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# Meanings and Models for Operations

## Examine the operations of addition, subtraction, multiplication, and division and their relationships to whole numbers. Work with area models for multiplication and division. Explore the use of two-color chips to model operations with positive and negative numbers.

In This Session
Part A: Meanings and Relationships of the Operations
Part B:
Area Models for Multiplication and Division
Part C: Colored-Chip Models
Homework
In this session, you will examine the operations of addition, subtraction, multiplication, and division and their relationships to whole numbers. You will also explore some of the laws that govern these operations and use mathematical models to reinforce the algorithms being discussed. See Note 1 below.

LEARNING OBJECTIVES
In this session, you will do the following:

• Understand and apply alternate interpretations for each of the four basic operations
• Understand subtraction as the inverse operation for addition, and division as the inverse operation for multiplication
• Understand subtraction as the addition of an inverse element
• Understand division as multiplication by an inverse element
• Examine and analyze models and manipulatives for representing operations with whole numbers and integers
• Examine and analyze alternative algorithms for operations with whole numbers

### Key Terms

Previously Introduced:

identity element

If I is an identity element for operation *, then a * I = I * a = a for all elements a in the set. The identity element for addition of real numbers is 0, and the identity element for multiplication of real numbers is 1.

inverse element

If b is the inverse element for a for operation *, then a * b = b * a = I, the identity element for that operation. The inverse for element a for addition is -a, because a + -a = -a + a = 0 for all values of a. The inverse for element a for multiplication is 1/a, because a * (1/a) = 1/a * a = 1 for all values of a except 0. Zero does not have an inverse for multiplication.

whole numbers

Whole numbers are the counting numbers and zero.

New in This Session:

algorithm

An algorithm is a recipe or a description of a mechanical set of steps for performing some task.

asymmetrical multiplication

An asymmetrical multiplication problem is one where the order of the operands is important. Switching the order of operands in this type of problem presents a different situation, even though the product is the same. For example, buying 10 tickets at \$5 each is quite different from buying 5 tickets at \$10 each, although the total cost (i.e., the product) is identical.

partitive division

A partitive division problem is one where you know the total number of groups, and you are trying to find the number of items in each group. If you have 30 popsicles and want to divide them equally among your 5 best friends, figuring out how many popsicles each person would get is a partitive division problem.

quotative division

A quotative division problem is one where you know the number of items in each group and are trying to find the number of groups. If you have 30 popsicles and want to give 5 popsicles to each person, figuring out the total number of people is a quotative division problem.

symmetrical multiplication

A symmetrical multiplication problem is one where the order of the operands is not important. Finding the area of a field that measures 150 feet by 50 feet, or finding the number of different sandwiches that can be made from 4 types of bread and 6 types of meat, are both symmetrical multiplication problems.