Finding Input-Output Rules
Date: 10/08/2003 at 09:49:20 From: Tammy and Jamie Subject: Number Patterns Hello, I am the parent of a 3rd grader who is having trouble in math. This week he's working on Patterns in Numbers. For example, In 12 20 35 Out 15 23 38 41 57 62 How can he explain the "changing" from in numbers to out numbers? He said he added 15 each time. Of course all were marked wrong. Please help us. Thanks, Tammy and Jamie
Date: 10/08/2003 at 09:57:32 From: Doctor Ian Subject: Re: Number Patterns Hi, The idea in a problem like this is to make up a rule that you can apply to a particular input number to get the corresponding output number. If you have the rule, making up the lists is pretty straightforward. For example, suppose my rule is 'multiply by 2', and my input list is 1, 3, 7, 12 To make the output list, I apply the rule to each input: Input -> Apply rule = Output 1 2 * 1 2 3 2 * 3 6 7 2 * 7 14 12 2 * 12 24 So my input and output lists look like this: In 1, 3, 7, 12 Out 2, 6, 14, 24 Does this make sense so far? What's less straightforward is to go in the other direction, i.e., to start with the two lists, and figure out what the rule is. Let's think about how we might do that. Suppose we're given the lists In 2, 4, 6, 9 Out 4, 16, 36, 81 The simplest kind of rule will look like add ___ Here's an example of that: In 4, 11, 15, 19 Out 6, 13, 17, 21 How can we find the rule? Try subtracting every input from its output: In 4, 11, 15, 19 Out 7, 14, 18, 22 Out - In 3, 3, 3, 3 If the difference is always the same, we know we have a rule of this type. Do you see why? Another simple rule would be the one I mentioned earlier: multiply by ___ In a case like that, we can find the rule by dividing the outputs by the inputs: In 1, 3, 7, 12 Out 2, 6, 14, 24 Out / In 2, 2, 2, 2 Let's try these with our mystery list: In 2, 4, 6, 9 Out 4, 16, 36, 81 Out - In 2, 8, 30, 72 <- Not a constant difference Out / In 2, 4, 6, 9 <- Not a constant ratio So neither of these rules will work. However, sometimes trying out simple rules can help you find a more complicated one! In this case, let's look only at the ratios: In 2, 4, 6, 9 <----- Out 4, 16, 36, 81 | These are the same! Out / In 2, 4, 6, 9 <----- What this is telling us is that if we divide the output by the input, we get the input again. That is, Out / In = In which means that Out = In * In In other words, the rule we're looking for is 'multiply the input by itself'. Finding these rules is more of an art than a science, and (as with most things) the more experience you get with them, the easier they become. There are particular techniques for dealing with certain kinds of patterns, but when you don't know what else to do, it's good to start with simple rules, like the ones I've discussed here. Often they'll work, but even when they don't, the results you get from trying them can give you ideas about other rules to try. I hope this helps! Write back if you'd like to talk more about this, or anything else. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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