EXPLORING PATTERNS, RELATIONS, AND FUNCTIONSAuthor(s): Charles P. GeerSource: The Arithmetic Teacher, Vol. 39, No. 9 (MAY 1992), pp. 19-21Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41195344 .

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EXPLORING

PATTERNS, RELOUONS, FUNCTIONS

Charles P. Geer

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teachers use NCTM's Curricu- lum and Evaluation Standards for School Mathematics (1989)

to develop programs that will prepare stu- dents for the twenty-first century, some are discovering that mathematics instruc- tion is going to be very different in the 1990s. Many previous programs placed a heavy emphasis on paper-and-pencil proficiency with computational skills and learning mathematics by memorizing rules. Because of advances in technology, new knowledge about how learning occurs, and the changing needs of business and industry, future programs will focus on mathematics with meaning, problem solv- ing, and higher-level cognitive skills.

In addition to these important changes, several previously neglected topics will receive increased attention during the 1990s. Some of the most interesting and important of these are patterns, relations, and functions. Owing to the widespread occurrence of patterns throughout mathematics, the importance of finding relations among these patterns, and the frequency with which mathematical pat- terns appear in everyday situations, these concepts are a central theme in all math- ematics. Fully to understand patterns, relations, and functions, students in the elementary school need to have a variety of experiences with these concepts and see their many applications in the real world.

Charles Geer teaches at Texas Tech University, Lubbock, TX 79409. He teaches graduate and under- graduate courses in mathematics education and serves as a mathematics consultant for school districts and service centers.

Students should learn to recognize and analyze patterns, describe patterns and relations, analyze functional relationships, and use patterns and functions to solve problems.

Many excellent materials are available as teachers introduce these important math- ematical ideas to students. Among the most interesting and motivating are domi- noes, playing cards, and the calendar. Each of these materials is readily available in large quantities and furnishes teachers with an almost endless source of activities for introducing patterns, relations, and func- tions.

Mathemagical activities combine the structure of mathematics with the enter- tainment value of a magic trick to produce seemingly impossible mathematical feats. The activities described in this article use dominoes, playing cards, and the calendar to give students mathematical experiences that involve patterns, relations, or func- tions. These activities reinforce basic com- putational skills and develop problem- solving strategies. They can also be extended to offer students informal expe- riences with the algebraic concepts of variable and expression. As students search for patterns, seek relations between their numbers and the final answer, and then learn that the mathematics involved in each activity is an intuitive application of simple functions, they are engaged in the types of mathematical experiences that need to be emphasized in the 1990s.

The following are three classroom-tested mathemagical activities guaranteed to amaze, entertain, and motivate upper-grade students. These activities were selected because they involve patterns, relations, or functions; are simple to master; can easily be taught to students; and have spectacular results.

MAY 1992 19

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Mathemagics with Dominoes Place a set of dominoes in a box and have students randomly pick one domino. Ask them to draw the domino they selected on a piece of paper, perform the mathemati- cal operations listed in table 1 , and tell you their answer. You can then instantly deter- mine the domino any student selected.

^^^^^^^^И :: * Teacher's Student's

instructions computations

1. Write the face value of the card you selected on a piece of paper. The aces and face cards have the following values: Ace = 1, J = 1 1, Q= 12, and К =13. 4

2. Double the face value. 8 3. Add3. 11 4. Multiply by 5. 55 5. If the card you selected

is a club, add 1; a heart, add 3; a diamond, add 2; or a spade, add 4. 56

6. Tell me your answer and I will show you the card you selected.

ВШИ NOVEMBER 1992

SUN MON TUE WED THU FRI SAT

1 2 3 4 5 6 7

8 9 Ht 4J 12 13 14 15 16 f'7 '$' 19 20 21 22 23 y 24 25 J 26 27 28 29 30 ^ S

Teacher's Student's instructions computations

1. Circle any two-by-two box of ft>ur dates on your calendar. 17, 18, 24, 25 2. Find the sum of the four numbers you circled« 84 3. Tell me your answer and I will show you the block of four

dates you selected.

Table 1 shows the teacher's instructions and student's computations for a 3-4 domino. By mentally subtracting 14 from the final answer, you can tell, draw, or match the domino any student selected. Your two- digit answer (34, after subtracting 1 4) tells the pips on each side of the domino; the tens digit (3) tells the number of pips on the left side, whereas the ones digit (4) tells the number of pips on the right side.

Mathemagics with Cards Walk around the classroom with a deck of cards and have students pick one card each from the deck. Ask them to make sure that no one sees their card, perform the math- ematical operations listed in table 2, and tell you their answer. For spectacular results, have an oversized deck of cards (7M χ 4.5"), ordered by suit and value, for yourself. After completing the mental calculations listed, you can instantly determine the card any student selected and then pull it out of your deck. Table 2 shows the teacher's instructions and student's computations for the four of clubs.

By mentally subtracting 15 from the final answer, you can match the card any

student selected. Your two- or three-digit answer (41, after subtracting 15) tells you the face value and suit of the card. The digit(s) in the tens and hundreds place(s) tells the face value of the card; you will have a number in the hundreds place only when the card's face value is ten or a face card. The ones digit tells you the suit of the card. A one in the ones place means that the suit of the card is a club, a two indicates a diamond, a three indicates a heart, and a four indicates a spade.

Mathemagics with Calendars Show a transparency of the current month ' s calendar on the overhead projector and then distribute copies of the calendar to students. Ask them to select and circle any two-by-two box of four dates, find the sum of these dates, and tell you their answer. You can then instantly locate on your transparency the block of dates any stu- dent selected. Table 3 shows a sample calendar with the "17, 18, 24, 25" block of dates circled. The teacher's instructions and student's computations involved in determining the sum of these dates are shown below the calendar.

By mentally dividing the number the student tells you by 4 and then subtracting 4 from the resulting answer, you can deter- mine and locate the box of dates selected by any student. Your answer (17 in the foregoing example) will always be the upper-left-hand date in the two-by-two box of dates.

Making Mathematical Discoveries After observing the activities described in this article, students will want to learn to do these "tricks." Teachers can then use each activity to create a high-quality les- son that allows students to make these discoveries and gives them experiences with patterns, relations, or functions.

To accomplish this goal, present one of the activities and tell several students the domino, card, or calendar dates they se- lected. Then explain to students that they can discover the mathematics involved in this activity by looking for patterns and finding a relationship between their final answer and the domino drawn, card se-

20 ARITHMETIC TEACHER

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lected, or calendar dates circled. Sample activities to do this exploration with domi- noes are discussed subsequently. Similar activities are possible with cards and the calendar.

Help students discover the relationship between their final answer and the domino selected by setting up a two-column chart. Table 4 shows a sample chart used to collect data for the "Mathemagics with Dominoes" activity. As students collect data and search for relationships between the numbers in the two columns, they will discover the mathematics involved in each activity. In making these discoveries, they are developing pattern-searching skills and finding relations between numbers.

Soon students will discover the rela- tionship between their final answer and the domino selected. Some will observe that the tens digit in their final answer is always one more than the number of pips on the left side of the domino and that the ones digit in the final answer is always four more than the number of pips on the right side of the domino. Others will see a different relationship between their final answer and the domino selected. They will form a two-digit number using the pips on the left side of the domino as the tens digit and the pips on the right side as the ones digit and discover that their final answer is always 14 more than the two-digit number formed by the pips on the left and right sides of the domino.

For students wanting to know why these "tricks" work or for teachers wishing to extend the activities to offer students in- troductory experiences involving variables and expressions, teacher ' s instructions and general cases for each activity are pre- sented in tables 5-7.

In table 5, χ is the variable for the number of pips on the left side of the domino and y is the variable for the num- ber of pips on the right side of the domino. The result of these instructions is always to multiply the number of pips on the domino's left side by 10 and add 14 to the total number. This computation makes it pos- sible to subtract 14 and then determine the domino selected.

In table 6, с is the variable for the face value of the card selected and s is the variable for the suit ofthat card. The result of these instructions is always to multiply the face value of the card by 1 0 and add 1 5 to the total number. This computation

MAY 1992

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instructions casé

1. Count the number of pips on the left side of the domino. χ

2. Multiply this number by 5/ 5x

3. Add.7. 5**7 4. Multiply by 2. 10* + 14 5. Add the number of

pips on the right side of the domino. l€br+14 + y-

^^^^^^^Я Teacher's General instructions case

1. Write the face value of the card you selected on a piece of paper. The aces and face cards have the following values: Ace«l,-J*'UvQ«i2, K*i3. с

2. Double the face value. 2c 3. Add3. 2c + 3 4. Multiply by 5. 10c +15 5. If the card you selected ; is a club, add 1; a heart, add 3; a diamond, add 2; or a spade, add 4. 10c + 15 + s

^^^^^^^J Teacher's Generel instructions case

1. Circle any two* by-two box of four dates on your calendar. a,b, c, d

2. Find the sum of the four numbers you circled. a + b + c + d**s

makes it possible to subtract 15 and then determine the card selected.

In table 7, a, b, c, and d are the variables for the four dates circled on the calendar and s is the variable for the sum of the four dates. The result of these instructions is always to find the sum of the four dates circled. It is then possible to divide the sum of the numbers by 4 and then subtract 4 from the resulting answer to find the small- est of these numbers. On the calendar, the smallest number will always be in the upper-left corner of the two-by-two box of dates circled.

Conclusion The three activities described in this article include only a few of the many "magic tricks" involving patterns, rela- tions, and functions available to the teacher of elementary school mathematics. With a little practice any teacher can perfect these and a number of other simple mathemagical activities. They can then offer their stu- dents interesting and entertaining math- ematical experiences while introducing patterns, relations, and functions.

Teachers who practice these activities before using them in class, actively in- volve students in their presentations, and use magical experiences to make math- ematics motivating and meaningful will find students asking for additional mathemagical performances. They will also discover that their students have a better understanding of patterns, relations, and functions and see interesting applica- tions of these important concepts.

Bibliography

Flexer, Roberta J. "Back to Basics the Magical Way." Arithmetic Teacher 11 (September 1979):22-26.

Frankenstein, Marilyn. "Using Mathematical Magic to Reinforce Problem-solving Methods." Math- ematics Teacher 11 (February 1984):96- 100.

Fraser, Don. Mathemagic. Palo Alto, Calif.: Dale Seymour Publications, 1985.

Gardner, Martin. Mathematics, Magic, and Mystery. New York: Dover Publications, 1956.

Gibson, Walter. Magic for All Ages. North Holly- wood, Calif.: Wilshire Book Co., 1980.

Howden, Hilde. "Patterns, Relationships, and Func- tions." Arithmetic Teacher 31 (November 1989): 18-24.

Mohr, Dean J., and Larry P. Leutzinger. "Magic in the Mathematics Classroom." Arithmetic Teacher 24 (April 1977):298-302.

National Council of Teachers of Mathematics. Cur- riculum and Evaluation Standards for School Mathematics. Reston, Va.: The Council, 1989. Щ

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