Students will look into instances that include division by whole numbers as repeated subtraction and Students apply their understanding of evaluating division expressions with fractions to problem solving. Students will: 
- model problem solutions utilizing both physical and model representations. 
- create problems involving cutting and measuring ribbons for various contexts, then generalize their answer from the ribbon problem to solve other problems. 

- How can mathematics help to quantify, compare, depict, and model numbers? 
- How are relationships represented mathematically? 
- What makes a tool and/or strategy suitable for a certain task? 

- Equivalent Fractions: Two or more fractions that represent the same amount. 
- Factor: A whole number that divides evenly into another whole number. 
- Fraction: A number expressible in the form a/b where a is a whole number and b is a positive whole number. 
- Numerical Expressions: Any combination of numbers and symbols that represent a mathematical relationship. 
- Unit Fraction: A fraction with a numerator of one.

- ribbon or string (precut to 72 inches for each group and one length of 68 inches)
- scissors
- ruler/yardstick
- three to four large bowls
- confectioner’s sugar
- measuring cups (include either a \(2 \over 3\)- or \(1 \over 3\)- cup)
- string, precut to a length of \(3 {3 \over 4} \) feet
- bucket of water
- tablespoon-measuring spoons
- liquid measuring cups
- copies of Problem-Solving Cards (M-5-2-3_Problem-Solving Cards and KEY)
- copies of Station Rotation Four Square (M-5-2-3_Station Rotation Four Square and KEY)
- copies of Station Rotation Task Cards (M-5-2-3_Station Rotation Task Cards)

- Examine students' workstation rotations to see if they understand how to solve a division problem using fractions. 
- Hold a group mini-conference on the Problems chart to determine if students can distinguish between when to answer a fraction problem with division and when to solve it using multiplication. 

Scaffolding, Active Engagement, Metacognition, and Modeling 
W: The lesson focuses on applying fraction multiplication and division to solve real-world problems. 
H: Use the ribbon-cutting problem to engage students in the lesson. Many problems, like this one, require fractional computations. 
E: Encourage students to solve real-world problems using fractions with a four-square workstation rotation activity. Students will work in groups, therefore they must work together to answer the problems offered at each workstation. 
R: Use problem-solving cards with groups to assess students' ability to choose the appropriate operation to solve fraction problems. 
E: Students must create a story problem involving fractional division. Evaluate these problems to see if students understand correctly when to divide in a fraction problem. 
T: Modify the lesson to match student needs as described in the Extension section. 
O: This lesson aims to improve students' abilities to answer division problems embedded in word problems. It is critical that students gain enough experience with physical models as they begin to create and enhance the method for dividing fractions. This lesson's problems do not need the interpretation of remainders. The expansion activity introduces participants to the concept of remainders in fractional division problems. 

Say, "Today, we'll look at how fractions can be used to solve problems in our daily lives." We worked multiplying fractions with paper folding and dividing fractions with manipulatives and visual models. Now, let's apply these methods to real-world problems. We will solve problems using physical models or manipulatives, then record the solutions in symbolic form." 

"Today, we are going to cut a ribbon. We have 68 feet of ribbon and need to cut equal lengths for a balloon display. Each ribbon that will be connected to a balloon must be \(8 {1 \over 2} \) feet long. How many ribbons can we cut from 68 feet?"

"I have 68 inches of ribbon." (Hold up the ribbon or string so that students may see it.) "Even though it isn't 68 feet, we can still use this physical model to help us. After we have solved the problem, we can change the units from inches to feet. How many -inch lengths of ribbon can I cut from this 68-inch length?" Accept possible guesses.

"I begin with a total of 68 inches and know how many inches are in each group: \(8 {1 \over 2} \). This is a grouping type division problem. I'll use my ruler to measure \(8 {1 \over 2} \) inches and cut, then measure another \(8 {1 \over 2} \) inches and cut. I still have some ribbon left, so I'll measure \(8 {1 \over 2} \) inches and cut again. This process will be repeated until there is just one piece left. "I'll measure it to ensure it's \(8 {1 \over 2} \) inches long." The ribbon should be correctly measured and cut.



"I cut eight pieces of ribbon that are each \(8 {1 \over 2} \) inches long. I remember the original challenge said that I had 68 feet, thus I can now answer the question: Each piece of ribbon cut for the balloons measured \(8 {1 \over 2} \) feet. This equation may be written as: 68 ÷ \(8 {1 \over 2} \) = 8 ribbons.”

"Here's another example. Michelle is making bookmarks for the local art and craft festival. She has 72 inches of ribbon to make bookmarks. Each bookmark is to be \(4 {1 \over 2} \) inches long. How many bookmarks can she make using the ribbon she has? I'm going to offer you and a partner 72 inches of string. This will represent Michelle's 72-inch ribbon. Your task will be to cut lengths of \(4 {1 \over 2} \) inches to determine how many bookmarks Michelle can create. Measure carefully!" The questions below can be used to provide formative assessment and feedback:

"What operation would you use to solve this problem?" (Division, as you are dividing up a whole into equal parts; or multiplication, because \(4 {1 \over 2} \) × ___ = 72.)
This presents an opportunity to reexamine the concept of fact families. 
"How many inches of ribbon are there altogether? Is this the dividend, division, or quotient?" (72 inches, dividend) 
"How do you think having a fraction in the problem affects the process to use when dividing?" (It makes the problem "messier" because you aren't dealing with whole numbers. You have parts, and you must ensure that you are measuring accurately, although knowing that adding \(1 \over 2\) and \(1 \over 2\) equals 1 may be helpful.) 
"What is your estimate for the number of bookmarks Michelle can make?" 
"What if you have leftover ribbon?" (Students may have measured incorrectly or had excess.)
"How is using a physical model helpful?" (You can see what happens when fractions are present. You are not required to perform the computation with numbers. When you finish, you will be able to "see" the answer. Unfortunately, a physical model may not always be practical.)

Station Rotation

Create workstations to demonstrate to students the various types of problems that involve fraction division. Use Station Cards to help students comprehend the tasks at each station (M-5-2-3_Station Rotation Task Cards). Also, ensure that each station has enough items for each group. Divide students into four groups. Give each group a Station Rotation Four Square (M-5-2-3_Station Rotation Four Square with KEY). Allow groups to work at each station for around seven to ten minutes. Any student who is having trouble can attend Station 4 and receive support from the teacher before returning to the rotation when ready.

Before students begin, remind them that the goal of the lesson is to comprehend division problems involving fractions. Each station will feature a physical model that students can use to help with the difficulty at that point. After working through the problem with the physical model, students must transfer the procedure symbolically to the Station Rotation Four Square and record the result. Make sure each student receives a Station Rotation Four Square to use as a recording sheet throughout the activity.

Station 1

"Sam was making several batches of chocolate fudge for a bake sale at school. He had only four cups of confectioner's sugar remaining. Each batch of chocolate fudge required \(2 \over 3\) cup of confectioner's sugar. "How many batches of fudge could Sam make with this amount of confectioner's sugar?" (4 ÷ \(2 \over 3\) = 6)

Station 2

"Miranda was sewing pillows for her new sofa. Every pillow requires \(3 \over 4\) yard of fabric. She had \(3 {3 \over 4} \) yards of fabric. "How many pillows can she make?" (\(3 {3 \over 4} \) ÷ \(3 \over 4\) = 5)

Station 3

"Chris intended to feed his newly planted vegetables. The instructions advised to mix \(1 {1 \over 2} \) fluid ounces of plant food per gallon of water. The bottle contained 18 fluid ounces of liquid plant food. "How many gallons of water are required to use all of the plant food?" (18 ÷ \(1 {1 \over 2} \) = 12)

Station 4

At this station, students will meet with you to discuss any issues they may be having and to ask any questions. Students can complete the fourth square of the Station Rotation Four Square (M-5-2-3_Station Rotation Four Square and KEY) while you conduct an informal assessment by answering the following questions about the problems at each station: 

"How were the problems similar?" (They all used fractional division.) 
"How do you know what information is important in a problem?" (The most significant information is the total amount and the number of groups required.) 
"What clues told you that you had to divide?" (Dividing a whole into equal parts.)
"What patterns are you starting to notice about division of fractions with problem solving?" (Dividing a whole number by a fraction results in a larger number.) 
"What physical model could you use to help solve the problem you just created?" (Draw a picture, fold some paper, use linking cubes, etc.) 
"Can you come up with a word problem that would require the division of fractions?"
Remind students that it can be difficult to decide whether to divide or multiply while solving a word problem. Knowing when to divide fractions in problem-solving situations is critical because this lesson is about dividing fractions. Students should be able to determine on their own which operation can be utilized to solve a story problem. Divide the problem-solving cards (M-5-2-3_Problem-Solving Cards with KEY) among the small groups. 

Explain to the groups that their task is to sort the problem-solving cards by the operations they can use to solve the story problems. They do not have to solve the problems. Then, in a group, students will utilize chart paper to generalize their understanding of when to use multiplication and when to use division in solving story problems. Visit each group and evaluate students' performance and interaction. Students will present their thoughts to the rest of the class. Identify misconceptions and clarify as needed. 

Here are some possible generalizations regarding when to multiply and divide:

Ask students to explain how utilizing related expressions can help them see story problems as division and multiplication problems. Use examples to help students understand. Students should write the corresponding division and multiplication expressions for each story problem on the problem-solving cards. Collect the problem-solving cards. 

"Now, each student group will create a story problem requiring fractional division. Your group will write the story problem on a piece of chart paper. Make sure the story problem is relevant to a real-world circumstance. On the back of the chart paper, you will demonstrate how you solved the story problem and write the associated division and multiplication expressions.”

Examine student comprehension using the story problems they placed on their chart paper. Provide feedback and clear any misconceptions. Hold a mini-conference with each group to discuss the following learning targets: 

"How is using a physical model helpful when dividing fractions?" (It explains what's going on with the fractions and why the quotients are what they are.) 
"How do you write an expression in symbolic form to show the division problem solved using a physical model?" (Divide the number of whole parts by the number of groups required.) 
"How do you know when to divide in problem solving?" (When a whole needs to be divided into equal parts.)

Extension: 

Routine: Emphasize the right use of vocabulary in lessons and classroom discussions. Allow students to collaborate with partners or small groups on various activities. Use warm-up or review tasks, such as the one below, to reinforce mathematical concepts and assess understanding. 
Give students two numbers (at least one of which is a fraction) and ask them to build a word problem involving fraction division. Students should switch problems with a partner, utilize a physical model to solve their problem, and develop a symbolic statement. You can instantly determine who may need further practice and who has mastered the skill.

Small Group: Gather students who may require further practice into a small group. To help students improve their skills, provide additional problems and highlight significant words. Make numerous manipulatives available so that students can physically perform the steps required to solve a word problem. Students should begin to build a fundamental understanding of the division of fractions. 

Additional Practice: Additional examples of word problems utilizing fractional division are shown below. 
Ribbons designed to represent first, second, and third place are used as awards at track meets. Each ribbon is made of five inches of material. With two yards of material, how many award ribbons can be made? Express the remainder as a fraction of an award ribbon. (2 yards = 72 inches; 72 ÷ 5 = \(14 {2 \over 5} \))
Ingredients were obtained for making batches of cookies. Each bag of sugar holds roughly ten cups. A batch of cookies requires \(2 {1 \over 3} \) cups of sugar. How many batches of cookies can be prepared using a single bag of sugar? Express the remaining sugar as a fraction of a batch of cookies. (10 ÷ \(2 {1 \over 3} \) = \(4 {2 \over 7} \))
Amina is in charge of providing candy to classmates. Each pound of candy has 100 pieces. A class of 25 people splits a \(1 \over 2\) pound of candy. How many pieces does each person receive? (100 × \(1 \over 2\) = 50 pieces of candy; 50 ÷ 25 = 2)
The pet store has 12 animals. If \(2 {1 \over 2} \) pounds of food are required to feed all of the animals, and each animal consumes the same quantity of food, how much food does each animal receive? (\(2 {1 \over 2} \) ÷ 12 = \(5 \over 24\)) 
Students who do not require additional support may utilize the problem-solving cards from the lesson. Students should answer each problem by working with a partner. These students do not need to use a physical model; instead, they can demonstrate how to solve the problem using a graphic, applying what they learned in the class to various problems. 

Expansion: "Sometimes when we divide using fractions, we end up with a remainder just as when we divide using whole numbers." Ask pairs of students to:
create a division problem in which one fraction is divided by another, and the result has a remainder. 
use a model to illustrate where the remainder comes from and what it signifies. 
try to simplify or divide their answer in a different method to get the result without a remainder, and demonstrate the mathematical steps and logic behind their calculations. 
If students struggle to conceive of a problem, consider providing one similar to \(1 \over 2\) ÷ \(1 \over 3\).