Maybe this method that I discovered from an exhaustive internet search is well-known to most, but maybe it’s not. So, I decided to post it just in case. That and, quite frankly, I am so freaking excited about it and my husband will have NO idea what I’m talking about if I decide to spew it all over him!

When I was in school a billion years ago, many things were taught as “guess-and-check.” For instance, factoring quadratics. (I learned an awesome no-fail method for that last fall from my cooperating teacher during my internship- I’ll post that sometime soon.) I don’t know about any of you, but I HATE guess-and-check. Besides, how do you even TEACH guess-and-check?

What prompted my diehard search for an easier way of finding function rules? My kiddos. I was out of work last week due to an emergency surgery (yeah, 4^{th} week of my first year teaching is a fabulous time to have your gall bladder removed!), so my sub covered the lesson involving finding function rules given a table. She’s completely competent (she’s my ESE teacher for one of my classes and she’s certified in math), so I don’t think it was a lack of instruction on the concept. She probably taught them the same exact way I would have: guess-and-check until you’re good enough to just “see” the rule. Well, the majority of my poor kids don’t get it- at all! So, today I was frustrated due to the gazillion questions during their end-of-unit test. Clearly, if they are all asking questions about the same test item something needs to be done.

Cue in research.

I found this explanation in a math forum. I’m so happy I did. It’s not the prettiest or clearest of explanations, so I’ll try to make it a bit prettier here.

Take, for instance, the below table.

Input, x |
Output, y |

1 |
7 |

2 |
11 |

3 |
15 |

4 |
19 |

5 |
23 |

The first thing you do is take 2 sets of numbers (ordered pairs):

(1,7) and (4,19) ; any 2 sets will work

Next, find the difference in x-values and y-values:

x-values: 4-1 =3

y-values: 19-7 =12

Notice that the difference in y’s is 4 times the difference in x’s:

3 * 4 = 12

Therefore, 4 is your multiplier for the rule.

So, we know that 4x plus or minus something = y.

From here, it’s easy to tell what we have to add or subtract to get y.

4(1) = 4

4+3=7

Thus, y= 4x + 3!

Again, this may be somewhat common knowledge in the math community at this point. However, I figured if I was clueless to this method then maybe (hopefully!) someone else is, too. At least, that is what I am going to tell myself to fall asleep tonight!

I cannot wait to show this to my kids tomorrow!!!

Your experience reminds me of when I learned about function patterns my first year as an undergraduate. All of a sudden I had a way to figure out what type of function generated a list of data — and even though it was very simple and obvious in retrospect, I was positive I’d never seen them before!

As another explanation of your method here, what you are doing is finding the equation of a line passing through two points by first calculating the slope (change in y divided by the change in x to get the ‘multiplier’), then plugging in one of the points to get the y-intercept!

And yet another way to think about this solution: The “steps” between my inputs are always 1, and the “steps” between my output are always 4. That is, I take 4 y-steps for every x-step; if I want a rule for the y-steps I can use dimensional analysis to say “4 y-steps per 1 x-step times X x-steps is Y y-steps” — and if I extend the data table back one value, I get input 0, output 3, so my starting y-step is 3. Thus, Y = 4X + 3 :)

Yes, exactly! It’s so funny that I just couldn’t think of the way to explain it to them without saying “slope” or using the formal method of finding it.

I was taught the guess and check way, so I have not come across this way of finding the rules before…interesting. I will have to play around with it. Thanks for the post!

Bless you! I’m furiously trying to help my daughter understand this (as she was not comprehending the lesson in school) and I am “math-dumb”. You have saved the day for both of us. Thank you!

That is so great! Thanks for letting me know!