## Fractional Clothesline

• Lesson
6-8
1

In this lesson, a string will be stretched across the classroom and various points will be marked for 0, 1, 2, 3, and 4. This classroom number line will be used to show that all proper fractions are grouped between 0 and 1, and that improper fractions or mixed numbers are all grouped above 1. Students clip index cards with various proper fractions, improper fractions, and mixed numbers on the clothesline to visually see groupings. Students then play an estimation game with groups using the same principle. Encouraging students to look at fractions in various ways will help foster their conceptual fraction sense.

Preparing for the Lesson

Several practice fractions have been provided on the second page of the Fractional Clothesline activity sheet. Create a class set of the fractions by gluing each fraction to an index card. You may want to adapt and use different fractions that your class has been struggling with or that you wish to add to extend your number line.

 Fractional Clothesline Activity Sheet

Additionally, create index cards for the number 0, 1, 2, 3, and 4, and stretch a string across the length of your classroom, attaching the ends to opposite walls with tape.

Student Number Line

Students should work in pairs or groups to allow for discussions. Pass out the 0, 1, 2, 3, and 4 index cards to any 5 students in your class. Have those students place their cards on the clothesline. Each of these whole numbers should be spaced an equal distance apart. As students place them, one or more of the cards may need to be adjusted to appear equally spaced. As a class, discuss whether the cards were spaced evenly and are agreed upon. The string and number line will cover the length of your classroom and should have several feet between the whole numbers. This will allow for other cards, representing fractions, to be placed on the number line.

Distribute the prepared fraction cards (one per student/pair/group). Once all the students have a card, tell them to arrange themselves in number line order, similar to the string. They will need to be in order from least to greatest, according to the value on their card. During this part, students will have to apply various strategies to compare and order themselves. You may remind students of strategies you’ve used to compare fractions. Several of the cards do not have common denominators and students will have to decide which is larger through common numerators, or based on decimal equivalents. Some students may organize themselves realizing that 3/4 is actually three 1/4's and figure out where to be. You may also have students use a calculator to verify their answers. Question students to help them explore their thinking. This portion of the lesson can be done quickly or, if you wish, extended to incorporate maximum comparison strategies, depending on students' experience with multiple fractions with different denominators.

In their groups, have students discuss briefly where the cards should be pinned on the clothesline. Once the groups have come to some conclusions, have the members of one group come up to the clothesline and pin up their index cards with paper clips or clothespins. For students who have equivalent numbers (e.g., an improper fraction that is equal to a mixed number), clip the cards together in a column so they are both in the same spot and still visible. Once all cards have been placed, have a whole-class discussion to decide as a group if any cards need to be moved, changed, or adjusted. Ask the class, “Do these cards look like they are in the proper spot? Are there any cards that need to be moved?”

As students are placing the cards, there are several ways to compare fractions. You can point out that fourths are dividing the distance from 0 to 1 into 4 equal lengths. This can also be done for thirds, fifths, etc. Then discuss how 2/3 compares to 3/4 or other fractions by using lengths of wholes instead of the usual circles and squares.

Ask students, "What strategies did you use to decide if a fractional card is closer to 0, 1/2, or a whole number?"

[Each multiple of the denominator is a whole number. If the numerator is close to the next multiple, it’s close to another whole value.]

Repeat this process for each group until all the index cards are placed. It is important to note that as more students begin to place cards, it will be come more obvious when previously placed cards are not in the correct spot. This is where discussion through the process can be invaluable. Lead your students in identifying whether some cards should be moved slightly so that the card currently being added will be placed accurately. Have students continue to adjust the cards on the clothesline until they are reasonable estimates.

Ask students, "What type of fractions are on the numberline that are greater than 1 whole?"

[Mixed numbers are greater than 1 whole and are a way of writing numbers so the whole and the part are seen separately.]

They may also notice that several mixed numbers and improper fractions overlap. It is important to have students use the words equivalent or equal when discussing this.

Clothesline Addition

The Clothesline Addition Game is a way for students to combine different amounts that are represented on the clothesline using estimation. To play the game, distribute the Fractional Clothesline Activity Sheet, and allow time for students to cut out the fraction cards on the page 2. Students should start the game by flipping over 2 cards from a face-down pile of fraction cards, and then estimating what they think the sum of the 2 cards will be. They should place a marker or penny on their sheet that shows where they think the sum is. For example, if they add 1/2 and 13/4, they will place their marker close to 21/4. Remind students to look at the lengths each fraction is represented by on the clothesline and to picture those lengths being placed end to end to create the sum. Once they agree on the answers, they repeat the process. If any students struggle, have them use calculators to double check their estimations. You may have students perform the operations according to a time limit or until the deck runs out of cards. As you move through the classroom, help students find exact answers for themselves if there are any disagreements.

Concluding the Lesson

After students have played several rounds of the Clothesline Addition Game, ask them to summarize the lesson and lead them in a discussion about their strategies for the game. Ask probing questions to find out how they thought of adding the fractions together.

Some key questions may include:

• How are you combining fractions with other fractions?
[Some students may see the fractions as lengths on the number line; some may see decimals; some may create equivalent fractions.]
• What strategy are you using when you have to add a mixed number with a proper fraction?
[Some may create improper fractions; some may estimate the fractions and whole numbers separately.]
• How can you tell if a proper fraction is closer to 1 whole or closer to 0?
[The closer the numerator value is to the denominator, the closer to 1 it becomes.]
• In an improper fraction, how can you tell which whole number it is closer to?
[Each multiple of the denominator is a whole number. If the numerator is closer to the next multiple, then it is closer to the next whole value. For example, in 5/4, the numerator is between 4 and 8, two multiples of the denominator. Since it is closer to 4 than to 8, it is closer to the whole number 1 than the whole number 2.]
• Are there any patterns you see in the placement of fractions on the number line?
[All proper fractions are greater than 0 and less than 1. Fractions beyond 1 involve a whole number and are actually mixed numbers or are improper fractions.]

Some students may think of using equivalent fractions while others may think of using lengths on the clothesline. Allow students to express their strategies and the patterns they noticed. It might also prove a useful tool to leave the clothesline up for a while following the lesson as a visible tool in the room.

• String long enough to stretch across classroom
• Index cards
• Paper clips or clothespins
• Calculator (optional)
• Scissors
• Pennies (or other markers)
• Envelopes to hold the cards
• Fractional Clothesline Activity Sheet

Assessments

1. As an entry task the next day, place the original 5 index cards on the clothesline. Give each student an index card and tell them to write a number on their card, that is not a whole number. Have students take turns placing another student's card on the line, discussing if it is in the correct spot.
2. In whole-class discussions, ask students to describe how they discovered their patterns. Encourage and validate a variety of appropriate responses.
3. Have students place the game cards in order from least to greatest on their desks.
4. Write on the board numbers that are not included in the fraction cards. Ask students to mark on their number lines where each number belongs.
5. Have students collaborate to write down situations in real life where they have used the numbers that are on the clothesline and relate how comparing or adding those numbers might apply to those situations.

Extensions

1. As students progress in their experiences, include decimals on the clothesline and in the game cards. As the year progresses, include percentages.
2. Create a clothesline for positive and negative integers. Have students play a game where they add positive and negative integers on a modified clothesline.
3. Place several fractions on the clothesline. Then, have students place the whole numbers.
4. Students can play the Fraction Game.
 Fraction Game

This tool allows students to individually practice working with relationships among fractions and ways of combining fractions. The object of this game is to get all of the markers to the right side of the game board, using as few cards as possible. This activity is a good extension of today's lesson, as it relies on number lines divided into specific fractional values.

Questions for Students

1. What is a fraction?

[Fractions are parts of wholes, whether part of a circle, part of a square, or part of a straight line. There is an amount that represents a whole piece (the denominator) and a fraction is a part (the numerator) of that whole.]

2. What is an improper fraction?

[Improper fractions are fractions with a numerator greater than the denominator. They are more than 3. 1 whole and include fractions of another whole.]

3. What is a mixed number?

[Mixed numbers are fractions that have regrouped the fractions pieces into a whole number and a proper fraction, which is less than another whole piece.]

4. How can you combine 13/4 with 11/2?

[Some students may point out an algorithm and actually add. Others may use decimals or consider adding the lengths of the line segments on the clothesline.]

Teacher Reflection

• Was your lesson developmentally appropriate? If not, what was inappropriate? What would you do to change it to make it more/less challenging?
• How did your lesson address auditory, tactile, and visual learning styles?
• Did you find it necessary to make adjustments while teaching the lesson? If so, what adjustments did you make? Were they effective?
• What worked with classroom behavior management? What didn't work? How would you change what didn’t work?
• What are alternative ways you can have students look at fractions to help their overall conceptual understanding?

### Fraction Game

3-5, 6-8
This applet allows students to individually practice working with relationships among fractions and ways of combining fractions.

### Learning Objectives

Students will:

• Create a number line with fractions, improper fractions, mixed numbers, and integers.
• Use estimation to practice combining various numbers.