- 09/08/13 | Adjusted: 03/15/16 | 1 file
- Grades 6

Mathematics Tasks

Fishing Adventures 2

Author: Illustrative Mathematics

Description
Files

What we like about this task

Mathematically:

Addresses standards: [standard 7.EE.B.4] and [standard MP.4]
Represents several aspects of modeling ([standard MP.4]), including defining variables to write an expression and create an inequality; computing with the model; and interpreting the results
Connects to computing with numbers in any form ([standard 7.EE.B.3])

In the classroom:

Correspondence between the symbols, the solutions, the number line, and the situation are highlighted in the solution
Allows the teacher to check for student understanding throughout students' work

Making the Shifts

How does this task exemplify the instructional Shifts required by CCSSM?

Focus	Belongs to the major work of seventh grade
Coherence	Builds on grade 6 work with expressions and equations; prepares students for more advanced modeling expectations
Rigor[tooltip Tasks will often target only one aspect of rigor.]	Conceptual Understanding: not targeted in this task Procedural Skill and Fluency: secondary in this task Application: primary in this task

Task

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry

Making the Shifts

How does this task exemplify the instructional Shifts required by CCSSM?

Focus	Belongs to the major work of seventh grade
Coherence	Builds on grade 6 work with expressions and equations; prepares students for more advanced modeling expectations
Rigor[tooltip Tasks will often target only one aspect of rigor.]	Conceptual Understanding: not targeted in this task Procedural Skill and Fluency: secondary in this task Application: primary in this task

Task

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry

Making the Shifts

How does this task exemplify the instructional Shifts required by CCSSM?

Focus	Belongs to the major work of seventh grade
Coherence	Builds on grade 6 work with expressions and equations; prepares students for more advanced modeling expectations
Rigor[tooltip Tasks will often target only one aspect of rigor.]	Conceptual Understanding: not targeted in this task Procedural Skill and Fluency: secondary in this task Application: primary in this task

Task

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set.

Several groups of people wish to rent a boat. Group

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set.

Several groups of people wish to rent a boat. Group

Making the Shifts

How does this task exemplify the instructional Shifts required by CCSSM?

Focus	Belongs to the major work of seventh grade
Coherence	Builds on grade 6 work with expressions and equations; prepares students for more advanced modeling expectations
Rigor[tooltip Tasks will often target only one aspect of rigor.]	Conceptual Understanding: not targeted in this task Procedural Skill and Fluency: secondary in this task Application: primary in this task

Task

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set.

Several groups of people wish to rent a boat. Group

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set.

Several groups of people wish to rent a boat. Group

Making the Shifts

How does this task exemplify the instructional Shifts required by CCSSM?

Focus	Belongs to the major work of seventh grade
Coherence	Builds on grade 6 work with expressions and equations; prepares students for more advanced modeling expectations
Rigor[tooltip Tasks will often target only one aspect of rigor.]	Conceptual Understanding: not targeted in this task Procedural Skill and Fluency: secondary in this task Application: primary in this task

Task

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry

Making the Shifts

How does this task exemplify the instructional Shifts required by CCSSM?

Focus	Belongs to the major work of seventh grade
Coherence	Builds on grade 6 work with expressions and equations; prepares students for more advanced modeling expectations
Rigor[tooltip Tasks will often target only one aspect of rigor.]	Conceptual Understanding: not targeted in this task Procedural Skill and Fluency: secondary in this task Application: primary in this task

Task

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set.

Several groups of people wish to rent a boat. Group

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set.

Several groups of people wish to rent a boat. Group

Making the Shifts

How does this task exemplify the instructional Shifts required by CCSSM?

Focus	Belongs to the major work of seventh grade
Coherence	Builds on grade 6 work with expressions and equations; prepares students for more advanced modeling expectations
Rigor[tooltip Tasks will often target only one aspect of rigor.]	Conceptual Understanding: not targeted in this task Procedural Skill and Fluency: secondary in this task Application: primary in this task

Task

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set.

Several groups of people wish to rent a boat. Group

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

$ has $8$ people. Which of the groups, if any, can safely rent a boat? What is the maximum number of people that may rent a boat?

Illustrative Mathematics Commentary and Solution

Commentary:

This task is the second in a series of three tasks that use inequalities in the same context at increasing complexity in 6th grade, 7th grade and in high school algebra. Students write and solve inequalities, and represent the solutions graphically. The progression of the content standards is [standard 6.EE.B.8] to[standard 7.EE.B.4] to [standard A-REI.D.12].

This particular task could be used for instruction or assessment.

Solution:

Let $p$ be the number of people in a group that wishes to rent a boat. Then

Because each group requires

Making the Shifts

How does this task exemplify the instructional Shifts required by CCSSM?

Focus	Belongs to the major work of seventh grade
Coherence	Builds on grade 6 work with expressions and equations; prepares students for more advanced modeling expectations
Rigor[tooltip Tasks will often target only one aspect of rigor.]	Conceptual Understanding: not targeted in this task Procedural Skill and Fluency: secondary in this task Application: primary in this task

Task

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry

Making the Shifts

How does this task exemplify the instructional Shifts required by CCSSM?

Focus	Belongs to the major work of seventh grade
Coherence	Builds on grade 6 work with expressions and equations; prepares students for more advanced modeling expectations
Rigor[tooltip Tasks will often target only one aspect of rigor.]	Conceptual Understanding: not targeted in this task Procedural Skill and Fluency: secondary in this task Application: primary in this task

Task

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set.

Several groups of people wish to rent a boat. Group

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set.

Several groups of people wish to rent a boat. Group

Making the Shifts

How does this task exemplify the instructional Shifts required by CCSSM?

Focus	Belongs to the major work of seventh grade
Coherence	Builds on grade 6 work with expressions and equations; prepares students for more advanced modeling expectations
Rigor[tooltip Tasks will often target only one aspect of rigor.]	Conceptual Understanding: not targeted in this task Procedural Skill and Fluency: secondary in this task Application: primary in this task

Task

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set.

Several groups of people wish to rent a boat. Group

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

A graph illustrating solutions is shown below. We observe that our solutions are values of $p$, listed below the number line and shown by the blue dots, so that the corresponding weights

We can find out which of the groups, if any, can safely rent a boat by substituting the number of people in each group for $p$ in our inequality. We see that

For Group

Making the Shifts

How does this task exemplify the instructional Shifts required by CCSSM?

Focus	Belongs to the major work of seventh grade
Coherence	Builds on grade 6 work with expressions and equations; prepares students for more advanced modeling expectations
Rigor[tooltip Tasks will often target only one aspect of rigor.]	Conceptual Understanding: not targeted in this task Procedural Skill and Fluency: secondary in this task Application: primary in this task

Task

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry

Making the Shifts

How does this task exemplify the instructional Shifts required by CCSSM?

Focus	Belongs to the major work of seventh grade
Coherence	Builds on grade 6 work with expressions and equations; prepares students for more advanced modeling expectations
Rigor[tooltip Tasks will often target only one aspect of rigor.]	Conceptual Understanding: not targeted in this task Procedural Skill and Fluency: secondary in this task Application: primary in this task

Task

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set.

Several groups of people wish to rent a boat. Group

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set.

Several groups of people wish to rent a boat. Group

Making the Shifts

How does this task exemplify the instructional Shifts required by CCSSM?

Focus	Belongs to the major work of seventh grade
Coherence	Builds on grade 6 work with expressions and equations; prepares students for more advanced modeling expectations
Rigor[tooltip Tasks will often target only one aspect of rigor.]	Conceptual Understanding: not targeted in this task Procedural Skill and Fluency: secondary in this task Application: primary in this task

Task

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set.

Several groups of people wish to rent a boat. Group

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

We find that both Group

Making the Shifts

How does this task exemplify the instructional Shifts required by CCSSM?

Focus	Belongs to the major work of seventh grade
Coherence	Builds on grade 6 work with expressions and equations; prepares students for more advanced modeling expectations
Rigor[tooltip Tasks will often target only one aspect of rigor.]	Conceptual Understanding: not targeted in this task Procedural Skill and Fluency: secondary in this task Application: primary in this task

Task

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry

Making the Shifts

How does this task exemplify the instructional Shifts required by CCSSM?

Focus	Belongs to the major work of seventh grade
Coherence	Builds on grade 6 work with expressions and equations; prepares students for more advanced modeling expectations
Rigor[tooltip Tasks will often target only one aspect of rigor.]	Conceptual Understanding: not targeted in this task Procedural Skill and Fluency: secondary in this task Application: primary in this task

Task

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set.

Several groups of people wish to rent a boat. Group

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set.

Several groups of people wish to rent a boat. Group

Making the Shifts

How does this task exemplify the instructional Shifts required by CCSSM?

Focus	Belongs to the major work of seventh grade
Coherence	Builds on grade 6 work with expressions and equations; prepares students for more advanced modeling expectations
Rigor[tooltip Tasks will often target only one aspect of rigor.]	Conceptual Understanding: not targeted in this task Procedural Skill and Fluency: secondary in this task Application: primary in this task

Task

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set.

Several groups of people wish to rent a boat. Group

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

$ exceeds the weight limit, and so cannot rent a boat.

To find the maximum number of people that may rent a boat, we solve our inequality for $p$:

$p≤6.25$

As we cannot have $.25$ person, we see that $6$ is the largest number of people that may rent a boat at once. This also matches our graph; since only integer values of $p$ makes sense, $6$ is the largest value of $p$ whose corresponding weight value lies below the limit of

Additional Thoughts

Note that simplifications like

Making the Shifts

How does this task exemplify the instructional Shifts required by CCSSM?

Focus	Belongs to the major work of seventh grade
Coherence	Builds on grade 6 work with expressions and equations; prepares students for more advanced modeling expectations
Rigor[tooltip Tasks will often target only one aspect of rigor.]	Conceptual Understanding: not targeted in this task Procedural Skill and Fluency: secondary in this task Application: primary in this task

Task

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry

Making the Shifts

How does this task exemplify the instructional Shifts required by CCSSM?

Focus	Belongs to the major work of seventh grade
Coherence	Builds on grade 6 work with expressions and equations; prepares students for more advanced modeling expectations
Rigor[tooltip Tasks will often target only one aspect of rigor.]	Conceptual Understanding: not targeted in this task Procedural Skill and Fluency: secondary in this task Application: primary in this task

Task

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set.

Several groups of people wish to rent a boat. Group

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set.

Several groups of people wish to rent a boat. Group

Making the Shifts

How does this task exemplify the instructional Shifts required by CCSSM?

Focus	Belongs to the major work of seventh grade
Coherence	Builds on grade 6 work with expressions and equations; prepares students for more advanced modeling expectations
Rigor[tooltip Tasks will often target only one aspect of rigor.]	Conceptual Understanding: not targeted in this task Procedural Skill and Fluency: secondary in this task Application: primary in this task

Task

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set.

Several groups of people wish to rent a boat. Group

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

nbsp;passengers standing at one end of the boat and all their gear piled at the other end of the boat. The right-hand side, $(150+10)p$, evokes a picture of $p

Making the Shifts

How does this task exemplify the instructional Shifts required by CCSSM?

Focus	Belongs to the major work of seventh grade
Coherence	Builds on grade 6 work with expressions and equations; prepares students for more advanced modeling expectations
Rigor[tooltip Tasks will often target only one aspect of rigor.]	Conceptual Understanding: not targeted in this task Procedural Skill and Fluency: secondary in this task Application: primary in this task

Task

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry

Making the Shifts

How does this task exemplify the instructional Shifts required by CCSSM?

Focus	Belongs to the major work of seventh grade
Coherence	Builds on grade 6 work with expressions and equations; prepares students for more advanced modeling expectations
Rigor[tooltip Tasks will often target only one aspect of rigor.]	Conceptual Understanding: not targeted in this task Procedural Skill and Fluency: secondary in this task Application: primary in this task

Task

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set.

Several groups of people wish to rent a boat. Group

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set.

Several groups of people wish to rent a boat. Group

Making the Shifts

How does this task exemplify the instructional Shifts required by CCSSM?

Focus	Belongs to the major work of seventh grade
Coherence	Builds on grade 6 work with expressions and equations; prepares students for more advanced modeling expectations
Rigor[tooltip Tasks will often target only one aspect of rigor.]	Conceptual Understanding: not targeted in this task Procedural Skill and Fluency: secondary in this task Application: primary in this task

Task

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set.

Several groups of people wish to rent a boat. Group

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

nbsp;passengers standing in the boat with each passenger clutching his or her own gear. The total weight is the same in either picture. So this task can also connect to [standard 7.EE.A.2]: "Understand that rewriting an expression in different forms in a problem can shed light on the problem and how the quantities in it are related."

Some students might take an arithmetic approach to this problem: First, subtracting

Making the Shifts

How does this task exemplify the instructional Shifts required by CCSSM?

Focus	Belongs to the major work of seventh grade
Coherence	Builds on grade 6 work with expressions and equations; prepares students for more advanced modeling expectations
Rigor[tooltip Tasks will often target only one aspect of rigor.]	Conceptual Understanding: not targeted in this task Procedural Skill and Fluency: secondary in this task Application: primary in this task

Task

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry

Making the Shifts

How does this task exemplify the instructional Shifts required by CCSSM?

Focus	Belongs to the major work of seventh grade
Coherence	Builds on grade 6 work with expressions and equations; prepares students for more advanced modeling expectations
Rigor[tooltip Tasks will often target only one aspect of rigor.]	Conceptual Understanding: not targeted in this task Procedural Skill and Fluency: secondary in this task Application: primary in this task

Task

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set.

Several groups of people wish to rent a boat. Group

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set.

Several groups of people wish to rent a boat. Group

Making the Shifts

How does this task exemplify the instructional Shifts required by CCSSM?

Focus	Belongs to the major work of seventh grade
Coherence	Builds on grade 6 work with expressions and equations; prepares students for more advanced modeling expectations
Rigor[tooltip Tasks will often target only one aspect of rigor.]	Conceptual Understanding: not targeted in this task Procedural Skill and Fluency: secondary in this task Application: primary in this task

Task

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set.

Several groups of people wish to rent a boat. Group

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

$ has $8$ people. Which of the groups, if any, can safely rent a boat? What is the maximum number of people that may rent a boat?

Illustrative Mathematics Commentary and Solution

Commentary:

This particular task could be used for instruction or assessment.

Solution:

Let $p$ be the number of people in a group that wishes to rent a boat. Then

Because each group requires

Making the Shifts

How does this task exemplify the instructional Shifts required by CCSSM?

Focus	Belongs to the major work of seventh grade
Coherence	Builds on grade 6 work with expressions and equations; prepares students for more advanced modeling expectations
Rigor[tooltip Tasks will often target only one aspect of rigor.]	Conceptual Understanding: not targeted in this task Procedural Skill and Fluency: secondary in this task Application: primary in this task

Task

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set.

Several groups of people wish to rent a boat. Group

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

A graph illustrating solutions is shown below. We observe that our solutions are values of $p$, listed below the number line and shown by the blue dots, so that the corresponding weights

We can find out which of the groups, if any, can safely rent a boat by substituting the number of people in each group for $p$ in our inequality. We see that

For Group

Making the Shifts

How does this task exemplify the instructional Shifts required by CCSSM?

Focus	Belongs to the major work of seventh grade
Coherence	Builds on grade 6 work with expressions and equations; prepares students for more advanced modeling expectations
Rigor[tooltip Tasks will often target only one aspect of rigor.]	Conceptual Understanding: not targeted in this task Procedural Skill and Fluency: secondary in this task Application: primary in this task

Task

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set.

Several groups of people wish to rent a boat. Group

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

We find that both Group

Making the Shifts

How does this task exemplify the instructional Shifts required by CCSSM?

Focus	Belongs to the major work of seventh grade
Coherence	Builds on grade 6 work with expressions and equations; prepares students for more advanced modeling expectations
Rigor[tooltip Tasks will often target only one aspect of rigor.]	Conceptual Understanding: not targeted in this task Procedural Skill and Fluency: secondary in this task Application: primary in this task

Task

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set.

Several groups of people wish to rent a boat. Group

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

$ exceeds the weight limit, and so cannot rent a boat.

To find the maximum number of people that may rent a boat, we solve our inequality for $p$:

$p≤6.25$

Additional Thoughts

Note that simplifications like

Making the Shifts

How does this task exemplify the instructional Shifts required by CCSSM?

Focus	Belongs to the major work of seventh grade
Coherence	Builds on grade 6 work with expressions and equations; prepares students for more advanced modeling expectations
Rigor[tooltip Tasks will often target only one aspect of rigor.]	Conceptual Understanding: not targeted in this task Procedural Skill and Fluency: secondary in this task Application: primary in this task

Task

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set.

Several groups of people wish to rent a boat. Group

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

nbsp;passengers standing at one end of the boat and all their gear piled at the other end of the boat. The right-hand side, $(150+10)p$, evokes a picture of $p

Making the Shifts

How does this task exemplify the instructional Shifts required by CCSSM?

Focus	Belongs to the major work of seventh grade
Coherence	Builds on grade 6 work with expressions and equations; prepares students for more advanced modeling expectations
Rigor[tooltip Tasks will often target only one aspect of rigor.]	Conceptual Understanding: not targeted in this task Procedural Skill and Fluency: secondary in this task Application: primary in this task

Task

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set.

Several groups of people wish to rent a boat. Group

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Some students might take an arithmetic approach to this problem: First, subtracting

Making the Shifts

How does this task exemplify the instructional Shifts required by CCSSM?

Focus	Belongs to the major work of seventh grade
Coherence	Builds on grade 6 work with expressions and equations; prepares students for more advanced modeling expectations
Rigor[tooltip Tasks will often target only one aspect of rigor.]	Conceptual Understanding: not targeted in this task Procedural Skill and Fluency: secondary in this task Application: primary in this task

Task

Fishing Adventures rents small fishing boats to tourists for day-long fishing trips. Each boat can only carry

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Create an inequality describing the restrictions on the number of people possible in a rented boat. Graph the solution set.

Several groups of people wish to rent a boat. Group

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Illustrative Mathematics Commentary and Solution

Commentary:

This particular task could be used for instruction or assessment.

Solution:

Let $p$ be the number of people in a group that wishes to rent a boat. Then

Because each group requires

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

A graph illustrating solutions is shown below. We observe that our solutions are values of $p$, listed below the number line and shown by the blue dots, so that the corresponding weights

We can find out which of the groups, if any, can safely rent a boat by substituting the number of people in each group for $p$ in our inequality. We see that

For Group

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

We find that both Group

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

To find the maximum number of people that may rent a boat, we solve our inequality for $p$:

$p≤6.25$

Additional Thoughts

Note that simplifications like

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

Some students might take an arithmetic approach to this problem: First, subtracting

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

Tonya and Chrissy are trying to understand the following story problem for 1 ÷ $\frac{2}{3}$.

There is $\frac{2}{3}$ cup of rice in one serving of rice. I ate 1 cup of rice. How many servings of rice did I eat?

Tonya says, "One cup of rice contains $\frac{2}{3}$ cup serving plus an additional $\frac{1}{3}$ cup of rice, so the answer should be $1\frac{1}{3}$ servings."

Chrissy says, "I heard someone say that the answer is $\frac{3}{2}$ or $1\frac{1}{2}$ servings."

Is Tonya correct or incorrect? Explain your reasoning. Support your explanation using this diagram.
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.
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, 6-8 The Number System.

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