Unable to typeset  PDF Format  Error 

Two Interpretations of Division
Task
 Maria cuts 12 feet of ribbon into 3 equal pieces so she can share it with her two sisters. How long is each piece?
 Maria has 12 feet of ribbon and wants to wrap some gifts. Each gift needs 3 feet of ribbon. How many gifts can she wrap using the ribbon?
IM Commentary
Both of the questions are solved by the division problem $12 \div 3$ but what happens to the ribbon is different in each case. In the first case, the number of pieces of ribbon is fixed at 3 and the question is asking "how long is each piece?" (12 feet divided into 3 pieces gives 4 feet per piece.) In the second question, the size of each piece is fixed and the question is "how many pieces does one get?" (12 feet divided by 3 feet per gift gives 4 gifts.) The problem can be solved with a drawing of a tape diagram or number line. For problem 1, the line must be divided into 3 equal parts. The second problem can be solved by successive subtraction of 3 feet to see how many times it fits in 12.
In this case it is particularly helpful for the teacher to require students to justify their answers with a diagram. The way in which a student represents the problem can reveal whether or not he or she really understands the distinction between the two types of division problems shown here.
Solutions
Solution: Tape Diagram

This question asks, "how long is each piece?" so is a "How many in each group?" division problem: $$3\times ?= 12$$
$12 \div 3 = 4$, so each child gets a piece of ribbon that is 4 feet long.

This question asks, "how many pieces does one get?" so is a "How many groups?" division problem: $$? \times 3= 12$$
$12 \div 3 = 4$, so Maria can wrap 4 gifts.
Solution: Number Line
 This question asks, "how long is each piece?" so is a "How many in each group?" division problem:
$$3\times ?= 12$$
$12 \div 3 = 4$, so each child gets a piece of ribbon that is 4 feet long.
This question asks, "how many pieces does one get?" so is a "How many groups?" division problem: $$?\times 3= 12$$
$12 \div 3 = 4$, so Maria can wrap 4 gifts.
Two Interpretations of Division
 Maria cuts 12 feet of ribbon into 3 equal pieces so she can share it with her two sisters. How long is each piece?
 Maria has 12 feet of ribbon and wants to wrap some gifts. Each gift needs 3 feet of ribbon. How many gifts can she wrap using the ribbon?
Comments
Log in to commentDixie Blackinton says:
almost 5 yearsJust a thought about division which may not be important in third grade but seems to help clarify division of fractions in later grades. When students are asked to record or report their answers in a sentence they are more likely to pay attention to units of measure. In division the units of measure depend on the context. In the first example, 12 feet of ribbon divided into 3 pieces. Each piece is 4 feet. (feet divided into 3 pieces = pieces of size 4 feet; feet divided 3 = feet) In the second example, 12 feet of ribbon divided into pieces of length 3 feet yields 4 pieces. (feet divided by feet = pieces) In addition and subtraction 10 feet add or subtract 6 feet equals so many feet. In multiplication 3 groups of 5 flowers equals 15 flowers. Not so in division.) In the example above, feet divided by feet = so many pieces.) This is easily explained by feet divided by feet per piece = pieces. This will be emphasized in later grades when studying rates. Oranges divided by oranges per case = cases while oranges divided into 6 cases = oranges per case.
joseph georgeson says:
about 5 yearsIn addition to representing these problems with a picture, tape diagram or number line, I feel it would be good to have students represent each interpretation of division as a "missing factor" multiplication problem. In the first example, since you know how many groups, the second factor would be missing: 3 times what equal 12. In the second example, since we know how much each group is, the multiplication would be: what multiplied by 3 is equal to 12. This is very helpful as students move to other than whole numbers.
It is the story that is important in understanding the meaning of the division or multiplication that is done. The numerical answer is the same but the interpretation and representation of the story is different.
Kristin says:
about 5 yearsI couldn't agree more, and I've added that to both solutions.
k8 Mathematics Coach says:
over 6 yearsI agree that this problem represents 2 different division situations but, as written, students could correctly answer these questions without demonstrating understanding and without any reference to the differences in the quotients, even though they are both "4."
I think it is important to add wording requiring representations and asking them to compare the results of both parts. How is it that they both have the same numerical result but the results are 2 very different things?
I just worry that teachers without a background of understanding high level tasks will attempt to mimic the wording of this problem, which, as it stands on paper, really isn't high level at all.
BanjoBen says:
over 6 yearsThanks for the input. The idea with this pair of problems is to have students justify their solutions with a model (such as a bar diagram, or number line). The student's solution, together with the model can be used to judge whether or not the child understands the meaning of the answer in each case.
I've added a bit to the commentary regarding this.