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“How Do You See It?
Discovering Mathematical Patterns and Sequences”

Keith Gaudet

Introduction

For the last three years I have been teaching math and science to middle school children. Teaching these kids has been a pleasure and a challenge. They are born from a variety of ethnical backgrounds, large families to small families, creative and simple. But one concept that surprised me was that a majority of these kids were not as perceptive in mathematics. They choose to take the simple or most obvious approach. Not that there is anything wrong with that approach but these students are not seeing the whole picture. This is crucial when dealing with mathematics and science as well. Scientists need to be able to see if there are other alternatives to a particular science experiment if the original one fails. With all this in mind, I have made looking for patterns my general underlying theme throughout the school year. Thus, this forms the basis for this paper.

My topic for this ATI curriculum is on the variety of mathematical patterns and sequences.  Though this curriculum will be designed for the middle school teacher, it includes ways to adapt it for any grade level. I have also been a firm believer in integrating all subject areas. It is also my intention to come up with ways to discover and use patterns and sequences in other subjects. I am planning on this paper to be set up in 4 major sections: Introduction to the Curriculum, Host School Academic Setting, Background Information, and Collection of Lesson Plans.

The introduction to the curriculum will inform the reader as to what the paper is about, the target audience of the paper and how the curriculum could be taught. The academic setting section is a general background of my assigned school, Truman Middle School. The section will break down the demographics in a variety of areas including: ethnic background, reduced meals vouchers, staff experience, and community profile. My Background Information is considered the heart of my paper. I shall first attempt to give a historical perspective on a variety of mathematical patterns and sequences. I broke down the sequences section into geometric and arithmetic. (Similar to what Jacobs' Mathematics: A Human Endeavor textbook does.) I have researched and recounted how the early mathematicians discovered these fascinating patterns. The next section is a collection of lesson plans. These are lessons that could be taught in conjunction with this unit. These lessons will not be in any particular order, for the order will be determined by what subject the teacher is discussing. These lesson plans will include New Mexico's state standards and benchmarks. Finally, I will wrap things up in my conclusion.

Truman Middle School, located at 9400 Benavides Rd. SW, is one of 25 middle schools that make up the Albuquerque Public Schools district. It is one of three middle schools that serve the West Mesa Cluster (John Adams and Jimmy Carter make up the other two middle schools). Truman’s enrollment was at 808 students during the 2000-2001 school year.  Hispanics make up for more than three quarters (83.7%) of the total student population. For all APS middle schools, Hispanics make up 49% of all ethnic backgrounds. The rest of Truman’s ethnic profile is as follows: Anglo (9.3%), Black (3.3%), Native American (2.6%), Asian (.6%) and other backgrounds (.5%).

Historically, Truman Middle School’s students receive free or reduced cost meals that are provided by APS Food Services. To qualify for free meals in the 1998-1999 school year, a family of four would have had to earn less than \$21,385.  To qualify for the reduced cost meals, a family would have had to earn between \$21,386 and \$30,433. The mean of Truman community’s income was \$28,641. Over of the students receive meal services. The district average among middle schools is less than half of the students.

Truman Middle School has a fairly young staff. The percentage of teachers with 0-5 years and 6-10 years of experience are both 33.3%. The APS middle school average percentages are 27.1% and 23.7%, respectively. Only 7.2% of Truman’s staff has 11-15 years of experience. The middle school average is double that of Truman’s, 14.7%. There is also a big disparity between teachers with more than 15 years experience at Truman and the rest of APS, 26% and 34.5% respectively. Nearly two-thirds of the staff has a bachelor’s degree. Percentages are roughly the same when comparing the number of male to female teachers at Truman. (All data is taken from the 2000-2001 APS School Research, Development and Accountability Division unless otherwise noted.)

The school’s mission statement is simple and straightforward: “Truman Middle School is a community of learners dedicated to developing Quality, Character, and Competency for success in the 21st century.” Truman has a very large number of support programs and activities aimed at students as well as the staff and community. Such programs include a school-wide bilingual program, special education, Title I School Wide Literacy Program, intramural sports, and the Middle School Cluster Initiative Activities.

Background Information

Before we can begin discussing and analyzing numerical sequences and patterns, let us examine the debate on what is a number. Mathematicians and historians alike have posed several questions regarding the basis of what numbers and number systems are, such as “Are numbers invented or discovered?” “Are numbers merely symbols or is it a language?” and “Are numbers actually existing because humans are existing?” Davis and Hersh use a religious theme to address the issue of mathematics being non-man made. They state the wonders of the cosmos as being the reason why mathematics works for humanity. On the other hand, Marcia Ascher, in her book Ethnomathematics, understands that mathematics is indeed a complex word to define but argues that mathematics is a Western philosophy rather than universal. However, she is careful to point out that this “is not to say that the ideas or concepts we deem mathematical do not exist in other cultures; it is rather that others do not distinguish them and class them together as we do.” In Mathematical Mysteries, Clawson does not make an attempt to define what a number is but instead he believes that numbers have three general uses. He states first, “numbers are used to find out how many objects are in a collection… [Secondly] to find the proper order of elements in a set…[and lastly] to simply give a unique name to an object.”   Consider early mathematician Pythagoras, “A number is the within of all things.”  Schimmel believes that “number systems are built according to different rhythms.” So where does all this bickering lead us to? On the Dictionary.com web site, it seems as if Clawson has the right idea. For our purposes, we will use the following general definition: A number is a symbol or word that is from an infinite counting set of positive and negative numbers. This brings us to discussing the counting numbers.

To make a valid point from above, that numbers are a symbol or word, Americans would use the counting numbers one, two, three and so on. The French uses un, deux, and trois. Eins, zwei, and drei are the first three German counting numbers. Babies would use their fingers to indicate how many. Stone age man, we suppose, would use stones and sticks to illustrate the same concept. Does this mean the world counts differently? Pragmatically speaking, yes. Comprehension, no. In the society we live in, we take our natural counting numbers for granted. We just assume that the first number is one (or uno, eins, un, or whatever language you choose) and continues on increasing by one. But since mathematicians must have some order to their lives, how do we justify this “order”? Giuseppe Peano was an Italian mathematician who helped shed some light on this. He established a set of axioms, statements that do not need proof because they are generally accepted. Peano's 5 Axioms can be stated as follows:

Axiom 1: One is a number
Axiom 2: If x is a number, then the successor of x is a number.
Axiom 3: No two numbers have the same successor.
Axiom 4: Two numbers whose successors are equal are themselves equal.
every number is in set A.

This system will help us establish what a number sequence is. A sequence is a set of numbers arranged in some particular order. These sets of numbers are called terms. Thus a number sequence can be defined as a series of numbers in which a successive number is arranged in a certain rule. To sum this all up, the natural counting numbers has a rule to determine its order. As a school-aged child would say, “just add one to the previous number to get the next number.”

Before we continue on with the different kinds of sequences, we must do some brief comparisons. Suppose one were to ask what the difference is between these two sets of numbers: {0,1,2,3,4} and {0,1,2,3,4…}? In the first set, it shows the first 5 natural counting numbers and it ends there. This is to say that this particular set of numbers has only these five numbers. By contrast, the second set also shows the 5 same numbers but shows three dots immediately following the 5th number. These three dots are called an ellipse. This represents that the numbers go on in that order in a continuous fashion. These two sets are called finite and infinite sequences, respectively.

A second comparison we must make is between a series and a sequence. Consider these two sets of numbers: {0,1,2,3,4} and {0+1+2+3+4}. As stated before, the first set is indeed called a sequence. It is a set of numbers in which each successive term follows another. The second set shows that these terms are being added to one another. These numbers represent the sum of the terms. In conclusion, the sums of a set of terms are called a series of numbers.

Let's now examine different kinds of sequences. In this curriculum we shall confine our efforts to arithmetic and geometric sequences. An arithmetic sequence is a sequence in which the difference between two successive terms is the same. This difference is often called the common difference. Evaluate the following sequence:

{2,4,6,8…}

As you can see, this is an infinite sequence that has a common difference of two.

{2 (+2), 4 (+2), 6 (+2), 8 (+2)…}

On a side historical note, the ancient Greeks would make note that this also an even sequence. They noted that an even number is a number that can be divided by 2 evenly, whereas an odd number cannot. Greeks were very fascinated with numbers. They made many comparisons such as even vs. odd, friendly vs. non-friendly numbers, perfect numbers vs. imperfect numbers, and harmonious numbers vs. imbalanced numbers. McLeish, in his book The Story of Numbers, continued the debate about numbers with a reference to Aristotle. Aristotle believed that “a number is a heap or multitude of units.” We will come back to examine more ideas the Greeks had further on in the paper.

Going back to our sequence, do arithmetic sequences always have to be increasing numerically? The answer is no. Arithmetic sequences can be increasing or decreasing in value. Remember, the only condition is that the difference between the two successive terms is constant. What if one received a problem in which they had to find the 10th term in an arithmetic sequence? Let's take our earlier sequence, {2,4,6,8…}.  We could simply continue the sequence (since we know it increases by two each time) until we count the 10th term. Instead there is a simple formula that allows us to come up with the answer directly. Let us state that term t is first term in the sequence and d represents the common difference.

X=t+(n-1)*d
So… X=2+(n-1)*2
X=2+2n-2
X=2n

Just substitute the n for 10(since we want the 10th term), and multiply by 2.

X=2*10=20

When Karl Gauss, a brilliant German mathematician, was 10 years old in the late 18th century he was presented a very difficult problem. His teacher, a stern and lazy man, wrote on the board the task that these young men had to perform. The problem was to add up all the numbers from 1 to 100. Knowing this would take his students time, the arrogant teacher began to go back to his seat and prepare himself for a long quiet day. As soon as he sat down, Gauss approached him and put the slate, a small board that these students used to do their work on, teacher's desk. All the students were shocked at how fast and seemingly effortlessly Gauss completed the problem. The teacher just glared at him. By the end of the school day, the last of the boys set his slate down. The teacher had a feeling that no person came up with the right answer. He began turning over each of the slates, each revealing a wrong answer. Finally he came to Gauss' slate. All the kids snickered as the teacher slowly turned his slate over. The teacher's ! What's even more surprising is that he had written very little besides the answer. How did young Gauss come up with the answer? Oddly, he did not have an equation like we have now. He did it purely on observation. Look at the series of numbers…

1 + 2 + 3 + 4 + 5 + 6 + …+ 94 + 95 + 96 + 97 + 98 + 99 + 100

Take the first and last term of the series, 1 and 100. Gauss says that this combination when adding together equals 101. Looking at another combination, 2 and 99 is also 101 and so forth. Thinking that this pattern repeated for all the others, he knew he had a certain number of 101's. The question was how many 101's were there? Knowing that exactly 50 pairs of numbers between 1 and 100, this led Gauss to just simply multiply 101 by 50= 5050. Gauss was brilliant for his age and became ever more brilliant as his life went on. Gauss' method is wonderful to look at but there still must be an easier way to figure out the sum of a finite arithmetic series. There is! We can solve this problem using the equation: sn = n(t1 + tn)/2
where
= the first term of the series and n = the term you are going up to.

So… S=100(1+100)/2
S=(100*101)/2
S=10100/2
S=5050

But what if the pattern does not increase or decrease by a common number? Let’s examine a new sequence of numbers.

{1, 3, 9, 27…}

This sequence of numbers seems to be increasing at a rate of 3, or tripling each time. Triple means to take three times as much as the other. So, 1 x 3 = 3, which is the second number in the sequence. Continue this pattern and get the next number in the sequence, 9 (3 x 3 = 9). This new kind of sequence is called a geometric sequence. A good way to define a geometric sequence would be that it is a sequence in which the ratio of two successive terms is constant. Thus, we say that the above sequence has a common ratio of 3; that is, we must multiply by 3 to get the next term in the sequence.

Like the arithmetic sequence, there is an easy way to figure out the nth term of a geometric sequence. To figure out the formula for this, we need to understand what is happening with the numbers in a geometric sequence. We said earlier that the terms seem to be increasing at a rate of 3. Another keen observer might say that the terms are exponentially increasing. Using exponents is the key to understanding the geometric formula. Consider the following:

No. in sequence Exponential form nth term

1 3(0) 1

3 3(1)  2

9 3(2)    3

27 3(3) 4

What this chart is showing is as the nth term is increasing by one the exponent in the base (3) is also increasing by one. Also notice that the nth term is not the same as the exponent but rather has a difference of one. With all this knowledge now, we can put the equation together as follows:

t= tn * r(n-1)
where t1 is the first term in the sequence
r is the common ratio
and n is the term position

With all this in mind, the 10th number in the sequence above is 19, 683.

This section so far has examined several ideas regarding sequences. However, there is one last idea that offers many surprises and challenges. We have examined both arithmetic and geometric sequences, where there is either a common difference or a common ratio. But what if there is a pattern within a sequence that has neither a common difference nor ratio? As mentioned earlier, the Greeks were quite fascinated with numbers. They saw sequences that are not arithmetic or geometric. The Greeks understood what was happening with the terms in the sequences but wanted to show visually what is occurring. Before the Greeks, the Babylonians came up with sequences written in clay tablets. One such sequence displayed was {1, 4, 9, 16, 25 …} The Babylonians knew what was occurring. Each term was being multiplied by its own term. (ig. 1 x 1 = 1, 2 x 2 = 4, 3 x 3 = 9, etc.)  We could even go back to what we mentioned earlier. This can be written in exponential notation. (1(2), 2(2), 3(2), etc.) We would say this as the numbers are being “squared.”  This explanation was not enough for the Greeks. They wanted to show visually why this is true. Since paper and pencil was out of the question, the Greeks used small stones or pebbles. Thus as Clawson puts it, “pebble notation.” It was a simple idea. One pebble represented the number one. Two pebbles would represent two and so on.  Figure 1 shows how they set up the Babylonian sequence. By looking at the shape that the Greeks formed, it was clear why we call 2(2) “squared.”   Thus, the set of numbers that the Babylonians came up with were called square numbers. Because the ancient Greeks used this “pebble notation” in geometric form, they saw a few intriguing patterns with the squares.   Most notably, they saw that the differences between the terms are the natural successive odd numbers.

Square #’s Sum of Odd #’s

4 1+3

9 1+3+5

16 1+3+5+7

Secondly, relating to this odd number phenomenon was that each square number using “pebble notation” formed an “L” shape. This “L” is called a gnomon (Figure 2). A gnomon was an ancient piece on a sundial used to measure time. A third interesting pattern within this visual form of the square numbers was that it formed yet another new sequence of numbers, the triangular numbers

The triangular numbers, if you look at their visual geometrical form, are just that. Figure 3 shows that using “pebble notation” these numbers take on a triangle shape. For example, visualize a set of bowling pins set up on a lane. Since there are 10 pins in a triangular fashion, 10 would be a triangular number. After sifting through these pebbles, the Greeks noticed that each of the triangular numbers is the sum of the consecutive natural numbers.

Triangular #’s Sum of Cons. #’s

3 1+2

6 1+2+3

10 1+2+3+4

So after all this, where does the square sequence come in place? Suppose we look carefully at the pebble notation of the square number 16. Drawing a diagonal line down left side of the middle of the square, we can split the pebbles into two separate triangles as shown in Figure 4. By counting the number of pebbles in the first triangle, we see that it totals 6. The opposite triangle on the right has 10. The sum of 6 and 10 is 16. Since 6 and 10 are triangular numbers, we can conclude another interesting rule about square numbers. The sum of two consecutive triangular numbers is equal to a square number!

Another geometric shape that forms using “pebble notation” is a rectangle. There are two major patterns with rectangular numbers. The following are the first 6 rectangular numbers:

{1, 2, 6, 12, 20, 30}

If we were to focus on the 5th rectangular number (20) using pebbles and divide the rectangle in a way similar to the square, we would notice that it also forms two separate triangles (Figure 5). The difference here, though, is that each triangle is identical. There are a total of 10 pebbles in each triangle. Does this pattern follow with other rectangular numbers? Yes, it does! We can now conclude that rectangular numbers are the sum of two identical triangular numbers. After closer examination of the rectangular numbers, it can also be said that rectangular numbers are the sum of a string of consecutive even numbers.

Rectangular Number Sum of Even Numbers

6 2 + 4

12 2 + 4 + 6

20 2 + 4 + 6 + 8

30 2 + 4 + 6 + 8 + 10

Clawson researched Greek mathematics and theory further and found that they noticed leftover pebbles. These pebbles are leftover because they do not form square, triangular, or rectangular shapes. Let’s consider the number 7. The number 7 does not form any of the aforementioned shapes. We can form a triangle and a square shape using 6 pebbles. The Greeks called these numbers prime numbers.

We have talked about arithmetic and geometric sequences. However, mathematical sequences do not end there. Pascal’s Triangle and Fibonacci’s Sequence are two of the most famous mathematical sequences.  Pascal’s Triangle starts off as follows:

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

This triangle was named after French mathematician Blaise Pascal. Although the Chinese discovered this pattern centuries before, Pascal was credited with the numerical discoveries within this mysterious triangle. These numerical patterns and sequences that Pascal discovered helped mathematicians develop and understand new properties of numbers. In addition, this triangle helped with algebraic and probability problems.

Leonardo of Pisa, an Italian mathematician, made an astonishing mathematical pattern involving the Chinese’s “Pascal’s Triangle.” This sequential pattern starts off like this:

{1, 1, 2, 3, 5, 8, 13, 21…}

Fibonacci, as Leonardo was called, saw that the terms in this sequence is the sum of the previous two terms.  For example, 1 +1 = 2, 2 + 3 = 5, 5 + 8 = 13 and so on. This sequence was very apparent in nature as well. Counting the scales on a pine cone, segments in a pineapple or leaves holds the Fibonacci’ Sequence true.

Mathematical patterns are indeed everywhere. If students can learn to recognize various patterns, it can open up a whole new world to them.

Unit Lesson Plans

These lesson plans can be modified in many ways. Among these modifications are special needs, grade level, LEP, and Bilingual. These lessons are adapted for Albuquerque Public School standards. Since these lessons are modifiable through all grade levels, there are no exact standards to follow. Depending on the grade level, all lessons will follow the same strand and content standard:

Strand V: Data Analysis, Statistics, and Probability

K-12 Content Standard: The student demonstrates an understanding of algebraic skills and concepts through experiences with meaningful mathematical problems that focus on discovering, describing, modeling, and generalizing patterns and functions, representing and analyzing relationships, and finding and supporting solutions.

These are in order based on how I would follow this lesson, but can be used in any order at the teacher’s discretion.

Lesson 1: “Puzzle Pieces” – This lesson works best with large tables rather than desks. Break the class into groups with approximately 3 or 4 students to a group. Hand out one 500 (or higher) piece-picture puzzle. Before the students open the boxes of the puzzles, explain that students are to come up with ways as a group to find the fastest and easiest way to put the puzzle together. These brainstorming ideas should be written down and will be shared at the end of the class period. This is an excellent activity that introduces the unit on patterns. This activity induces the students’ critical thinking skills, collaboration with their peers, and teaches about teamwork. I have never came across a group that completely finishes a puzzle. Those groups that come up with good ideas to start the puzzle tend to complete the border of the puzzle. Perhaps a small corner of the puzzle is completed as well. Allow time for a brief discussion on their ideas and time to clean up the room. This project works well in block periods, roughly 90 minutes in time.

Lesson 2: “Finding Patterns Within the Multiplication Table” – This activity can be done either individually or in pairs. Special needs students should be paired with a regular education student. Pass out a copy of the blank multiplication table. There is one included at the end of this unit. There are two tasks at hand for the students. The first one is for students to fill out the table correctly. It is very important that this is done accurately. Any one number that is wrong will hamper the student’s efforts with the second task. This is a wonderful activity to refresh their memory on the multiplication facts. Once the table is completed, students are to find as many patterns as they can find in the table. Have color pencils or markers around; this helps the students make their patterns more visible. For an added exercise, students can explain their patterns by writing them out on the backside of the paper. Have the students set a goal of finding at least 15 or more patterns in the time given. The time will vary based on the level of your class. This is a very engaging activity. Students have complained that they wanted more time to find more patterns. Offer extra credit for more discovered patterns as homework. When discussing and sharing patterns as a class, have an overhead transparency of the times chart. Invite students to come up to the projector and share their ideas.

Lesson 3: “M&M™ Pebble Notation” – This project allows students to explore various sequences the way the ancient Greeks did. Instead of using pebbles or small stones, get some bags of M&M’s ™ and distribute a small amount (25) of M&Ms to each student. If this is too costly or difficult for the teacher, round two-color counters will work just as well. (Only the students can’t have fun eating them afterwards.) Have students examine several of the sequences that were mentioned in the background information section of this unit. Sequences that are worth examining with these M&M’s or counters are triangular, square, rectangular and prime numbers. Students should be drawing what they see and the patterns within them. This is a great one-day class activity.

Lesson 4: “Sequences Using Geoboards” – This lesson is another variation of Lesson 3 except students are exploring these sequences using geoboards.

Lesson 5: “Computer Lab Time” – Students can search the web for information on numerical sequences. See the Web Sites list at the end of this unit for some useful sites. The task at hand for the students is to find information on the Internet that shows and explains what number sequences are and some examples of the sequences. Students must search and record at least 3 sites and the information contained within them. Before signing on to the computers, teachers must make sure that each student has signed their school’s policy on computer use. Alternate assignments must be given to those students who do not have such agreement signed.

Lesson 6: “Explaining Sequences Using Power Point” – This can be a cumulating activity. Students would head to a computer lab that has Mircosoft Power Point. Students will design a Power Point presentation that will sum up what they have learned in the unit on mathematical sequences.

Documentation

Bibliography

Ascher, Marcia. Ethnomathics: A Multicultural View of Mathematical Ideas. Belmont, CA: Chapman & Hall/CRC, 1991.

Clawson, Calvin C. Mathematical Mysteries: The Beauty and Magic of Numbers. New York: Plenum Press, 1996.

Davis, Philip and Reuben Hersh. The Mathematical Experience. Boston: Houghton Mifflin Co., 1981

Mcleish, John. The Story of Numbers. New York: Fawcett Columbine, 1991.

Jacobs, Harold R. Mathematics: A Human Endeavor 3rd edition. New York: W. H. Freeman and Company, 1994.

Reimer, Luetta and Wilbert. Mathematicians are People, Too: Stories from the Lives of Great Mathematicians. Palo Alto, California: Dale Seymour Publications, 1990.

Schimmel, Annemarie. The Mystery of Numbers. New York: Oxford University Press, 1993.

Voolich, Erica Dakin.  A Peek Into Math of the Past. New Jersey: Dale Seymour Publications, 2001.

Vorderman, Carol. Reader’s Digest How Math Works. New York: The Readers’ Digest Association, 1996.

Web Sites

- Wonderful site on Pascal’s Triangle and patterns within it.

- Have students try the sequence activity. This is a fun way for students to explore patterns that have larger numbers within them.

- A collection of math patterns and relationships lesson plans.

A short and simple site talking about the history of Greek counting.

- A few sample worksheets on sequences can be found here. More can be created on a subscription basis.

- Cool math posters you can order.

Figure 1: Square numbers arranged by “pebble notation”

Figure 2: “L-shape” gnome in square numbers

Figure 3: Triangular Numbers using “Pebble Notation”

Figure 4:

Diagonal through square number 16 revealing two consecutive triangular numbers (6 and 10).

Figure 5: Diagonal line through rectangular number 20. This reveals 2 identical triangles. (10 and 10)

 X 0 1 2 3 4 5 6 7 8 9 10 11 12 1 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 1 2 3 4 5 6 7 8 9 10 11 12 3 0 2 4 6 8 10 12 14 16 18 20 22 24 4 0 4 8 12 16 20 24 28 32 36 40 44 48 5 0 5 10 15 20 25 30 35 40 45 50 55 60 6 0 6 12 18 24 30 36 42 48 54 60 66 72 7 0 7 14 21 28 35 42 49 56 63 70 77 84 8 0 8 16 24 32 40 48 56 64 72 80 88 96 9 0 9 18 27 36 45 54 63 72 81 90 99 108 10 0 10 20 30 40 50 60 70 80 90 100 110 120 11 0 11 22 33 44 55 66 77 88 99 110 121 132 12 0 12 24 36 48 60 72 84 96 108 120 132 144