 Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum            Session 4, Part A:
Meanings and Relationships of the Operations

In This Part: Addition | Subtraction | Multiplication | Division

All of the meanings of multiplication can be used for division, since if the product and one of the factors is known, division can be used to find the other factor. But for the asymmetrical example of equal groups, the process feels different depending on which factor is known -- the multiplier or the number in each group.

As you will see, there are two very different concepts of division:

 • 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. • 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.

The following example demonstrates the distinction between the two types of division problems: Note 6  Partitive: Quotative: 12 3 = 4
 12 3 = 4 Partition into 3 groups.
 Repeatedly subtract 3. There are 4 in each group.
 There are 4 groups of 3 in 12. Twelve apples, 3 bags -- how many in each?
 Twelve apples, 3 in a bag -- how many bags?       Problem A4 a. Draw a diagram that represents 15 3 as a partitive problem. b. Draw another diagram that represents 15 3 as a quotative problem. c. Write a problem for each diagram. Problem A5 a. Which type of division, quotative or partitive, would be most efficient for computing 100 50? Why? b. Which would you use for 100 2?   Video Segment In this video segment, Susan and Jeanne explore the different notions of quotative and partitive division problems. They challenge their understanding with new insights. Watch this segment after you've completed Problems A4 and A5. Think about which method is easier to do in a particular division problem. If you are using a VCR, you can find this segment on the session video approximately 4 minutes and 9 seconds after the Annenberg Media logo.    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 a. Write a problem that uses the computation 43 4 and gives 10 as the correct answer. b. Write a problem that uses the computation 43 4 and gives 11 as the correct answer. c. Write a problem that uses the computation 43 4 and gives 10.75 as the correct answer.

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 12 2 = 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.   Session 4: Index | Notes | Solutions | Video