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Pattern Sequences

Team B

Kim Eschler, Michelle Gonzales, Alysia McIntosh, and Megan Rowland

MTH213

September 1, 2011

Lee Claycomb

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Pattern Sequences

“Mathematics has been described as the study of patterns,” (Billstein, Libeskind, & Lott,

2010, p. 22). The ability to recognize and create patterns is a skill taught in early education and

kindergarten. Patterns are the first math lesson children learn because of the necessary

fundamentals and simplicity that young children can understand. The ability to analyze patterns

and relationships within mathematic equations or sequences develops as comprehension

expands. “A sequence is an ordered arrangement of numbers, figures, or objects,” (Billstein,

Libeskind, & Lott, 2010, p. 26). Three types of pattern sequences are arithmetic, Fibonacci, and

geometric. These pattern sequence lessons can be difficult for students with limited mathematic

equation experience. Some examples in this paper are best understood by at least third grade

level students.

Learning tools such as songs, games, puzzles, etc. are excellent aids for reinforcement

and reaching students who have varying learning abilities. Teachers decide to implement these

tactics based on personal preference and class dynamic. Every student should have a math

journal to record definitions and write down example problems. A math journal is a great

reinforcement and study tool. The lessons in this paper incorporate recording vocabulary and

examples in math journals for memory and reference. This paper includes a lesson for each type

of pattern sequence, and lessons for teaching additional examples.

Arithmetic Sequence

In class: put a simple pattern on the board. (2, 4, 6. 8, __, __, __, __)

Ask the students to raise their hand when they see the pattern. Call on one student

to come up to the board and finish the pattern

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Put another pattern on the board. Use one that is a little harder and uses

subtraction 18, 15, 12, 9, __, __, __, __.

Follow the procedure used for the first example. Have the child tell the class how

he or she came up with his or her answer.

If there is not a student who feels comfortable with completing the pattern

explain to the students how he or she can find the answer to the pattern by finding

the common difference.

The common difference is the number that can be added or subtracted from each

number in the pattern to produce the next number in the pattern. For instance the

difference between 18 and 15 is 3. The difference between 15 and 12 is 3. The

difference between 12 and 9 is 3, and so on. Proving that subtracting 3 from each

number in this pattern will complete the pattern.

Explain that this particular kind of sequence is an arithmetic sequence. Tell the

students to make a journal entry: “Arithmetic sequence is when there is a set

number (or common difference) added or subtracted from each number to provide

the next number in the sequence,” (Billstein, Libeskind, & Lott, 2010, p. 26).

Take a look back at the first sequence solved by the students. Indicate 2 are added

to discover the next number in the sequence.

Write another example: 1, 5, 15, 20, __, __, __, __.

Complete the sequence with class discussion.

Ask for volunteer to write 4 numbers in a new sequence that has a common

difference between each number. Then ask a second volunteer to write the next 4

numbers in the sequence.

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Repeat this task until the students feel comfortable with the idea of arithmetic

sequence.

Fibonacci Sequence

Write this sequence on the board: 1, 1, 2, 3, 5, 8, __, __.

Ask the students if they recognize a pattern.

If a student does recognize a pattern allow him or her to explain.

Proceed with further explanation.

Change the commas to addition symbols and ask students if they can recognize a pattern.

Write 0 + 1 on the board. Write the answer, 1.

Write 1 + 1 on the board. Students answer, write 2.

Write 1+2 on the board. Students answer, write 3, etc.

Ask the students what the next equation should be by a raise of hand.

Call on more students to continue the pattern. This will extend the pattern to 1, 1, 2, 3, 5,

8, 13.

Cover rules again, ask the class to make another journal entry. Fibonacci sequence: “The

infinite sequence of numbers in which each number is the sum of the previous two”

(dictionary.com).

Geometric Sequence

Tell students the goal is to find what would come next in this set of numbers: 1, 3, 9, 27.

By raise of hands, ask students if they see a pattern.

Call on a student to ask what was added, subtracted, multiplied, or divided to obtain the

next number.

Ask students what tool they could use to figure this out.

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Ask a student to tell the class what the pattern is.

Prove the pattern on the board asking specific students for answers along the way. 1

multiplied by 3 is 3. 3 multiplied by 3 is 9. 9 multiplied by 3 is 27.

Ask what will be the next product in the sequence. Call on a student to come up the board

and solve 27 multiplied by 3. The answer is 81. Do the same two more times.

Make up additional problems with multiplication and division. Call on students to answer

until the concept seems clear such as 1, 2, 4, 16, and 32, 16, 8.

Cover rules again, ask the class to make another journal entry. Following the definition

and example of arithmetic sequence they will define geometric sequence as follows: Each

successive term is obtained from its predecessor by multiplying [or dividing] by a fixed

nonzero number,” (Billstein, Libeskind, & Lott, 2010, p. 32). Have them include one of

the sample questions as an example.

Problem Solving

After an initial lesson is given, review such as mathematic vocabulary and additional

problem solving is necessary to reinforce concepts. Solving sequences can increase in difficulty

and complexity. A clear understanding of multiplication and exponents are necessary; for

example Geometry with Michelle, the final sequence solution in this section.

Arithmetic with Alysia

Write the following sequence on the board 2, 5, 8, 11, 14, __, __, __

Ask the students to raise a hand once they see the pattern in the sequence.

Have students refer to math journals. Ask the students what kind of sequence this is.

(arithmetic)

Ask students, by raise of hand, to identify the fixed number in the pattern. (3)

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Complete the sequence by calling on students to identify the missing numbers.

The correct sequence is : 2, 5, 8, 11, 14, 17, 20, 23.

Fibonacci with Megan

Write the pattern 1, 2, 3, 5, 8, 13, ___, ___, ___on the board.

Ask the students to see if the recognize the pattern.

Have students refer to math journals. Ask the students what kind of sequence this is.

(Fibonacci)

Ask the students to offer his or her input on how the pattern is building.

If the pattern is unrecognizable start at the beginning and work through each number.

1 + 2 is 3, 2 + 3 is 5 and so on

This will help the students to realize how the pattern is building.

If the pattern is recognized, ask the students, by raise of hand, to provide the next

number in the pattern.

Confirm that the pattern is understood by explaining how each number is found.

Then ask the students to, individually, determine the next three numbers

By volunteer select three students to give the next three numbers in the pattern.

The correct sequence is : 1, 2, 3, 5, 8, 13, 21, 34, 55.

Geometric with Kim

Write 6400, 3200, 1600, 800, ____, ____, ____ on the board.

By raise of hands ask students if they see a pattern.

Have students refer to math journals. Ask the students type of sequence this is.

(geometric)

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Confirm with students what mathematical process is creating the sequence. On the board

work 6400 divided by 2, for the result of 3200.

Ask a student to come up to the board and continue to prove the pattern. 3200 divided by

2. Ask a second student if the answer is correct. Do the same with 1600.

Once at 800 ask student to work independently to solve the missing three numbers.

Call on three more students to come up to the board and fill in the missing numbers. Once

all numbers are filled in, ask the class if the answers are correct.

The correct sequence is: 6400, 3200, 1600, 800, 400, 200, 100.

Geometric with Michelle

Write the numbers 1, 8, 27, 64, __, __, __ on the board.

Ask the students if they can see a pattern.

Allow the children to explain the reasoning behind their thought process.

Ask the children, using multiplication, what are some numbers that will give the product

of 1.

The correct response is: 1 x 1. Write 1 x 1 on the board.

Ask the children to do the same for the number 8.

The correct responses are 1 x 8 and 2 x 4. Write it on the board below 1 x 1.

Ask the children to do the same for 27.

The correct responses are 1 x 27 and 3 x 9. Write it on the board below 1 x 8 and 2 x 4.

Ask the children if they can see a pattern forming.

Regardless of the response, ask the children what numbers can be used for the product

64.

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The correct responses are: 1 x 64, 2 x 32, 4 x 16, and 8 x 8. Write these answers below

the previous equations.

Ask the children, by raise of hand, to choose a response that they think could be used to

create a pattern of multiplication problems.

A child may choose the 1 x n option. Technically, they would be correct. If this is the

initial response ask another child by raise of hand to choose another pattern. Writing the

equations below the prior will help create a table that should help the children recognize

the pattern more easily.

The correct pattern will look like this:

1 x 1

2 x 4

3 x 9

4 x 16

Ask the children, by raise of hand, what the next equation in the sequence would be.

It will be obvious that the next equation will begin 5 x n. The question is what is n? If

there is no response, ask the children to, by raise of hand, note any patterns that they

might see.

Children who understand exponents will see a pattern. The first number of the equation is

squared, creating the second number (1 squared is 1, 1 x 1; 2 squared is 4, 2 x 4; 3

squared is 9, 3 x 9; etc.).

The final step is the realization that 2 x 4 = 2 x 2 x 2 = 8. Or 2 cubed (to the third power).

The final equation will be n to the 3rd = n position.

Solve the equations for remaining numbers of the sequence.

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The sequence for seven positions is: 1, 8, 27, 64, 125, 216, 343.

Conclusion

Pattern discovery can be fun math lesson at any age. An understanding of the differences

in arithmetic, Fibonacci, and geometric sequencing should help students identify and solve

pattern sequencing math problems. Pattern sequencing is essential to understanding the logic of

math and how math proves itself correct.

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References

Ask.com. (2011). Askkids.com. Retrieved on August 28, 2011 from

http://answers.askkids.com/Math/what_is_a_geometric_sequence

Billstein, R., Libeskind, S., and Lott, J. (2010). A problem solving approach to mathematics for

elementary school teachers (10th ed.). Boston, MA: Pearson Addison Wesley.

Dictionary.com (unkn, 2011) Fibonacci sequence. Retrieved on August 31, 2011 from

http://dictionary.reference.com/browse/Fibonacci+sequence.

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Addendum

Turnitin.com (2011) Team Paper. Retrieved on August 31, 2011 from

https://api.turnitin.com/newreport.asp?oid=199679547&svr=2&session-

id=d84326828dcebcc82640ad789931ba48&lang=en_us&r=66.1822668416004

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Write Point. (2011) Team Paper. Retrieved on August 31, 2011 from

https://ecampus.phoenix.edu/classroom/ic/cwe/PaperList.aspx.