Sequences and Series

A few days ago I posted a card sort I did to start my unit on Sequences. I figured I’d share my entire packet for Sequences (2016-10-31-sequences [doc] 2016-10-31-sequences [pdf]) in case it’s any help for you. This was designed for our standard precalculus classes, and I have to say it worked pretty well.

  1. It allows for kids playing with math at the start (card sort, some visual patterns, a 3-act)
  2. It doesn’t “tell” kids anything. They discover everything. And sometimes asks for a couple different ways to do things.
  3. Kids messing up notation is always an issue with sequences. So I introduced notation way after kids started playing around with sequences — and got a handle on them. I did still see some kids confuse n and a_n (the term number and the value of the term), but it wasn’t a huge number. I actually was pretty proud of that.

Note: Problem #17 on page 23 is actually ill-posed. So I didn’t have my kids work on it. I have a replacement warm-up question (included at the bottom of this post) which worked wonders!

Also in the packet, there is a blank page (on page 4) which simply says “A Pixel Puzzle.” For that, I used Dan Meyer’s 3-Act.

pixel.png

For me, the 3-Act was a mixed bag. Mainly because I thought it would be easier than it ended up being for my kids, so I didn’t plan much about how to help groups get started. At the beginning of the 3-Act, kids asked great questions, but not the one I was hoping for (when does the pixel hit the border). So I didn’t have a smooth way to deal with that. And I anticipated it would take less time than it actually did, so the 3-Act got split up between multiple classes. And when kids were working, I don’t think I was strong at facilitating them working. I should have added in a wee bit more structure to help kids out. Even if it’s something as simple as coming up with ways to get kids to think about making a table or a graph — and what would be useful/important to include in the table/graph. And I didn’t have a good way for kids to share their findings. So I’d need to think about the close to the 3-Act better. I’d give myself a “C.” Would I do it again? Yes. I saw the potential in it. It got kids thinking about sequences visually. It had kids thinking temporally. And had them relate tables/graphs/equations. All good things!

One last thing I want to include in this post… This is related to the ill-posed Problem #17 on page 23 I made note of before. There is one type of question which tends to flummox kids when it comes to geometric sequences. It reads something like “The third term in a geometric sequence is 6 and the seventh term is 96. Come up with a formula for the nth term in this sequence.”

The reason this is tricky is because there are two possible sequences! (The common ratio could be 2 or -2.)

Thus, I created this warm-up (.docx) for my kids to check if they truly understood what we were doing. The conversations were incredible. Some groups were done in 7 minutes… Others had a solid 15 minutes of discussion.

sequence.png

 

 

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4 comments

      1. Thanks so much! I really appreciate the comments in the document with links and teaching tips. How long do you spend on this handout in class? I typically cover this material in about 2-3 days and then go on to series and summation and then Pascal’s Triangle and the binomial theorem and the whole unit lasts about 11-12 class days (3 weeks using our schedule). I use some of the same examples. I particularly like the one when subdividing the circle yields the sequence 1,2,4,8,16,31. I used to use that as the reason we need to prove formulas using mathematical induction. Unfortunately, I rarely have time to include that in the curriculum now. (Do you do proof by mathematical induction in Precalc?) I also love the fractal pattern of squares which yields so many interesting sequences.

      2. I can’t remember exactly. This handout in particular, with the card sort and with the 3-Act, probably took 10 or more class days. They are doing all the discovering — and it takes them a while to figure out and make clear a lot of the ideas. We don’t do induction.

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