Category Archives: functions

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












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


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!!!

Tagged ,

Ramblings of a HS Math Teacher

Productive Struggle

Teachers Working Together to make our Struggles Productive

Just My Lucky Life

Life after infertility, adoption, and gastric bypass surgery

King of the hills

Ebenezer's blog!


Are you pondering what I'm pondering?

PPerfect Squares

Simple roots everywhere

Quantum Progress

Raising the energy level of my physics classes

War and Piecewise Functions

a math teacher blog. all opinions are my own.

It's all math.

Teach it. Learn it. Live it.

being less helpful

an intermittent blog on Math Education

the space between

Living, loving and DIYing in the now, that little space between what has been and what will be.

Math and more Math

Heather Simmons


Just another site

I Speak Math

Integrating Technology and Mathematics

Continuous Everywhere but Differentiable Nowhere

I have no idea why I picked this blog name, but there's no turning back now

%d bloggers like this: