This lesson introduces students to the concept of fractions and how they can be compared. (Mastery is not expected until grade five.) Students will: 
- to compare fractions, use concrete models. 
- compare fractions using benchmarks. 
- order fractions using concrete models, benchmarks, and parallel number lines. 
- investigate fair-sharing situations involving unit fractions.

- How are relationships represented mathematically?
- How can mathematics help us communicate more effectively?
- How can mathematics help to quantify, compare, depict, and model numbers?
- What does it mean to analyze and estimate numerical quantities?
- What makes a tool and/or strategy suitable for a certain task?

- Denominator: In a fraction, the number or quantity below the fraction bar. Tells the number of equal parts into which a whole is divided. 
- Fraction: Notation used to represent part of a whole or part of a group by telling the number of equal parts in the whole (denominator), and the number of those parts being described (numerator). 
- Inequality: A mathematical sentence that contains a symbol (<. >, ≤ , ≥, or ≠) in which the terms on either side of the symbol are unequal. 
- Mixed Number: The sum of a whole number and a fraction. 
- Numerator: In a fraction, the number or quantity above the fraction bar. Tells the number of parts of a whole being described. 
- Unit Fraction: A fraction with a numerator of 1.

- fraction circles
- chips
- overhead fraction circles
- paper strips to fold (1 inch by 11 inches)
- parallel fraction lines
- Parallel Number Line worksheet (M-4-3-1_Parallel Number Line Sheet)
- Fair Share Problems worksheet (M-4-3-1_Fair Share Problems and KEY)
- number cubes
- fraction flash cards
- Lesson 1 Assessment (M-4-3-1_Lesson 1 Assessment and KEY)
- Fraction Circle Template (M-4-3-1_Fraction Circle Template)
- Fraction Strip Template (M-4-3-1_Fraction Strip Template)
- Expansion Resource worksheet (M-4-3-1_Expansion Resource Sheet)

- Your observations throughout lesson activities and classroom discussions will aid in determining student proficiency in comparing fractions. 
- Have students finish the Lesson 1 Assessment. 
- Use the comparisons made during the Small Group Extension assignment to evaluate student progress. 

Scaffolding, Active Engagement, Modeling, and Formative Assessment 
W: Students will review fraction concepts and vocabulary. 
H: Students use manipulatives to compare fractions and determine which one is greater. 
E: Students analyze benchmark fractions, including 0, \(1 \over 2\) , and 1. 
R: During the benchmark fraction task, students evaluate what they learned and identify patterns for fraction comparisons. 
E: Students' understanding is assessed through conversations, questions, and the Lesson 1 Assessment. 
T: The lesson can be modified to match the needs of different students. Students who need more experience can benefit from comparing fractions with the same denominator. Students can take the concept of comparing fractions a step further by using improper fractions. 
O: Students use several fraction manipulatives to compare and order fractions. 

Note: Blank templates for fraction strips and fraction circles are available if required (M-4-3-1_Fraction Circle Template and M-4-3-1_Fraction Strip Template). 

Say to the class, "Let's go over some of what we know about fractions." A fraction represents part of a whole unit or a collection of objects. Fractional parts of a whole unit must be equal in size." 

Draw the following images on the board.


Write the following on the board.

Ask a student to name the fraction for the shaded part of each circle.

Begin by asking "Which is more?" questions using the samples on the board. Note that figures A and C demonstrate two representations of one-half that are equal, despite the fact that the numerator and denominator in figure C are bigger than those in figure A. Students should also explain in their own words how they know that figure B represents a fraction that is less than the others, as well as what it would take to make it equal to or greater than half. Encourage students to show their reasoning with fraction circles, chips, 1-inch color tiles, images, or number lines.

“Which is more, \(5 \over 12\) or \(7 \over 12\) of a dozen eggs?” ( \(7 \over 12\) )

“Which is more, \(1 \over 2\) or \(1 \over 4\) of a pie?” ( \(1 \over 2\) )

“Which is more, \(5 \over 6\) or \(5 \over 12\) of an energy bar?” ( \(5 \over 6\) ) 

“Which is more, \(1 \over 6\) or \(2 \over 3\) of a pizza?" ( \(2 \over 3\) )

Students' replies will provide a quick evaluation of their ability to compare and order fractions. Introduce the word benchmark fractions to students. Explain that benchmark fractions are commonly used fractions. Provide them with the following information: "We will be comparing fractions to 0, \(1 \over 2\) , and 1." 

Students should fold a sheet of paper to form a three-column benchmark chart and record their responses in it.

ANSWER KEY

On the board or overhead transparency, write the following fractions:

Students should analyze each fraction in relation to 0, \(1 \over 2\) , and 1. Set up parallel number lines (M-4-3-1_Parallel Number Line Sheet) or fraction circles for students to use. Ask, 

"Which fractions are closest to 0? 
Which fractions are closest to \(1 \over 2\) ? 
Which fractions are closest to 1? 
Do you notice any patterns among the fractions in each column?”
Assist students in identifying and describing the following patterns among the fractions in each column:

Fractions closest to 0 have numerators that are much smaller than their denominators. 
Fractions close to \(1 \over 2\) have denominators that are approximately twice as great as their numerators, or numerators that are half as great as their denominators. 
Fractions closest to 1 have numerators and denominators that are about equal. 
Write the following groups of fractions on the board:

Have students discuss how to order each group of fractions from least to greatest. Have students use the Parallel Number Line worksheet.

ANSWER KEY: 

When ordering unit fractions from least to greatest, the fraction with the largest denominator has the smallest value.

Note: Make sure students understand that the above pattern only applies when the numerators are all 1. 

Let's say to the students, "Now that you can order fractions, we are going to compare fractions using the symbols <, >, or =." Write and draw the following on the board:

Make sure to show out that the fraction strips are of the same length.



Request a student volunteer to circle \(2 \over 3\) and \(3 \over 8\) on the fraction strips.

Ask the question, "Which fraction is greater?" Ask another student to place the appropriate symbol in the box.

For additional practice, write the following problems on the board (without symbols).



Give students a few minutes to complete the practice problems. Call on students to provide responses and discuss any misunderstandings. 

"Now that we know more about comparing fractions we will look at this problem." Introduce a "fair-share" problem. Students should work in pair with a partner. 

Jenny, Marcus, and Renee bought two one-foot-long submarine sandwiches. They want to distribute the sandwiches equally. How much should each of the three friends receive? Draw a picture of the problem's context on a chart paper. 

Ask students to spend a few minutes discussing with their partners how to divide the two sandwiches equally among three people.

“How much does each friend get?” ( \(1 \over 3\) of each sandwich or \(2 \over 3\) of the total sandwiches)
“How is it possible for three people to share two sandwiches?” (They divided each sandwich by thirds to make 6 parts and took 2 parts each.)

Encourage students to use any items they find useful for comparing fractions (folded paper strips, linked cubes) and to make drawings to help them. It is important that students create a method to demonstrate the problem. Do not offer materials with fractions already labeled. Making and sketching fractional pieces allows students to consider the meaning of fractions. 

For extra practice, have students fill out the Fair Share Problems worksheet (M-4-3-1_Fair Share Problems and KEY). 

When students have finished all of the tasks, discuss the results and clarify any confusion. Then, have them all complete the Lesson 1 Assessment (M-4-3-1_Lesson 1 Assessment and KEY).

Extension:

Routine: Make a set of fraction cards and randomly choose two. Have students choose the greater or lesser fraction of the pair. Repeat the activity on a daily basis or to teach students how to compare fractions throughout the school year. 

Small Group: Divide the class into pairs. Give every student a number cube. Ask them to write a fraction with a 7 denominator and a blank numerator. Have students roll the number cube and use the result as the numerator of their fraction. Encourage students to compare and record every pair of fractions. Repeat the task with other denominators greater than 6.

Expansion: Ask students to determine which of the benchmark fractions, 0, \(1 \over 2\) , or 1, the provided fractions are nearer. Include fractions that are equally between the two benchmarks for students to discuss (for example, \(1 \over 4\) and \(3 \over 4\) are equidistant from \(1 \over 2\) . A resource worksheet is available to assist students in organizing their work (M-4-3-1_Expansion Resource Sheet).
To extend the concept, list various improper fractions and ask students to determine which wholes or halves each is nearer to. For example, \(8 \over 5\) is between 1 and 2, but closer to 2 or \(1 {1 \over 2} \) . (For further examples, use \(13 \over 12\) , \(9 \over 5\) , \(8 \over 6\) ).