Teaching Percentage: Ideas and Suggestions

Teaching Percentage: Ideas and SuggestionsAuthor(s): David J. GlatzerSource: The Arithmetic Teacher, Vol. 31, No. 6 (February 1984), pp. 24-26Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41191004 .

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Teaching Percentage: Ideas and Suggestions

By David J. Glatzer

How much of a tip should I leave for this meal? This everyday question is a challenge for many individuals. The lack of an understanding of the concept of percentage is at the heart of this situation. This article offers ideas and suggestions for teaching the concept of percentage, recognition of patterns, percentage as a special ratio, ways of finding percentages, and fractional percentages.

In extending the initial question about the tip, it is possible to devise the following technique (based on a 5- percent sales tax):

Restaurant check:

Hamburger $ 1.85 French fries .85 Soda .50

$ 3.20 Tax .16

^ $ 3.36 J

Tip (triple the tax) .48 ̂

With a 5-percent tax rate and the assumption of a 15-percent tip, all the individual has to do is triple the tax. Thus, the tip we get is $0.48 (we might round it to $0.50). It is interesting to note that some people are skeptical about this procedure. "Are you sure this always works?" is not an uncom- mon question. Some people use a method involving 10 percent and then half of the resulting dollar amount:

David Glatzer is director of mathematics for grades K-12 in the West Orange School Dis- trict, West Orange, NJ 07052. He has taught middle school and senior high school mathematics. He is active in the Association of Math- ematics Teachers of New Jersey and is the current chairman of NCTM's Regional Serv- ices Committee.

10 percent of $3.20 = $0.32.

15 percent tip = $0.32 + ^($0.32) = $0.32 + $0.16 = $0.48.

Still others solve the problem by doing the calculation the "long way":

$ 3.20 X0.15 1600 320

$0.4800

Again, we get $0.48 for the tip. These three solutions illustrate vari-

ous ways of understanding percentages. The real issue is to teach percentage in a manner that emphasizes the basic concepts and allows students to apply useful patterns.

The Concept of Percent If you ask fourth or fifth graders what scores they received on a ten-question spelling quiz, you are likely to hear responses such as "I got a hundred" or "I got an eighty." These answers show that an intuitive concept of percentage is present before formal introduction of the topic in the curriculum. Textbooks and curriculum guides show that percentage is usually intro- duced in grade six.

The typical textbook presentation has the following objectives: • The concept of percentage • Changing fractions and decimals to percentages

• Changing percentages to decimals or fractions

• Using decimals and fractions to find percentages of numbers

• Finding a percentage

• Solving percentage problems and devising applications

The emphasis seems to be on mechan- ical procedures. Although many exercises are devoted to "conversions," the development generally does not give sufficient attention to the basic concept and to the use of patterns associated with percentages.

Students need to explore and discuss questions such as the following:

1. Can you have 100 percent atten- dance in class?

2. Can you have 200 percent atten- dance in class?

3. Can a price increase 40 percent? 4. Can a price increase 100 percent? 5. Can a price increase 300 percent? 6. Can a price decrease 50 percent? 7. Can a price decrease 200 percent? 8. What does it mean to have 10 per-

cent unemployment? 9. What does it mean to have 10.5%

unemployment? These questions can help students de- velop a real understanding of the concept of percentage. This understanding can be enhanced by instruction that incorporates a spiral approach. Initial class discussions can begin early in the school year; there is no need to confine these introductory concepts to the percentage unit. In addition, simple illustrations can be used before sixth grade.

The concept of percentage can be illustrated through examples found at the local supermarket. Students can collect and discuss illustrations such as the following: 1. Florida orange juice . . . 100 per-

cent pure

24 Arithmetic Teacher



2. Paper towels . . . now 20 percent more absorbent

3. Mixed nuts . . . less than 50 percent peanuts

4. Nose drops . . . ' percent child's strength

Newspapers and magazines are also excellent resources for students to use in learning about percentages. Teach- ers can encourage students to collect illustrations representing different applications: money, sports, nutrition, income tax, inflation, and weather re- ports. In addition to these areas, students can be encouraged to find examples of percentage increases and decreases, fractional percentages (in- cluding percentages of less than 1), and percentages greater than 100.

Recognition of patterns in percentages Mathematics can be considered as the study of patterns. Percentage is a topic in which patterns are ever present. Teachers need to show students that a great many percentage problems are based on patterns that employ easier problems.

For example, without pencil or calculator, quickly answer the following question:

What is 18% of 50?

Although several approaches are possible, how many readers "switched" to "50% of 18" to come up with the answer of 9? We should expose our students to this important pattern:

A%ofB = B% of A

This relationship can be shown through use of proportions or the commutative property for multiplica- tion. It is surprising that textbooks do not present this "neat" relationship.

Other patterns include these:

1. (K • A)% of В = K(A% ofB). Example: 20% of 18 is twice 10% of

18. 2. (A + C)% of В = (A% of B) + (C%

of£). Example: 60% of 42 = 50% of 42 +

10% of 42.

These patterns need to be presented, not because of a desire for extreme

formality, but because the patterns are very powerful devices in solving everyday problems. Of course, emphasis should be placed on patterns associated with place value: 100 percent of a number; 10 percent; 1 percent; 0.1 percent; and 1000 percent.

A close connection exists between an appreciation of patterns and the ability to estimate in doing work with percentages. One effective strategy is to ask students to circle exercises that they can do mentally. Consider the following: 50% of 86 2% of 800 17% of 10 25% of 88 56% of 89 150% of 40 15% of 60 11% of 50 1.5% of 1000

This activity can be repeated several times during the year so that students can show progress in being able to apply patterns and estimation.

Percentage as a special ratio

It is important for teachers to stress percentage as a special ratio involving parts per hundred. The idea of sixty minutes per hour can be related to the idea of five parts per hundred. Fur- thermore, problems involving "equal ratios" need to include more examples with 100 as the "new" denominator. For example:

Find the missing number that will make equal ratios:

3 ? 2 ? 7 ? "Ш=НЮ 5

= КЮ 25~=НЮ

Additional emphasis can be given by asking students to give the ratio ex- pressed as a percentage: 1. 3 red marbles out of a total of 5 marbles Ratio: 3 to 5 (60 percent of the mar-

bles are red) 2. 2 shaded squares out of a total of 8 congruent squares Ratio: 1 to 4 (25 percent of the

squares are shaded) This activity can be done in grades four and five in connection with simple percentages (100 percent, 50 percent, 25 percent, 10 percent).

Special cases

Students often have difficulty with the

case requiring them to find the percentage: 14 is what percentage of 21; 6 is what percentage of 4. Consider the following example:

What percentage of the letters in MISSISSIPPI are Ss?

We can show students that this problem requires a consideration of specif- ic ratios equal to 4/11. This task can be done using a series of MISSISSIP- PIs for which students count the total number of letters and the number of Ss.

Total Ss letters

1 MISSISSIPPI 4 11 2 MISSISSIPPIS 8 22 3 MISSISSIPPIS 12 33

9 MISSISSIPPIS 36 99

Now, students should be asked, "Why is it a good idea to stop at 9 MISSISSIPPIS?" At this point, the idea of looking for an equal ratio with a denominator of 100 (or very close to 100) should be obvious. Of course, the answer is approximately 36 percent.

This activity can lead to consideration of solving these problems with the "proportion" method. At the same time, teachers should show students that this process is the same as finding the decimal equivalent for the fraction and then converting the result to a percentage. Students should also recognize the relationship between ratios and percentages as in the following problem:

Team A has a record of seven wins out of eleven games. Team В has won six out of ten games. Which team has a better record?

Realizing that calculators will be used for typical percentage applications, teachers should stress the need to understand the process of using a calculator to solve a proportion. Fur- thermore, instruction should include work on estimation and ways of checking the reasonableness of results. A good deal of time should be spent on problems requiring no work:

20 is what percentage of 20? 20 is what percentage of 40?

February 1984 25



20 is what percentage of 200? 20 is what percentage of 10?

Fractional percentages Some students experience difficulty in working with fractional percentages and percentages less than 1 percent. With newspaper headlines telling us about a 4'9 percent increase in prices," it is essential for students to have a clear understanding in this area. This topic should be offered within a practical context. Consider the following example of a 6.5 percent rate of unemployment for 1000 work- ers: 6 percent rate would mean 60 people out of work 6.5 percent would mean people out of work 7 percent rate would mean 70 people out of work

Students are often asked to do the

following kinds of problems: 1. Write 0.005 as a percentage. 2. Write 0.3% as a decimal. 3. 6 is what percentage of 800?

Although these exercises are accept- able, the skills should be related to estimation. Students must appreciate the significance of the results. For example, in the third problem, it is important to see the significance of deriving 0.0075 on a calculator. Knowing that 1 percent of 800 is 8 should result in an understanding that the answer is less than 1 percent. Of course, this approach requires contin- uous reinforcement.

Summary Other problems that need to be ad- dressed include percentage of increase or decrease, discounts, and simple and compound interest.

In conclusion, teachers can use a number of strategies in the teaching of percentage. Short- and long-range planning should provide for -

1 . extensive emphasis on concept development and relevance to everyday situations;

2. a spiral approach in which the topics are considered throughout the year;

3. less emphasis on rote procedures in isolation from the concept of percentage;

4. an early introduction to the concept of percentage, with some work in grades four and five;

5. use of several resources to supple- ment the textbook; and

6. emphasis on estimation.

These suggestions may help teachers give students a better understanding of a critical part of the mathematics curriculum, m

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Teaching Percentage: Ideas and Suggestions