When teaching partitive and quotative division, the type of division comes down to the context as described in a word problem. With partitive division, also called sharing division, a word problem is asking you to perform division to find out how many items go in each group. For example, if you have 30 chocolate bars and 10 people, how many chocolate bars does each person get? With quotative division, the object is to find out how many groups can be formed with a prescribed group size. For example, if there are 20 books and each person needs two books, how many people will get books? Teachers do not need to teach students the terms "partitive"and "quotative," but they should guide students to be able to solve both forms of real-world division problems.

## Multiplication and Division

Teachers should review multiplication and division facts on the board or orally and reaffirm the relationship between multiplication and division. Students should understand that when they multiply the quotient and divisor together, they will get the dividend as an answer. For example, 63 divided by 9 equals 7, and 7 times 9 equals 63. Once students have a firm grasp on quotative and partitive division, the teacher can demonstrate the relationship between fractions and division. The numerator of a fraction is like the dividend and the denominator is like the divisor.

## Role Playing

Teachers can introduce partitive and quotative division by acting out real-world scenarios for the whole class using props and student volunteers. For example, for partitive division, the teacher can call up four volunteers to the front of the class; each student represents one group. The teacher has 16 items and starts passing out one prop to each student, while asking the class how many items each student will receive. To act out a quotative real-world scenario, the teacher can tell the class that each person needs two socks and that she has 12. She starts passing out two socks to each student while asking the class how many students (or groups) will receive the appropriate amount of socks. For each role-play, the teacher should follow up by writing the relevant equation on the board.

## Worksheets

Students can complete worksheets that include both quotative and partitive division word problems. For the initial worksheets, teachers should show students how to illustrate each scenario. For example, for a question that says, "Two people were given eight hot dogs; how many hot dogs does each person get?", students can draw eight hot dogs and underneath draw two people, each with four hot dogs surrounding them. Once they understand the concepts, they can simply complete word problems without drawing.

## Recognizing Division Problems

If students are doing a sheet with only division problems, they have a fairly simple task of coming up with the correct answer. But, if they are given a sheet with multiplication and division problems, they may have issues figuring out the correct operation. Teachers should read word problems aloud and see if the class can identify whether the problem requires multiplication, division, or even addition or subtraction. "Per," "groups of" and "shared equally" are examples of keywords that denote division. The Aversboro Math Resources page offers a list of keywords for each operation that teachers can teach their students.

#### References

- University of the Phillipines, The International Online Journal: Middle School Children's Understanding of Algebraic Functions as Quotients
- "Turkish Online Journal of Qualitative Inquiry"; Collaborative Action Research: Teaching of Multiplication and Division in the Second Grade; Eda Vula et al.; 2011
- Sefton-Ash Publishers: Division
- Contexts For Learning - Mathematics: Introducing Fractions

#### Photo Credits

- Comstock Images/Comstock/Getty Images