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If the number in each group is known, and you are trying to find the number of groups, then the problem is referred to as a quotative division problem. Quotative division may also be called measurement, or repeated subtraction. You are, in effect, counting or measuring the number of times you can subtract the divisor from the dividend. Long division (remember long division?!) uses this concept.

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If the number of groups is known, and you are trying to find the number in each group, then the problem is referred to as a partitive division problem. Partitive division may also be called equal groups, or sharing and distribution. You are, in effect, partitioning the dividend into the number of groups indicated by the divisor and then counting the number of items in each of the groups.

a. 

Draw a diagram that represents 153 as a partitive problem.

b. 

Draw another diagram that represents 153 as a quotative problem.

c. 

Write a problem for each diagram.

Problem A5

When division problems do not work out evenly, the context of the problem dictates the answer. Sometimes we may need to round the answer up or down to the next integer, and sometimes we may need the exact decimal value of the division.

Problem A6

Another important concept to remember, especially when working with rational numbers, is that division can be thought of in terms of multiplying by the inverse. This can be particularly useful when dividing by fractions. Thus, we could show that 122 = 12 • 1/2, where 1/2 is the multiplicative inverse of 2, and 12(1/2) = 12 • 2, where 2 is the multiplicative inverse of 1/2. In these cases, you can see that the multiplicative inverse of every number except 0 is the reciprocal of that number, and that the product of a number and its reciprocal is 1.