Terms and Rules

Date: 12/14/2001 at 09:55:30
From: William Johnson
Subject: Terms and rules

My step daughter is in sixth grade and she has been doing a pattern 
journal where she has two columns of numbers: the first column is the 
n, and the second colum is the term, and she has to find the rule 
(e.g. n2), etc. Is there some place I can find out the basics for this 
concept?

Date: 12/14/2001 at 23:17:02
From: Doctor Peterson
Subject: Re: Terms and rules

Hi, William.

I don't know that there is much to say in general about this kind of 
problem. A lot depends on the level of difficulty; most likely she has 
been given a set of problems that all have a similar kind of pattern, 
so that a method can be developed for solving them. The problems can 
differ in the presentation (whether consecutive terms are given, for 
example), and in the complexity of the rule (a simple multiplication, 
a more complicated calculation from n, a recursive rule - based on the 
previous term - or even a weird trick rule like "the number of letters 
in the English word for the number n"). For that reason, it would be 
very helpful if you could send us a couple sample questions so we 
could help more specifically.

I don't see any good examples in our archives at the elementary level 
- probably just because we've never felt that any one such answer was 
useful to help others. Let's try a few samples. Here's an easy one:

     n | term
    ---+------
     1 | 3
     2 | 6
     3 | 9
     4 | 12

Here you may just be able to see that the terms are a column in a 
multiplication table (or, more simply, that all the terms are 
multiples of 3); or you might look at the differences between 
successive terms (a very useful method at a higher level, called 
"finite differences") and see that the terms are "skip-counting" by 
3's, a clue that the rule involves multiplication by 3. However you 
see it, the rule (using "*" for the multiplication sign) is

    3 * n

Here's a slightly more complicated one:

     n | term
    ---+------
     1 | 6
     2 | 11
     3 | 16
     4 | 21

Here the differences are all 5, suggesting multiplication by 5. You 
might want to add a new column to the table so you can compare the 
given term with 5n:

     n | 5n | term
    ---+----+------
     1 | 5  | 6
     2 | 10 | 11
     3 | 15 | 16
     4 | 20 | 21

Now you can see that each term is one more than 5n, so the rule is

    5n + 1

Now they might get still more complicated:

     n | term
    ---+------
     1 | 2
     2 | 5
     3 | 10
     4 | 17

Here the differences aren't all the same (3, 5, 7), so something 
besides multiplication and addition must be going on. I happen to know 
that when the differences are themselves increasing regularly (by 2 
each time in this case) that there is a square involved; try adding a 
column for the square of n and see if you can figure it out.

This just gives a small taste of what these puzzles might be like. 
They can get much harder. Actually, in many harder cases I think the 
problems can be very unfair, because in reality (when you don't know 
ahead of time what kind of rule to expect) the rule could be 
absolutely anything, such as "the nth number on a page of random 
numbers I found in a book"! It's really just guesswork, and sometimes 
it's really hard to say which of several possible answers is what the 
author of the problem might have had in mind. But at your daughter's 
level you don't have to worry about that; it may be mostly just a 
matter of recognizing familiar patterns such as skip counting or 
squares. It's only where I sit, seeing problems completely out of 
context, and having to make a wild guess as to what sort of rule to 
look for, that these problems are always challenging.

If your problems aren't like these, and you need more help, feel free 
to show them to me, so I can suggest ways to handle them.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   

Date: 12/15/2001 at 13:39:31
From: William Johnson
Subject: Terms and rules

Thank you, Dr. Peterson. That is exactly what I was talking about and 
you gave me great examples. One question I do remember was connected 
pentagons 1 = 5, sides -2 = 8 - 3 = 11 etc. where n is the number of 
pentagons and the term is the number of sides exposed, and then we 
needed to find the rule for 100 pentagons. Your answer definitely 
helped.

Thanks, 
William F. Johnson

Date: 12/15/2001 at 22:30:49
From: Doctor Peterson
Subject: Re: Terms and rules

Hi, William.

My answer is most helpful if you are just given a list of terms, and 
have to guess the rule. Your example is actually easier in some 
respects, and harder in others. It's common to just make a table of 
terms based on the geometry of the problem, and then try to guess a 
rule from that. But how do you know that the rule is really the right 
one, when it's not just an arbitrary list of terms, but one generated 
by a real situation? You really need to find some reason for 
connecting the rule to the objects you are counting. For that reason, 
I recommend trying to find a rule in the counting process itself.

How do you count the exposed sides? You start with 5; then when you 
add a second pentagon, you add 5 more sides, but one side from each 
pentagon becomes "hidden"; so the new number is 5 + (5 - 2) = 8. When 
you add another (presumably always on the "opposite" side of the 
pentagon, to avoid overlapping with those already placed), you again 
add 5 and take away 2; so the rule seems to be that you start with 5 
and add 3 for each pentagon after the first. This leads directly to 
the rule, with no need to guess.

So you can use the guessing approach if you want, and I suspect many 
texts and teachers expect that, but it's much more satisfying to skip 
the tables and really know you have the right answer!

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/