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).
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.
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 is allowed, is allowed, but so are and and . (And for reasons students will discover, things like 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.
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.
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 . 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… 
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.]
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.
 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.)
There is a famous, well-known problem in the world of “rich math tasks” that involves taking an n x n x n cube and painting the outside of it. Then you break apart the large cube into unit cubes (see image below cribbed from here for n=2 and n=3):
Notice that some of the unit cubes have 3 painted faces, some have 2 painted faces, some have 1 painted face, and some have 0 painted faces.
The standard question is: For an n x n x n cube, how many of the unit cubes have 3 painted faces, 2 painted faces, 1 painted face, and 0 painted faces.
[In case you aren’t sure what I mean, for a 3 x 3 x 3 cube, there are 8 unit cubes with 3 painted faces, 12 unit cubes with 2 painted faces, 6 unit cubes with 1 painted face, and 1 unit cube with 0 painted faces.]
Earlier this year, I worked with a middle school student on this question. It was great fun, and so many insights were had. This problem comes highly recommended!
Today we had some in house professional development, and a colleague/teacher shared the problem with us, but he presented an insight I had never seen before that was lovely and mindblowing.
Spoiler alert: I’m about to give some of the fun away. So only jump below / keep reading if you’re okay with some some spoilers.
During this school year, we now have occasional 90 minute blocks with our classes. I was trying to decide what to do a couple weeks ago with my precalculus class, and stumbled upon the embryo of a good idea. Kids playing with blocks to create 3D sequences. (This idea was inspired by Fawn Nguyen’s site Visual Patterns.)
I got blocks from our lower school math coach. I told kids (either working individually or in pairs) to play around with them until they found a pattern that looked interesting to them. I didn’t want them thinking about the sequence yet… I wanted them to create patterns that looked neat. The only restriction I put on them is that the pattern had to be three dimensional. If it could be represented in two dimensions, I didn’t want to see it.
They made some really nice sequences! Here are a random set of 4 to look at:
I then had students work on filling out this form. It asks them to articulate their “rule” (for building up the sequence) and has them attempt to come up with both explicit and recursive forms to get the nth term. I make it clear to them that if they can’t get the formulae, I’ll give them full marks as long as they show a serious attempt. (Some of the sequences they built involve some mathematical hoops they might not be able to traverse… for example, one group needed to find which is lovely, but not something they are going to easily figure out.
If I had time, I’d love to do two more things with this.
(1) I think it would be neat to take the photographs of one person’s sequence and give them to another person, to see what they figured out for the explicit and recursive definitions for these sequences. Why? Not only is it sharing more publicly the sequence the kids created, but many of them got a bit stuck on an explicit formula that they do have the capabilities to find, but couldn’t. I think a fresh pair of eyes, and a conversation, could be beneficial for both the original sequence creator and the new person approaching the sequence. (Additionally, there are often many ways to look at these sequences, so even if both got the same formula, there is a good chance they came up with it in different ways.)
(2) Students created a table with the first 5 terms of the sequence in it. I’d love for students to extend the table to 7 or 8 terms in the sequence, and then have students work on finding the first differences, the second differences, the third differences, etc. If students understand that having the same first difference means they have a linear relationship, having the same second difference means they have a quadratic relationship, having the same third difference means they have a cubic relationship, etc., then students who got stuck will have a new tool in their arsenal to find the explicit formula for the sequence. If, for example, they had 5, 9, 15, 23, …, and saw a common second difference, they could do the following:
Since they suspect the relationship is quadratic, they could say: . And then they’d be hunting for the to make this the correct quadratic for our sequence. And then use the following three equations, they could come up with the .
In fact, this is an awesome thing to revisit when we get to matrices to solve systems of three variables!!!
UPDATE: One more thought before I lose it! What if I gave students the numerical sequence (e.g. 5, 9, 15, 23) expressed either written out as a list, written out as an explicit formula, or written out as a recursive formula, and had them generate a visual sequence to match it. I’d love to see how many different and interesting sequences might be created that go along with a single sequence!
I’m mad I didn’t actually take photos or record any of today’s precalculus lesson. Apologies. But even though this is going to be a textheavy post, I think it’s pretty awesome.
TL;DR: We fistbumped in precalculus. It was awesome. Super complex math got done.
One of the things I’m working on is improving my questioning this year. One of my strengths is scaffolding, but sometimes — in my desire to be super overzealously prepared — I scaffold too much. Today we had our first “long block” (90 minutes) in Advanced Precalculus. This is how class unfolded, after our warm-up.
I asked students to fistbump everyone at their table.
How many fistbumps did you just do?
How many fistbumps just happened in class?
Then I showed them there were many fantastical ways to fistbump besides the standard “clink knuckles” method. Blow it up. Snail. Squid. Turkey.  That was a random impromtu aside. But now, next class, I must show my kids the following video:
If you do this in your class, you should definitely have this video queued up. 
Then: everyone had 20 seconds of individual think time for this question:
If you wanted to devise an efficient way for everyone in the class to fistbump everyone else in the class, what would that way be?
Kids asked what “efficient” meant. I said “it should be as quick as possible, with the least chance of someone not actually fistbumping someone else.” Now you, friend, take a guess. I have 14 kids in my class. How long do you think it would take my kids to fistbump everyone else with an efficient strategy!
Seriously… reader… take a guess! Good. I’ll reveal the answer in a bit.
After the 20 seconds of individual time, each group shared with each other, and had to converge upon their proposal to the class. We went around. The four groups had three ideas:
Line everyone up. The first person fistbumps with everyone else, then leaves. Then the second person fistbumps with everyone else, then leaves. And so forth.
We have four groups in our class. The first group goes around and fistbumps with the members of other three groups in order. Then the second group does that with the remaining groups. And so forth.
Do the exact same thing as proposal #1, except as you don’t wait until the first person is done fistbumping everyone else. As soon as the first person is done fistbumping with the second and third person (and continues on down the line), the second person starts fistbumping down the line. And so forth.
(I had also anticipated students talking about getting in a “circle” and having one person fistbump with everyone, then another person, then another, etc. It’s organized, but not very efficient. One thing kids asked: can we all fistbump each other at the same time, in one giant mass of fists? I nixed that. I also had kids ask if you could fistbump with both hands simultaneously — to two different people. I said yes! But I didn’t give enough time for students to devise something super efficient with that so that never got turned into a proposal.)
As a class, we decided proposal #3 was going to be the most efficient. So I had them all file into the hallway and try out their fistbump method. I got my stopwatch out. And they went at it, after organizing themselves.
You may wonder what all of this has to do with math. That’s coming. This was just the setup. I honestly think by this point in the class, some kids were wondering what the heck we were doing this for…
So how long did it take them?
Yup. Under 12 seconds! I! Was! In! Awe!
Then each group got out a giant whiteboard and markers and answered the following questions:
How many fistbumps did you just do? What was the average time per fistbump?
Once they answered that question, they called me over to discuss their findings with me. Then I had two extensions:
We have 998 students at our school. How many fistbumps would that be? How long would it take, if we used our efficient method and assumed the same average time per fistbump? 
And from this point on, you have to facilitate class based on what your kids are doing. Some advice?
Advice 1: If kids are struggling, have ’em start noticing patterns about the number of handshakes for smaller numbers of people. Two people? Three people? Four people? Continue working up. Make a table. Look for patterns.
Advice 2: If kids have seen the “rainbow method” or some variation (see below), have them think about the difference between an even number of things being added and an odd number of things being added.
Advice 3: Have kids work on coming up with a single formula that works for even and odd numbers of things being added. Then have them explain why that formula works.
Advice 4: Lead kids to the idea of “double counting”: if we have 4 people, then have each person fistbump with everyone else. Since each person fistbumped with 3 people, there were a total of 4*3=12 fistbumps. However we’re double counting in this, so there are really only 6 fistbumps. (If kids don’t see the doublecounting, have a group of four act it out.)
Advice 5: If a group needs an assist, have individual members circulate to other groups and gather ideas, and then return and share what they found.
I loved doing this activity. Kids got into it. They felt ownership and camaraderie. Kids were up and moving. Because we had a long block, kids had time to play and productively struggle with the ideas. And most importantly: I didn’t overscaffold. I built up motivation and then sprung a good open-ended question for kids to work on.
 If you don’t know what I’m talking about, clearly you’ve never hung out with middle school students.
 queue is such a strange word, right? 80% of the letters are unnecessary. “q” is the same pronunciation as “queue.”
 The answer is around 18 hours. What I loved is that when a group got that — after we got 12 seconds for our class — I was like “come ON guys? does that make sense? it would take almost a whole day with no breaks? REALLY?” I wanted them to see the answer was kinda absurd. But it is right, because although it might seem absurd on the surface, each time we add more people, we’re making the number of handshakes grow pretty darn fast! (Follow up? How fast? Let’s make a graph! Ohhhh, quadratic? PRETTY! And grows super quickly for higher numbers, unlike linear graphs.) Turns out the answer is much shorter than 18 hours. I had a misconception that someone helped me see on the betterQs blogpost! I liked admitting to my class i was wrong!
Today in Precalculus I went on a bit of a 7 minute digression, talking about continued fractions. You see, a recursive problem showed up (we’re doing sequences): Write out the first five terms of the following sequence:
So obviously they go like: ,, , , and
So great. Awesome. NOT. Booooring. So I showed them the decimal expansions:
WHOA! This is getting closer and closer to 2… Weiiiird…
And then I say I can show them this will continue, and we can find a way to show that [where the pattern continues forever] will practically become 2.
DIGRESSION WHICH IS ACTUALLY WHY I WANTED TO BLOG ABOUT THIS
To do this, I start with something else. I don’t know why, but I really wanted to show them a continued fraction first, to get the point across easier than with the square root. This was the continued fraction.
I went through a frenetic mini-lecture, and I think I had about 40% of the kids along with me for the whole ride. I’m not sure… maybe? But later a kid came by my office, and I thought of a better way to show it. Hence, this blogpost, to show you. (I have seen teachers use this method when teaching substitution when solving systems of equations… but I have never used it myself. I’m dumb! This is awesome!) This is what I did when showing the kid how to think about this in my office.
First I took a small piece of paper and I wrote the infinite fraction on it.
Then I flipped it over and on the back wrote what it equaled… Our unknown that we were trying to solve for.
I emphasized that that card itself represented the value of that fraction. The front and back are both different ways to express the same (unknown) quantity we were looking for.
Then I took a big sheet of paper and wrote where I left the denominator blank. And then I put the small card (fraction side up) in the denominator of the fraction…
And I said… what does this whole thing equal?
And without too much thinking, the student gave me the answer…
Yup. We’ve seen that infinite fraction before. That is !
THAT FLIP IS THE COOLEST THING EVER FOR A MATH TEACHER. That flip was the single thing that made me want to blog about this.
Now you have an equation that you can solve for … and is what you’re trying to find the value of. This equation can easily be turned into a quadratic, and when you solve it you get (yes, the Golden Ratio). And it turns out that is close to what we might have predicted…
Because in class, we (by hand) calculated the first few terms of where … and we saw:
And when I drew a numberline on the board, plotted 1, then 2, then 1.5, then 1.66666666, then 1.6, then 1.625, we saw that the numbers bounced back and forth… and they seemed to be getting closer and closer to a single number… And yes, that single number is about 1.618.
BACK TO OUR REGULARLY SCHEDULED PROGRAM
So after I showed them how to calculate the crazy infinite fraction, I went back to the problem at hand… What is ?
Then we can say
And even simply by inspection, we can see that is a solution to this!
 What’s neat is that yesterday I introduced the notion of a recursive sequence that relies on the previous two terms. So soon I can show them the Fibonacci sequence (1,1,2,3,5,8,13,…). What does that have to do with any of this? Well let’s look at the exact values of where .