Session 4, Part A:
Meanings and Relationships of the Operations

In This Part: Addition | Subtraction | Multiplication | Division

There are several ways in which we can think of multiplication:

 • Multiplication is often thought of as repeated addition of equal groups. While this definition works for some sets of numbers, it is not particularly intuitive or meaningful when we think of multiplying 3 by 1/2, for example, or 5 by -2. In such cases, it may be helpful to widen the idea of grouping to include evaluation of part of a group. This concept is related to partitioning (which, in turn, is related to division). For example, three groups of five students can be read as 3 • 5, or 15 students, while half a group of 10 stars can be represented as 1/2 • 10, or 5 stars. These are examples of partitioning; each one of the three groups of five is part of the group of 15, and the group of 5 stars is part of the group of 10. • A second concept of multiplication is that of rate or price. For example, if a car travels four hours at 50 miles per hour, then it travels a total of 4 • 50, or 200 miles; if CDs cost eight dollars each, then three CDs will cost 3 • \$8, or \$24. • A third concept of multiplication is that of multiplicative comparison. For example, let's say that Sara has four CDs, Joanne has three times as many as Sara, and Sylvia has half as many as Sara. Thus, Joanne has 3 • 4, or 12 CDs, and Sylvia has 1/2 • 4, or 2 CDs.

Two additional situations require multiplication:

 • Finding the area of a rectangle using rectangular arrays. For example, an array with three rows by five each will have 3 • 5, or 15, square units in all. This model is often used to introduce multiplication. • Finding the number of possible combinations using a Cartesian product. For example, with two shirts and three pairs of pants, you could have 2 • 3, or 6, different shirt-pant combinations.

In the first three scenarios, one factor was clearly the multiplier (the number of groups), and the other factor was clearly the number being multiplied (the number of items or individuals in each group). These types of problems are called asymmetrical, because the factors are so different; exchanging the roles of the factors results in an entirely different scenario. For example, 3 CDs at \$8 each is different from 8 CDs at \$3 each, even though the "answer" is the same.

The two last scenarios present two examples of symmetrical problems, because it isn't important which of the two factors is the multiplier. The quantities are interchangeable.

The asymmetrical nature of some problems explains why the commutative law of multiplication is less intuitive than the commutative law of addition. The multiplication problem 3 • 4, interpreted as a grouping problem, means three groups of four items. This corresponds to the addition problem 4 + 4 + 4. In contrast, the multiplication problem 4 • 3, also interpreted as a grouping problem, means four groups of three items, which corresponds to the addition problem 3 + 3 + 3 + 3. It may not be obvious at first glance that 4 + 4 + 4 and 3 + 3 + 3 + 3 would give the same sum, or why. Manipulatives and other visual clues discussed later in this session can be helpful in showing the relationship between the two operations.

One of the most important facts about the operations of multiplication and division is that the units of the quantities being multiplied or divided do not have to be the same for the operations to function properly. You can multiply 2 tens by 3 ones, and the result will be 6 tens. Likewise, you can divide 6 tens by 3 ones and get 2 tens. The unit for the answer is found in the same way that you found the numerical answer. For example, if you divided the numbers, then you must divide the units as well. Similarly, if you multiplied the numbers, then you must multiply the units as well.
Note 5

Problem A3

Consider the following multiplication situations. For each one, identify the multiplication problem, the units involved, whether the problem is symmetrical or asymmetrical, and which multiplication concept it is demonstrating:

 a. Dinner at the Ritz costs 4 times as much as dinner at the Savoy. My bill at the Savoy was \$10. What would dinner cost me at the Ritz? b. Bob & Jimmy's Ice Cream offers 6 different ice cream flavors and 5 different sundae toppings. How many different kinds of sundaes can be made using 1 flavor of ice cream and 1 topping per sundae? c. My shower flows at 3 gallons per minute. How much water would a 6-minute shower use? d. There are 9 content sessions in this course, and you have completed 1/3 of them. How many sessions have you completed? e. My yard is 20 meters wide and 33 meters from front to back. What is its area?

 Session 4: Index | Notes | Solutions | Video