IX.  Analysis of Trends and Patterns

Section 1 Introduction

Patterns and trends are found very frequently in the physical and social sciences, as well as in mathematics. When they occur it is often useful to identify them precisely because they pave the way for predictions about future events. For example, here is the pattern of average minimum temperatures for certain months in New York City.

Based on this pattern, approximately what would you expect the average minimum temperatures to be for January, April, July, and October of 1976?

Patterns and trends are found in almost all areas where regular, periodic observation is made. This includes changes in the distances between heavenly bodies; sales records of products ranging from shoes to automobiles; rainfall, wind velocity and other meteorological measures; and even human statistics such as birth, suicide and health patterns. Generally the cycles are more complex and erratic than the simple example above. Nevertheless, by systematically analyzing groups of observations, consistencies are often found which help bring organization to large bodies of facts, and furnish more comprehensible, usable pictures of the universe in which we live.

The problems in this chapter give you practice in identifying patterns and trends among numbers and letters. Not only will your grasp of patterns increase, but because patterns are really recurring relationships, working the problems will also improve your overall skill in analyzing relationships. Furthermore, many students have reported that by going through the problems in this chapter they developed greater confidence and skill in performing arithmetic operations.

Section 2 Sample Problem

This series of letters follows a certain pattern. Try to discover the pattern and write the 3 letters which should come next.

If you are using this book in a class, one student should solve the problem aloud at the chalkboard.

There is one more part to this problem. Describe in your own words the pattern of the letters. You may find this a little difficult at first. However, if you were able to decide which 3 letters came next, then you have discovered the pattern.

Write the pattern description below (or on the chalkboard).

Pattern description:

Section 3 Problem Solution

Original Problem

Problem Solution

Section 4 Sample Problem

In this problem, numbers are arranged according to a pattern. Identify the pattern, decide which three numbers should come next, and write the pattern description. In a class one student should work the problem aloud at the chalkboard.

Pattern description:

Section 5 Problem Solution

Original Problem

Problem Solution

Section 6 Sample Problem

Decide which 3 numbers should come next in this series and write the pattern description. In a class, one student should work the problem aloud at the chalkboard.

Pattern description:

Section 7 Problem Solution

Original Problem

Problem Solution

Section 8 Writing The Pattern Description

In the last section the problem solver wrote this pattern description:

Pattern description: Add 5, subtract 3, add 5, subtract 3, etc.

There are various other ways to phrase this same idea. For example, here is a second way.

Pattern description: Alternately add 5 and subtract 3.

Any phraseology which fully expresses this notion is equally good. All that’s important is that the pattern description show the basic principle underlying the pattern.

Sometimes different problem solvers will analyze the same pattern in different ways. Here is another way to approach the problem. Notice the numbers that are underlined.

These numbers form the series: 2 4 6 8. This is a series which simply increases by two each time.

Now look at the remaining numbers.

These numbers form the series: 7 9 11 13. This series also increases by two each time.

A person could look at the original problem as two separate, alternating series—one starting with 2, the other starting with 7—and both increasing by two. From this point of view a good pattern description would be:

Pattern description: Two alternating series of numbers, each increasing by two.

If in working the problems in this chapter, you and your partner come up with different pattern descriptions, first check that they are both really correct. If they are, you will see that they are actually two ways of looking at the same pattern, and both ways will lead to the same answers in filling the blanks.

For some of the problems which you will work later in the chapter, phrasing an accurate pattern description will greatly challenge your verbalizing skills—exercising and strengthening them. The talent to paint in words things that are seen and felt is the distinction of successful writers and public speakers. Teachers are also greatly dependent on this skill if they are to be effective. For example, an excellent gymnastics teacher is able to articulate clearly the positions, the movements, the twists and turns through which a student must direct his body. You have seen in earlier chapters how important vocalizing is in teaching verbal and mathematical problem solving. Teaching in any area means communicating. In fact, for the rest of your life, in both professional and social settings, you will often need to explain things to people. The better you do it, the greater your chances are for vocational advancement and personal happiness.

The Boyer Commission of the Carnegie Foundation states in its report Reinventing Undergraduate Education (1998):

For the problems in this chapter, if you can fill the blanks then you know what the pattern is. If the pattern is somewhat complex, don’t expect to be able to describe it in just five or ten words. It may take 25 or 30 words—maybe 3 sentences—to describe the pattern fully enough so that someone else can understand it from your description. Take all the time and space necessary to do the job of describing the pattern well.

Section 9 Sample Problem

Write the next 3 entries in this series and the pattern description. In a class, one student should work this problem aloud at the chalkboard.

Pattern description:

Section 10 Problem Solution

In reading this problem analysis take special note of three points which may help you in your own problem solving:

1. One hypothesis considered, checked, rejected, and another one formulated. At first, the problem solver thought the letters and numbers were related. But after pursuing this for a while and not finding any relationship, she decided to deal with the letters and the numbers separately.

2. Confusion, error, checking, writing the alphabet, and correction. In analyzing the letter pattern, the problem solver got confused and made an error. However, good problem solvers continually re-check work that they are in any way unsure about. In re-checking, the problem solver wrote out the alphabet rather than trying to analyze the pattern completely in her own mind. She knew this would lead to greater accuracy, and it allowed her to find and correct her mistake.

3. Blanks for letters filled first. The problem solver filled the blanks with the appropriate letters before starting to analyze the numbers. She didn’t depend on her memory to recall the letters later. Instead, she wrote them as soon as she figured them out.

Original Problem

Problem Solution

Section 11 Sample Problem

This problem is a little different than the others. However, there is a systematic trend which you can discover through careful analysis, and then use to fill the blanks.

Pattern description:

Section 12 Problem Solution

In reading this solution, notice how the problem solver came to identify the trend in small, gradual steps. During her initial reading of the problem, she compared the entries, but only noticed that something was changing near the end of each entry. She didn’t even realize that each element involved exactly the same letters. Then she carefully compared the entries again, focusing on the sections that seemed to be changing, and she obtained new information about the types of changes that were occurring. In this way—by making many comparisons and carefully noting the differences from one entry to the next—she was able to pinpoint the trend.

It is generally necessary to make numerous comparisons because your mind can absorb only a limited amount of information at one time. You make comparisons and learn something about the problem. This helps you decide which comparisons to make next. You make more comparisons and you learn more about the problem. Gradually, by noting the similarities, the differences and the changes among the entries, you get a picture of all the relationships existing in the problem. The heart of this process is numerous, careful comparisons.

Use this same method in working later problems. Continue to make comparisons until you are sure you understand the relationships completely.

Original Problem

Problem Solution

Section 13 Summary

A good problem solver begins one of these problems by reading along in the series, looking for patterns. He identifies similarities and differences among the entries, and makes mental notes of relationships that he sees. For example, he may observe that the series is composed of letters alternating with numbers, that the numbers seem to be increasing, and that the letters appear to be moving backwards in the alphabet.

As the problem solver gains familiarity with the series, he tries to specify precisely the underlying pattern. He attempts to formulate in his mind some rule which explains how the letters or numbers change in going from one to the next. When he thinks he has found the rule, he checks it against the entire series to be sure it is completely correct. If any part of the series doesn’t agree with his rule, he changes his rule until it portrays the entire series accurately.

Finally, when the problem solver is sure his rule is correct, he uses it to fill the blanks, and then writes his rule in the space designated “pattern description.”

An important point in working these problems is that the final rule that you formulate and use to fill the blanks must be valid for the entire series. It cannot be only approximately correct. Each of the problems in this chapter was devised according to a definite pattern, and your task is to find a rule which fits (or describes) the pattern.

Quite possibly, as the problems become more difficult, there will be some for which you cannot find the pattern. This does not reflect poor reasoning ability on your part. If you systematically analyze a problem—carefully comparing the elements and looking for relationships—and you still cannot find the overall pattern, it simply means that you have not had background experience with the type of pattern used in that problem. However, if you formulate a sloppy rule—one which does not fit the series perfectly—and if you do not realize that your rule is incorrect because you did not check it carefully—then this is a sure sign of poor reasoning ability. The most serious error that you can make in working these problems is to fill the blanks incorrectly because the rule you formulated does not precisely fit the series. This is pure carelessness. It is much better to leave a question unanswered, than to answer it incorrectly and not recognize your error. The heart of good reasoning is being very careful in formulating your rule, and then thoroughly checking it to determine whether it fits the facts perfectly.

The activities of a good problem solver described above are very similar to the steps preceding many scientific breakthroughs. A scientist such as Charles Darwin observes some facts, gets an idea, collects more facts, formulates the idea precisely into an hypothesis, collects still more facts which either support the hypothesis or lead to a new one, and gradually “discovers” a valid, scientific law. The problems in this chapter can be regarded as miniature problems in scientific analysis and discovery.

Quiz Yourself

1.   What is the most serious error that you can make in working the problems in this chapter?

2.   Describe in your own words the approach used by good problem solvers in working trend problems.

Problems in Identifying Patterns

Work in pairs. If there is a certain problem that you cannot solve after several repeated attempts, ask one of the other pairs for a hint. If that also fails ask your instructor for help.

1.   2 7 10 15 18 23 26 31 34 39 ___ ___ ___

Pattern description:

2.   A B A B B A B A B B A B ___ ___ ___

Pattern description:

3.   9 a 8 c 7 e 6 ___ ___ ___

Pattern description:

4.   9 12 11 14 13 16 15 18 ___ ___ ___

Pattern description:

5.   B A D C F E H G___ ___ ___

Pattern description:

6.   Q Q L Q Q Q Q L L L Q Q L Q Q Q Q L L L Q ___ ___ ___

Pattern description:

7.   27 24 22 19 17 14 12 9 ___ ___ ___

Pattern description:

8.   A Z B Y C X D ___ ___ ___

Pattern description:

9.   32 27 29 24 26 21 23 ___ ___ ___

Pattern description:

10.   1 12 121 1212 12121 _____ _____

Pattern description:

11.  8 10 13 17 22 28 35 ___ ___ ___

Pattern description:

12.  147 144 137 141 138 131 135 132 125 ___ ___ ___

Pattern description:

13.  A Z C X E V G ___ ___ ___

Pattern description:

14.  J 1 P 3 M 5 J 8 P 1 M 3 J 5 P 8 M 1 J 3 ___ ___ ___

Pattern description:

15.  A1 B2 D4 726 J10 C3 A1 G7 N14 M13 E___ A___ B___ ___3 ___16

Pattern description:

16.  5 10 20 40 80 160 ___ ___ ___

Pattern description:

17.  13R 16P 20N 25L 31J ___ ___ ___

Pattern description:

18.  120 115 109 102 94 ___ ___ ___

Pattern description:

19.  A B C___ A 1 2 3 2 1 A B C B A 1 ___3 2 1___ ___C B A

Pattern description:

20.  B M N ___ B 2 13 14 13 2 B ___ N M B 2 ___ 14 13 2 ___ ___N M B

Pattern description:

21.  2 6 18 54 162 ___ ___

Pattern description:

22.  A2 3B B4 3A C2 3D D4 3C E2 ___ ___ ___

Pattern description:

23.  A A B B D D G G K ___ ___ ___

Pattern description:

24.  A C E C E G E G I G ___ ___ ___

Pattern description:

25.  64 32 16 8 4 ___ ___ ___ ___

Pattern description:

26.  1 11 20 ___ 1 A ___T K A 1 ___ 20 11 ___ ___ K T K ___

Pattern description:

27.  ACEG GACE EGAC CEGA ___ ___

Pattern description:

28.  XOXOOOO OXOXOOO OOXOXOO _____ _____

Pattern description:

29.  2 4 3 6 5 10 9 18 17 ___ ___

Pattern description:

30.  49 48 46 43 39 34 ___ ___ ___

Pattern description:

31.  8 9 7 10 6 11 5 12 ___ ___ ___

Pattern description:

32.  1 3 6 8 16 18 36 ___ ___ ___

Pattern description:

33.  5AA 10BB 12CD 17DG 19EK ___ ___

Pattern description:

34.  D G F H K J L O N P S R ___ ___ ___

Pattern description:

35.  1 26 24 120 ___ 5040 ___

Pattern description:

36.  1 34 7 11 18 2 30 32 5 8 13 5 12 ___ 3 18 ___

Pattern description:

37.  15 4 19 5 80 85 11 2 13 63 ___ 4 ___ 14 ___ 8 29

Pattern description:

38.  2 3 5 8 13 21 ___ ___ ___

Pattern description:

39.  1 2 3 6 11 20 37 ___ ___ ___

Pattern description:

40.  5000 6000 5900 6900 6800 7800 7700 _____ _____ _____

Pattern description:

41.  13 18 20 19 24 26 25 30 32 ___ ___ ___

Pattern description:

42.  4 14 21 26 36 43 48 58 65 ___ ___ ___

Pattern description:

43.  7 3 4 8 4 5 10 6 7 14 10 11 22 ___ ___ ___

Pattern description:

44.  d a a e c c f e e ___ ___

Pattern description:

45.  c i o d j p e k q ___ ___

Pattern description:

46.  d g h e j k f m n ___ ___

Pattern description:

47.  u n g t m f s ___ ___

Pattern description: