
 09/08/13  Adjusted: 01/10/17  1 file
 Grades 6
Cup of Rice
 Description
 Files
What we like about this task
Mathematically:
 Addresses standards: 6.NS.A.1, 5.NF.B.7, and MP.3
 Involves concepts, procedure, and application of fraction division – all required by standard 6.NS.A.1
 Devotes attention to a mathematically important case (dividend equal to 1)
 Builds on fraction division work from fifth grade (see 5.NF.B.7)
 Engages students in constructing viable arguments and critiquing the arguments of others (MP.3)
In the classroom:
 Uses visual models to support understanding
 Allows for individual or group work
 Encourages students to share their developing thinking

Making the Shifts
How does this task exemplify the instructional Shifts required by CCSSM?
Focus Belongs to the major work of sixth grade Coherence Addresses the culminating standard in the progression of fraction operations; prepares for rational arithmetic in grade 7 Rigor^{} Conceptual Understanding: primary in this task
Procedural Skill and Fluency: secondary in this task
Application: primary in this task 
Task
Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.
One serving of rice is $\frac{2}{3}$ 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 $\frac{2}{3}$cup serving of rice in 1 cup, and there is $\frac{1}{3}$ cup of rice left over, so the answer should be $1\frac{1}{3}$."
Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ = $1\frac{1}{2}$. Which answer is right?"
Is the answer $1\frac{1}{3}$ or $1\frac{1}{2}$? Explain your reasoning using the diagram.

Illustrative Mathematics Commentary and Solution
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, $\frac{1}{3}$ is the remainder with units "cups of rice" and $\frac{1}{2}$ has units "servings", which is what the problem is asking for.
Solution:
In Tonya's solution of $1\frac{1}{3}$, she correctly notices that there is one $\frac{2}{3}$cup serving of rice in $1$ cup, and there is $\frac{1}{3}$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 $\frac{1}{3}$ cup of rice? The answer is "$\frac{1}{3}$ cup of rice is $\frac{1}{2}$ of a serving."
It would be correct to say, "There is one serving of rice with $\frac{1}{3}$ cup of rice left over," but to interpret the quotient $1\frac{1}{2}$, the units for the $1$ and the units for the $\frac{1}{2}$ must be the same:
There are $1\frac{1}{2}$ servings in $1$ cup of rice if each serving is $\frac{2}{3}$ cup.

Additional Thoughts
The quotient chosen for this problem, $1 ÷ \frac{2}{3} = \frac{3}{2}$, sheds light on the fact that dividing is multiplying by the reciprocal. Once students understand a quotient like $1 ÷ \frac{2}{3} = \frac{3}{2}$, they can think about a problem like $\frac{3}{4} ÷ \frac{2}{3}$ by taking $\frac{3}{4}$ of the known quotient $1 ÷ \frac{2}{3}$. That is, $\frac{3}{4} ÷ \frac{2}{3} = \frac{3}{4} × 1 ÷ \frac{2}{3} = \frac{3}{4} × \frac{3}{2}$.
For more insight into the expectations for fraction division, read pages 5 and 6 of the progression document, 68 The Number System.