Expressing equally divided quantities as fractions: Lesson 1
Comparing fractions: Lesson 3-4
Number lines and sequencing of fractions: Lesson 5
Adding and subtracting fractions with the same denominator: Lesson 6-7
Practice: Lesson 8
 

Lesson 2

RELATIONSHIP BETWEEN A UNIT FRACTION AND A PROPER FRACTION

A. Objectives

Students will be able to:

  1. Understand that a unit fraction is one part of a whole that has been equally divided into parts.
  2. Understand that a proper fraction is one or more parts of a whole that is equal to less than one whole.
  3. Understand the relationship between a unit fraction and a proper fraction.
  4. Know how many unit fractions are needed to make a given proper fraction.

B. Instructional Content and Activities

Relationship between a unit fraction and a proper fraction

Understanding the relationship between a unit fraction and a proper fraction is important for a better understanding of fractions. The main focus of this lesson is understanding how many unit fractions are needed to make a given proper fraction, and expressing that fact as a fraction.

First, provide equally divided figures as shown below and express the shaded areas as fractions.

Example 1

Students should use equal division to find out how many 1/4's are needed to make 3/4. Provide a 1-meter paper tape and let them divide it to 4 equal parts. Talk about one part, its relationship to the whole, and how to express that relationship as a fraction (what number over what number).

One part is 1/4 of the whole tape and is therefore 1/4 meter. Point out that

  • 2 parts of 1/4 meter each is the same as 2/4 meter
  • 3 parts of 1/4 meter is 3/4 meter
  • 4 parts of 1/4 meter can be expressed as 4/4 meter

In other words, starting with 1/4 meter as a unit, students can discover that 2, 3, and 4 times 1/4 meter is the same as 2/4 meter, 3/4 meter, and 4/4 meter, respectively, through such activities. Help them express and recognize the relationship between a unit fraction and a proper fraction from the previous fact.

Two 1/4's is 2/4 so 2/4 is 2 times 1/4
Three 1/4's is 3/4 3/4 is 3 times 1/4
Four 1/4's is 4/4 4/4 is 4 times 1/4

Relationship between a proper fraction and a unit fraction

Use three equally divided continuous measures as shown below to study the relationship between a proper fraction and a unit fraction.

171

First, have students figure out how to express the shaded area of the first bar (what number over what number) in relation to the whole bar.

Example 2

Get them to express their understanding through a statement such as the following: "One of 6 equally divided parts is 1 over 6 or 1/6."

Example 3 Example 4

Do overlapping activities with the other two figures to help students discover the relationship of each shaded area (2/6 and 4/6) to the starting unit 1/6. Then, ask the following questions:

  • What is 2 times 1/6?
  • How many 1/6's does it take to make 4/6?

Have students solve the following problems in a similar way. Provide circles equally divided into 8 parts and get them to express the shaded areas as fractions.

Example 5

One of 8 equal pieces is 1/8 of a whole. Have students do overlapping activities to find out how many 1/8's are in 3/8, 5/8, and 6/8, and also what is 3 times 1/8, 5 times 1/8, and 6 times 1/8.

C. Teaching Tips

  • When dividing a measure and expressing it as a fraction, students may make errors in the dividing process, as in the example below. ("Slicing" the circle in equal widths does not result in equal-size parts.)

    Example 6
  • Complete understanding of equal division is necessary for accurate understanding of a fraction.
  • When teaching that 3/4 is the same as three 1/4's, provide a concrete example of an equally divided object, do overlapping activities, and make sure that students understand that 3/4 is 3 times 1/4.
Copyright © 2007 by Janice Grow-Maienza