Division

Division is a binary operation which can be described two ways -- partitive and quotative division.

Partitive division: If the number of groups is known, and you need to find how many in each group, then the problem is a partitive division problem. It is also called "equal groups" or the "sharing" method.

Quotative division: If the number in each group is known, and you are trying to find the number of groups, then the problem is a quotative division problem. This is also called the "measurement" method. You are counting or measuring the number of times you can subtract the divisor from the dividend. This is related to the repeated subtraction interpretation of division.

Annenberg learner.org video

Division Algorithm Theorem A = dq + r A is the original amount; d is the divisor; q is the quotient; and r is the remainder with 1 < r < d.

This theorem is just common sense -- for example, if we are walking to a certain point, we make so many steps of a repeated length in order to reach the point. We may land on the point exactly, or fall short of one full step to finish. In this way, our goal "point" lies between two multiples of our step length. Any leftover piece is a remainder. If the remainder is longer than our step, then we make one more step. Therefore any remainder must be less than our step length.

Teachers often require students to use this theorem to check their long division problems. Since long division is taking a whole and placing it into several groups of the same size (sometimes with a remainder), we can go backwards by multiplying the number of groups by the group size and adding the remainder back to reconstruct the whole.