Aha moment, and frustrations with the Algebra 2 Textbooks.

I guess I have to thank the Dana Center for developing their Algebra II sequence.  Without it, I might still have been blind to something so simple, that I can't believe I never saw it before.  The only consolation is that when I showed it to other teachers in the building, their reaction was very similar.  Why? Because the Dana Center suggests starting Algebra II with sequences and series.  I kind of had my doubts, but our district decided to follow this outline, so I dove right in.  Well maybe not right in.  I spend a couple of days doing some other basic Algebra I concepts which included the point-slope equation. And that's where my aha moment came, in fact it didn't really come until the act of teaching.  I had for years gone through the process of just showing these formulas that are provided for them on the End-Of-Instruction exam.  

I always knew that the arithmetic sequence gives us a line straight line of dots like this:

But what I failed to see until now was that the textbooks and the testing companies all have this formula "wrong".   Mathematically what they have is correct, but in the scope of what the students' already know, it is wrong.  The aha moment came because I had been reviewing the point slope equation just the day before starting sequences. Piggybacking off of the point-slope equation, which is covered in Algebra I, if not sooner, the arithmetic sequence  equation should  look like this:

It's a subtle difference, but to me it makes all the difference in the world.  Now my students don't have to memorize one more formula, now they can utilize something they are familiar with.  The common difference is now just the slope between two points, and the point is just  

Now the students can apply what they know about lines and functions and utilize it for this special case of the arithmetic sequence which deals only with integer inputs.  

And the cool part is, when they see a problem where they are asked to find the rule for the nth term, like this:

They don't have to set it up as solving a system of equations, (as the textbook suggest them to do) they can look at this a two points (4,31) and (10,85) where they can find the slope (which will be the common difference) and then use one of the points to complete the nth term equation.Since the "new" arithmetic sequence equation can be written for ANY point.

I'm not sure my kids appreciated the "aha" moment as much as I did, but it gave me a whole new perspective on how to approach arithmetic sequences.  In the past the arithmetic sequence stuff is buried in the back of the textbook so by the time we got there the thought of point-slope has long be buried under a pile of other mathematical concepts like rational functions  logs, conic sections.  I'm just glad I was able to see this so clearly now.