Lesson 1
EXPRESSING EQUALLY DIVIDED QUANTITIES AS FRACTIONS
A. Objectives
Students will be able to:
- Equally divide a number of specific objects into groups.
- Represent a part of the whole as a fraction, using an equally divided set of objects.
B. Instructional Content and Activities
Equally dividing an object
In the second grade, students learned to equally divide a continuous measure and to express a part as a fraction. Based on this, the main goal here is to equally divide a discrete measure and to express each part as 1 over the number of equal parts.
First, review what students learned in the second grade. Provide a picture that illustrates the equal division of a continuous measure and let them express the division as a fraction. Remind them that 1/4 is 1 of 4 equally divided parts of a whole. Provide a problem situation by having them fold and cut a sheet of paper into 4 equal parts.
Equally dividing a quantity
Next, get students to think about how to find the number of pieces of paper that is 1/4 of 8 colored papers. ("To find 1/4 of 8 papers, how many equal piles of paper would you need to make?") Be sure they understand that here 1/4 is 1 out of 4 equally divided portions, and that the batch of 8 papers needs to be equally divided into 4 groups. Talk about how to divide 8 items into 4 groups and then practice using actual pieces of paper.
Expressing an equally divided quantity as a fraction
Remind students that a fraction is written as one number over another number. To describe 1 of the 4 equally divided groups they made from the 8 papers, they would write 1 over 4 or 1/4. Since each group has 2 papers, 1/4 of 8 papers is 2 papers.
Finding the value of a fraction through equal division
Help students understand how to find out 1/3 of 12 by equal division. Provide a picture of 12 identical objects, and talk about what they should do to find 1/3 of 12. It is better to use a magnetic board or overhead projector with a specific problem such as the following:
Manuel, Sophie, and Jay had 12 star-shaped cookies, which they shared equally and then ate. This means that Manuel had 1/3 of the 12 cookies. To see how many cookies Manuel ate, find out what number is 1/3 of 12.
The fraction 1/3 means 1 out of 3 equally divided parts, so 12 needs to be equally divided into 3 groups. Have students divide the 12 stars into 3 equal groups.
Each of the 3 equally divided groups is 1/3 of the total. Thus, 1/3 of 12 is the number in 1 group, or 4. So Manuel had 4 of the 12 cookies.
Next, have students figure out 1/6 of 18 by equal division. Provide a picture of 18 identical objects such as the following:
Here, a specific problem condition also can be provided. To figure out 1/6 of 18, students must first realize that 18 needs to be equally divided by 6. That is, they should divide the 18 circles into 6 equal groups. One of 6 groups is 1/6 of the total, and thus 1/6 of 18 is the number in 1 group (3).
Now, get students to figure out 1/5 of 15 by using similar steps with the figure below.
1/5 of 15 is 1 of the 5 equally divided portions in the figure. Since each portion contains 3 sections, students should understand that 1/5 of 15 is 3.
Next have students figure out 1/4 of 20. Provide a problem with a specific situation or set of objects and have students solve it.
C. Teaching Tip
- The material can be considered as a whole, that is, 1 for the division of a continuous measure, but the concept might be hard for that of a discrete measure. Thus, point out that one of the parts obtained from equal division of a discrete measure can be expressed as a fraction, just as for a continuous measure, and understand the fraction concept for a discrete measure. Figuring out some over some of a number, ratio fractions will be studied at fourth grade.