The Two Interpretations of Division Problems

There are two interpretation of division: the "How many groups?" interpretation and the "how many in each group?" interpretation. With the "How many groups?" interpretation of division, A ÷ B means the number of Bs that are in A––that is, the number of groups when A objects are divided into groups with B objects in each group. With the "How many in each group?" interpretation of division, A ÷ B means the number of objects in each group when A objects are divided equally among B groups. Watch the video below to review how to solve some simple divsion problems using picture representations and keep this process in mind as you read this blog post.

How Many Groups?

As shown below, the division problem 8 ÷ 2 can be translated as a "How Many Groups?" problem by writing an equivalent multiplication problem. In this case, "8 ÷ 2 = ?" can be rewritten as "? x 2 = 8". In this format, the multiplicand (? in our case) is considered to be the number of groups that are solution is asking for. Our multiplier (2 in this case) is the amount of cups in one popcorn ball. Finally the product (8) is the amount of cups that the number of groups we are looking for will produce. In simpler terms, our problem reads, "What number times 2 equals 8?"

Once you have figured out how to rewrite division expressions as multiplication expressions using whole numbers, the same methodology follows for fractions.

There are a few ways in which you can solve these types of problems including the use of tables, strip diagrams, double number lines, and/or annotated equations. All of these ways are listed in the chart below.

Practice Problem: In your math journals, write a “How Many Groups?” word problem for ⅔ ÷ ¾ and solve the problem with the aid of a strip diagram, table, and a double number line. Be sure to show your work.

How Many in One Group?

The second interpretation of division lies in the "How Many in One Group?" problems. For example, take the problem 6 ÷ 2. In order to demonstrate a "How Many in One Group" problem, we will say that 2 identical containers hold exactly 6 cups of popcorn. Our task is to determine how many cups of popcorn are contained within 1 of those containers. Therefore, we will rewrite the expression 6 ÷ 2 as a multiplication expression. As a result, our new expression will be 2 x ? = 6. The multiplicand (2) represents how many groups we have in our problem in terms of containers. The multiplier, or ?, is how many cups of popcorn are contained in each group or container. The product, or 6, is how many cups are held in 2 containers. In simpler terms, our equation reads as "2 times what equals 6?"

Notice the difference in how this equation reads compared to the "How Many Groups?" problem from before.

Much like the first interpretation, once you have determined how to rewrite the division expressions as multiplication expression for whole numbers, the same methodology is true for fractions (as shown below).

Practice Problem: In your math journals, write a "How Many in One Group?" word problem for 5 ÷ ⅓ with the aid of a strip diagram, table, and a double number line. Be sure to show your work.