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Finding a Pattern

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Strategy 1 - Finding a Pattern bd10409_.wmf (27742 bytes)

Finding a pattern in a set of numbers can help you organize data.  

Look at this set of numbers and see if you can figure out what comes next to complete the pattern.

2, 6, 12, 20, ___, ___, ___  What is the pattern?  How did you know what the pattern was in the numbers?  Can you make a similar pattern?   Try your pattern on a friend.

Here are some other patterns to try: 

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1, 2, 4, 8, ___, ___, ___ 

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1, 2, 4, 7, 11, ___, ___, ___

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2, 3, 5, 9, ___, ___, ___

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1, 5, 25, 125, ___, ___, ___

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3, 12, 48, 192, ___, ___, ___

Now, let's find some more patterns.  These might get a bit challenging, so you may work with a partner to solve them. 

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James began writing a book.  At the end of the first week, he'd written 10 pages.  By the end of the second week, he'd written 6 more pages, for a total of 16 pages.  At the end of the third week, he had a total of 23 pages and by the end of the fourth week he had 31 pages completed in his book.  If he continues writing at this same rate, how many pages will his book have at the end of the seventh week?

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Mary's five friends began an exercise group.  They decided to walk along a trail each day.   On the first day, they walked 2/3 of the trail.  On the second day, they walked 3/5 of the trail.  On the third day, they walked 4/7 and on the fourth day 5/9 of the trail.  If this pattern continues, how far will Mary and her friends walk on the tenth day?

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Jason began a weight-training program.  The first week, he lifted 12 pounds.  For the following three weeks, he lifted 13, 14.5, and 16.5 pounds.  If he continues this pattern, during which week will he lift more than 50 pounds?

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Susie decided to join a stamp-collecting club.  During the first month, she collected 48 stamps.  During the next three months, she collected 72, 96, and 120 stamps.  If she continues collecting stamps at this same rate, during which month will she have more than 200 stamps in her collection?

Patterns have been a part of mathematics for a very long time.   There are famous mathematicians who discovered patterns that are still used today.   For example, Leonardo Fibonacci discovered the Fibonacci sequence.  In this pattern, the first six numbers are: 1, 1, 2, 3, 5, 8.  Work with a friend to find the next 5 numbers in this sequence.  Write down the numbers that follow in the set and explain the pattern to your partner.

Architect Le Corbusier used the Fibonacci sequence: 1, 1, 2, 3, 5, 8, ... when he was designing buildings.  He applied Fibonacci's pattern for determining the length and width of each room so that the room sizes were related to each other.  For example, a rectangular room with a width of 3 meters would have a length of 5 meters.  A rectangular room with a width of 5 meters would have a length of 8 meters.  What would be the length of a rectangular room with a width of 13 meters using the Fibonacci sequence?

If you liked the Fibonacci Pattern, here are a few websites which give more information and examples:

Easier Fibonacci Puzzles - http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibpuzzles.html

    At this site, you can see examples of Fibonacci patterns in real-life problems.  Try a few and see if you can discover the same relationship that Fibonacci made famous.

Fibonacci Spirals - http://students.bath.ac.uk/ma1caab/spirals.html

    This site has many pictures that were created using the Fibonacci pattern.  Take a look at these patterns and read about the artist who created them.

Another famous, Blaise Pascal, from France, discovered a pattern of numbers called Pascal's Triangle.    Take a look at a description of Pascal's Triangle at  http://mathforum.com/workshops/usi/pascal/pascal_elemdisc.html and http://mathforum.com/workshops/usi/pascal/pascal_middisc.html   to learn how this pattern works.   Once you understand the pattern in Pascal's Triangle, you can begin the next activity.

Here is an example of Pascal's Triangle to use for the following activity:

 

 

 

 

 

 

 

 1

 

 

 

 

 

 

 

 

  Row 1

 

 

 

 

 

 

 1

 

 1

 

 

 

 

 

 

 

  Row 2

 

 

 

 

 

 1

 

 2

 

 1

 

 

 

 

 

 

  Row 3

 

 

 

 

  1

 3

 3

 

 1

 

 

 

 

 

  Row 4

 

 

 

 1

 

 4

 

 6

 

 4

 

 1

 

 

 

 

  Row 5

 

 

 1

 

 5

 

10

 

10

 

 5

 

 1

 

 

 

  Row 6

 

 1

 

 6

 

15

 

20

 

15

 

 6

 

 1

 

 

  Row 7

 1

 

 7

 

21

 

35

 

35

 

21

 

 7

 

 1

 

  Row 8
 1

 

 8

 

28

 

56

 

70

 

56

 

28

 

 8

 

 1

  Row 9

Complete each of the following tables and describe the pattern:

Problem #1 Problem #2
Row Sum of All the Numbers in This Row Row Sum of the First Three Numbers in This Row
1 1 1 1
2 2 2 2
3 4 3 4
4 8 4 7
5 _________ 5 _________
6 _________ 6 _________
7 _________ 7 _________
8 _________ 8 _________
... ... ... ...
10 _________ 10 _________
... ... ... ...
15 ________ 15 _________
... ... ... ...
18 ________ 18 _________
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