Multiplication & Division Problem Structures (Word Problems)

Researchers have defined and classified multiplication and division problems in a number of different ways based on their semantic structure—that is, how the relationships are expressed in words. There are two broad categories: asymmetrical and symmetrical problems. There are three subcategories of asymmetrical problems: (1) equal grouping, (2) rate, and (3) multiplicative compare. There are two subcategories of symmetrical problems: (1) rectangular array and (2) Cartesian product.

Asymmetrical Problems

In asymmetrical situations, the quantities play different roles. For example: There are 15 cars in the parking lot and each car has 4 tires. How many tires are there in all?, 4 represents the amount in one group, 15 represents the number of groups and also acts as the multiplier. These roles are not interchangeable. If you switch the numbers (4 cars with 15 tires each) you have a different problem: the number of things in one group is 15 and the number of groups is now 4. (In an asymmetrical problem the answer is the same regardless of the role of the quantities, but that is not obvious to children.)

1. Equal Grouping

In equal grouping multiplication problems, one factor tells the number of things in a group and the other factor tells the number of equal-size groups. This second factor acts as a multiplier.

Example:

There are four basketball teams at the tournament and each team has five players. How many players
are at the tournament?, the factor 5 indicates the number of players in one group and the factor 4 indicates the number of equal groups of 5. The problem would be written as 4 x 5 = ?

Two situations result in problems being classified as an equal grouping division—either the number in each group is unknown or the number of groups is unknown. These two types of division situations are referred to as partitive division and quotitive division, respectively.

Partitive Division

Here is a partitive division problem: 

Twenty-four apples need to be placed into eight paper sacks. How many apples will you put in each sack if you want the same number in each sack?

The action involved in partitive division problems is one of dividing or partitioning a set into a predetermined number of groups. If students model this situation, 24 objects are evenly distributed into 8 different paper sacks or groups.

Quotitive Division

In quotitive division problems (sometimes referred to as repeated subtraction problems) the number of objects in each group is known, but the number of groups is unknown. 

For example: 

I have 24 apples. How many paper sacks will I be able to fill if I put 3 apples into each sack?

The action involved in quotitive division is one of subtracting out predetermined amounts. If asked to model this problem, students usually repeatedly subtract 3 objects from a group of 24 objects and then count the number of groups of 3 they removed (i.e., 8).

2. Rate

Rate problems involve a rate—a special type of ratio in which two different quantities or things are compared. Common rates are miles per gallon, wages per hour, and points per game. Rates are frequently expressed as unit rates—that is, one of the quantities in the ratio is given as a unit (e.g., price per single pound or miles per single hour). In rate problems, one number identifies the unit rate and the other tells the number of sets and acts as the multiplier. 

For example:

Concert tickets cost seven dollars each. How much will it cost for a family of four to attend the concert?

the unit rate is seven dollars per single ticket (seven to one) and the multiplier is four.

Rate problems can also be expressed as division situations. Here is a partitive division rate problem (size of one group is unknown): On the Hollingers’ trip to New York City, they drove 400 miles and used 12 gallons of gasoline. How many miles per gallon did they average? Quotitive division problems (number of equal groups is unknown) are also common in this category: Jasmine spent $108 on some new CDs. Each CD cost $18. How many did she buy?

3. Multiplicative Compare

The third type of asymmetrical problem is multiplicative compare, also called a scalar problem. Here, one number identifies the quantity in one group or set while the other number is the comparison factor.

For example:

Catherine read twelve books. Elizabeth read four times as many. How many books did Elizabeth read?, 

Here the number 12 tells us the amount in a group and the number 4 tells us how many of these groups are needed. 

In multiplicative compare division problems, either the amount in each group or the comparison factor is missing. 

The following problems are multiplicative compare division problems:

Elizabeth read 48 books during summer vacation. This is four times as many as Catherine. How many books did Catherine read during summer vacation?

Elizabeth read 48 books during summer vacation. Catherine read 12 books during summer vacation. How many times greater is the number of books Elizabeth read compared with the number of books Catherine read?

Symmetrical Problems

In symmetrical situations, on the other hand, the quantities have interchangeable roles. In symmetrical

multiplication problems it is not clear which factor is the multiplier. For example, in the problem What is the area of a room that is 10 feet by 12 feet?, either number can be the width or the length and either can be used as the multiplier.

1. Rectangular Array (Area)

The rectangular array problem, a subcategory of symmetrical problem commonly known as an area problem, is often used to introduce the idea of multiplication. Students are presented with an array (e.g., three by four) and asked to label the two sides of the array and determine the total number of square units in the array:

In rectangular array problems the role of the factors is interchangeable. For instance, when finding the area of the array above, neither the three nor the four is clearly the multiplier.

2. Cartesian Product

The other subcategory of symmetrical problems, known as Cartesian product problems, involves two sets and the pairing of elements between the sets. These problems entail a number of combinations.

For example: 

Pete’s Deli stocks four types of cold cuts and two types of cheese. How many different sandwiches consisting of one type of meat and one type of cheese are possible?

In these problems, like the rectangular array problems, neither of the two factors is clearly the multiplier.

Students tend to use tree diagrams to solve Cartesian product problems, in order to help them find all the combinations. For example, for the problem above, a student might use a diagram like this:

However, this diagram doesn’t highlight how multiplication can be used to determine the number of combinations. Teachers often have to point out the relationship between the number of objects to be combined (4 and 2) and the final number of combinations (4 ⨉ 2 = 8).

Another way to solve Cartesian product problems is to construct a rectangular array like the one below. The number of columns (4) and the number of rows (2) are the factors in the problem.

Math Matters: Understanding the math you teach

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