Fibonacci numbers: fun and fundamentals for the slow learner

Fibonacci numbers: fun and fundamentals for the slow learnerAuthor(s): SONJA LOFTUSSource: The Arithmetic Teacher, Vol. 17, No. 3 (MARCH 1970), pp. 204-208Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41186170 .

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Fibonacci numbers: fun and fundamentals for the slow learner SONJA LOFTUS

Sonja Loftus is a teacher at Washington Junior High School in Salinas, California. She is doing graduate work at San Jose State College.

A he problem of making mathematics more alive and more interesting for the slow learner is a challenge for teachers. Too often the slow learner is asked to repeat and repeat the same material year after year. If we as teachers can find materials to stimulate students, it can be fun for us too. This paper describes my experiences with teaching slow learners Fibonacci numbers, a topic new to me.

The article, "Fibonacci Numbers and the Slow Learner," by James C. Curl, was given to me with a request for my reactions [1].* After reading the article, I was

* Numerals in brackets refer to the bibliography at the end of the article.

fascinated to the point of wanting to ex- periment with this new tool. I was fearful of the new topic, but I found it not at all hard to discover the pattern in the Fibo- nacci sequence, 1, 1, 2, 3, 5, 8, .... The third number is obtained by adding the first two; in general the next number is obtained by adding together the two numbers immediately preceding it. To work with Fibonacci numbers, each number is represented by a code, Fn (read F sub n). Thus Fx = 1, F2 = 1, Fs = 2, F4 = 3, etc. Finding patterns in the Fibonacci numbers and expressing the patterns in the Fibonacci code supplies excitement and charm to the problem.

At the time, I had one section of slow seventh-graders. To get a broader experi-

Table 1 Sequences

Part I Form Sequence jrom Given Rule Rule

Fill in the blanks, using the rule on the right- Remember n is any number, but start with hand side. 0, 1,2, etc., unless the problem says otherwise. 1. 3, 5, , , , , 1. In + 3 2. 0, 4, , , , , 2. An 3. , , , , , 3. n - 3, starting with n = 5 4. 1, , , , 4. 3/7 + 1 5. , , , , 5. n + 1

Part 2 Determine the Rule jrom Given Sequence Rule

1. 1,2, 3,4, 5, 6, ... 2. 0, 1,2,3,4,5, ... 3. 2, 4, 6, 8, 10, ... 4. 1, 3, 5, 7, 9, ... 5. 3,6,9, 12, 15, ...

204 The Arithmetic Teacher /March 1970



Table 2 Counting Numbers and Codes

Complete each of the following charts: Counting Counting Counting

1. Code Numbers 2. Code Numbers 3. Code Numbers

Ci 1 £i 2 O, 1 C2 2 £2 4 O2 3 C3 3 Еъ 6 О., 5 C4 4 £48 O4 7 C5 £5 О 5 C6 £6 O6

mental base, and to gain a companion in the venture, I invited Mr. Bill Head, a fellow team teacher and the head of our mathematics department, to try the Fibo- nacci numbers with his eighth-graders, who were also of low ability.

Rather than jump immediately into Fi- bonacci numbers with the students, I started by reviewing the counting numbers and building needed background about sequences of numbers. This was accomplished through worksheet problems that guided the student to discover a rule for finding additional terms in a sequence or asked him to take a given rule and write out the sequence. Table 1 displays samples from some of the first worksheets.

Working with known sequences allowed me to gradually introduce the idea of a code. For counting numbers, we used С with an appropriate subscript. Thus, C2 represents 2, C5 represents 5. For the odd numbers we use O, and for the even numbers E, each with proper subscripts. E' stands for 2, the first even number; E>2 stands for 4, the second even number; similarly Ox stands for 1, the first odd number; О 2 for 3, the second odd number. Simple patterns are useful here, O2 = C:h E4 = ОГ) - 1 . Tables 2 and 3 are samples from worksheets on this material.

Once most of the class seemed to be at ease with codes, I introduced the Fibonacci series on the board and asked the students to guess the next number. By the time I got to 5, nearly all could see what numbers needed to be added. Next, I asked them

to say how to get the next number in the sequence. At first, verbalizing the rule seemed nigh impossible for many of them, but all were anxious to try.

In class we did the first twenty-five Fi- bonacci numbers (the first twenty-five terms in the Fibonacci sequence) as an oral exercise. Most of the students needed that much help to gain the confidence to proceed on their own. The assignment that night was to continue the work started in class, getting twenty-five more Fibonacci numbers. I gave the thirtieth, fortieth, and fiftieth numbers as checkpoints. These in- termediate goals encouraged them and urged some to go back and make correc- tions.

Generating the Fibonacci numbers is

Table 3 Practice with counting number codes

(Note how this material enriches the student's experience with squares of numbers as well as codes.)

1. Fill in the blank. a) In Table 2, Column 1 , the С stands for b) The rule for the /?th number (C,,)is c) In Table 2, Column 2, the E stands for

2. Find the sums: a) C, + C2= a) b) C, + С 2 + С, = b) c) С, + С2 + . . . + С, = с)

3. Squares: a) С? = а) b) On2= b) c) Е;?= с)

4. Sums of squares: a) С,2 + С,2 = а) b) С,2 + С22 + С82 = Ь) c) С,2 + С/ + . . . + С7 = с)

Excellence in Mathematics Education - For ЛИ 205



Table 4 The first fifty Fibonacci Numbers

Fibonacci Code Fibonacci Number Fibonacci Code Fibonacci Number

Fl 1 F26 121393 F2 1 F27 196418 F3 2 F28 317811 F4 3 F29 514229 F5 5 F30 832040 F6 8 F31 1346269 F7 13 F32 2178309 F8 21 F33 3524578 F9 34 F34 5702887 Fio 55 F35 9227465 Fu 89 F36 14930352 F12 144 F37 24157817 F13 233 F38 39088169 Fu 377 F39 63245986 F16 610 F40 102334155 Fie 987 F41 165580141 F17 1597 F42 267914296 Fie 2584 F43 433494437 F19 4181 F44 701408733 F20 6765 F45 1134903170 F21 10946 F46 1836311903 F22 17711 F47 2971215073 F23 28657 F48 4807526976 F24 46368 F49 7778742049 F25 75025 F50 12586269025

nothing but a monstrous series of addition problems. Yet the same students who balked at adding three-digit numbers would repeatedly go back over their work until they had satisfactorily completed the assignment. Table 4 suggests the magnitude of that first Fibonacci number assignment. I hope it also offers checkpoints for interested teachers.

The next day I gave the pupils a dittoed list of the first fifty Fibonacci numbers with the codes (table 4). We spent the day working with the idea of code and the meaning and the usefulness of Fn notation. I asked the students to observe that F1 + F2 = F3, and F 2 + F3 = F4. Soon we were able

to state the general rule for generating the next Fibonacci number, namely, Fn+2 = Fn+1 + Fn, n > 1. We then started looking for other patterns. I suggested 1 + 1 = 3-l;l + l + 2 = 5-l. After trying a few more cases, we translated our results into code, getting Fx + F2 = F4 - 1; Fx + F2 + F3 = F5 - 1; Fx + F2 + F3 + F4 = F6 - 1 . Soon the class could verbalize the rule, and then we worte it in terms of n, F, + F2 + F3 + ..- + Fn = Fn+2 - 1.

I will not attempt to outline each day's work. In general I taught most of the topics outlined by Mr. Curl. The emphasis was always on discovery. Our motto was "look for patterns." Once a pattern was

Table 5 An example of a student worksheet

Directions: Complete each column. Watch for patterns and be very careful in your arithmetic. 2 5

1 Fibonacci 3 4 Factor answers Code Numbers Squares Sum of Squares in column 4

Fi 1 1 F!2 = 1 1 F2 1 1 F!2 + F22 = 2 1-2 F3 2 4 Fi2 + F22 + F32 = 6 2-3 F4 3 9 Fi2 + F22 -}- F32 + F42 = 15 3-5 F5

206 The Arithmetic Teacher /March 1970



discovered, we verbalized the rule and finally wrote it with the code notation. Assignments both for homework and classwork were ditto sheets designed to encourage discovery [2]. Table 5 displays a student worksheet designed to encourage discovery of the Fibonacci identity F2 + F2 + F2 + ••• + Fu2 = F1() • Fn and the associated general rule F2 + F2 + • • • + p2 - p . p

When most of the students had gained a fair facility with the Fibonacci numbers, I introduced the Lucas numbers. These offered very little more difficulty and gave a wealth of additional results. For the slower student, the Lucas sequence, 1, 3, 4, 7, 11, 18 . . ., offered a chance to gain further confidence by working with more basic patterns again. The more interested and able students were anxious to look for relations that involved numbers of both kinds. A list of some of the relations that ultimately came from the students' work are in table 6. Their results were guided by worksheets that encouraged discovery. The teacher was guided in her choice of Fibonacci identities by well-known results [2].

As one works with these materials, op- portunities to talk about the standard topics continue to appear. I insisted that the students call "things" by a correct mathema- tical name, hoping to increase their voca- bulary and improve communication in the explanations. Practice with factoring de- veloped as we used trivial factors, prime factors, factoring for a particular purpose, and squaring. The last was surprisingly difficult for the students to understand.

From the first rule listed in table 6, neg-

Table 6 Typical Fibonacci and Lucas identities

1. FJ - Fn+k-Fn-k = {-'Y+k-Fj? 2. Fi + F2 + F3 + • • • + Fn = Fn+2 - 1 3. Ft + F3 + F5 + • • • + F2„_, = F2n 4. F2 + F4 + F6 + - • • + F2n = F2n+l - 1 5. Fj» + F22 + F3» + - ■ - + F„2 = Fn,Fn+ì 6. U + U + U + • • • + Ln = LH+2 - 3 7. F2n = FnLn

ative numbers come into use. Subtract Ftì+k * Fn-¡, from Fn2, and the answer will be positive or negative depending on whether n + к is even or odd. To a problem such as 24 - 25, the class reaction was to declare that it couldn't be done. After a short discussion, they could see that with numbers and ideas it was pos- sible, if we noted that it was not the same answer as 25 - 24. We discussed what happens when one borrows money, when temperatures go below zero, and so on, and decided such numbers could be called negative numbers.

When working with substitution of var- iables and algebraic phrases such as n + k, 2n - I, the students suspected these might be connected with algebra. This caused a noticeable increase in effort. It impressed and excited them to think that they were learning things that even the "smart kids" hadn't had. Algebra is a word with strange fascination. It carries a certain amount of status to say "we are learning algebra"; and slow learners need some status.

Obviously, the four basic operations were in profuse use, yet the usual reluc- tance to do the work was gone. One student commented, "It is better than dividing and plain multiplying. It is really good for me."

I found it was worthwhile to point out errors to individuals as they worked during class time. Everyone wanted to be correct. In prior work, if a problem had been done once, right or wrong, interest was gone. Now each student took pride in being correct.

Although many topics were difficult, one of the most difficult and most essential skills used in doing any of the work on the Fibonacci numbers was distinguishing between the subscript of the number and the number itself. The prior discussion of the counting numbers with codes such as Cu Оц, E*, helped make the work on the Fibonacci numbers much more exciting.

Prior to this, for these classes, numbers were numbers to be added or subtracted, and the students never thought of the num-

Excellence in Mathematics Education - For All 207



ber patterns and relationships that exist. This is perhaps the greatest contribution of the Fibonacci and Lucas numbers. So much more learning can occur when the students have fun seeing fascinating nu- merical relationships. The unit taught the students that it takes determination and work to get things done. As one student put it when asked to express his feelings about the unit: "They are fun, they make you think."

In general, the students' reactions were very good, both in the written comments that we requested at the end of the unit, and in their classwork. As one girl put it:

I like Fibonacci and Lucas numbers because they are different from anything else I've learned. It's a challenge for me and something different. It's much easier in a way than the other math.

Also, one boy's comment in his broken English:

My feeling is good because I learn a lot of thing in Fibonacci number, multiplication, sub- traction, addition, division, and other things in this class. I am glad to learn about this problem.

One of the biggest thrills was when a seventh-grade student discovered the Lucas number identity Ln2 = L2n + 2(-l)w. I am not suggesting that he came up with

the general rule, but he saw the relation- ship, although it was not part of the assignment. This particular student, who had a reputation for not doing any work either in or out of class, produced a great amount of work during the unit. He also had better attendance, a better attitude, and his comment is noteworthy.

I like it because I am learning about it and when I am absent I catch up with my work and I am enjoying it and I think it is fun working on these numbers.

He did indeed do all of the work he missed, often completing his make-up work and the daily assignment before leaving class.

We think that the unit was successful and worthwhile. We urge teachers to ex- periment with it for not only low ability students, but also for the average and above average. One can readily vary the approach and material to fit the learning level of the group.

Bibliography 1. Curl, James C. "Fibonacci Numbers and the

Slow Learner." The Fibonacci Quarterly 6 (October 1968): 266-74.

2. Hoggatt, Verner E., Jr. Fibonacci and Lucas Numbers. Boston: Houghton-Mifflin Co., 1969.

Letters to the editor [Continued from p. 195.]

plans, just in case we needed them. I received no suggestions until the day before the desig- nated first "investigation day," when a bright third-grade girl said, "But I don't want to investigate math. I want to investigate art." There was such a twinkle in her eye that I suspected she was remembering my comment of some time before that math played a part in nearly all aspects of life, so I had to follow through.

After school I enlisted the aid of the school librarian and found a number of wonderful new art books. One actually had the structural geometric shapes superimposed on the masterpieces. Another was called Taking a Walk with a Line and led into geometric shapes in modern design. The children were fascinated. It was not long before the children were bouncing in their seats, begging to try their hand at it. By this time, it

208

was almost time for P.E., so I sent a note to our very cooperative principal and distributed sheets of 12" x 18" newsprint. My mistake was in not giving them better paper, for without exception, and with very little help, they set to work with pencils, rulers, and crayons. By lunch time we had some of the best pictures I have seen produced by children of that age.

We did not continue to have such dramatic or concrete results from our "investigation days," but all were stimulating and successful. One fourth-grade boy asked to investigate computers, and was simply thrilled when he and a third- grade girl almost simultaneously discovered base two.

Best of all, of course, the children really learned how much fun math can be. - Edith Warner, Atsugi, Japan

The Arithmetic Teacher /March 1970



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Fibonacci numbers: fun and fundamentals for the slow learner