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A Much Belated Overview of My Advanced Geometry Curriculum

I recently got an email from someone who saw some of my many posts on geometry (you can see all my posts about geometry by clicking here). I realized I never shared them formally and everything is a bit scattered. So I’m going to try to include a few resources here. But the real joy is in all the blogposts, honestly.

I taught Advanced Geometry at my school for two years (2014-2016), and I wrote the curriculum with a good friend and dear colleague. We both hadn’t taught geometry before and decided we’d do a super deep dive and come up with a sequencing that made sense to us, and that prioritized conjecturing and noticing. In fact, we were so excited by this process that we shared our thinking both about how we built up the curriculum but also how we collaborated at a conference. Below are our slides, but you can also click here and go to the slideshow and read some of our presenter notes for each slide for more detail.

We were super intentional about everything. We carefully thought through how we wanted to motivate everything, and we didn’t want to give anything away throughout the course. In other words, we wanted kids to do all the heavy lifting and to be the mathematicians that we knew they could be.

Below is a word document with all our skills/topics (you can download the .docx file here: All Topic Lists Combined). The order might seem a little strange (we end, for example, the year with triangle congruence), but it worked for us! Everything was done on purpose (in this case, congruence is just a special case of similarity… so that came beforehand, along with trig which is all about exploiting similarity!). We eschewed two-column proofs for different forms (paragraph proofs, flowchart proofs, and anything else that showed logical reasoning).

Oh wait! For some reason our work on Area and Volume didn’t have a topic list. And I just looked and my core packet for Area and Volume derivations (where kids just figure things out on their own) has handdrawn images in it, but I didn’t scan a PDF of those. Well, at some point in the future if I remember, I’ll try to write a post to share that. (We did it after kids learned trigonometry, so they had a lot of flexibility. For example, I think kids came up with like 6 different methods to find the area of a trapezoid when they were asked to create a formula and justify it!)

I hope this is helpful for anyone trying to think through geometry. As I said before, the best thing might be to just read the blogposts, but this is a bit of an overview.

“How I Have Changed the Way I Have Thought about Myself and the World”

In the past few years, my school started a “senior speech” program. Five seniors get up and talk to all students and faculty in grades 9-12 about their lives. Coming out as gay. Mental health challenges they face. Racism they face. Code-switching. Being a conservative in a liberal environment. Bullying. Friendship. They usually make me tear up. (Stupid onions.) After each one, I think to myself: “Wow, if I were a ninth grader hearing these seniors be raw and honest and vulnerable, it would help me on a path to become brave and speak my truth.” I also think: “Why don’t we collect these all and publish them somewhere?” I don’t want them lost.

One of my students this year told me I could put her senior speech on my blog. So here it is, for posterity.

How I Have Changed the Way I Have Thought about Myself and the World

 

When I was a lot younger, I was more hyperactive than I am now and I was always doing something to take up my time. I mainly always liked to keep myself out of my house because then I wouldn’t have to be in the middle of what was going on inside.

There was a lot of fighting that happened between my parents. They’d get physical with each other or throw things at each other. They never seemed to think of my brother and I being there.

I realize now that I was fearful of what was going on, that it put a lot of stress on me. When I was younger my thoughts were not so developed. When I would be with the rest of my family, my brother would be there to protect me. He could not prevent me from seeing or hearing certain things, but he would take my mind off of it the best he could. I don’t know what I figured the behavior between my parents meant or was- but if they were loving to me, I forgot about it and moved on. This was the way life went for me for many years. It was a contorted way of thinking.

At this time I also felt that the person I was defined by was my home life and my immediate family. I felt this way yet I didn’t think moments when my parents were violent towards each other or embarrassing my brother and I were moments that meant anything. I was always taught to let it pass and start new. That way of thinking ran me into trouble. I took what was happening in my home outside of it and I would have unkind thoughts/ engage in unkind behavior towards people I loved and cared for. Afterwards, whether I identified if it was mean or not, I would move on. I didn’t think it said anything about who I was.

Then in seventh grade I was in Ms. Oster’s health class and she showed us this speech by an author named George Saunders. It’s titled Advice to Graduates, and it is centered around the questions why aren’t we kinder? and how do we become more kind? It touched on the idea that becoming kinder comes with age but there are ways you can speed up the process. I didn’t feel like It wasn’t as if I hadn’t heard these words before, or as though this was a new concept to me but they processed differently this time and I began to question who I was.

I realized that I had always thought I was a really nice person but if I wasn’t genuinely acting kind, meaning my actions weren’t speaking for themselves, how could I be a nice person? This caused me to think back to things I had done and ask myself a series of questions like: Did I do the right thing? What did I feel that made me act the way I did? What caused that feeling to arise? Why could that feeling have been brought up?

Seeing that my actions didn’t reflect who I knew I was, I continued in life being very analytical of instinctual tendencies, thoughts, and behaviors. This way of questioning took over and continues to this day. Questioning myself allowed for me to realize the roots of my actions so I can allow myself to change. If I can identify the root, then maybe I can change the way I do things.

And so, through this process of questioning, I came across the concept of time and how I spend my time. I thought to myself what I spend my time doing makes me who I am. I thought, what had I been spending my time doing? School came to mind. I go to school most days out of the year and I am in school most hours out of each of those days. Again, it wasn’t that I didn’t know these facts before this moment but I had never paid attention to the concept. When I began to pay attention I realized that school was a place that nurtured me in all positive ways and I was taking it for granted. I could look to the building as a safe place, the teachers as people who could listen and give me advice and, my peers as those who stand equal to me in our journey to understand and live with ourselves. I am very grateful that I am able to attend school, have teachers who care for their students, students who engage deeply in class conversation, and a place that I can actually call my home.

At this point in my life I feel I have come a long way. Since 7th grade I have realized that a lot of what I was seeing when I was younger was my mom struggling with her mental health. In 8th grade child services took me from living at my mom’s house. They had been in and out of my life until this past summer when I permanently started living with my dad. It has been difficult accepting that my mom was not able to guide me through big developmental parts of my life but I love my mother and I am so thankful she is the woman I call my mom. Now here’s a George Saunders quote that is important to me.

“If we’re going to become kinder, that process has to include taking ourselves seriously — as doers, as accomplishers, as dreamers. We have to do that, to be our best selves.”

Now also, Over these years of high school I have come to understand that change is inevitable and a big part of our job is to adapt, learn, and accept it. Through that process I am continuously discovering my best self.

OAME 2019: The Teacher Voice

In August of 2018, I got a message from my friend Mary Bourassa who asked me if I was interested in being a featured speaker at OAME (the annual math teachers conference in Ottawa). I was absolutely going to say no, because I’m terrified of public speaking and I wasn’t sure I had anything of value to say to other teachers…

I know the irony of being scared of public speaking and being a teacher, but I know a lot of other teachers also feel this. But my biggest fear was just not being good enough. I wrote to Mary:

The truth is I can’t help feeling like this might be too big a leap for me. I don’t know if I could do what Fawn or Julie to me when I hear them talk, or do what Chris Shore or John Stevens do when they present. And I don’t want to commit unless I knew I wouldn’t be wasting anyone’s time. 

She replied:

Firstly, and most importantly, you would not be wasting anyone’s time.

I am a big proponent of elevating classroom teachers and giving them a voice and I hope that you will find yours for this event. 

As someone who feels like an evangelist of the online math teacher community, I’m always saying to people hesitant to dip their toes into the water that their voices and perspectives are important and valuable. And when I say this, I mean it with every fiber of my being. So why was I doubting the value of my own voice? I agreed to do the talk as long as I could do it with a collaborator and friend. It was a 75 minute talk (and an associated 75 minute workshop) and planning that individually seemed so not fun. But I thought working collaboratively would be so much more intellectually fun! So I dragged my friend Mattie Baker (@stoodle) into the presentation. We brainstormed for ages, but in the end, finally decided that the idea of “The Teacher Voice” was exactly what we needed to talk about.

It’s now over, so I’ll start with the ending. The lecture hall of 200 people for our talk was almost full. And the 75 minute talk went fabulously. I had to save some tweets for posterity.

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And OMG, we got a standing ovation. That was unexpected. And people were crying. That too was unexpected. I am not someone who feels proud about things easily. I usually focus on all that went wrong or ways I could have been better. But when we took a bow at the end and people stood up, my heart was bursting. All the work Mattie and I put into the talk for the previous 10 months, the weekends we sacrificed to write and practice and edit felt purposeful because at least for some teachers in the audience, our message was at least temporarily valuable. Weirdly my fear of public speaking disappeared the day of the talk after we had a solid rehearsal the previous day, and my fear of wasting peoples’ time disappeared after the talk ended and people came up to say such nice things to us.

The talk was broken into two parts. First, Mattie and I shared something we each did in our classrooms that was inspired by other teachers, and then adopted by other teachers. We wanted the audience to have something concrete to walk away with in case the rest of our talk didn’t resonate with them. We were breaking down the silos of our classrooms. Second, we each talked about the emotional life of a teacher. We wanted to break down the silos of our emotional worlds. There were so many messages we included in this part of the talk. Mattie shared his first year in teaching, which he previously shared on the Story Collider podcast. But here is one takeaway from my section of the talk:

Teaching is hard. We are going to feel bad. We’re going to be bad. And that’s okay. It’s okay to not love what you’re doing all the time. I’ve never met a teacher who is putting themselves out there in the important but hard ways who does. But we can be brought closer as we become vulnerable and share these things and realize we aren’t all alone in this.

And a second related takeaway:

Often times, we’re so critical about ourselves, we think of all that isn’t going right, all that we aren’t doing… that we lose sight of all that we are…. It’s so easy to be critical of yourself, to set the bar high, to see all the ways you’re not succeeding.

You see yourself in one way. But the reality of the situation is: We aren’t really all that good at seeing ourselves. That’s my big realization, and it only took twelve years. When we’re down and think we suck, yeah, we probably definitely maybe can be doing better. But hell if we aren’t already doing good, and we need to acknowledge that and spread it. We need to believe our friends when they tell us that our ideas our good, that something we did was good… we need to believe our kids when they say something spontaneous and positive about something happening in the classroom… and… we need to be sharing the good and the positive that we see in others. We need to help others see how important they are to you. We need to give cupcakes, send the random email, prop each other up, and help others see how they make your life better.

The talk focused on the hard times in teaching, and what we do when we hit them. At one point I asked the audience to share their coping strategies at the low points. I promised I would share them online, so here are what the audience typed. It’s amazing how similar the responses are…

(All the references to “coffee” in quotations comes from part of Mattie’s talk. You can interpret that to be getting a drink at happy hour.)

I shared my coping strategies afterward, and so many of them were covered by what people in the audience typed! Except for those people who talked about exercise and running and the gym and other evil things like that. Some of my favorites!

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In the talk, I also shared the Explore Math project that I do with students. The website that I created for the project is here: https://explore-math.weebly.com/

I posted about it early on when I first started it, but haven’t done any additional posts on how I’ve changed it or how it’s evolved or what I’ve noticed when doing it with different grades (sorry, I should). The posts are here, here, and here. The most important thing I can suggest is that you need to adapt it to work for your kids and your school. For example, this year I tried this in 10th grade and it wasn’t as successful overall because I think the kids needed more structure and hand-holding. So I’m going to take that into account for next year.

Two teachers shared their experiences with the project, which I couldn’t fit into the talk. So I’m posting them here in case it entices you to do the project or some variation in your classroom.

 

 

At the conference, Mattie and I also gave a 75 minute workshop on the online math teacher community designed for people who were interested in joining in but didn’t know how (our slides for that are here).

 

Tiny Game Re: Euler’s Number

I’m teaching Algebra 2 this year and the other teacher and I decided that we should introduce e to our kids. The reason it’s challenging is that it’s hard to motivate in any real way. You can do compound interest, but that doesn’t do much for you in terms of highlighting how important the number is. [1] I asked on Twitter for some help, and I got a ton of amazing responses (read them all here). My mind was blown. This year, though, I didn’t have time to execute my plan that I outlined at the bottom of that post. So here’s what I did:

  1. The core part of what I did to get the number to pop up was to use @lukeselfwalker’s Desmos activity. I like it for so many reasons, but I’ll list a few here. It starts by “building up” a more and more complicated polynomial of the form (1+\frac{x}{n})^n, but in a super concrete way so kids can see the polynomial for different n-values. It shows why the x-intercept travels more and more left as you increase n, so when you finally (in the class discussion) talk about what happens when n goes to infinity, you can have kids understand this is how to “build” a horizontal asymptote. It gets kid saying trying to articulate sentences like “this number is increasing, but slower and slower” (when talking about the value of the polynomial when x=1. And they see how this polynomial gets to look more and more like an exponential function as you increase the value of n. If you want to introduce e, this is one fantastic way to do it.
  2. A few days later, I had everyone put their stuff down and take only a calculator with them. They paired up. (If someone didn’t have a pair, it would be fine… they just sit out the first round.) On the count of three, both people say a number between 0 and 5. (I reinforce the number doesn’t have to be an integer, so it can be 4.5 or something.)Then using their calculators, they calculate their score: they take their number and raise it to their competitor’s number. The winner has the higher number. (If it’s a tie, they go again until there is a winner.)

    Then the loser is done. They “tag” along with the winner and cheer them on as they find another winner to play. This goes on. By the end, you have the class divided into two groups each cheering on one person. (I learned this game this year as an ice breaker for a large group… it’s awesome. This is the best youtube video I could find showing it.)

    Finally there is a class winner.

    So I then went up against them.

    And when we both said our numbers, I said: e.

    The class groans, realizing it was all a trick and I was going to win. We did the calculations. I obviously won.

    We sit down and I show them on my laptop how this works:

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    The red graph is my score, for any student number chosen (e^x).
    The blue graph is the student score, for any student number chosen (x^e).

    Clearly I will always win, except for if my opponent picks e.

    I tell kids they can win money off of their parents by playing this game for quarters, losing a few times, and then doing a triple or nothing contest where they then play 2.718. WINNER WINNER CHICKEN DINNER!

  3. After this, I show kids these additionally cool things (from the blogpost), saying I just learned them and don’t know why they work (yet), but that’s what makes them so intriguing to me! And more importantly, they all seem to have nothing to do with one another, but e pops up in all of them!pic1pic2pic3

    I re-emphasize e is a number like \pi and I showed them this to explain that it pops up in all these places in math that seem to have nothing to do with that polynomial we saw. And that even though we don’t have time to explore e in depth, that I wanted them to get a glimpse of why it was important enough to have a mathematical constant for it, and why their calculators have built in e and ln. 

That is all. I honestly really just wrote this just because I was excited by the “game” I made out of one of the properties of and wanted to archive it so I would remember it. (And in case someone out there in the blogoversesphere might want to try it.)

UPDATE: Coconspirator in math teaching at my school, Tom James (blogs here) created the checkerboard experiment using some code. You can access the code/alter the code here. The darker the square, the more times the number for the square has been called by the random number generator. And with some updates, you can make more squares! In the future, we can give this to kids and have them figure out an approximation for e.

 

[1] And introducing it with compound interest means you have to assume 100% interest compounded continuously. Where are you going to get 100% interest?!?!

Archiving some gems from Twitter (April 2019)

I have seen a lot of great stuff on twitter lately, and I’ve missed a lot too, I’m sure. I wanted to just archive some of the things that I’ve saved so they don’t disappear! I also think it might be a benefit for someone who reads this who isn’t on twitter or missed some of these tweets. But that’s just a side benefit. I’m writing this for me!!!

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Desmos writes interesting job descriptions when they have openings. When someone pointed that out to them, they mentioned that this article on reducing unconscious bias helped informed how they write their job descriptions. It’s pretty great and I highly recommend it if you’re hiring. I have thought a lot about “fit” in the past few years when doing hiring, but it’s tricky to think about it well. I have come to recognize that someone entering our department needs to be open and willing to collaborate and compromise, but also have sympathetic pedagogical beliefs with what our department values (and can’t compromise on those). One way I have tried to avoid it is thinking about these things:

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But also I have found it harder to balance these thoughts, which I admittedly have a lot:

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Not quite those things, but similar thoughts that get at my own personal views on the what persona/personality traits make an effective teacher. Which I tend to think mirror my own traits. But that’s only because I have these traits because I think they make an effective teacher. But I have worked with enough amazing teachers to know that amazing teachers come in all personas! Just like amazing students don’t all have to have the same personas. But this type of bias is something I am trying to be super cognizant about when on hiring committees.

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I saved this just because I like the question and wanted to work on it. And I can see all kinds of extensions. A formula for n circles? What about spheres? I’m guessing (without working on this problem yet) that this is a classic “low entry point, high ceiling” type problem.

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I just really liked this quotation, and I need to think about the ways that students can see themselves in the mathematics they do. It is part of a larger thing I want to do which is “humanize math” — but I’m not very good at making it a core part of what I do in the classroom. Small bits here and there humanize and expand what kids think about math, but I’m not there yet. I want to one year leave the classroom and know that kids have looked in the mirror and saw something. (It kind of reminds me in a super literal way of how Elissa Miller put a mirror in her classroom, and I think on the bottom she wrote “mathematician.”)

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Okay, I love this so much. If you’ve never seen it before, it a great trick. You have someone pick any number between 1 and 63 secretly. They just point to the cards that number is on. In about three seconds, I can tell you your number.

I actually made a set of these cards where the numbers are more jumbled up, so kids don’t see a pattern to it. I do put the powers of 2 in one of the four corners though to make things easier for me. Oh wait, have I said too much?

If you don’t know this trick, or how or why it works, I’m sure you can google it. But I’m going to recommend the awesome book “Math Girls Talk About Integers” (there are a lot of great “Math Girls” books out there, so make sure you get the Integer one.

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Not only is the book awesome (and great for kids to read), but it breaks down this trick so well. *Shivers with joy*

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I was excited with Karen Uhlenbeck won this year’s Abel Prize, the first woman to win it ever! I had my kids read this article in the NYTimes about it, and write down three notes about the article. We started the next class with a “popcorn sharing” of what people wrote down. (I also said that although I liked the article, it was a bit dense and thought it could have been written more lucidly.) One thing that came up in both classes I did this in was what a “minimal surface” was — so I told kids it is a surface with minimal area.

I then showed my kids this short youtube video:

And explained that bubbles, though not “central” to all higher level mathematics, do come up. And then I gave them a question. I’m too lazy to type it out, but watch the first 1 minute and 45 seconds of this video (https://www.youtube.com/watch?v=dAyDi1aa40E) and you’ll see it. Then we talked about some basic solutions. And THEN I revealed the best answer was the answer shown in the video we all watched together.

Of course @toddf9 (Todd Feitelson) used this as inspiration to create his own bubble thingies:

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but he also explained how he made them…

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and then he EVEN created an awesome desmos activity on this very problem, which I want to archive here for use later: https://teacher.desmos.com/activitybuilder/custom/5cb50bed4dcd045435210d29

(Oh! And Mike Lawler (@mikeandallie) made a mobius strip bubble!)

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Dylan Kane wrote a nice blogpost about calling on students (and the “popsicle sticks of destiny” — though he doesn’t call them that). My favorite line is this simple question that isn’t about right or wrong:

  • After students attempt a problem in groups, or reflect on an idea and share with partners, I call on students asking, “How did your group approach the problem?” or “What is something useful that you or your partner shared?”

It’s so obvious, but even after so many years of teaching, I forget to ask things like this. Or my curriculum isn’t group problem solving based enough for things like this to make sense asking. Or whatever.

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There’s nothing special about this one… I’ve read it a few places before and it always makes me laugh.

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Questions are good. I might have a kid read this at the start of the year and then have a short conversation about why we’re reading it.

It will get at the problematic idea of “obvious,” and when and how learning happens and more importantly when and how learning doesn’t happen.

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In case you didn’t know, Desmos has a list of all their mathematicians they use when they anonymize in Activity Builder.

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https://docs.google.com/document/d/1OY-8dk6vYW1Cags8E6_v3I8YZ-RYROzgsCauW5CZt9w/edit

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I can imagine putting this picture on a geometry test as a bonus question and asking them why it makes math teachers all angsty… Plus it made me chuckle!

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I’m so not here yet. Anyone who knows me as a teacher will probably know I’ll probably never get here. I’m such a stickler for making the use of every second of classtime.

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Crystal Lancour (@lancour28) tweeted out a slide from a session led by Robert Berry (NCTM president) which had this very powerful slide:

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Four rights of the learner in the mathematics classroom

  1. The right to be confused and to share their confusions with each other and the teacher
  2. The right to claim a mistake
  3. The right to speak, listen, and be heard
  4. The right to write, do, and represent only what makes sense to you

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Love the idea of using marbles/paint to draw parabolas (click here to go to the original tweet and watch the video — it’s not a static picture).

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Bree Pickford-Murray (@btwnthenumbers) gave a talk at NCTM about a team-taught math and humanities course called “Math and Democracy.” Not only did she share her slides (like *right after* the talk) but also she links to her entire curriculum in a google folder. SUPERSTAR!!!

I’ve gone to a few talks about math and gerrymandering (both at MoMATH and NYU) and listened to a number of supreme court oral arguments on these cases. It’s fascinating!

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I just finished teaching “shape of a graph” in calculus. But I wish I had developed some activities like this, to make it interactive:

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I’ve literally been preparing to give a talk next month for… months now. And this one stupid tweet summarized the talk. Thanks.

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I have so many more things I can post, but I’m now tired. So this will be the end.

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Clothesline Math – Logarithm Style

I remember when I first heard about Clothesline Math, I was excited by all the possibilities. And in a few conference sessions with Chris Shore, I saw there was so much more than I had even imagined that one could do with it!

It’s basically a number line, that’s all. But it’s a nice public giant number line which can get kids talking. Today I came back from spring break and before break, students learned about logarithms. However I wanted to have them recall what precisely logarithms were… so I created a quick Clothesline Math activity.

I hung a string in the classroom. I highlighted it in yellow because you can’t really see it in the photo…

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I then showed them this slide – explaining the string is a number line…

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I then showed them this slide, which explains what they have to do if they get two of the same number. (I brought cute little clothespins, but mini binder clips or paperclips would have worked just as well):

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And then I gave them the rules of play:

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I handed out the cards and let kids go. It was nice to see they didn’t get tripped up as a class on too many of them, but I got to listen to debates over a few trickier ones, which we collectively resolved at the end.

Here are the cards I handed out: .DOC FORM: 2019-04-01 Clothesline Math – Logarithms

Here is a picture of some of the cards. The two on the left are average level of difficulty. The two in the middle caused my kids to pause… it took them time to think things through (they haven’t learned any log properties yet). The one on the right doesn’t belong on the number one (it is undefined) and the kid who got that card immediately knew that. Huzzah!

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Here’s a picture of the numberline at the end.

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And… that’s it!

I was excited to try it out as a quick review activity. And it worked perfectly for that!

 

(Other things of note: Mary Bourassa made a clothesline math for log properties and shares that here. The author of Give Me A Sine blog does something similar here, but has kids create the cards. I couldn’t find anything with basic log expressions — so I made ’em and am sharing them in this post. Chris Hunter has a nice tarsia puzzle that sticks with basic log expressions here, but I wanted to try out clothesline math so I didn’t use that!) But if anyone has others out there involving logs, I’d love to see them in the comments!)

The challenge that “e” poses

The Launch

Last week I met up with my co-teacher in Algebra 2. We’re working on our unit of exponential functions and logarithms, and we were talking about spending a short amount of time introducing “e” to our kids. Personally, this question has haunted me because when I taught Algebra 2 at the start of my career, I couldn’t ever find a motivation for it — except for interest being compounded continuously. That never quite sat well with me because you have to assume that you have an interest rate of 100%.

Like (1+\frac{100\%}{n})^n as n gets larger and larger.

But where are you getting an interest rate of 100%?!? It isn’t a terrible way to introduce “e” (getting kids to understand the structure of that equation above, there’s a lot of deep thinking that goes on in there). There is also the idea that there is a limiting value for that expression above — instead of the value just going up infinitely — that can be exploited and discussed.

But I never thought “e” and “ln” really belonged in Algebra 2, precisely because I couldn’t motivate them in a way that was intellectually satisfactory. So I tweeted out:

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Little did I know I was going to get so many responses! I wanted to archive them here, which is why I’m writing this post, but then share where I’m landing on this whole “e” thing right now.

Ways To Introduce “e”

  1. @retaneri linked to this question from “Play With Your Math” I love that site, but hadn’t worked on this problem before.
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    Interestingly, @bowmanimal and @averypickford came up with the same problem to share with me! Apparently the answer is to break up the number into a bunch of es … I haven’t figured out why this works yet.  But for 25, if I understand this correctly, I think it means 25=e+e+e+e+e+e+e+e+e+e+0.196986e which has a product of e^{9.196986}. (Or to be super precise: e^{25/e}.)
    And @benjamindickman shared some articles about this:pic10.png
  2. @mikeandallie shared with me this approach by throwing darts. Which includes this gem:
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    and this instantiation:
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  3. @jensilvermath suggested just looking at (1+1/n)^n for larger and larger n, without reference to a limit. Have kids make predictions about what is going to happen and why, and then let them explore it. At first, I was like “hmmm, would this work?” but I love the idea of kids stumbling upon and wrestling with (1+almost zero number)^(super huge number) might be tricky. Does it have to be a huge result? What data could they collect? What would they do with it?

    And @LukeSelfwalker shared this simple but stunning Desmos activity which gets kids to see how polynomials can start approximating exponential functions — a beautiful visual connection to all of this.

  4. Of course @dandersod showed me a connection between Pascal’s Triangle and e, which I didn’t know about (or if I did, I totally forgot). He sent me this link:
    pic5.png
    Whaaaa?! But okay! WHOA?!
  5. @roughlynormal suggests:
    pic6.png
    I have to think about this… Basically the differences relate to the derivative… But I did a quick 5 minute look with a google spreadsheet and I couldn’t make this work. I think for it to work, you need to divide the first differences by the change in x, and also divide the second differences by the change in x, and look for them to be equal. In other words, the derivative…pic7.png
  6. @jdyer gave this gem which I’ve never heard of before:
    “You have a full glass with 1 liter of water. You take gulps from the glass; each gulp is a random real number of liters from 0 to 1. On average, how many gulps do you expect it to take before the glass is empty?”

    And his discrete version is you start with 100 and kids take away a random number (generated from 0 to 100) per step.

  7. @bowmanimal also talked about this “game”:
    pic8.png
    Whoa! TOTALLY new to me. Since we can’t put up e fingers, we could play the game where we each pick a number (doesn’t have to be an integer) from 0 to 5. I had to see this for myself!
    pic9.pngThe red line is always greater than the blue line (I think… I want to prove it algebraically!). [Note: related, this set of tweets on x^y=y^x which @BenjaminASmith alerted me to]. And @mathillustrated shared this amazing presentation (read it!) on scaffolding and formalizing this game with kids.
  8. @CmonMattTHINK shared this fact which I LUUURVE but forgot about:
    The probability of a random permutation of n objects being a derangement (no object remaining in its original position) approaches 1/e as n->oo.And @DavidKButlerUoA shared a wonderful presentation he made on where the derangement formula comes from.
  9. @bobloch also shared with me a fact I don’t remember ever learning! e shows up in the harmonic series.sum_{n=1}^{\infty} \frac{1}{n} = 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+...Apparently, if we look at which n values bring this series to an additional integer, we get:pic11.png

    And if you go further and further, and take ratios of these n values, you get a better and better approximation for e. I calculated 227/83 and got about 2.735.

  10. @mathgeek76 reminded me of the chapter in Steve Strogatz’s excellent book The Joy of X which talks about how you find the right partner, mathematically.
    pic12.png
    The answer involves 1/e. (Spoiler here.)

And then a wonderful conversation about where “e” belongs in a high school curriculum popped up on my feed, launched off by this tweet:
pic13.png
I should say this was Steve’s launching tweet, but he was open to thoughts on both sides of the discussion that ensued. My favorite part of the conversation was when someone (I can’t remember who) brought up \pi and when that should be introduced and why.

My conclusions

First and foremost, I knew there was a lot to e — it bridges continuous and discontinuous phenomena. It is important in calculus and (to some degree) combinatorics. There’s a lot I don’t fully understand yet about what people shared with me in their tweets — and I have to work through them to see their connections. I do see lots of connections but I haven’t worked through the math of any of these things to draw them in fully.

Second, in my musings on this, I think I’ve come to recognize that why I have always found e so fascinating is that is keeps on popping up in unexpected places. In my mathematical career, my jaw has hit the floor a number of times when I see it suddenly emerge when I never thought there would be a connection. I mean, think of the first time you saw e^{i\theta}=\cos(\theta)+i\sin(\theta). ‘Nuff said.

So if I want my kids to see what I see when I encounter e, that’s what I want them to encounter. A surprise. That it can pop up in totally unexpected places, and you  might not initially know why, but it eventually can become clear.

I don’t think I have time to pull this together for this year, but here’s what I’ve decided I want to do in some future year… I want to have a period of time where kids are told “this is a problem solving day” (or set of days). And I give the class different problems that will result in them approximating e… but they don’t know it. Like Pascal’s triangle… they can just do the calculations and see e pop out. Or the chess board experiment. Or taking random numbers away from 100. Or the harmonic series. Or the product challenge. Or the x^y vs y^x game.  Or the compound interest problem. And have them work on them. They all seem unrelated. Yes, this is contrived. Yes, I’m telling them what to do in many of them.

But BOOM. Soon this e number that keeps popping up in all these unrelated problems.

For each of these, they are going to get numbers close to e. And for me, I can say

This, my friends, this is what is so beautiful about this number for me. It is a universal constant. It is like \pi. It pops up in so many unexpected places. There is an underlying structure to why this is all happening, why it pops up everywhere. That is going to start to be revealed in calculus, but that’s only the bare beginnings… It goes much deeper. But I wanted you to get the experience of wonderment and have something that’s you know is true that begs the question why… WHY? WHAAAA? WHYYYYYYY! Because this desire to know, to figure out why something is true when you know it must be and it feels too unbelievable to be true, that’s a feeling mathematicians get that drive them forward in their work. And making those connections, and we know they must exist, it’s awesome when it happens. So yeah… I wanted to introduce you to this important number e which we’ll just take as that… an important number that we’ll get to play around with like we do \pi… but know it’s more than just a number. For you, now, it’s a question that’s begging an answer.

A realistic ending to an idealist exhortation from me to my students

Okay, I don’t know if I could really pull this off. But I’d love to.