- Identify the total amount, the size of one group, and the number of groups in the situation. - Match those three roles to the numbers in the division equation.

1. The \(2\) represents the total yards of ribbon available. 2. The \(\frac{1}{5}\) represents the yards of ribbon needed for one badge. 3. The quotient \(10\) represents the number of badges that can be made.

\(2\) is the total ribbon, \(\frac{1}{5}\) yard is the amount per badge, and \(10\) is the number of badges.

- Decide whether the unknown is the number of groups or the size of each group. - Match the total amount and the number of equal shares to a division expression.

1. The total amount is \(\frac{1}{4}\) gallon and it is split into \(2\) equal shares, so the equation is \(\frac{1}{4}\div2=\frac{1}{8}\). 2. The quotient \(\frac{1}{8}\) is the amount of paint used on each poster.

\(\frac{1}{4}\div2=\frac{1}{8}\). Each poster gets \(\frac{1}{8}\) gallon.

- In panel a), compare the full shaded amount with the left shaded share. - In panel b), find the size of one small group and count how many such groups fit in both wholes. - Match the roles in each picture to the roles in each equation.

1. Panel a) starts with \(\frac{1}{4}\) of a whole, split into \(2\) equal shares of \(\frac{1}{8}\), so it matches A. 2. Panel b) has \(2\) separate wholes split into one-fourth-size groups, so it matches B.

a) A) \(\frac{1}{4}\div2=\frac{1}{8}\) b) B) \(2\div\frac{1}{4}=8\)

- Identify the total amount and the size of one bag. - Ask whether the unknown is the size of each group or the number of groups.

1. The total amount is \(4\) pounds, and each group has size \(\frac{1}{8}\) pound. 2. Dividing total amount by group size finds the number of groups. 3. Therefore \(32\) is the number of bags that can be filled.

The quotient \(32\) represents \(32\) bags.

- Ask what the division is counting in the story. - Distinguish the unit for the size of one piece from the unit for the number of pieces.

1. The division asks how many \(\frac{1}{4}\)-foot groups fit in \(6\) feet. 2. Each foot contains \(4\) quarter-foot pieces, so \(6\times4=24\). 3. The quotient is a count of pieces, not a length in feet.

\(6\div\frac{1}{4}=24\), representing \(24\) pieces.

- Identify which quantity is already known: group count or group size. - Ask what the division is trying to find in this story.

1. The number of sections is already known to be \(4\), so the unknown is the length of each section. 2. \(\frac{1}{6}\div4=\frac{1}{24}\). 3. The quotient means each section lasts \(\frac{1}{24}\) minute.

The quotient is the length of each section: \(\frac{1}{24}\) minute.

- Identify the total amount and the capacity of one bottle. - A correct group count multiplied by the group size should rebuild the total.

1. The division asks how many groups of size \(\frac{1}{4}\) gallon fit in \(3\) gallons. 2. There are \(4\) quarter-gallons in each gallon, so there are \(12\) bottles. 3. Check: \(12\times\frac{1}{4}=3\) gallons.

\(12\) bottles; the check is \(12\times\frac{1}{4}=3\).

- Match the dividend to the total amount in each story. - Match the divisor to either a number of shares or a group size.

1. In A, \(\frac{1}{8}\) pound is the total and \(3\) is the number of equal shares, so it matches \(\frac{1}{8}\div3\). 2. The quotient is \(\frac{1}{24}\) pound per student. 3. Story B would use \(3\div\frac{1}{8}\), not the given expression.

A; the quotient is \(\frac{1}{24}\) pound per student.

- Identify which number counts groups and which number measures one group. - Interpret the multiplication check as putting all equal shares back together.

1. The \(8\) is the number of equal snack cups. 2. The \(\frac{1}{16}\) pound is the amount in each cup. 3. The multiplication check combines \(8\) equal cup amounts to recover the original \(\frac{1}{2}\) pound.

There are \(8\) cups, each gets \(\frac{1}{16}\) pound, and the check shows all shares total \(\frac{1}{2}\) pound.

- The dividend should be the total amount being divided. - The divisor should be the size of one group when the unknown is the number of groups.

1. The total length is \(4\) miles and the size of one section is \(\frac{1}{7}\) mile, so the equation is \(4\div\frac{1}{7}\). 2. Seven sections fit in each mile, so \(4\times7=28\). 3. The quotient counts the sections.

\(4\div\frac{1}{7}=28\), so there are \(28\) sections.

- Identify what one group represents in the situation. - The quotient should use a counting unit, not a length unit.

1. The total tape is \(5\) yards, and each group is a \(\frac{1}{2}\)-yard length. 2. The quotient counts how many half-yard lengths fit in the total. 3. Therefore the crew can cut \(10\) pieces.

The crew can cut \(10\) half-yard pieces of tape.

- Separate the measurement unit of one portion from the counting unit of the quotient. - Ask what the result is counting in the situation.

1. The division asks how many groups of size \(\frac{1}{5}\) cup fit in \(5\) cups. 2. There are \(25\) such groups. 3. The quotient is a count of sample cups, so the correct unit is sample cups, not cups of liquid.

The number \(25\) is correct because twenty-five \(\frac{1}{5}\)-cup portions fit in \(5\) cups. The quotient counts containers, so the correct result is \(25\) sample cups, not \(25\) cups of liquid.

- Decide whether the divisor tells the number of equal groups or the size of each group. - The quotient should describe what one equal share looks like. - Check whether your answer has the same kind of unit as the original distance.

1. The situation is \(\frac{1}{5}\div3\). 2. Dividing one fifth into \(3\) equal parts gives \(\frac{1}{15}\). 3. The quotient is a length because it describes the size of each of the \(3\) equal drills.

\(\frac{1}{5}\div3=\frac{1}{15}\). Each drill is \(\frac{1}{15}\) mile long, so the quotient is a length.

- Decide whether the whole number in the expression is a number of groups or an amount being divided. - In a sharing situation, the divisor can tell how many equal shares are made. - Check that your quotient’s unit matches what the question asks for.

1. In \(\frac{1}{4}\div6\), the \(6\) is the number of equal groups, so question a) matches. 2. Divide the unit fraction into \(6\) equal shares: \(\frac{1}{4}\div6=\frac{1}{24}\). 3. The quotient means each person gets \(\frac{1}{24}\) pound. 4. Question b) would be represented by \(6\div\frac{1}{4}\).

a); each person gets \(\frac{1}{24}\) pound.

- Identify the total amount, the size of one bag, and the number of bags. - The quotient should name how many groups of the divisor size fit in the total. - A multiplication check should combine all groups to recover the original amount.

1. The quotient \(30\) is the number of \(\frac{1}{6}\)-pound bags that can be filled from \(5\) pounds. 2. Each of the \(30\) bags contains \(\frac{1}{6}\) pound. 3. Multiplying the number of bags by the amount per bag gives \(30\times\frac{1}{6}=5\) pounds, which reconstructs the original total.

\(30\) is the number of bags. The check shows that \(30\) bags at \(\frac{1}{6}\) pound each contain the original \(5\) pounds.

- Identify the total area and the area covered by one tile. - The division should count how many groups of the tile-size area fit in the total. - The quotient should be a count, not an area measurement.

1. The total area is \(2\) square feet, and each tile covers \(\frac{1}{7}\) square foot. 2. Counting how many tile-sized areas fit in the total gives \(2\div\frac{1}{7}=14\). 3. The quotient \(14\) is the number of tiles.

\(2\div\frac{1}{7}=14\). The quotient is the number of tiles.

- Decide whether the whole number is a number of equal shares or an amount being divided. - The quotient in this expression should describe the size of one share. - Check which story keeps the fraction as the total amount being shared.

1. In \(\frac{1}{4}\div8\), the whole number \(8\) tells how many equal shares are made from one fourth, so story a) matches. 2. Dividing one fourth into \(8\) equal parts gives \(\frac{1}{32}\). 3. The quotient means each person receives \(\frac{1}{32}\) of a whole pizza. 4. Story b) would use \(8\div\frac{1}{4}\).

a); each person receives \(\frac{1}{32}\) of a whole pizza.

- Identify the total length and the length of one group. - The quotient counts how many groups of that size fit in the total. - The inverse multiplication should rebuild the original total length.

1. The divisor \(\frac{1}{7}\) yard is the size of one piece, so the division asks how many pieces of that size fit in \(4\) yards. 2. The quotient \(28\) is the number of pieces. 3. The check \(28\times\frac{1}{7}=4\) combines the \(28\) equal piece lengths and reconstructs the original \(4\)-yard roll.

The quotient \(28\) is the number of pieces. The multiplication check shows that \(28\) pieces of \(\frac{1}{7}\) yard make \(4\) yards.

- In panel a), read the shaded fraction before thinking about the four equal shares. - In panel b), determine the size of one section and count how many such groups fit in all the separate wholes. - Use the meaning of each quotient to decide the order of the numbers in the equation.

1. In panel a), \(\frac{1}{2}\) of a whole is split into \(4\) equal shares. One share is \(\frac{1}{8}\), so \(\frac{1}{2}\div4=\frac{1}{8}\). The quotient is a share size. 2. In panel b), there are \(4\) separate wholes, each split into halves. There are \(8\) half-size groups, so \(4\div\frac{1}{2}=8\). The quotient is a number of groups.

a) \(\frac{1}{2}\div4=\frac{1}{8}\); the quotient is the size of one share. b) \(4\div\frac{1}{2}=8\); the quotient is the number of groups.

- In the bar model, identify the full shaded fraction and the size of one marked share. - In the circle model, name the size of one serving and count how many such servings fit in both wholes. - Match each context by asking whether it describes sharing an amount or counting groups of a given size.

1. The bar model has \(2\) of \(6\) parts shaded, so the starting amount is \(\frac{1}{3}\). The left shaded part is one of \(2\) equal shares, so it matches Context 1 and shows \(\frac{1}{3}\div2=\frac{1}{6}\). 2. The two circles are \(2\) wholes, each split into thirds. They match Context 2 and show \(2\div\frac{1}{3}=6\).

Bar model: Context 1, \(\frac{1}{3}\div2=\frac{1}{6}\). Circle model: Context 2, \(2\div\frac{1}{3}=6\).

- In panel a), compare the full shaded amount with the left shaded share. - In panel b), count the separate wholes and the equal groups in each whole. - Decide whether each quotient is a share size or a group count.

1. In panel a), \(3\) of \(12\) counters are shaded, so the starting amount is \(\frac{1}{4}\) of the collection. 2. One shaded counter is \(\frac{1}{12}\) of the collection, and there are \(3\) equal shares. Therefore panel a) shows \(\frac{1}{4}\div3=\frac{1}{12}\). 3. In panel b), there are \(3\) separate wholes, and each whole contains \(4\) one-fourth groups. 4. There are \(12\) one-fourth groups altogether, so panel b) shows \(3\div\frac{1}{4}=12\). 5. Jordan reversed a sharing situation and a grouping situation.

a) \(\frac{1}{4}\div3=\frac{1}{12}\) b) \(3\div\frac{1}{4}=12\) Jordan reversed the roles: panel a) asks for a share size, while panel b) counts one-fourth-size groups.

- Combine the two fourths first and find the remaining fraction. - Ask: if \(3\) feet is one half of the whole, what division expression finds the whole? - State what each number in the division equation measures or represents.

1. The underground and painted parts total \(\frac{1}{4}+\frac{1}{4}=\frac{1}{2}\), so the unpainted part is \(\frac{1}{2}\) of the whole. 2. Since \(3\,\text{ft}\) is \(\frac{1}{2}\) of the whole, the division equation is \(3\div\frac{1}{2}=6\). 3. The dividend is the known \(3\)-foot unpainted portion, the divisor says that portion is one half of the whole, and the quotient is the entire post length.

a) \(\frac{1}{2}\) b) \(3\div\frac{1}{2}=6\). The \(3\,\text{ft}\) dividend is the known half-post length, \(\frac{1}{2}\) is the fraction of the whole that length represents, and \(6\,\text{ft}\) is the whole post length.

- Rename one half in fourths and find the remaining fraction. - Use the known length as the dividend and the remaining unit fraction as the divisor. - Interpret the quotient as the whole route, not as a number of unrelated pieces.

1. The forest and meadow account for \(\frac{1}{2}+\frac{1}{4}=\frac{2}{4}+\frac{1}{4}=\frac{3}{4}\), so the level path is \(\frac{1}{4}\) of the route. 2. Since \(2\,\text{mi}\) is \(\frac{1}{4}\) of the whole, \(2\div\frac{1}{4}=8\). 3. The dividend is the known level-path length, the divisor is the fraction of the whole route represented by that path, and the quotient is the complete route length.

a) \(\frac{1}{4}\) b) \(2\div\frac{1}{4}=8\). The \(2\,\text{mi}\) dividend is the known quarter-route length, \(\frac{1}{4}\) is the part of the whole route it represents, and \(8\,\text{mi}\) is the whole route.

- In each situation, identify the total amount first. - Decide whether the unknown is a share size or a number of equal groups. - Use the context to interpret the units of each quotient.

1. In a), the total is \(\frac{1}{3}\) pound and there are \(5\) equal shares: \(\frac{1}{3}\div5=\frac{1}{15}\). Each student gets \(\frac{1}{15}\) pound. 2. In b), the total is \(5\) pounds and each group is \(\frac{1}{3}\) pound: \(5\div\frac{1}{3}=15\). There are \(15\) bags.

a) \(\frac{1}{3}\div5=\frac{1}{15}\) pound per student. b) \(5\div\frac{1}{3}=15\) bags.

- Decide whether each story is finding the size of one share or the number of groups. - Match the total and divisor to the correct division order.

1. Story A uses \(\frac{1}{6}\div4=\frac{1}{24}\), so it matches. 2. Story B uses \(4\div\frac{1}{6}=24\), which counts pieces instead of finding a share size.

A; Story B has quotient \(24\).

- In each story, decide whether the divisor is a group size or a number of groups. - Ask whether the quotient should be a count of containers or the size of one share. - The order of the numbers in a division expression changes the meaning.

1. Situation 1 is \(2\div\frac{1}{8}=16\). The quotient counts \(16\) bottles. 2. Situation 2 is \(\frac{1}{8}\div2=\frac{1}{16}\). The quotient is the \(\frac{1}{16}\)-gallon share for each lab. 3. The same numbers play different roles: the first divisor is a group size, while the second divisor is a number of equal groups.

1) \(2\div\frac{1}{8}=16\): \(16\) bottles. 2) \(\frac{1}{8}\div2=\frac{1}{16}\): \(\frac{1}{16}\) gallon per lab.

- For each expression, ask what amount is being divided and what the divisor represents. - Decide whether you are counting groups of a given size or finding the size of one equal share. - Keep track of whether the answer should be a count or a measurement.

1. In a), \(5\div\frac{1}{4}=20\). It can mean the number of \(\frac{1}{4}\)-yard pieces that fit in \(5\) yards, so the quotient is a count of \(20\) pieces. 2. In b), \(\frac{1}{4}\div5=\frac{1}{20}\). It can mean splitting \(\frac{1}{4}\) yard into \(5\) equal pieces, so the quotient is a length of \(\frac{1}{20}\) yard per piece. 3. Reversing the dividend and divisor changes what is being found.

a) \(20\) pieces. This quotient counts how many \(\frac{1}{4}\)-yard pieces fit in \(5\) yards. b) \(\frac{1}{20}\) yard per piece. This quotient is a length because \(\frac{1}{4}\) yard is being split into \(5\) equal shares.

- Pay attention to whether the divisor is the whole number \(3\) or the fraction \(\frac{1}{3}\). - Think of a story where a fixed fractional amount is split into a stated number of equal shares. - The quotient should describe the size of one share.

1. In \(\frac{1}{7}\div3\), the divisor \(3\) is a whole-number count, not a group size of \(\frac{1}{3}\). 2. A matching story is: one seventh of a pan of food is shared equally among \(3\) people. 3. Each share is \(\frac{1}{7}\div3=\frac{1}{21}\) of the pan.

Arjun's interpretation is incorrect because the divisor is the whole number \(3\), not \(\frac{1}{3}\). A correct story is: one seventh of a pan of food is shared equally among \(3\) people. Each person receives \(\frac{1}{21}\) of the pan.

- Think of a situation where a whole amount is separated into groups with a fractional size. - Make sure the divisor describes the size of each group, not the number of groups. - The quotient should be a count of how many such groups fit in the whole amount.

1. One valid question is: “A \(6\)-yard roll of ribbon is cut into pieces that are each \(\frac{1}{3}\) yard long. How many pieces are made?” 2. Each yard contains \(3\) one-third-yard pieces, so \(6\) yards contain \(6\times3=18\) pieces. 3. The quotient \(18\) counts pieces, not yards.

Answers will vary. A valid question must use \(6\) whole units as the total, \(\frac{1}{3}\) unit as the size of each group, and interpret \(18\) as the number of groups.

- Decide whether each question is counting pieces or finding the size of one share. - The same numbers can appear in different positions depending on their roles. - Use the model to distinguish three wholes from one fifth of a whole.

1. For a), count fifths in \(3\) wholes: \(3\div\frac{1}{5}=15\). 2. For b), split one fifth into \(3\) equal shares: \(\frac{1}{5}\div3=\frac{1}{15}\). 3. The first quotient is a count of pieces, while the second quotient is the size of one share.

a) \(3\div\frac{1}{5}=15\). b) \(\frac{1}{5}\div3=\frac{1}{15}\).

- Decide whether the divisor is a number of equal groups or the size of each group. - Ask whether the quotient should describe a share size or a group count. - Keep the order of dividend and divisor tied to their roles in the situation.

1. In a), the whole-number divisor \(2\) tells how many equal shares are made from \(\frac{1}{6}\). The quotient \(\frac{1}{12}\) can represent the size of one share. 2. In b), the unit-fraction divisor \(\frac{1}{6}\) tells the size of each group being counted within \(2\) wholes. The quotient \(12\) can represent the number of one-sixth-size groups. 3. The equations use the same numbers but answer different kinds of questions.

a) finds the size of one of \(2\) equal shares of \(\frac{1}{6}\), giving \(\frac{1}{12}\). b) counts how many sixths fit in \(2\) wholes, giving \(12\).

- Choose a context where a whole-number amount is separated into equal groups with a unit-fraction size. - Match each number to total amount, group size, or number of groups. - The quotient should be a count rather than another length in this type of division.

1. One valid situation is cutting \(4\) yards of cord into pieces that are each \(\frac{1}{9}\) yard long. 2. The dividend \(4\) is the total number of yards of cord. 3. The divisor \(\frac{1}{9}\) yard is the length of one piece. 4. The quotient \(36\) is the number of pieces that can be cut.

Answers will vary. A valid situation must identify \(4\) as a total amount, \(\frac{1}{9}\) as the size of each group with matching units, and \(36\) as the number of groups.

- Match the total amount to the dividend and the size of one counted group to the divisor. - Ask whether the quotient should be a count or a length. - Reversing a division expression changes the roles of the quantities.

1. Counting pieces of size \(\frac{1}{8}\) yard in \(3\) yards is represented by \(3\div\frac{1}{8}\). 2. Each yard contains \(8\) eighth-yard pieces, so \(3\div\frac{1}{8}=24\). 3. Reversing the expression to \(\frac{1}{8}\div3\) would mean splitting one eighth of a yard into \(3\) equal shares, not counting pieces in \(3\) yards.

\(3\div\frac{1}{8}=24\). Reversing the expression would find a share size instead of a piece count.

- Decide whether the original question is counting groups of a given size or finding the size of one equal share. - Keep the total amount as the dividend when counting how many portions fit. - For the second expression, reverse the roles by turning the whole number into a count of equal shares.

1. Counting quarter-pizza portions in \(3\) pizzas uses \(3\div\frac{1}{4}=12\), so \(12\) portions can be served. 2. The other expression \(\frac{1}{4}\div3=\frac{1}{12}\) could answer: “If \(\frac{1}{4}\) pizza is shared equally among \(3\) people, how much pizza does each person receive?” 3. In that question, each person receives \(\frac{1}{12}\) pizza.

Use \(3\div\frac{1}{4}=12\), so \(12\) portions can be served. A question for \(\frac{1}{4}\div3\) is sharing one quarter pizza equally among three people, giving \(\frac{1}{12}\) pizza each.

- Identify whether the divisor tells a number of equal shares or the size of each group. - Think about what the quotient counts or measures in each situation. - Compare the roles of the numbers, not just the numerical answers.

1. \(\frac{1}{9}\div4=\frac{1}{36}\). This can describe the size of one share when one ninth of a whole is split equally among \(4\) people. 2. \(4\div\frac{1}{9}=36\). This can describe the number of one-ninth-size groups contained in \(4\) wholes. 3. The first quotient is a share size. The second quotient is a group count, so the same numbers play different roles in the two situations.

\(\frac{1}{9}\div4=\frac{1}{36}\) and \(4\div\frac{1}{9}=36\). The first quotient is a share size; the second is a group count.

- Tell a story in which a fixed fractional amount is split among a whole-number count of equal shares. - Use the story to predict whether one share should be larger or smaller than the original amount. - Think about how many equal pieces the whole would have after the split.

1. A matching story is: one tenth of a pan of food is shared equally among \(5\) people. 2. Each share must be smaller than the original \(\frac{1}{10}\), so \(\frac{1}{2}\) cannot be reasonable. 3. Splitting one tenth into \(5\) equal parts gives \(\frac{1}{50}\). 4. Therefore \(\frac{1}{10}\div5=\frac{1}{50}\).

Hana's reasoning is incorrect. A matching story is one tenth of a pan shared equally among \(5\) people; each share must be smaller than \(\frac{1}{10}\), not larger. The correct quotient is \(\frac{1}{50}\).

- Ask whether each division is counting groups of a known size or finding the size of one share. - A count and a measurement use different kinds of units. - Match the quotient unit to the question the division answers.

1. In a), \(6\div\frac{1}{2}=12\) counts how many half-mile groups fit in \(6\) miles, so \(12\) is a count of groups, not a distance of \(12\) miles per share. 2. In b), \(\frac{1}{2}\div6=\frac{1}{12}\) gives the size of one share when half a mile is split among \(6\) groups, so \(\frac{1}{12}\) mile is a distance, not a number of groups. 3. The quotient's meaning depends on whether division is counting groups or finding share size.

a) \(12\) is the number of half-mile groups in \(6\) miles. b) \(\frac{1}{12}\) mile is the size of one of \(6\) equal shares of half a mile.

- Rewrite \(\frac{1}{6}\) in twenty-fourths. - Think about how many copies of \(\frac{1}{24}\) make the full path section. - Use the multiplication relationship first, then connect it to division.

1. Rewrite \(\frac{1}{6}=\frac{4}{24}\). The path section is \(4\) copies of \(\frac{1}{24}\,\text{mile}\). 2. So \(n=4\) because \(4\times\frac{1}{24}=\frac{1}{6}\). 3. The matching division equation is \(\frac{1}{6}\div4=\frac{1}{24}\). The number \(4\) is the number of equal pieces.

\(n=4\). Since \(4\times\frac{1}{24}=\frac{1}{6}\), the matching division equation is \(\frac{1}{6}\div4=\frac{1}{24}\). The \(4\) tells how many equal pieces were made.

- Identify which quantity is the total amount being shared. - Identify which number tells how many equal shares are made. - Put those roles in the correct order in the division expression.

1. The total amount being shared is \(\frac{1}{9}\) pound, so it must be the dividend. 2. The \(4\) represents the number of equal shares, so it is the divisor. 3. The correct equation is \(\frac{1}{9}\div4=\frac{1}{36}\). 4. Each person receives \(\frac{1}{36}\) pound.

Eva reversed the quantities: \(\frac{1}{9}\) pound is the total being shared, so it is the dividend, and \(4\) is the number of equal shares, so it is the divisor. The correct equation is \(\frac{1}{9}\div4=\frac{1}{36}\), so each person gets \(\frac{1}{36}\) pound.

- Decide whether each story is finding a share size or counting groups of a known size. - Put the total amount in the dividend position. - Check that the quotient’s unit matches what the story asks for.

1. Story a) is equal sharing: \(\frac{1}{8}\div6=\frac{1}{48}\). The quotient is the number of gallons in each jar. 2. Story b) counts jars of a fixed size: \(6\div\frac{1}{8}=48\). The quotient is the number of jars. 3. The order changes because the whole number is a group count in a) but the total amount in b).

a) \(\frac{1}{8}\div6=\frac{1}{48}\) gallon per jar. b) \(6\div\frac{1}{8}=48\) jars.