Multiplication and division often involve grouping or partitioning collections of countable objects. Problems of this type involve three quantities as illustrated in the following example:
Jill has 5 pages of stickers. There are 6 stickers on each page. All together she has 30 stickers.
The three quantities in the problem are the number of pages, the number of stickers on each page, and the total number of stickers. When solving problems, any one of the three quantities can be unknown. When the total is unknown, the problem is a multiplication problem. When the number of pages, or groups, is unknown, the problem is called a Measurement Division problem. When the number of stickers on each page, or the size of the group, is unknown, the problem is called a Partitive Division problem. Here is an example of each type:
Multiplication: (Whole unknown) Sue has 4 bags of marbles. There are 5 marbles in each bag. How many marbles does Sue have all together?
Measurement Division: (Number of groups unknown) Sue has 20 marbles. She puts them in bags with 5 marbles in each. How many bags did she use?
Partitive Division: (Size of group unknown) Sue has 20 marbles. She wants to share them equally among 4 friends. How many marbles will each friend receive?
In each of these problem types, the numbers all represent a countable quantity. In another type of problem, the Multiplicative Comparison, this is not the case. Multiplicative Comparison problems involve a comparison of two quantities in which one is described in terms of how many times larger it is than the other. Once again, any of the three items (the product, the size of the referent, the multiplier) can be unknown. For example:
Multiplicative Comparison (Product unknown): Carlos picked 4 peaches. Juanita picked 5 times as many. How many peaches did Juanita pick?
Multiplicative Comparison (Referent unknown): Juanita picked 20 peaches. She picked 5 times as many as Carlos. How many peaches did Carlos pick?
Multiplicative Comparison (Multiplier unknown): Juanita picked 20 peaches and Carlos picked 4. How many times as many peaches did Juanita pick as Carlos?
Children need plenty of experience with real life situations involving repeated addition to reach a target and repeated subtraction from a given quantity, making the connection between multiplication and division. They should experience sharing and sorting sets of countable objects into equal subsets before they meet the formal language and symbols for division.