3 in negative 15. However, there are opposite groups of 3 in negative 15, resulting in an answer of negative 5.

The symbols for integers are the symbols for whole numbers, which are the ten digits in addition to the + and – signs. The + (positive) and – (negative) signs are the new addition to symbolize integers. The positive sign represents numerical values to the right of zero on the number line and the negative sign represents values to the left of zero. The symbol for negative and the symbol for subtraction often confuse students since they are similar in style. However, using the negative and positive signs in the superscript position in front of a number will help students differentiate between the integer symbols and the operation symbols.

The positive symbol also confuses students since we do not always write it before the integer. Just like not writing the decimal point at the end of a whole number or integer, we do not always write the positive sign with integers. This way of sometimes using the positive symbol and sometimes not using the symbol should not be ignored or just touched upon. Students need time adjusting to and practicing both ways of expressing positive integers before their study of Algebra.