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/
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