Interesting Quadratic Relationship - Tape Measurer

Spurred on by What can you do with this: Annuli by Dan Meyer,

I handed out tickets to the students as they entered my classroom on the first day of class in my Algebra II class, on each ticket, I write a number for their desk, that they have to find in the room.  After some of the obligatory first day stuff, I have them inquire about the roll of tickets and through a joint process we slowly come up with a plan of action on how to measure the ticket roll and estimate the number tickets.  The exercise goes fairly well, with the sole purpose to get them to start thinking of math as to answer questions of curiosity.   Anyhow as an extension, I was thinking about these big 30 meter tape measurers we have in our science department, and wondered if we could come up with a relationship between the number of turns it takes to roll in the tape and the reading on the tape measurer.   We collected the data a few days ago, but haven't done anything with the data yet.  However, I took a sneak peak  at the data and it provides a very strong quadratic relationship.

I plan to have the students plot the data and see if they can developed a quadratic function from data by choosing 3 points and solving a system of equations.  Obviously not quite as accurate as doing a full blown quadratic regression on the calculator, but hopefully it give them a feel for solving a system of equations as well as being able to make some predictions with their data.  Anyhow those are my thoughts at the moment.

In addition, I think I would like them to think about WHY this is a quadratic relation.  I'm not sure how well that exploration will go, but I am curious what they think.  Comments???

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.

It's time to start.

Well, after 10 years of teaching and what seems as hundreds of hours lurking on other math blogs, I've decided to start my own.  I'm not sure that I have that much to add to the math world, but if nothing else it will give me a place to document some of the things I do.  And with all the push towards Common Core, it seems like now is a good time to put in my two cents worth when I can.  I may also start a 180 blog soon although at this point it might be a 150 blog, but we will see, as this is all new to me.

Just a little background on me, as stated I completed my 10th year of teaching.  Prior to teaching I spent some 20 plus years in both engineering (fun) and sales (not so fun).  Although sometimes, I feel like every day of teaching is like another day in sales where I'm continually trying to convince a room of 30 some customers that what we have here is the latest, greatest product.

Currently I'm teaching  3 sections of PreAP Algebra II, 1 section of AP Statistics,  1 section of AP Physics C, and  after school is out I coach (well maybe manage is a better term) our school's FIRST Robotics Team, Team 2359, the RoboLobos.

Hence the name of the blog:  MSRP (math, statistics, robots, physics),  Hopefully most of you caught the play on the term (manufacturer's suggested retail price), and the idea that teaching is priceless.

Anyhow, we will see how this goes.