Operational Concepts

[In each description below, when the term “set”, or “sets” is used, it refers to a collection or group of objects that can be counted, or it may refer to a quantifiable measure of some attribute.]

Addition a + b

• Conceptually, addition is the joining of two sets of objects.

Subtraction a – b 

• Take away – A single set is given and the situation calls for a part of the set to be removed.
• Comparison – Two sets are given and the situation asks how much larger (or smaller) quantitatively one set is than the other.
• Missing addend – Two quantities are known, or two sets are given. The situation asks how much needs to be added to the smaller set to make it as large as the larger set.

Multiplication a  b

• Repeated addition – The first factor indicates how many times the second factor is to be used as an addend.
• Rectangular array – Two or more sets are arranged in rows with the same number of elements in each row, forming a rectangular arrangement. The first factor tells how many are in each row/column and the second factor indicates the number in each row/column. The total number of elements in the array is desired.
• Combinations – Each element in one set is paired in some way with each element in another set. The total number of possible pairs, or combinations, consisting of one element from each set is desired.

Division a ÷ b

• Repeated subtraction – A set is given and the situation asks for the number of smaller sets that are contained in (or can be subtracted from) the given set when the amount to be subtracted (the divisor) is given. If, after subtracting amounts equal to the divisor the remainder is smaller than the divisor, that remainder is compared to the divisor to determine the fractional part of another set (or whole) that can be subtracted. The quotient is the number of equal sets, plus any fractional part of another set, that can be subtracted.
• Partitive – A set is given and the condition calls for partitioning the given set into a whole number of equal subsets. The quotient is the amount in each of the equal subsets after the partitioning is completed.
• Ratio or comparison – A set is given and the condition calls for dividing by an amount larger than what is given. Since the larger number (the divisor) cannot be taken away from the smaller number (the dividend), or the larger number (the divisor) may not be a whole number, we must resort to comparing the two numbers by division. The dividend is compared to the divisor as a ratio and the quotient is determined by asking, “The dividend is what part of the divisor?” Your answer will be a fraction less than one.