Finding a Rule That Fits an Input-Output Problem

Date: 08/26/2006 at 17:48:48
From: Diane
Subject: What strategy do I use to help my child with input/output?

My daughter has this input/output problem:

Input        Output
  12           10
               16
  32  
  62           35
  70           39
  86
     
She has to state the rule and fill in the missing numbers.  These
numbers do not seem to line up in a pattern.  I don't have a clue
where to even begin.

Date: 08/26/2006 at 23:06:36
From: Doctor Peterson
Subject: Re: What strategy do I use to help my child with input/output?

Hi, Diane.

The first step in your strategy, I think, would be to find out from
your daughter what kinds of "rules" they have been using, perhaps
looking at examples from her text or worksheets.  For me, unable to do
that, the puzzle is a lot harder--in general, there are infinitely 
many possible "patterns" one could choose, so with no context it's 
just a guessing game.  And I see some puzzles that look a lot like
this one with much different rules.

But I've seen enough of these to know that for younger children,
typically the rule involves one or two operations, often an addition
and a multiplication (which, in algebra, would be called a linear
function, but here might be called a "two-step rule" or something like
that).

I don't know whether teachers present any specific strategy, or just
expect children to make a lot of guesses (maybe based on the examples
they've seen) and so get in a lot of arithmetic practice while trying
out ideas.  But I have found a very useful strategy.

The idea behind it is that if you multiply by something and then add
something to the result (algebraically written as "y = ax + b"), then
CHANGES in x are multiplied by the same thing, and the addition
doesn't affect the change.  For example, if the pattern were y = 4x+3,
then when x=2, y=11, and when x=5, y=23.  The difference between 2 and
5 is 3; the difference between 11 and 23 is 12, which is 4 times as
much.  The fact that we added 3 to each product doesn't affect the
change.  So we can find the multiplier by comparing the differences.
In your example, you might make a table like this (looking only at the
lines in which both numbers are known):

  Input Change Output Change
    12           10
    62    50     35     25
    70     8     39      4

That is, 62-12 = 50 (the change in the input), and 35-10 = 25 (the
change in the output for the same pair).

Notice that the change in the output is always half the change in the
input.  That tells us that the multiplier is 1/2.  So the rule starts
with "multiply by 1/2" (which is the same as "divide by 2").

Can you see how to find the rest of the rule?  See if you can work it
out.  Here are some examples of this sort of problem, in which I
finished the task (or gave a strong hint):

  Multi-Step Patterns
    http://mathforum.org/library/drmath/view/64042.html 

  Terms and Rules
    http://mathforum.org/library/drmath/view/58001.html 

If you have any further questions, feel free to write back.  I'd be
interested to hear from you whether your daughter has been taught any
strategy, and how her teacher says to solve this problem.  There are
probably many ways to approach it; mine is that of a mathematician,
and there may well be one that is more natural to a kid.

To tell the truth, I initially solved this one by a slightly different
method.  I focused my attention on the biggest example, 70 -> 39, and
looked for the closest simple multiplier.  Since half of 70 is 35, it
was a good guess that the rule might be to take half, and then add 4.
Then I just checked the other examples to see whether that worked for
all of them.  The idea behind this method is that multiplication by
big numbers "swamps" any addition; so you can see the multiplicative
behavior by looking at big numbers.  That's an insight that comes from
years of experience; but kids might be able to see it too.


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