### Fractions and Ribbons

 •  CC.2.1.5.C.2 - Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

 •  M05.A-F.2 - Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

 •  M05.A-F.2.1.1 - Solve word problems involving division of whole numbers leading to answers in the form of fractions (including mixed numbers). •  M05.A-F.2.1.2 - Multiply a fraction (including mixed numbers) by a fraction. •  M05.A-F.2.1.4 - Divide unit fractions by whole numbers and whole numbers by unit fractions.

• 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.

Students extend their understanding of evaluating division expressions involving fractions to problem solving. Students will:

• model the solution of problems using both physical models and symbolic representations.
• generate problems that involve cutting and measuring ribbons for different situations and generalize their solution from the ribbon problem to solve additional problems.

• How is mathematics used to quantify, compare, represent, and model numbers?
• How are relationships represented mathematically?
• What makes a tool and/or strategy appropriate for a given task?

60–90 minutes

Say, “Today we are going to see how fractions can be used to solve problems that we may find in our everyday lives. We have practiced how to multiply fractions using paper folding and how to divide fractions using manipulatives and visual models. Now, let’s apply these techniques to real-world problems. We will create the solutions to problems with physical models or manipulatives and then record the solutions in symbolic form.”

“Today we are going to perform a ribbon-cutting task. We have 68 feet of ribbon, and we need to cut equal lengths of ribbon for a balloon display. Each ribbon that will be tied to a balloon has to be  feet long. How many ribbons can we cut from 68 feet?”

“I have 68 inches of ribbon.” (Hold up the ribbon or string for students to see.) “Even though it is not 68 feet, we can still use this physical model to help us. After we complete the process of solving the problem, we can change the units from inches to feet. Looking at this length of 68 inches, how many -inch pieces of ribbon do you think I can cut?” Accept possible guesses.

“I start with a total of 68 inches, and I know how many inches are in each group: . This is a grouping type division problem. I will take my ruler, measure  inches and cut; then I will measure another  inches and cut. I still have more ribbon left, so I will measure  inches and cut again. I will repeat this process until I have one piece left. I will measure it to be sure it is  inches long.” Care should be taken to accurately measure and cut the ribbon.

“I cut eight pieces of ribbon that are each  inches long. I remember that the original problem said I had 68 feet, so I can now answer the question: Each piece of ribbon cut for the balloons was  feet long. We can write this as an equation: 68 ÷  = 8 ribbons.”

“Here is another example. Michelle is creating bookmarks for the local arts and crafts fair. She has 72 inches of ribbon to make some bookmarks. Each bookmark is to be  inches in length. How many bookmarks can she make using the ribbon she has? I am going to give you and a partner 72 inches of string. This will represent the 72 inches of ribbon Michelle has. Your task will be to cut lengths of  inches to find out how many bookmarks Michelle can make. Measure carefully!” The following questions can be used to provide formative assessment and feedback:

• “What operation would you use to solve this problem?” (Division, because you are breaking up a whole into equal parts; or multiplication, because .)
This is an opportunity to revisit the concept of fact families.
• “What is the total number of inches of ribbon? Is this the dividend, divisor, 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 are not dealing with whole numbers. You have parts, and you have to be careful that you are measuring accurately, although knowing that adding  and  equals 1 might 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 may have extra.)
• “How is using a physical model helpful?” (You can visualize what happens when fractions are involved. You do not necessarily have to do the computation with numbers. You can “see” what the answer is when you are done. Unfortunately, a physical model might not always be practical.)

Station Rotation

Create workstations to show students different types of problems that would require the division of fractions. Use Station Cards so students understand the task at each station (M-5-2-3_Station Rotation Task Cards.docx). Also, be sure enough materials are available at each station for each group. Divide students into four groups. Give each group a Station Rotation Four Square (M-5-2-3_Station Rotation Four Square and KEY.docx). Allow groups to work at each station for approximately seven to ten minutes. Any student who is experiencing difficulty can visit Station 4 and receive the necessary help from the teacher and then return to the rotation when ready.

Before students begin, remind them that the focus of the lesson is to understand division problems involving fractions. Each station will have a physical model students can use to help solve the problem at that station. After students work through the problem using the physical model, they are to transfer the process symbolically to the Station Rotation Four Square  and record the answer. Be sure each student is given a Station Rotation Four Square to use as a recording sheet throughout the activity.

Station 1

“Sam was baking several batches of chocolate fudge for a bake sale at school. He only had four cups of confectioner’s sugar left. Each batch of chocolate fudge needed  cup of confectioner’s sugar. How many batches of fudge could Sam make with this amount of confectioner’s sugar?” ()

Station 2

“Miranda was sewing pillows for her new couch. Each pillow required  yard of fabric. She had  yards of fabric. How many pillows could she make?” ()

Station 3

“Chris wanted to feed his newly planted vegetables. The directions said to mix  fluid ounces of plant food per gallon of water. There were 18 fluid ounces of liquid plant food in the bottle. How many gallons of water are necessary to use all of the plant food?” ()

Station 4

At this station, students will meet with you to review any complications they may be experiencing and to have any questions answered. Students can be working on the fourth square of the Station Rotation Four Square (M-5-2-3_Station Rotation Four Square and KEY.docx) while you are giving an informal assessment by asking the following questions related to the problems at each station:

• How were the problems similar?” (All used division of fractions.)
• “How do you know what information is important in a problem?” (The important information is the amount of the whole and the number of groups needed.)
• “What clues told you that you had to divide?” (Splitting 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 makes a larger number.)
• “What physical model could you use to help solve the problem you just created?” (Draw a picture, do paper-folding, use linking cubes, etc.)
• “Can you come up with a word problem that would require the division of fractions?”

Remind students that sometimes deciding whether you should divide or multiply to solve a word problem can be tricky. Because this lesson is about dividing fractions in problem-solving situations, knowing when to divide is important. Students should be able to distinguish on their own what type of operation can be used to solve a story problem. In small groups, give each group a set of problem-solving cards (M-5-2-3_Problem-Solving Cards and KEY.docx).

Explain to the groups that their task will be to sort the problem-solving cards according to the operations they can use to solve the story problems. They do not need to solve the problems. Then, as a group, students will use chart paper to generalize their understanding of when to use multiplication and when to use division in solving story problems. Visit each group and assess student performance and interaction. Students will share their ideas with the rest of the class. Identify misconceptions and clarify when necessary.

Here are some possible ideas for generalizations about when to multiply and when to divide:

 When to Multiply When to Divide When calculating area When increasing by a factor of a certain amount When finding the whole using many equal parts When sharing equally When putting into groups When figuring how many times a container needs to be filled, if you know the total amount of material and size of the container, When determining equal intervals When figuring the whole and know some of the parts

Ask students if they can generalize how using related expressions can help them look at story problems as both division and multiplication problems. Guide student understanding with examples if necessary. Have students write the corresponding division expression and the corresponding multiplication expression for each story problem on the problem-solving cards. Collect the problem-solving cards.

“Each student group will now generate a story problem that requires division of fractions. Your group will record the story problem on a piece of chart paper. Be sure your story problem relates to a real-world scenario. On the back of the chart paper, you will show how could you solve the story problem and write the corresponding division expression and the corresponding multiplication expression.”

Assess student understanding using the story problems students posted on their chart paper. Provide feedback and clarify any misconceptions. Hold a mini-conference with each group to discuss these targets of learning:

• “How is using a physical model helpful when dividing fractions?” (It helps you understand what is really 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?” (Take the number of whole parts divided by the number of groups you need.)
• “How do you know when to divide in problem solving?” (When a whole needs to be split up into equal groups.)

Extension:

• Routine: Emphasize proper use of vocabulary in lessons and classroom discussions. Allow students to work with partners or in small groups during some activities. Use warm-up or review activities, such as the one that follows, to reinforce mathematical concepts and check for understanding.

Give students two numbers (including at least one fraction) and have them create a word problem that requires division of fractions. Have students switch problems with a partner, use a physical model to solve their problem, and write an expression in symbolic form. You can quickly assess who may still need additional practice and who has mastered the skill.

• Small Group: Pull students who may need additional practice into a small group. Use additional problems and highlight key words to help students become more proficient. Make various manipulatives available so students can physically do the process needed to solve a word problem. It is important for students to begin to develop a fundamental understanding of the division of fractions.
• Additional Practice:More examples of word problems involving division with fractions follow.
• Awards for a track meet are ribbons that are decorated to indicate first, second, and third place. Each ribbon is made from five inches of material. If two yards of material are available, how many award ribbons can be made? Express the remainder as a fractional part of an award ribbon. (2 yards = 72 inches; )
• Ingredients were purchased to bake batches of cookies. Each bag of sugar contains approximately ten cups. A batch of cookies requires cups of sugar. How many batches of cookies can be made from one bag of sugar? Express the remaining amount of sugar as a fraction of a batch of cookies. (  )
• Amina is in charge of distributing candy to members of the class. A pound of candy contains 100 pieces. A class of 25 shares   pound of candy. How many pieces does each person receive? ()
• At the pet store there are 12 animals. If  pounds of food are required to feed all the animals, and each animal eats the same amount of food, how many pounds of food does each animal receive? ()

Students who do not need additional support can use the problem-solving cards from the lesson. Working with a partner, students should solve each problem. These students do not need to use a physical model but rather could show how to solve the problem with a picture, applying what they learned in the lesson to different 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 where a fraction is being divided by another fraction and the answer has a remainder.
• use a model to show where the remainder came from and explain what it represents.
• try to simplify their answer or divide in another way to get the answer in a form without a remainder and show the mathematical steps and reasoning for their calculations.

If students have trouble thinking of a problem, you can provide one similar to .

• Active Engagement
• Metacognition
• Scaffolding
• Modeling

 W: The focus of the lesson is using fraction multiplication and division in real-world problems. H: Use the ribbon-cutting problem to help draw students into the lesson. This problem and many others require calculations with fractions. E: Let students experience real-world problem solving with fractions by having them participate in a four-square workstation rotation activity. Students will work in groups, so they must work as a team to solve the problem presented at each workstation. R: Use problem-solving cards with groups of students to see whether they are able to decide which operation to choose to solve a fraction problem. E: Students must create a story problem that requires division by fractions. Evaluate these problems to determine if students correctly understand when to use division in a fraction problem. T: Tailor the lesson to meet student needs by using modifications suggested in the Extension section. O: This lesson is designed to further develop students’ ability to solve division problems embedded in word problems. It is important that students have enough practice using physical models as they begin to develop and strengthen the algorithm used for dividing fractions. The problems in this lesson do not require the interpretation of remainders. The expansion activity provides exposure to the concept of remainders in fractional division problems.

• Observe students during the workstation rotation to determine if they understand how to solve a division problem with fractions.
• Hold a group mini-conference regarding the Problems chart to see if students can tell the difference between when to solve a fraction problem with division and when to solve it with multiplication.