# Sequences and Series: An Exploratory Unit

I had great ambitions to do a lot of schoolwork this summer. Instead I started, abandoned, and restarted a unit for a course that I’m teaching next year. That’s about it. It’s a new course for me: Advanced Precalculus. The other teacher and I have decided to totally mess around with the ordering of topics, and we put sequences and series as the second unit. Our department is also trying to integrate more problem solving in the curriculum, and so I tried to make this unit involve as much problem solving as possible [1]. I like that we’ll be doing it early in the year, because I want them to see immediately that we are not going to be focusing on plug-and-chug but real thinking.

Those of you who know me know that I am a pretty traditional teacher, and I have gotten in the habit of creating guided worksheets as a structural backbone for a lot of my classes. This is the first time I’m creating an entire guided unit. It isn’t flashy or have a good hook. It’s simply a scaffolded way to help kids think in an increasingly abstract way. It also gets at almost all the standard things in a sequences and series unit (except for recursive equations, which I threw out). To put it out there: I would never say that what I do is inherently engaging for my kids. But it does get kids talking. I guess what I mean to say is: these packets/worksheets that I tend to create don’t make kids like/love math, but it does get them to think about math. I’m not great at the former, but I’m definitely getting better at the latter.

The last thing I have to say is that although it may look pretty traditional (the questions), try to think about the packet if you were a student and you were in a class going through it. It builds up elegantly, in my opinion. The motivation for sequences comes out of a series of IQ-test-ish puzzles, and the motivation for series comes out of a lottery problem. No formula is given to students. There are connections drawn to graphs, and to a few geometric visualizations of sequences and series. Students are asked to conjecture and defend their conjecture at various times.

I’m including two copies below. The packet with my teacher notes, and the packet without my teacher notes.

With Teacher Notes

Without Teacher Notes (Blank)

[Word version of this to download: .docx… to see my teacher notes, go to “Review” and go to “Final Showing Markup”]

Huge thanks in the creation of this goes to @JackieB, who went through a lot of it page by page and gave excellent suggestions! Precalculus guru! Also I included a few blogposts at the end of the document which I stole wholesale from or adapted in my own way.

Lastly, yes, I know this is a long packet. Usually I think classes do this whole unit in single week, and there’s no way we’ll be done with it in that time. It’s an experiment. From what I’ve heard from teachers everywhere, sequences and series always get short shrift in precalculus classes because they come at the end of the year. But I think there is so much depth and abstract thinking that can be brought out of a unit properly done. I’m super nervous, but we’ll see if this is an experiment that fails or not.

[1] I’m liberally defining problem solving as having students deal with situations they have never dealt with before, and generalizing from those situations. But I understand I am giving them A LOT of scaffolding with which to do it.

1. Kelley Clark @kkssclark says:

What a phenomenal collection of information! This must have taken you forever. I’ll bet you can’t wait to actually use it.

Personally, I love sequences and series. We teach them in Alg II, then in Trig, and again in Pre-Calculus. We are making a decision this year as to whether to leave them in Trig or Pre-Calc, but we need to free up some room for conics somewhere.

Your unit flows so well, building on itself the whole way through. Thanks for sharing it!

2. Thanks! It did take a surprisingly HUGE amount of time. I don’t know, but probably total 15 or 20 hours.

3. I also always put s&s at the start of the course (two year course including calc and more for my IB students). The reason: it gets them into the habit early on of thinking about patterns, drawing conjectures, abstracting to general formulas. There is so much there that we can build on later when we do functions, calculus, etc. I find myself constantly saying “think back to sequences and series…”. And with my higher levels, this is where I also introduce proof by induction. It fits nicely with s&s AND is required knowledge for most other topics in the course.
Thanks for sharing all this rich material!

4. Holy swear word batman! This is awesome. Sam, I’m totally downloading that docx and using it. I may not even wait for my students to show up. This is also making me think about where to put sequences and series in my own pre-calc curriculum (and/or calc bc). Thanks!

1. Thanks! I’m pretty proud of it, but we’ll see how it goes once I actually USE it. I’m excited/nervous to see if I can make it work in class… some group, some individual, some homework… Mainly I’m worried about debriefing and having that go smoothly and not take AN INSANE AMOUNT OF TIME and going over everything. But I’m learning how to be less teacher-centered and this is something I know I have to practice doing (even if I do it poorly for a (long) while).

One thing I thought of that might be cool is that now that we’re doing seq. & series as our second unit, when we do rational functions, we can treat something like (2x^2-5x+1)/(-3x^3-1) as two sequences (the top polynomial sequence, and the bottom polynomial sequence). And then we can analyze the quotient of the sequences… Or something. Clearly it’s not fleshed out. But I like the idea that we can use sequences to talk about growth/rate of change… So maybe that might be an interesting way to go about end behavior of rational functions?

5. I’m not going to send you a check, but I should.

It’s like you were there in class with Gauss when he told that teacher to SUCK IT!

6. mr bombastic says:

Nice job. I like the tiled diagrams at the start – especially where you can visually see a squared term and a linear term. Also liked the proof without words. I have a couple of items for consideration.

On parts 3 & 4 you may want to provide a table or part of a graph and ask them for the 40th term or something. I think this reinforces what you have already stated about these just being linear or exponential functions. It also adds a little variety to a page full of words.

I would definately set up the tables in section 5 with the first row labeled “n” and the second row labeled “Sn” – again to reinforce that this is just a linear or exponential function with the first row being the independent variable. I explicitly tell students to think of the Sn notation as meaning exactly the same thing as S(n).

Several times I have just asked what is 1 + 2 + … + 100 with no buildup. Usually takes quite a few hints, but depending on the class, it can be fairly productive. Also have you done the rainbow method on this? Add the first & last (draw loop), then add the 2nd and 2nd to last (draw loop), etc.

On part 8 you might want them to write the same sum using several variations of summation notation (i.e. use different lower limits for the same sum). I found this to be a very helpful way to get students to understand how the notation works. It also triggers questions like, “can the lower limit be negative or a fraction”.

1. Thanks so much for the awesome feedback. You were so kind to go so carefully through it! I really like your thoughts/suggestions and I will look more carefully in the morning about changing things.

7. Bombastic’s comment matches the one from the Common Core standards: sequences are functions. So many kids lose track because of the a_n notation, especially if the same sort of notation is used for series sum (some books use t_n and S_n). This is also good prep work for calculus. Since the integral or derivative of a function is another function, the prep work is for a summation or difference rule to be another function.

Lastly, starting sequences from 0 instead of 1 allows you to make good comparisons between arithmetic sequences and linear functions, and between geometric sequences and exponential functions. Death to all these formulas that have (n-1) in them, I sez.

8. Lisa says:

I just found this and am adapting it to my class. I love it. They love it. Thank you so much for sharing. I love how it seems to naturally and logically raise student confidence too. I will consider moving s&s earlier in the school year, because I see how this can build their confidence and ability to see patterns for the other units.

1. OMG thank you so much for your comment. You don’t know how happy you just made me!!! I’m so grateful you decided to comment! It’s maybe my favorite thing about the blog… when someone says they could use something I post and it works! Because why reinvent the wheel?!

9. Dan Dudley says:

Thanks for sharing. It has been a couple of years on since you first posted, how has it been since then? I really like the idea of changing the order of things to have them fit into the curriculum better. I have a couple of topics that are at the end of the year that I moved earlier because they can have some “review” material included while introducing new topics. I really hate that almost every Pre-Calc book starts with Algebra II material and I feel that the kids get lulled into a false sense of “this course is going to be easy,” then they hit trig functions and they aren’t prepared.

10. Wow Sam. This is brilliant and exactly what I was looking for. Thank you for taking the time to make this so others can benefit from it. I love it!

11. The “aha!” moments are everywhere within this. I love it. Thank you!