Welcome back to high school! School started this past Monday for me and after four years teaching middle school I am back in high school! Although I do miss my sweet middle school kids and my colleagues at my last school, I must admit that I love teaching high school students again. And I LOVE THE MATH. I’m not sure if it’s so great for me because it is high school math, or if it is just nice to change things up after a while. Whatever the reason, I am extremely happy right now. I love my new school and everyone in it. I love my students already, and I love the subjects that I am teaching. Life is crazy busy, but good. Please don’t pinch me, because I don’t want to wake up from all of this amazingness.

Not that everything is smooth sailing, of course, because I am a NEWBIE. I get still get confused (or physically lost) almost everyday. I don’t know everyone’s names or what school seminars I’m supposed to go to (or again, WHERE), or how I’m supposed to do things, or where I’m supposed to get things. I can’t wait until I just know things!

As exciting as it is to teach a new level, and new subjects, there are some unavoidable pitfalls as well. Enter Algebra 2. My first unit is recursively defined sequences. I knew (and had even been forewarned) that subscripts would be difficult for them. But, before I actually taught it I didn’t know know how difficult or how to do it better. Experience really is the best teacher. So, day one was rather rough, but I adjusted and (hopefully) day two was much better for them. Next year I will know that this topic will take two days, and I will be able to structure it so that they do not go home freaked out the first day. Yes, I feel bad about that.

Next year, I think I will not even introduce subscripts until day two. I may just write **sequence = previous term + common difference or x common ratio**. Tables and graphs have really been our friends. I will need to create a homework that has patterns, tables, graphs, and then just ask them to write in words how to find the subsequent terms. Then, day two I will let them help me develop the notation so that the sequence definition will be more efficient to write (and subsequently put into the graphing calculators). Also, the graphs of sequences on the TI-84 are just treacherous. The zooms are nuts and the window option is so complicated. Eli promised me that they were working on recursively defined sequences in Desmos. I am just praying it is finished before next August so we won’t all have to suffer the TI graphing torture again next year. (Side note: I also didn’t know that Alg2 students weren’t proficient at the graphing calculator. I’m not either, and I was hoping that they could teach me. It may be good that we will all learn together though!) This year, I taught them how to enter the sequences in the TI-84 and we leaned on the tables, but the graphs were a mess. I made all of my graphs in Google Spreadsheets so I could show them what was happening in the long run. Thank goodness for spreadsheets!

I created an activity using Skittles for recursively defined sequences that decay. It went pretty well and I hope to blog about it in the next couple of days.

Thanks again to Sam for his AMAZING sequence and series packet! I leaned on it heavily during these two days and plan on using parts of it to make this new day one homework assignment that I am dreaming up in my head. As I have said many times before, I love this village.

Julie–

I wonder if your students would enjoy doing some graphic/visual sequences first. No numbers, just get a feel for the concept. Good luck with Algebra 2!

As in, draw the next step without telling me how many blocks would be in the next step?

Yup. Triangular #’s, square #’s as stacked dots. Or drawing a square, then drawing another square within w/ vertices at the midpts, etc. That’s one of my faves: perimeter and area of the squares have a nice pattern. Plus, Ss love how cool it looks.

That does sound cool! Do you have any pics of ones your students have done? I would love to see them. 🙂

If you give me your email, I can send a worksheet that I compiled with sequence problems and some other resources I have.

Suggestion:

You write

“sequence = previous term + common difference or x common ratio”

but this is supposed to be a way of describing the current term of the sequence, so it is better to write

“this term = previous term + common difference or x common ratio”

Then you can introduce the index variable n this way:

“term(n) = term(n-1) + common difference or x common ratio”

It is sometimes (usually) better to use “look ahead” and write

“term(n+1) = term(n) + common difference or x common ratio”

In either case it is now a very small step to the subscript form.

Examples are essential, such as term(n+1) = term(n) + 2.5

and term(n+1) = 1.2 x term(n)

I personally would avoid terms like common difference and common ratio until the end, as they are specific to arithmetic and geometric sequences. Don’t overlook the Fibonacci sequence term(n+1) = term(n) + term(n-1)

and the sum of integers term(n+1) = term(n) + (n+1)

and so on.

Wow. These are ALL excellent improvements on my idea! And I adore the Fibonacci sequence so I love that you included it here. Thank you so much!

Julie,

I haven’t picked up my TI-84 in five years now, but I think Zoom:Stat would help fit the window. Give it a shot anyway!

Jen

Do you have to turn the plots ON to Zoom Stat?

More:

It sounds as if sequences on graphing calculators was something the designers did as an afterthought. I would use pencil and paper! Squared paper if absolutely necessary.

Yes. I used mainly paper and pencil. But when investigating what happened over the long run (limits), the calc was helpful. I would love to avoid it all together. It’s so complicated and students will never remember it in the long run. I would love a better method.

If the sequence has a limit then loosely speaking we are at the point where there is no change from one term to the next. This means that you can replace both term(n) and term(n+1) by the same “thing”, call it L.

So, in the case of the geometric sequence term(n+1) = 0.2 x term(n) you get L = 0.2L, which clearly gives L = 0. But we knew that anyway.

If we had a slightly more interesting sequence, say

term(n+1) = 0.2 x term(n) + 0.5

then the limit is more interesting,

L = 0.2L + 0.5,

which gives the limit as L = 0.5/0.8 = 0.625

A few hand calculations and a plot should make this believable.

I did something similar with Skittles Friday. The students saw the limit by doing the activity and putting the numbers into a table and graph. We are going to do further calculations on Monday. I will blog about it soon!

Last year was my year to switch schools and take on a grade level I’d never taught before. I spent all of last week (our pre-planning week) being thankful that I knew where everything was and that I’m not having to start at square one with my lesson plans! All of the newness can be exhilarating and exhausting at the same time. I hope you have some friendly colleagues looking out for you!

Thanks Alisan! And YES. I have an amazing math department of EIGHT math teachers. I still can’t believe this sometimes. Two other math teachers are new as well, so it’s nice not going through this alone. 🙂

Julie – good luck in your new school! I hope you will join us for #alg2chat some Monday evening. Re recursive sequences: A student I tutored last year was using the CPM Core Connections Algebra textbook, which approached sequences in a very intuitive way; she was in 8th grade and had a better understanding of how to write them and work with them than many of my high school students. If you can get your hands on one of their textbooks, there might be some useful content there.

Thanks so much Wendy! I have the Alg2 and Geo. Will look for the Alg1 book on Mon at school. 🙂

I use a fun problem called the checkerboard problem from NCTM that is a cool recursive. If you email I will try to send it to you from work Monday.

Nikki

That sounds great! Thank you so much!

Here is a real life situation: Simple control of the speed of an electric motor and flywheel.

I am going to use a computer programming style, it’s more readable.

The counter n is a time variable.

rpm(n+1) = o.8 x rpm(n) + 1.3 x volts . . . . . x is times

Question 1 : 120 volts, what is the steady state speed of the motor?

Question 2 : how long (how many time steps) to reach 99% of steady state speed.

Question 3 : What can you say about the relationship between voltage and steady state speed?

( This is a discrete model of a first order linear dynamic system – and that sounds quite ferocious)