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# Cup of Rice

Alignments to Content Standards: 6.NS.A.1

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.

#### Julie says:

(Weird that in preview, comments show up by job title. If it doesn't say my name, I'm Julie Wright.)

I highly recommend this problem. My sixth graders were approximately evenly divided on which was the right answer, and it made for a great discussion. We were able to highlight the correct thinking in the 1 1/3 answer (that it is 1 serving plus 1/3 cup) and find exactly where it broke down. I had the explainer go slowly and we did thumbs up for agreement or thumbs down for disagreement with each statement as we went along. One interesting thing is many of the 1 1/2 answerers disagreed with the statement "The cup of rice is a serving of rice plus an additional 1/3 cup of rice" until we broke down both parts and they realized that statement was true after all.

The pictures were very illuminating as the students talked about their thinking. It might be worth setting up a number line model for this, too.

#### Cam says:

Agreed, it is a little weird. On your user profile, there's a little checkbox you can unclick so that it doesn't use this as your descriptor (or you could just change your descriptor to be your name).

Thanks for the feedback on this task!

#### St. Christopher School says:

Thanks for pointing that out! I worried that if $1 \frac12$ comes from the teacher, students will think it is correct because the teacher said so rather than for the reasons given in the solution. I did, however, reword it. How does the current formulation work?