Cup of Rice

Task

Tonya and Chrissy are trying to understand the following story problem for $1 \div \frac23$:

One serving of rice is $\frac23$ of a cup. I ate 1 cup of rice. How many servings of rice did I eat?

To solve the problem, Tonya and Chrissy draw a diagram divided into three equal pieces, and shade two of those pieces.

Tonya says, “There is one $\frac23$-cup serving of rice in 1 cup, and there is $\frac13$ cup of rice left over, so the answer should be $1 \frac13$.”

Chrissy says, “I heard someone say that the answer is $\frac32 = 1 \frac12$. Which answer is right?”

Is the answer $1 \frac13$ or $1 \frac12$? Explain your reasoning using the diagram.

IM Commentary

One common mistake students make when dividing fractions using visuals is the confusion between remainder and the fractional part of a mixed number answer. In this problem, $\frac13$ is the remainder with units “cups of rice” and $\frac12$ has units “servings”, which is what the problem is asking for.

To see an annotated version of this and other Illustrative Mathematics tasks as well as other Common Core aligned resources, visit Achieve the Core.

Task based on a problem by Sybilla Beckmann, Mathematics for Elementary Teachers, Pearson 2010.

Solution

In Tonya’s solution of $1 \frac13$, she correctly notices that there is one $\frac23$-cup serving of rice in 1 cup, and there is $\frac13$ cup of rice left over. But she is mixing up the quantities of servings and cups in her answer. The question becomes how many servings is $\frac13$ cup of rice? The answer is “$\frac13$ cup of rice is $\frac12$ of a serving.”

It would be correct to say, "There is one serving of rice with $\frac13$ cup of rice left over," but to interpret the quotient $1\frac12$, the units for the 1 and the units for the $\frac12$ must be the same:

There are $1\frac12$ servings in 1 cup of rice if each serving is $\frac23$ cup.