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!!!
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!