Home Chapter 18 Summary

Summary

MAIN IDEAS

The position taken in this chapter is that fraction concepts and probably fraction symbolism are prior ideas to decimals. As a result, the most prominent idea pervading all of the content is that decimals are simply another way to write fractions. Nearly every decimal idea can be related to existing fraction ideas. If students can capitalize on this connection, existing ideas can be expanded rather than having to construct disconnected new concepts.

Decimals: An Alternative Symbol System for Base-Ten Fractions

The connection between fractions and decimals is first made with base-ten fraction models. (That terminology is unique to this book.) The basic approach is three-fold. First, extend familiar fraction concepts through the use of base-ten fraction models. The idea is to familiarize students with tenths, hundredths, and thousandths using models similar to those they have used for common fractions. Second, and somewhat independently, the place-value system is extended to the right. If this concept is successful, there is no "first" place but rather an infinite string of positional values. A piece in any one position makes ten to the right. Finally, these two ideas are combined: fractions modeled with base-ten models and written as 45/100 or 4/10 + 5/100, for example, are translated with the help of a model on the place-value chart, resulting in a decimal form of the fraction.

Characterization of the Decimal Point

In traditional school textbooks, the decimal point is sometimes described as a symbol to separate the whole numbers from the fractions. I have offered a slightly different characterization that, while not in conflict with the separation idea, is perhaps a bit more powerful. The decimal point is described in this book as a symbol that identifies the unit position - the units are always in the position just to the left of the decimal. The large eyes on the smiley-faced decimal "look up" toward the name of the units. This is admittedly quite corny but I have found that it is effective with both kids and teachers. The approach allows one to make sense out of expressions such as \$3.5 billion or 25.2 thousand where the units are not ones or singles but billions or thousands. The idea is applied to the metric system (Figure 18.6) and also to the definition of percent (Figure 18.14). With this characterization of the decimal, to convert units, say from meters to centimeters, the decimal is simply repositioned to "look at" or to designate centimeters as the new unit. No multiplication or division is involved in this type of unit change. A similar approach is applied to percents.

Familiar Fraction/Decimal Equivalents

A serious effort should be made using models and a problem-solving approach to have children connect the common fraction and decimal representations for halves, thirds, fourths, fifths, and eighths. (Tenths and hundredths are much more obvious.) These are referred to as familiar fractions - the ones used most frequently in everyday life. The approach of converting fraction to decimal through division of numerator by denominator is strongly discouraged. I have even avoided using the equivalent fraction concept where 3/4 is first seen as equivalent to 75/100 through a symbolic manipulation and then written as a decimal. Rather, I have encouraged students to find how 3/4 or other familiar fractions can be modeled using a base-ten fraction model. These models can then be translated directly and even physically (in some instances) to a base-ten place value representation or decimal equivalent. (See Figures 18.8 and 18.9.)

This set of the familiar fractions, along with tenths, means that no number between zero and one is more than five hundredths from a familiar fraction. For thinking about percentages, estimation of decimal computation, and mental computation, the use of a familiar fraction is frequently an incredible advantage over the decimal representation.

For example, estimating 74.23% of 145 is difficult, but 3/4 of 145 is close to 3 x 36 or about 108 - mentally and quickly. (The actual product of 0.7423 x 145 is 107.6335.) While you may not always use a fraction equivalent in a mental computation or estimate, the option expands the possibilities considerably. Further, the connection of decimals to familiar fraction concepts enhances our number sense with decimal numeration.

Percents Are Another Name for Hundredths

Capitalizing on the role of the decimal as identifying the units place, allows us to think easily about percent as an alternative name for hundredths. As presented in the text, 0.04, for example, is four hundredths; that is, 4.0 hundredths of a whole or of one unit, although it is never written in the latter form. Therefore, if the term "hundredth" is replaced by the synonym "percent," then 0.04 (whole units) = 4.0 hundredths = 4.0 %. In this way the rule "To change a decimal to a percent, shift the decimal point two places to the right (or multiply by 100)" is completely avoided. Again, I have chosen this approach to maximize mathematical connections instead of introducing new ideas and rules unnecessarily. It is an approach that will be new to your students and will not likely be found in student textbooks.

Percent Problems

It is in solving percent problems that the connection between fraction and decimals proves to be extremely valuable. Every problem involving percents can be seen to be one of the three types of parts-and-whole problems that were explored in Chapter 16. Figure 18.15 illustrates the connection.

Students get bogged down with percent problems due to rules about shifting decimals and trying to decide if they should multiply or divide. The rules overcome intuition and there is no clear way to think through or analyze the situation. While there may be some merit to setting up proportions (covered in the next chapter), that approach is not helpful for mental computation or estimation. The percent problems that students get in school should, for a long time, involve percentages equivalent to the familiar fractions. The other quantities involved should be compatible with the fraction equivalents. All percent problems should be in the most realistic language possible, even taken from directly from newspapers and magazine articles or TV. Most importantly, students should be required to draw models to explain their analysis of the problem. The fraction connection to decimals makes all of this possible.

When the percents are not equivalent to a familiar fraction and/or the other quantities in the exercise are not numerically compatible, then estimates can be made by substituting a close familiar fraction. In that sense, 86% might be "rounded" to 7/8 or 87.5% - a whole new meaning to rounding that was earlier introduced with whole numbers. Exact computations come last and should get the least emphasis.

Decimal Computation

The most important thing we can do with decimal computation is stress estimation. The rule of counting decimal places is completely outdated for a computation such as 73.69 x 2.04. The answer is about 2 x 73 1/2 or 147. If there is a need for a more accurate result, a calculator would certainly accompany that need. For 3.6 x 0.7, the corresponding whole number product, 36 x 7, provides the correct digits: 252. Estimation quickly places the decimal - 2.52.