Lesson 2
RELATIONSHIP BETWEEN A UNIT FRACTION AND A PROPER FRACTION
A. Objectives
Students will be able to:
 Understand that a unit fraction is one part of a whole that has been equally divided into parts.
 Understand that a proper fraction is one or more parts of a whole that is equal to less than one whole.
 Understand the relationship between a unit fraction and a proper fraction.
 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.
Students should use equal division to find out how many 1/4's are needed to make 3/4. Provide a 1meter 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.
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.
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."
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.
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 equalsize parts.)
 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.