Summary:

To understand the partitive model of division (fair sharing when the number of groups is known, but the amount to be shared is not), the third-graders at the Wilson Elementary School in Mequon, Wisconsin, use various problem-solving strategies to arrive at unit costs. In this lesson, they use household items brought to class by their teacher, Lorraine Bahr. With no traditional algorithm to rely on, students are encouraged to draw pictures, guess and check, make charts, or act out to determine the cost per ounce or cost per item of a product. Before beginning the unit cost assignment, the class discusses what information it needs to determine what a one-ounce serving of cereal costs. One group uses "guess and check" and "draw a picture" problem solving strategies to determine the cost of one pair of socks, based on the cost of a three-pack, and also the cost of one plastic zippered bag, based on the cost of a box of twenty. The students share their reasoning and build on the reasoning of others to conceptualize the meaning of the division operation.

Standard Connection:

(Practice Standard)—Common Core Practice Standard #2—Reason abstractly and quantitatively—is in evidence in the group discussions of unit costs for a single plastic bag and a pair of socks. Students have no notions of quotients, dividends, and divisors, and no trusted division algorithm to use. They have a partitive division problem but no concept or context for thinking about division. Rather, these students rely on inventive abstract reasoning to conclude that distributing amounts across the total number of items produces a unit price. Once the students de-contextualize their problems and abstract the essence of them, they are asked to re-contextualize their problems to determine if the quantitative elements of their problems make sense.

(Content Standard)—Operations and Algebraic Thinking 3.OA—is the domain that encompasses the content of this lesson. These third-graders are just learning what it means to represent division problems as partitive distributions. This is an early foray into problems involving multiplication and division. The cost per unit problems they have encountered will help them "interpret whole-number quotients of whole numbers" and will help them describe contexts in which a number of shares can be expressed in a partitive division model or the number of groups can be expressed in a quotitive (measurement) division model.

Questions to Consider:

How would you explain to students the difference between the partitive and quotitive models of division? Why is it important for teachers to understand the difference between these two models? Discuss the pros and cons of giving third-graders division problems with only integral solutions.

2. Reason abstractly and quantitatively

3.OA Operations and Algebraic Thinking