Comparing Fractions

A student's first introduction to fractions might not offer much opportunity to represent and compare fractional quantities. Typically, the presentation of fractional concepts is limited to pie diagrams. That is, students write a fraction such as 1/4 and then draw the pie diagram for it. This is done one fraction at a time. It is difficult for students to see the effects of a simple process such as dividing fractions by 2 (or multiplying by 1/2) if you use just paper and pencils.

This lesson enables the teacher and students to explore different fractional representations using the spreadsheet. The spreadsheet can be used to calculate fractions and to represent them using pie diagrams. The advantage of an exercise like this is that students can visualize comparisons (e.g., 3/4 vs. 7/8) or see the effects of a pattern (e.g., repeated division of a fraction).

Classroom Discussion and Activities

Computer Lab Activities

Classroom Discussion and Activities

• Comparing Fractions

 

Write a pair of basic fractions on the board such as 3/4 and 7/8. Ask the students which fraction is bigger and discuss how a drawing might help them visualize the difference. Be sure to discuss the fact that just because a denominator is bigger, it doesn't mean that the fraction is bigger. In fact, fractions are tricky in that way. We need to think carefully about fractions before we decide which one is bigger or smaller. Drawings can help our understanding of this.

 

It is also important to see how a fraction relates to its "complement"-together they make one whole. Thus, 1/4 and 3/4 are complements because they add up to one whole. (This idea must be understood in order to make pie charts in the spreadsheet.)

 

• The Effects of Patterns

 

Another important idea is the effect of a pattern on a fraction. What happens if we continually divide a fraction by a number? What happens if we divide 1/2 by 2 and then divide that number by 2 and so on? What happens to the size of the fraction? How is that expressed in fractional numbers? (In this case, the size of the denominator continually doubles.)

 

• Building Tables

We can build tables for our fractions (use Grid Worksheets). In the table below, students can practice writing the fraction, its complement, and the total.

 

In this table, the students can practice writing the formula for determining the complement and then copy that formula to the right. They can also show how they would create a pattern. In the bottom of the table below, they will divide the fraction by 2.

 

• Algorithms for Dividing Fractions (optional)

This algorithm will only make sense if students are comfortable in applying different operations to fractions. If you are just introducing fractions, these algorithms may be confusing. Keep them in mind when you teach division of fractions.

 

Computer Lab Activities

• Set up the Spreadsheet

To prevent confusion, students should start a new spreadsheet file by first converting all of the cells to fractions. This can be done by using the FORMAT and then CELLS option. Then select NUMBER and then FRACTION. Be sure that they have two place values for the fractions (select the option that looks like this: # ??/??).

 

• Enter Data from Classroom Worksheets

Have students enter all of the data from their worksheets. Make sure to monitor how well they enter the formulas. The file compfrac.xls shows a model of a file. The first worksheet shows how two fractions are compared. The second worksheet in the workbook labeled "Patterns" shows the effect of a fraction divided by 2. Be sure to limit how many times you divide. Excel will limit fractions to denominators with two place values. In this case, you can't go beyond 1/64th. You also can't represent fractions very well with a pie diagram if they are smaller than 1/64.

 

Once the formulas are created, students can instantly see the effects of using different fractions. The document compfrac.xls uses labels from the table of fractions. By making the labels equal to the fraction shown in the pie chart, they change automatically if you change the first fraction in the table. Just try it. Change the 1/2 in the first fraction cell to 1/3. You will notice that all of the labels and all of the pie diagrams change automatically. This illustrates the effect of dividing by 2 on a number of different fractions at once. The changing labels also give you the opportunity to discuss the pattern in the denominators.

 

• Discuss the Comparisons and Patterns

You may want to do these exercises as a group. That way you can discuss the effects of representing fractions and making changes.

 

• Make Transparencies for Class

You may want to print a few of the pie charts for discussion in class. Make transparencies from the printed copies. They can be used as overlays if each chart is the same size.