# Fish Tanks

Alignments to Content Standards: 3.OA.A.2 Suppose there are 4 tanks and 3 fish in each tank. The total number of fish in this situation can be expressed as $4 \times 3 = 12$.

1. Describe what is meant in this situation by $12 \div 3 = 4$
2. Describe what is meant in this situation by $12 \div 4 = 3$

## IM Commentary

The purpose of this task is to provide an opportunity for students to demonstrate that they can interpret within a given context the meaning of whole number quotients. Since this particular context is an asymmetric model of multiplication, there are two distinct interpretations of the quotients that can arise.

The given context is an asymmetric model of multiplication in the sense that 4 tanks with 3 fish per tank is different than 3 tanks with 4 fish per tank, even though in both cases there are a total of 12 fish. The convention used here is that the number of groups is listed first and the number of objects in each group is listed second, although in this case the context determines whether 3 or 4 is the group (respectively, number of objects in each group). Although these terms are not mentioned in the standards, the number of groups is sometimes called the multiplier and the number of objects in each group is called the mutiplicand.

Note: It is important that students can make sense of both of these interpretations of division arising from a single multiplication context with the factors and quotients all whole numbers. These interpretations will be extended to contexts with fractions and decimal in later grades.

## Solution

The students' language may vary. What is important is the structure. The student needs to match what is known (either the number of groups or the amount in each group) and what is being determined by the quotient operation in this context.

1. Describe what is meant in this situation by 12 ÷ 3 = 4

Solution: There are 12 fish. If 3 fish are put in each tank there will be 4 tanks.

Comment: This is a measurement model of division, the number of objects in each group is known, and we are determining the number of groups.

2. Describe what is meant in this situation by 12 ÷ 4 = 3

Solution: There are 12 fish. If we want to partition (share) them equally among 4 tanks, we will end up with 3 fish in each tank.

Comment: This a sharing model of division, the number of groups is known and we are trying to determine the number in each group.