## Hallelujah! Finding two-step function rules from tables

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, 4th 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!!!

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