Some Nice Twitter/Bloggy/Desmos Things

I love how creative I am with my blog titles. Meh. I realized I “favorite” tweets on twitter a lot when I want to save them for later, because they are awesome. But as I was looking through them recently, I was like: I should put some of these in a blogpost for others. And so one day if I’m looking for something, I can actually find it by searching my blog (something I do way too often) instead of scrolling forever on twitter.

@rawrdimus shared this applet he made on Desmos for helping kids to understand the idea of a derivative as “slope-iness.” What I like about it is (a) you only get a small line segment instead of the whole tangent line (the whole line would be distracting), and (b) that kids can drag the slider for a and get a sense for what’s happening and how that relates to what’s being plotted, and (c) that kids can then make a prediction where the next point will be (and then drag the slider to see if their thinking was correct.


Related to this is something many people worked on earlier this year based on a tweet I wrote (I wanted a surfer or skier to be travelling on a curve, and the surfboard/ski to be the tangent line)… an updated version of this was posted by @lustomatical…


My friend @pispeak posted a nice calculus puzzle that I enjoyed thinking through and solving: “Found this cool question below online (for a challenge) but got stuck…thoughts? help? @samjshah @calcdave @stoodle #mtbos “The line y=0 is tangent to both x^2 and x^3. But there exists another line tangent to both curves. What is the equation of that line?””

I don’t know this teacher, but I like the idea of doing this. Maybe next year I can make it a goal to do write one positive note to each student. Something heartfelt and genuine. A student met with me before school to talk about a “math exploration” she was going to do, and I loved how into her idea she was. I can totally write so many notes saying good things like that to my kids. Like this teacher, doing stuff like this will make me feel good.


@mikeandallie retweeted a link to a page that explains the unsolvability of the quintic without needing all that abstract algebra. I forgot to dig into this page. But OMG it looks like it’s going to be aweeeeeesome.

@bowmanimal wrote a freaking amazing blogpost about something he did in stats class before winter break. I still am reeling with how awesome it was. The question: “How can we use basic statistics to examine and tell apart writing styles? What do statistics about your own writing say about your style?”  Doesn’t get you excited? Trust me, click on the link and read how he does this. I don’t often come across lessons that I’m desperate to teach, but this is one of them. It also clearly comes from a master curriculum designer.

@dandersod wrote a blogpost ages ago about how to turn a graph into a 3D printed object. I desperately loved it, and had our tech integrator teach me how to do this on our school’s 3D printer.


I wanted to have my precalc kids make mathematical ornaments based on beautiful polar or parametric graphs they tinker around with/discover (maybe have a christMATH tree? haha sorry)… but the timing wasn’t right this year for ornaments (we do polar in the spring). But I still want to make this a reality this year. I hope I remember!!!

@fermatslibrary tweeted out this picture:


I love it because I remember doing something two years ago with my geometry class, arguing that we don’t need cosine and tangent, and that having sine is enough. We showed that we could have done all of trig with just sine. But then we talked about why having cosine and tangent in the mix makes our lives easier. I love this chart because it clearly illustrates what life would be like if we didn’t have multiple trig functions. (On a side note, I wonder what kids would notice and wonder about this chart if they hadn’t ever seen or heard of trig before. Like a middle school kid or a late elementary school kid.)

@mzbat (don’t know who this is) wrote a riff on my fav Carly Rae Jepsen, which I feel often enough:

hey i just met you
and this is crazy
but could this meeting
be an email maybe



POP! Popcorn Optimization Problem

I’m in the middle of optimization in my calculus class now. I had a “long block” (every seven school days, I see a class once for 90 minutes) and for the second half of that long block, I like to do something slightly different. Since I knew my kids hadn’t seen or done the traditional “box optimization problem” in precalculus (since I taught them last year also!), I decided to do that.

This might jog your memory if you don’t know what I’m talking about. You take a piece of paper. You cut out four squares (the same size squares) from the corners. You then fold up the four flaps and tape the box shut. There you go!


You can probably tell that the box’s volume is going to be based on the original paper you start with, and the size of the square you decide to cut out. The question is: what’s the largest volume you can get for this box. 

If you cut out a teeeny tiiiiny square, you’re going to have a very large base for the box, but almost no height. And if you cut out a giant square, you’re going to have a large height but a teeeeeeeny tiiny base. And somewhere between a teeeeeny tiny square and a giant square is going to be the perfect square to cut out which will give you the largest volume.

So the question is: given a specific piece of paper, what size square do you need to cut out to get the maximum volume.

This question has been done to death in middle school classes, in Algebra II classes, in Precalculus classes, and in Calculus classes. And I recognize that this post is just another rehashing of the same old problem. But I remember reading about a teacher who did a variation of this by including popcorn. And I wanted to do the same. No surprise, when I looked it up, it was dear Fawn. But I had such a lovely time in class today watching this unfold that I wanted to share the specific sheet I made up for kids to do this.

[2018-01-31 Popcorn Activity .docx version to download]

Teacher Moves / Outline

This activity requires students knowing and using the quadratic formula. My kids (standard level calculus) are pretty weak with algebra, so I started the class with a “do now” that had kids use the QF. So I recommend that.

Show kids the popcorn. (I had two different flavors.) Show your excitement about the activity. (I was genuinely excited!) Get this psyched. Hand out the worksheet but nothing else.

Put a three minute timer on the board. Explain the problem. Show kids a piece of cardstock with 4 squares drawn on it. Show kids a second piece of cardstock with those same four squares cut out and the flaps folded up so it looks like a box (but untaped, so you can unfold it too). Tell them the volume they create is the amount of popcorn they are going to get. And that you aren’t going to overfill their boxes — just to the brim. Tell them they have 3 minutes to work with their partner to come up with the best size square they want to cut out. And they are not allowed to do any calculations. Just visual estimation. 

At that point, give cardstock, ruler, scissors, and tape to kids. Do not let kids start until you press “GO” on the timer. Then… GO!

After three minutes, my kids were done. They measured the side length of the square they cut out and recorded it on the worksheet. They then cut and taped. They weren’t allowed to get their popcorn until they did one more thing… some math…


It was super important to me that kids didn’t measure anything, except for the side length of the square, to do these problems. Why? Because this is where I want kids to recognize the side length of the square is the height (that was obvious to all my kids), but also that when calculating the length and width, they were going to be doing 216-2x and 279-2x (where x was their side length). Only a few kids didn’t get the 2bit (they only subtracted x), but I sent them back to their seats to rethink their length and width and they immediately got it. It was actually awesome to hear their OOOOOOHHHHH moment. But yeah, no measuring. They have to use their brainzzz to come up with the length and width with what they are given!

Only after checking their volume with me, and I said it was correct, could they fill their boxes with popcorn.

As an aside, when writing this activity, I had to decide what level of scaffolding I wanted to give for this. I decided not much. So I didn’t include any diagrams. (Well, I did put two on the very last page of the worksheet in case a kid needed some additional help. Turns out no one did.) I also initially wrote the worksheet to be in inches, but then changed to centimeters, and then after thinking a bit more, I changed to millimeters. Why? So kids don’t have to deal with fractions (inches) or decimals (centimeters), and we could keep our eye on the prize. It also made the volume huge — and so kids would have to do a little work to get the correct window when graphing.

At this point, I sent them back to their seats with popcorn in their box to then solve the general case. Close to the end of class, I posted the different volumes students got by estimation (it was a tiny class today… kids were absent or at sports).


Overall, I spent about 35 minutes on this in class. One pair finished completely. All the others are at the place where they are in the middle of the calculus work (close to being done).

Submit a Proposal for TMC18! You, yes you!

I’m on the organizing committee for Twitter Math Camp. If you don’t know what it is, you should know it’s the greatest professional development I’ve been to. And I have gone to it every year since it started. So check out the website ( The key aspect is that it’s a grassroots conference that was started by math teachers for math teachers (and related educators). By people who were passionate about their classrooms and just wanted to get together with each other. Here are some fun pictures from TMC last year:


Below I have an invitation to submit a proposal to talk at TMC this summer. There are three options: a short 30 minute session, a regular 60 minute session, and leading a  6 hour multi-day session. If you want to come to TMC and haven’t considered giving a talk, I want you to take a moment and think “well… if I did put myself out there, this is what I would talk about… this is what I know.” If you’re a first year teacher, it could be a session called “If I could do it over” and talk about what you learned, to help other early career teachers. If you’re a math coach, it could be about how to wrangle your more challenging teachers and getting them on your side. If you’re an experienced teacher, it could be about how you design your quadratics unit or how you bring outside speakers to the classroom or … I’m just asking you to consider leading a session.

We in the online math teaching community and at TMC believe that everyone has things of value to share, and we can all learn from each other. TMC is a welcoming place, and if you’re scared of presenting, you’ll know that you’ll be doing it at a small conference to a small and friendly audience (anywhere from 5 to 20 people, usually). It’s a place to just put yourself out there! I personally am terrified of public speaking, but it was at TMC that I first put myself out there, and it turned out to be so much fun to design and implement my sessions, and just a lot less scary than I thought. And I did it with someone else, which made it more fun! So yeah, I’d love for you think about it. Think about what you know, think about what you have to say, think about what you’re strong at… and if you think you don’t have anything, I’d argue you’re being too hard on yourself. We all have things of value to share. And we all can learn from each other.

Now without further ado…


We are starting to gear up for TMC18, which will be at St. Ignatius High School in Cleveland, OH  (map is here) from July 19-22, 2018. We are looking forward to a great event! Part of what makes TMC special is the wonderful presentations we have from math teachers who are facing the same challenges that we all are. 

To get an idea of what the community is interested in hearing about and/or learning about we set up a Google Doc ( It’s a GDoc for people to list their interests and someone who might be good to present that topic. The form is still open for editing, so if you have an idea of what you’d like to see someone else present as you’re writing your own proposal, feel free to add it! 

This conference is by teachers, for teachers. That means we need you to present. Yes, you! In the past nearly everyone who submitted on time was accepted, however, we cannot guarantee that will be the case. We do know that we need 10-12 morning sessions (these sessions are held 3 consecutive mornings for 2 hours each morning) and 12 sessions at each afternoon slot (12 half hour sessions that will be on Thursday, July 19 and 48 one hour sessions that will be either Thursday, July 19, Friday, July 20, or Saturday, July 21). That means we are looking for somewhere around 70 sessions for TMC18. We are requesting that if you are applying to speak for a 30 or 60 minute session that there are no more than 2 speakers and if you are applying for a morning session that there are no more than 3 speakers.

What can you share that you do in your classroom that others can learn from? Presentations can be anything from a strategy you use to how you organize your entire curriculum. Anything someone has ever asked you about is something worth sharing. And that thing that no one has asked about but you wish they would? That’s worth sharing too. Once you’ve decided on a topic, come up with a title and description and submit the form. The description you submit now is the one that will go into the program, so make sure it is clear and enticing. Please make sure that people can tell the difference between your session and one that may be similar. For example, is your session an Intro to Desmos session or one for power users? This helps us build a better schedule and helps you pick the sessions that will be most helpful to you!

If you have an idea for something short (between 5 and 15 minutes) to share, plan on doing a My Favorite. Those will be submitted at a later date.

The deadline for submitting your TMC Speaker Proposal is January 15, 2018 at 11:59 pm Eastern time. This is a firm deadline since we will reserve spots for all presenters before we begin to open registration on February 1st.

Thank you for your interest!

Team TMC – Lisa Henry, Lead Organizer, Mary Bourassa, Tina Cardone, James Cleveland, Cortni Muir, Jami Packer, David Sabol, Sam Shah, and Glenn Waddell

A Secret Handshake


In one of my Advanced Precalculus classes last week, I saw a group of three students successfully figure something out. To celebrate, one group member taught his group how to do a three person handshake which was elaborate and awesomesauce.

Yeah, it moved me. Those are the things that get me.

Because what it showed me, in that moment, was how solid a group that was. They came to a collective understanding. They were having fun together — being wonderfully silly. And they were celebrating their success. It was a sign that the group had gone beyond being three people working together; they had created some sort of synergy. It was a lovely instantiation of that synergy.

Sadly for them, two days later, I changed groups. (My groups stay with each other for 6-7 weeks, usually.)

But inspired by this group, I changed what I did when I had kids sit with their new group members (in all four of my classes). They all said “Hi!” but then I dramatically and mysteriously had them hush up, and I showed them the first 11 seconds of this video.

They were entranced. So I asked if they wanted to see it again. They did, so I showed them the first 11 seconds again. They all thought they were going to learn that handshake. Fools! FOOLS!

Instead, I told them about how amazing it was to see this precalculus group develop their own handshake. I shared with them what that handshake meant to me, an outside observer… what it said about their group to me.

And then… I gave kids 3 minutes to develop their own group handshake together. The only thing I said was that the handshake had to involve everyone from the group. (Of course, this took 4 minutes, but saying you are giving them 3 minutes gets them working together very quickly.)

Now I’ll be honest. I thought this could go either way. I thought kids might be hesitant to do something corny/dorky like this, and it would be a huge flop. But in all four classes, every group did it. [1] And they were doing SUPER COOL THINGS including pounding the table, incorporating fistbumps, incorporating dance moves, and creating beautifully symmetric hand formations. It was super fun to watch. And some kids wanted to share their handshakes publicly, so those who were comfortable (and that was most of them) demonstrated their handshakes for the rest of the class.

What is going to happen with this?

I don’t know. Maybe nothing. At the very least, it was a great quick way to get kids working as a team on something when switching groups. And if the group can use it when celebrating a collective success, it will make visible and public what fun and friendly groupwork can look like. And that might just inspire other groups to do the same. I like an atmosphere where kids are propping each other up, patting each other on the back, and see themselves working as a team. And the more structures that I can develop that promote this [2], the better.


[1] Okay, one of my calculus classes was a little less enthusiastic as the others, but they all did it too! I didn’t get the same JUMP RIGHT IN feeling in all groups I got from the other classes. Some groups had it, but not all.

[2] Like the hotel bells

The DIY Math Curriculum: Simple tricks to make creating your own material feel less onerous

Math teacher friend Bowman Dickson and I presented a session at NCTM in Orlando on Friday. I have never given a public talk about math teaching before. Well, that’s not precisely true. I’ve led a couple of sessions at Math for America on the online math teacher community known as #MTBoS (as part of a larger thing that MfA was doing for new teachers). And at TMC, I have led some workshops. But this felt more official. The program committee for the Orlando meeting contacted me about presenting, and it wasn’t a workshop but a talk. And upon advice from a friend who said “you need to do this because it terrifies you,” I decided to do it. But only if my friend Bowman would do it with me. And of course he did.

This post is going to share the talk. If you scroll to the bottom, you’ll get access to the slides and the handout.

Title: The DIY Math Curriculum: Simple tricks to make creating your own material feel less onerous

AbstractDon’t like the way the textbook approaches a concept but are intimidated by creating your own content? Bowman and Sam both write their own content from scratch. We’ll share the simple lesson-design tricks we use to write investigations that lead to vibrant discussions and a-ha moments. You will leave ready and excited to write your own content!

Hack #1: Old Problem, New Problem
The Important takeaway:
This is the simplest of all the hacks. You might already do this naturally, and textbooks sometimes have questions that switch what students are traditionally given and what they are asked to find. If you’re hankering to see if students have gotten what they’re doing conceptually, mix things up. Just look at a problem and see if you can’t refurbish it by maybe giving them some information and “the answer” and asking them for some other piece of information that they traditionally are given. When you do this, kids will think harder, talk a heck of a lot more with each other (because the problem is more abstract), and you’ll often have many different responses that lead to great whole class conversations.

My favorite slides (one content, one funny):


Relevant blogposts:

  • Give students right triangles and have them associate the correct trigonometry equation that corresponds with those right triangles:
  • Come up with the equation for a parabola given a focus and directrix, and the backwards question:
  • Give students definite integrals and signed areas but missing the function, and see what functions they can draw:
  • Play Rational Function Headbandz with students, where students have a rational function (or trig! or logarithm! or whatever!) on their forehead so they can’t see it, but they ask each other yes and no questions to determine the equation of the graph:
  • Students use protractors to attack forwards and backward questions on inverse trigonometry on the unit circle:
  • Instead of giving students visual patterns and ask them to come up with the sequence, why not have them come up with their own visual pattern using blocks?:
  • Mathematical Iron Chef using group-sized student whiteboards:

Hack #2: Thinking Before Mathing
The Important takeaway: 
Too often, mathematical notation and premature abstractness get in the way of student thinking instead of being the tool for efficiency and communication that it is for those of us that already understand the concept. Let students play around with ideas in their heads, with their own framing, and own vocabulary, before you develop abstract structures. Let them do it their own, inefficient way before you show a better, more efficient, “correct” mathematical way – the right way won’t stick unless they’ve created something in their brain to stick it to!

My favorite slides (one content, one funny):


Relevant blogposts:

Hack #3: Make Math Magical Again
The Important takeaway: 
This hack takes some time, but it is worth it. You are trying to build up a moment of surprise and curiosity for kids – something that will make them want to learn more. (It’s like watching a magic trick. You’re in awe, but you desperately know how the trick was performed because magic isn’t real.) You have to think about something you find interesting and really dig deep to figure out for yourself why it is interesting. That takes some thinking! But once you find the answer, I’ve found it often points directly to a way to get kids to appreciate that thing. Often times, I’ve found that having kids explore uninteresting things is powerful because it gives context for the interesting outcome (e.g. appreciating that the complex solutions to polynomials when plotted aren’t that interesting, but solutions to xn=1 are interesting). Also, like in magic, misdirection can also work. Have kids think they are working on one thing, but actually have them accidentally stumble upon another thing can be powerful (e.g. algebraically finding properties of very different looking trig equations like x-intercepts and vertical asymptotes, but as students work, they find out the very different looking equations actually produce the same graph).

My favorite slides (one content, one funny):


Relevant Blogposts:

Hack #4: Toss ‘Em An Anchor
The Important takeaway: 
Math instruction doesn’t always need to go from skill to practice to application. Instead, application to some interesting context, whether that be abstract or “real world” can actually drive student learning, and help them learn the more mundane skills and contexts. Great anchors are both natural to the mathematical context, and sticky – tangible, novel, memorable, easy to refer back to.

My favorite slides (one content, one funny):


Relevant Blogposts:

Photos of Me and Bowman Presenting:



A photo of Bowman, me, and my colleague who came to support me!


Some Tweets about the Presentation:


This slideshow requires JavaScript.


NCTM Orlando Handout PDF

Slides (with one taken out…):

Exploding Dots! Global Math Week 2017!

Hi all,

Life is getting away from me with some tough personal stuff. So I haven’t been as active with the online math teacher community/twitter/blogging/etc. for a while, and I sadly probably I won’t be for a while.

That being said, I really wish I could participate in this initiative that Raj Shah (no relation!) shared with me a while ago. But because of life stuff I might not be able to. But one of the biggest things I want to do is bring joy into the math classroom as a core value, and this does that. And I love the idea of a collective joyful math moment for students and teachers all around the world! I’ve done a bit of exploration with this initiative — exploding dots — and I think it’s fabulous and full of wonderment. What it takes? At minimum, 15 minutes of classtime! I highly recommend you reading the guest post I asked Raj to write (below), and joining in this worldwide effort to celebrate the interestingness of mathematics!




The Global Math Project is an invitation to students, teachers, and communities everywhere to actively foster their sense of wonder and to enjoy truly uplifting mathematics. Math is a human endeavor: It’s about thinking creatively, exploring patterns, explaining structure, and solving real problems. The Global Math Project will share a unifying, joyful experience of mathematics with people all across the world.
Our aim is to thrill 1 million students, teachers, and adults with an engaging piece of mathematics and to initiate a fundamental paradigm shift in how the world perceives and enjoys mathematics during one special week each year. We are calling it Global Math Week.
This year, Global Math Week will be held from October 10–17. The focus of Global Math Week 2017 is the story of Exploding Dots™ which was developed by Global Math Project founding team member James Tanton, Ph.D.
Exploding Dots is an “astounding mathematical story that starts at the very beginning of mathematics — it assumes nothing — and swiftly takes you on a wondrous journey through grade school arithmetic, polynomial algebra, and infinite sums to unsolved problems baffling mathematicians to this day.”
The Exploding Dots story will work in any classroom, with a variety of learning styles. It’s an easy to understand mathematical model that brings context and understanding to a wide array of mathematical concepts from K-12 including:
  • place value
  • standard algorithms for addition, subtraction, multiplication, and long division
  • integers
  • algebra
  • polynomial division
  • infinite sums
  • and more!
Teachers routinely call Exploding Dots “mind-blowing”!
“I am still amazed by this. Exploding Dots has changed my fifth grade class forever!” – Jo Anna F.
“This makes me WANT to teach algebra!” – Kristin K.
“YES!” Hands up in the air in triumph! Decades of believing I couldn’t do math—poof! Exploded!”  – Jennifer P.
Join us for Global Math Week, October 10 – 17, 2017!
During Global Math Week, teachers and other math leaders are asked to commit to spending from 15-minutes to one class period on Exploding Dots and to share their students’ experience with the Global Math Project community through social media.
You can join the movement in four easy steps:
1) See Exploding Dots for yourself
Here’s a brief overview:
2) Register to Participate at
3) Conduct an introductory Exploding Dots experience with your students during Global Math Week
All videos, lesson guides, handouts are available for free at Since everything is available online, inspired students (and teachers) can continue to explore on their own.
4) Share your experience on Twitter during Global Math Week using #gmw2017
That’s it!
The power of the global math education community is truly astounding. To date, over 4,000 teachers have registered to participate in Global Math Week (#gmw2017) and they have pledged to share Exploding Dots with over 560,000 kids from over 100 countries! We already over half-way to our goal
Help us reach and thrill a one million students!
The Global Math Project is a collaboration among math professionals from around the world. Spearheaded by popular speaker, author, and mathematician James Tanton, partner organizations include the American Institute of Mathematics,, Math Plus Academy, and the National Museum of Mathematics.

Bragging about my school

This is a milestone for me. I have been at my school for ten years, and this is the start of my eleventh. It’s the only school I’ve worked at. That’s a testament to my school, but more specifically, to my colleagues.

Last year, my school’s awesome director of communications contacted the math department to let us know that the one issue of the magazine she publishes four times a year was going to focus on math. And she wasn’t kidding! The cover of the magazine had most of my multivariable calculus kids on it (thinking deeply at the math-art show I helped put on last year)!


One of my favorite things is that the feature article with an alliterative title, Making Math Meaningful, was simply the transcript of a roundtable discussion we had. A bunch of math teachers got in a room around a big table, and we were led by our director of communications who had done her research and come with some questions. There was a digital recorder in the center of the table. And through talking with carefully crafted prompts, we got to think deeply and collectively about our own practice. I can’t even tell you how interesting it was to listen to my colleagues during that facilitated conversation, and how proud I was to be in a school with such like-minded folks that I have the opportunity to learn from. (If you’re a department chair or academic dean, consider doing this!)

I wish I could just post a PDF of the article for you to read, but alas, the whole magazine is online but can’t be downloaded. Here are two quotations to whet your appetite:

quote 0

quote 1.PNG

So if you want to read about a department that is doing strong work moving towards inquiry-based learning, and read the words of real teachers having a real conversation playing off of each other, I highly recommend you:

  1. Go to this site
  2. Make the magazine full screen
  3. Read pages 18 to 29

That is all!