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.


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.




2, 4, 6, 8, what do we appreciate? A Card Sort!

This year I’m teaching both Advanced Precalculus and Standard Precalculus. (Totally confusing on a daily basis? Yup.) And I’m working with two other teachers to write the Standard Precalculus curriculum from scratch. Of course this is something that is daunting, but I love to do when I have the time and like-minded colleagues.

I was in charge of spearheading our sequences and series unit. In this post, I want to briefly share how we started the unit. Instead of diving right in, or doing something intense, I wanted to gently get some good conversations percolating. So I handed out this set of cards:


and gave them this set of instructions:


I debated having kids use Desmos for the card sort, but since I have kids work in groups (mostly groups of 3), and I wanted the entire group to be working together, and I wanted them to actually physically move and shuffle cards, I decided to use physical cards. I also had all kids stand up while doing the card sort. I had a feeling that would be magical, in terms of getting kids talking, moving, and engaging with each other (even thought they were all at the same table), and it was! So I highly recommend that.

These cards end up having three different types: arithmetic, geometric, and recursive.

Most kids got the arithmetic sequences quickly, but it was interesting to watch them struggle with the geometric and recursive. There were great conversations, and because I demanded the next number for each of the sequences, kids had to really think through what the pattern was (and in geometric sequences, how to find the common ratio). I had thought that kids would finish this really quickly, but I was totally wrong. It took about 20 minutes. So plan accordingly. (A few groups needed to do a little bit more at the end of class, so I had them take a photo of their card sort and use that photo to finish things up!)

One note: Card H which has the sequence 0,0,0,0,0,0 fits all three categories. So it’s great fun to watch kids try to place it.

I wanted to share this activity because I haven’t really done many card sorts before —  and I was so pleased that this particular one generated productive conversations. So I need to keep this teaching tool in my arsenal for generating conversations about something new. (Example: I just thought of giving a bunch of graphs of rational functions to kids on cards, before we start that unit, and say “find different ways to sort these!” There are so many features, so that could lead to so many different ways to sort the cards. I suspect that Desmos would be good for that particular card sort, since there would be many different ways to sort those pictures, and I’d want to project the different ways kids did it to the entire class. I bet through that sort, we could actually recognize vertical asymptotes, horizontal asymptotes, oblique asymptotes, and holes!)

Here is the .docx (2016-10-31-card-sort-for-sequences) and .pdf (2016-10-31-card-sort-for-sequences) for the cards.

I will try to write up some more about my sequences and series unit soon!

Binomial Expansion

This is going to be a super short blogpost. But I’m excited about a visualization I came up with today — as I was working on a lesson — for showing why Pascal’s Triangle works the way it does with binomial expansions.


I’m sure that someone has come up with this visualization before. It feels so obvious to me now. That that didn’t make me any less excited about coming up with it! I immediately showed it to two other teachers because I was so enthralled by it. #GEEKOUT

I am thinking how powerful a gif this would be. Start out with 1. Have two arrows emanate from that 1 (one arrow saying times x and one arrow saying times y) and then it generates the next row: 1x    1y. And again, two arrows emanate out of both 1x and the 1y (arrows saying times x and times y). And generating 1x^2    1xy     1xy     1y^2. Then then a “bloop” noise as the like terms combine so we see 1x^2     2xy     1y^2.

And this continues for 5 or so rows, as this sinks in.

Then at the very end, some light wind chime twinkling music comes up and all the variables disappear (while the coefficients stay the same).

Of course good color choices have to be made.

Who’s up for the challenge?

Okay, I’m guessing something similar to this already exists. So feel free to just pass that along to me. Now feel free to go back to your regularly scheduled program.

Visualizing Standard Deviation

A few days I got an email from someone (Jeremy Jones) who wanted me to look at their video on standard deviation. And then today, I was working with Mattie Baker at a coffeeshop. He was thinking about exactly the same thing — how to get standard deviation to make some sort of conceptual sense to his kids. He said they get that it’s a measure of spread, but he was wondering how to get them to see how it differs from the range of a data set (which also is a measure of spread).

Of course I was hitting a wall with my own work, so I started thinking about this. While watching Jeremy Jones’s video, I started thinking of what was happening graphically/visually with standard deviation.And I had an insight I never really had before.

So I made an applet to show others this insight! I link to the applet below, but first, the idea…

Let’s say we had the numbers 6, 7, 7, 7, 11. What is the standard deviation?

First I calculate the mean and plot/graph all five numbers. Then I create “squares” from the numbers to the mean:


The area of those squares is a visual representation of how far each point is from the mean.[1] So the total areas of all those five rainbow squares is a measure of how far the entire data set is from the mean.

Let’s add the area of all those squares together to create a massive square.


As I said, this total area is a measure of how far the entire data set is from the mean. How spread out the data is from the mean.

Now we are going to equalize this. We’re going to create five equal smaller squares which have an area that matches the big square.


We’re, in essence, “equalizing” the five rainbow colored squares so they are all equal. The side length of one of these small, blue, equal squares is the standard deviation of the data set. So instead of having five small rainbow colored squares with different measures from the mean, the five equal blue squares are like the average square distance from the mean. Instead of having five different numbers to represent how spread out the data is from the mean, this equalizing process lets us have a single average number. That’s the standard deviation.



I’m not totally clear on everything, but this visualization and typing this out has really help me grok standard deviation better than I had before.

I created a geogebra applet. You can either drag the red points up and down (for the five points in the data set), or manually enter the five numbers.

My recommendation is something like this:

  1. {4, 4, 4, 4, 4}. Make a prediction for what the standard deviation will be. Then set the five numbers and look at what you see. What is the standard deviation? Were you right?
  2. {8, 8, 8, 8, 8}. Make a prediction for what the standard deviation will be. Then set the five numbers and look at what you see. What is the standard deviation? Were you right?
  3. Set the five numbers to {2, 4, 4, 4, 6} and look at what you see. What is the standard deviation?
  4. Consider the number {5, 7, 7, 7, 9}. Make a prediction if the standard deviation will be higher or lower or the same as the standard deviation in #3. Then set the five numbers to {5, 7, 7, 7, 9} and look what you see. What is the standard deviation? Were you right?
  5. Consider the numbers {3, 7, 7, 7, 11}. Make a prediction if the standard deviation will be higher or lower or the same as the standard deviation in #4. Explain your thinking. Then set the five numbers to {3, 7, 7, 7, 11} and look at what you see. What is the standard deviation? Were you right?
  6. Consider the numbers {3, 6, 7, 8, 11}. Make a prediction if the standard deviation will be higher or lower or the same as the standard deviation in #5. Explain your thinking. Then set the five numbers to {3, 6, 7, 8, 11} and look at what you see. What is the standard deviation? Were you right?
  7. What do you think the standard deviation of {4, 8, 8, 8, 12} be? Why? Check your answer with the applet.
  8. Can you come up with a different data set which matches the standard deviation in #6? Explain how you know it will work.
  9. Set the five numbers to {4, 4, 4, 4, 4}. Initially there are no squares visible. The standard deviation is 0. Now drag one of the numbers (red dots in the applet) up. Describe what the squares look like when they appear? Eventually drag that number to 15. What do you notice about the standard deviation? Use your understanding of what happened to describe how a single outlier in a data set can affect the standard deviation

Okay, I literally just whipped the applet up in 35 minutes, and only spent the last 15 minutes coming up with these scaffolded questions. I’m sure it could be better. But I enjoyed thinking through this! It has helped me get a geometric/visual sense of standard deviation.


Now time to eat dinner!!!

 Update: a few people have pointed out that the n in the denominator of the standard deviation formula should be n-1. However that would be for the standard deviation formula if you’re taking a sample of a population. This post is if you have an entire population and you’re figuring out the standard deviation for it. 

[1] One might ask why square the distance to the mean, instead of taking the straight up distance to the mean (so the absolute value of each number minus the mean). The answer gets a bit involved I think, but the short answer to my understanding is: the square function is “nice” and easy to work with, while an absolute value function is “not nice” because of the cusp.

Good Conversations and Nominations, Part II

This is a short continuation of the last blogpost.

In Advanced Precalculus, I start the year with kids working on a packet with a bunch of combinatorics/counting problems. There is no teaching. The kids discuss. You can hear me asking why a lot. Kids have procedures down, and they have intuition, but they can’t explain why they’re doing what they’re doing. For example, in the following questions…


…students pretty quickly write (4)(3)=12 and (4)(3)(5)=60 for the answers. But they just sort of know to multiply. And great conversations, and multiple visual representations pop up, when kids are asked “why multiply? why not add? why not do something else? convince me multiplication works.”

Now, similar to my standard Precalculus class (blogged in Nominations, Part I, inspired by Kathryn Belmonte), I had my kids critique each others’s writings. And I collected a writeup they did and gave them feedback.

But what I want to share today is a different way to use the “Nomination” structure. Last night I had kids work on the following question:


Today I had kids in a group exchange their notebooks clockwise. They read someone else’s explanations. They didn’t return the notebooks. Instead, I threw this slide up:


I was nervous. Would anyone want to give a shoutout to someone else’s work? Was this going to be a failed experiment? Instead, it was awesome. About a third of the class’s hands went in the air. These people wanted to share someone else’s work they found commendable. And so I threw four different writeups under the document projector, and had the nominator explain what they appreciated about the writeup. As we were talking through the problem, we saw similarities and differences in the solutions. And there were a-ha moments! I thought it was pretty awesome.

(Thought: I need to get candy for the classroom, and give some to the nominator and nominee!)

The best part — something Kathryn Belmonte noted when presenting this idea to math teachers — is that kids now see what makes a good writeup, and what their colleagues are doing. Their colleagues are setting the bar.


I also wanted to quickly share one of my favorite combinatorics problems, because of all the great discussion it promotes. Especially with someone I did this year. This is a problem kids get before learning about combinations and permutations.


Working in groups, almost all finish part (a). The different approaches kids take, and different ways they represent/codify/record information in part (a), is great fodder for discussion. Almost inevitably, kids work on part (b). They think they get the right answer. And then I shoot them down and have them continue to think.

This year was no different.

But I did do something slightly different this year, after each group attempted part (b). I gave them three wrong solutions to part (b).


The three wrong approaches were:

And it was awesome. Kids weren’t allowed to say “you’re wrong, let me show you know to do it.” The whole goal was to really take the different wrong approaches on their own terms. And though many students immediately saw the error in part (a), many struggled to find the errors in (b) and (c) and I loved watching them grapple and come through victorious.

And with that, time to zzz.

Quadratic Play

CAVEAT: There isn’t any deep math in this post. There aren’t any lessons or lesson ideas. I was just playing with quadratics today and below includes some of my play.

I’ve been struggling with coming up with a precalculus unit on polynomials that makes some sort of coherent sense. You see, what’s fascinating about precalculus polynomials is that to get at the fundamental theorem of blahblahblah (every nth degree polynomial has n roots, as long as you count nonreal roots as well as double/triple/etc. roots), one needs to start allowing inputs to be non-real numbers. To me, this means that we can always break up a polynomial into n factors — even if some of those factors are non-real. This took up many hours, and hopefully I’ll post about some of how I’m getting at this idea in an organic way… If I can figure that way out…

However more recently in my play, I had a nice realization.

In precalculus, I want students to realize that all quadratics are factorable — as long as you are allowed to factor them over complex numbers instead of integers. (What this means is that (x-2)(x+5) is allowed, (x-5.2)(x-1.2) is allowed, but so are (x-i)(x+i) and (x-5+2i)(x-5-2i) and (x-\sqrt{2}+\sqrt{7}i)(x-\sqrt{2}-\sqrt{7}i). (And for reasons students will discover, things like (x+i)(x+2) won’t work — at least not for our definition of polynomials which has real coefficients.)

So here’s the realization… As I started playing with this, I realized that if a student has any parabola written in vertex form, they can simply use a sum or difference of squares to put it in factored form in one step. I know this isn’t deep. Algebraically it’s trivial. But it’s something I never really recognized until I allowed myself to play.


I mean, it’s possibly (probable, even) that when I taught Algebra II ages ago, I saw this. But I definitely forgot this, because I got such a wonderful a ha moment when I saw this!

And seeing this, since students know that all quadratics can be written in vertex form, they can see how they can quickly go from vertex form to factored form.


Another observation I had… assuming student will have previously figured out why non-real roots to quadratics must come in pairs (if p+qi is a root, so is p-qi): We can use the box/area method to find the factoring for any not-nice quadratic.


And we can see at the bottom that regardless of which value of b you choose, you get the same factoring.

I wasn’t sure if this would also work if the roots of the quadratic were real… I suspected it would because I didn’t violate any laws of math when I did the work above. But I had to see it for myself:


As soon as I started doing the math, I saw what beautiful thing was going to happen. Our value for b was going to be imaginary! Which made a+bi a real value. So lovely. So so so lovely.


Finally, I wanted to see what the connection between the algebraic work when completing the square and the visual work with the area model. It turns out to be quite nice. The “square” part turns out to be associated with the real part of the roots, and the remaining part is the square associated with the imaginary part of the roots.

pic 8.png


Will any of this make it’s way into my unit on polynomials? I have no idea. I’m doubtful much of it will. But it still surprises me how I can be amused by something I think I understand well.

Very rarely, I get asked how I come up with ideas for my worksheets. It’s a tough thing to answer — a process I should probably pay attention to. But one thing I know is part of my process for some of them: just playing around. Even with objects that are the most familiar to you. I love asking myself questions. For example, today I wondered if there was a way to factor any quadratic without using completing the square explicitly or the quadratic formula. That came in the middle of me trying to figure out how I can get students who have an understanding of quadratics from Algebra II to get a deeper understanding of quadratics in Precalculus. Which meant I was thinking a lot about imaginary numbers.

That’s what got me playing today.


Parabolas: Focus and Directrix

I am teaching conics now. I usually skip teaching anything about parabolas in depth because… well, they do so much with quadratics in Algebra II… and I would rather devote my time to something new. However this year I’m teaching with another teacher who did cover parabolas. So I had to learn what a focus and directrix is. I mean, I knew ages ago, but who needs to keep that kind of information in your head?!


For those who aren’t in the know, for me the big idea is that we can conceptualize a parabola as the result of graphing the algebraic equation y=ax^2+bx+c. But there is a second way to concieve of the same mathematical object: with a geometric argument.

If you have a piece of paper with single point drawn, and a single line (that doesn’t contain the point) drawn, those two objects uniquely define a parabola.

That’s a pretty awesome thing, once I started thinking about it. An alternative way to view something that I only ever think about in the standard “graph a quadratic” way!

The Forwards Question

So given a point and a line, how can we draw this parabola? Here is how…


The point is the blue X. The line is the black line. We want to drag the red point along this vertical line so that the distance from the blue point to the red point is equal to the distance between the red point and the black line. So we use a ruler, some trial and error, and find that red point belongs somewhere here… [1]


And then we leave that red dot there, and start again with another vertical line. And find another point on that vertical line which has the same property!


And again and again and again. Until you have created a whole bunch of red points. Those form a parabola.


I’m still not 100% sure how I’m going to introduce this notion to my kids. I’m pretty sure I’m going to give each kid a printed paper that looks like


And ask them where to place the red dot… And then see if they can find a more efficient way than using a ruler and guessing a checking. (Paper fold! See it? If not, read the footnote.) I will probably do this as a warmup one day — and then have kids go “whaaaaat is this for?” and I’ll shrug and say “Wish I knew, kids…” and then move on not referencing this.

And then the next day for the warmup, I’ll find a way to have the whole class collect points for the same blue point and black line… We’ll generate the locus of all these points which are equidistant from the blue point and perpendicular distance to the black line… and lo and behold… the parabola. And then we’ll do the patty paper folding thing down in the footnote video.

So… Yeah. Now we have an obvious place to go…

The Backwards Question

Here it is: Given a parabola, can you find the defining point and line? (The fancy mathematical words for these defining objects are “the focus” and “the directrix.”)

And so I created a sheet to have my kids figure out how to find these objects given a parabola. [Note: I haven’t used the sheet. I haven’t even worked out the sheet and made a key. I just whipped it up now! So apologies for any errors, if any.]

2016-04-25 Parabolas [docx form]

Now to be perfectly perfectly honest, there are two things about this sheet I hate.

(1) I give footnote 1.

(2) I give 3c. In fact, partly I think giving 3a is a bit much as is.

Both give away too much. So why didn’t I change it? Do I not have confidence in my kids?

No. It’s because I wasn’t even planning on introducing parabolas. And now I got sucked into them — learning all about them — and I am excited to share some of this stuff with my kids. But I don’t have the time for this. The fact that I’m going to give about a day for parabolas is more than I was planning… so I have to keep things a bit on the crisper side.

What else would I change if I had more time? I would have kids think about if this works for an “upsidedown” parabola. And also have them use what they know about inverse functions to apply this to “sideways” parabolas.

I honestly don’t know if I’m going to use this in class. I probably will because I took the time to make it, and I kinda got excited when I was figuring out for myself all this focus/directrix stuff. I pretty much took this definition of a parabola and figured all this out myself — and I hope kids get the same joy. But have I convinced myself that kids need to learn about a parabola other than there is this other way to “create” them that isn’t algebraic? Is there a “big idea” hidden in this worksheet? I don’t think so. This may be a one-time use worksheet.


[1] Now in actually, there is an easy geometric way to find that red point. It involves a simple paper fold. Fold the blue point to the point on the directrix below the red point. What that crease intersects the vertical line is where the red dot should be. Perpendicular bisectors FTW! And you can do a quick patty paper demonstration of this to create a parabola! (We did this in my class last year, for parabolas, hyperbolas, and ellipses, thanks to Tina C.)