Desmos + Statistics = Happiness

Sunday – a quiet evening before President’s Day – checking out twitter – not looking for trouble – and then,

Wait..what’s this?  Standard Deviation?  It was my birthday this past Saturday, and the Desmos folks knew exactly what to get me as a present.  Abandon all plans, it’s time to play.  A lesson I picked up from Daren Starnes (of The Practice of Statistics fame) is a favorite of mine when looking at scatterplots.  In the past, Fathom had been the tool of choice, but now it’s time to fly with Desmos.  There are a few nuggets from AP Statistics here, and efforts to build conceptual understanding.


Click the icon to the right to open a Desmos document, which contains a table of data from The Practice of Statistics.  In you are playing along at home, this data set comes from page 194 of TPS5e and shows the body mess and resting metabolic rate of 12 adult female subjects. One of the points is “moveable” – find the ghosted point, give it a drag, and observe the change in the LSRL (least-squares regression line) – explore changes and think about what it means to be an “influential” point.

Next, click the “Means” folder to activate it.  Here, we are given a vertical line and horizontal line, representing the means of the explanatory (x) and response (y) variables. Note the intersection of these lines.  Having AP students buy into the importance of the (x-bar, y-bar) point in regression beyond a memorized fact is tricky in this unit.  Drag the point, play, and hopefully we can develop the idea that this landmark point always lies on the LSRL.

Another “fact” from this unit which can easily wind up in the “just memorize it” bin is this formula which brings together slope, correlation, and standard deviation:

The formula is given on the exam, with b1 acting as the slope, so even memorizing it isn’t required, but we can develop a “feel” for the formula by looking at its components.

Click the “Means plus Std Devs” Folder and two new lines appear. we have moved one standard deviation in each direction for the x and y variables. Note that the intersection of these new lines is no longer on the LSRL. But it’s pretty close…seems like there is something going on here.

Ask students to play with the moveable point, and observe how close the rise comes to the intersection point. Can it ever reach the intersection? Can we ever over-shoot it? In the “Rise Over Run” folder, we can then verify that the slope of the LSRL can be found by taking a “rise” of one standard deviation of y, dividing by a “run” of one standard deviation of x, and multiplying by the correlation coefficient, r.

There’s other great stuff happening in the Desmos universe as well.

1.  This summer brings the 4th edition of Twitter Math Camp, to be held at Harvey Mudd College in California. I’m thrilled to have latched onto a team leading a morning session on Desmos. Consider coming out for the free PD event, and join myself, Michael Fenton, Jed Butler, and Glenn Waddell for what promise to be awesome mornings. To be honest, I feel the Ringo of this crew….

2. Can’t make it to the west coast this summer? Join me at the ISTE conference in Philadelphia, where I will present a learning session: “Rethink Math Class with the Desmos Graphing Calculator“. Bring your own device and join in the fun!

3. Are you new to the world of Desmos? Michael Fenton has organized an outstanding series of challenges, with 3 difficulty levels, to help you learn by doing. Try them out – they promise to get you think about how you and your students approach relationships.

4. Merry GIFSmos everybody!  The team at Desmos has developed GIFSmos to let you build your own animated gifs from Desmos files. EDIT – as Eli noted in the comments, credit for GIFSmos goes to Chris Lusto.  Thanks for being so awesome, Chris!

Odds and Ends from a New Semester

A new semester for me started last week.  Lots of excitement, new faces, new classes, and much going on professionally. So many feelings coming from all directions….


Last fall, I participated in an Ingite session with the awesome folks from the Math Forum at the Association of Math Teachers of New Jersey conference.  Thanks to Suzanne for her kind invitation to share with a group of math leaders I have admired from afar.  The videos from the Ignite are finally available. Enjoy my flailing arms in a talk related to a blog post from long ago on math phrases I’d like to expunge.

While you are on YouTube, check out talks from the rest of the panel especially Max Ray and Annie Fetter. I marvel at Max’s ability to weave a math story cleanly and effectively about a theme. And Annie always provides ideas I need to try the next day in my math class.


This new semester brings me 3 new classes, which are 3 completely different preps: AP Statistics, Academic Prob/Stat (which is co-taught) and CP Pre-Calculus. Are there 3 classes which could be any more different? I feel troubled when I have a schedule like this, since I feel like none of the classes get the attention they deserves, and I spend most of my time chasing my tail. Also, CP Pre-Calc is a course I have never taught before. Does anyone else have trouble becoming invested in developing inquiry lessons for a course, when you know you may never teach it again?  Tough days….


For my new niece, who may be born any second now!


Our school adjusted the bell schedule this year; the result being structured time in the middle of the day where all students have lunch and/or study time. This semester, I am organizing Hatters’ Discovery Time – an opportunity for teachers to share their passions with students – modeled after the Genius Hour concept.  So far, our No Place for Hate team has facilitated an open discussion on race – a well-attended event with productive, positive discussions. This month, one of our science teachers who is also an EMT will share lessons on medical diagnoses, while a fellow math teacher offers an introduction to juggling. My session at the end of February will feature codebreaking, inspired the movie The Imitation Game, and just in time for Oscar season. Sessions on medidation and woodworking are also in the hopper. Such great ideas from my colleagues and hope students enjoy the offerings!


The last month has brought a flurry of professional plans for the summer and beyond. I’ve been invited to attend the AP Statistics reading for a 4th year, which is one of the highlight of my year…it’s summer camp for the Stats Kids!  Late in June, I have been accepted to share Desmos ideas for the classroom at the ISTE conference in my hometown of Philadelphia. And plans for Twitter Math Camp in July are taking shape: a team of 3 math professionals whose work I all admire and I will team to present morning sessions on “Next Steps with Desmos”, while I have been accepted to present a session on tackling those “tricky statistics concepts”.  Should be a blast in LA.

But, today also brought rejection from NCTM, as my session on Standards-Based Grading in Algebra 1 was rejected for 2 regionals. Last year, I was rejected for a Desmos session, and the year before for a session on encouraging writing skills in math class. At this point, I really think I am being punked.  But congrats to all of my friends and colleagues who have been accepted, I’ll be curled up at home with a box of cookies while you all enjoy the conference……

All is not lost though. I think it’s time to put the SBG talk away, as I have done it at PASCD and a local conference now. I never got around to editing the video of this talk from our fall ATMOPAV conference, but I’ll share it now for old time’s sake.  Enjoy!

Adding Distributions of Simulated Data

The current chapter on expected value and combining distributions in AP Statistics is one of my favorites for a number of reasons.  First, we have the opportunity to play games and analyze them…if you can’t make this fun, you are doing something wrong. Second, it often feels like the first time in the course we are doing some heavy lifting. Until now, we have discussed ideas like sampling, scatterplots and describing distributions – nothing really “new”, though we are certainly taking a much deeper dive.

The section on combining distributions contains a number of “major league” ideas; non-negotiable concepts which help build the engine for hypothesis testing later.  The activity I’m sharing today will focus on these facts:

  • The variance of the sum of independent, random variables is the sum of their variances.
  • The variance of the difference of independent, random variables is the sum of their variances.
  • The sum of normal variables is also normal.

First, we need to have student “buy in” that variances add. Then we have the strange second fact: how can it be that we ADD variances, when we are subtracting random variables? In this activity, we’ll look at large samples, and what happens when we add and subtract these samples. Since many students taking AP Stats have the SAT on their brain, and there is a natural need to add and subtract these variables, we have a meaningful context for exploration.


The printable classroom instructions for this activity are given at the bottom of this post.

photoTo begin, students use their graphing calculator to generate 200 simulated SAT math scores, using the “randnorm” feature on their TI calculators, and using the fact that section scores have an approximately normal distribution with mean 500 and standard deviation 110. Note – some older, non-silver edition TI-84’s won’t be happy with this, and a few students had to downgrade and use a sample of size 50 instead. There are a few issues with realism here: SAT section scores are always multiples of 10, which randnorm doesn’t “know”, and occasionally we will get a score below 200 or above 800, which are outside the possible range of scores. Also, there is a clear dependence on SAT section scores (higher math scores are associated with higher verbal scores, and vice versa), and here we are treating them independently.  But since our intent is to observe behavior of distributions, and not reach conclusions about actual SAT scores, we can live with this. In my class, no student questioned this as problematic.

Repeat the simulation in another column to simulate verbal scores. Then, for both columns, compute and record the sample mean and standard deviation. For my simulated data, we have the following:

It’s time to pause and make sure all students are clear on what we are simulating. We now have 200 students with paired data – the math and verbal score for each. Like most students, our simulated students would like to know their overall score, so adding math and verbal scores is natural. I help students write this command in a new column, then let them loose with the remaining instructions on both sides of the paper.

Students had little trouble finding the sum of the math and verbal scores, and computing the summary statistics. For my sample data, we have:

As students work through this, I want to make sure they are making connections to the notes they have already taken on combining distributions. I visited each student group (my students sit in groups of 4) to discuss their findings. Most groups could quickly identify that the means add, but what about those standard deviations? By now, if my students have taken good notes, they know that standard deviations don’t add, and that variances should. I leave groups with the task of verifying that the variances add.

Here’s the beautiful thing: students who immediately tell me that they “checked” the variances and verified the addition get the evil eye from me. In this simuation, students should find that the variances are “close” to adding, but not quite.  At the end of the acitivity, I ask students to conjecture why the addition is a “not quite” – even after I have beat into them that variances add.  There are two main reasons for this, and I was happy that a number of students sniffed these out.

  1. We are dealing with samples, not populations. There is inherent variability in the samples which causes the sample variances to not behave nicely.
  2. Variances add – but only if distributions are independent. Here, even though we created large random samples, there is still some small dependence. And while we don’t specifically cover the formula for dependent distributions in AP Stats, it’s worth discussing.

Next, it’s time to look at the differences.  Here’s students are asked to subtract math and verbal scores, compute the summary statistics, and compare the sum and differences. This was a nice way to go back and re-visit center, shape and spread.


CENTER: Sums are centered around 1000, while differences are centered around zero.

SHAPE: Both distributions appear approximately normal.

SPREAD: The sum and difference distributions appear to have similar variability.

And this idea that the spread, and standard deviation, will be similar for both the sum and difference, can be also be explained by looking at the range of each population distribution.

  • For the sums, the max score is 1600 (800 M and 800 V), with a min of 400 (200 each)
  • For the differences, the max score is +600 (800M and 200 V), with a min of -600 (200M and 800 V).

Here, we can see that both distributiuons has the same range.

From start to finish, this exploration took about 30-40 minutes, and was worthwhile for verifying and developing understanding of the facts for combining distributions.  The student instructions and video notes students take beforehand are given below.  Enjoy!

Thoughts on Teacher Preparation

Just before winter break, I received an e-mail from a former student who was now teaching in a private school in Maryland. This is his first year teaching AP Statistics, and he was looking for some nuggets of wisdom and advice from someone he trusted.  Drew was one of my favorite students, mainly because he can keep up with my warped sense of humor, and it’s thrilling to add another former student to my teaching “tree”.  The conversation began with a comment I suppose is universal to new teachers. (note – many quotes here paraphrased by my weak memory).

Nobody told me it was going to be this much work.

Yep. Drink up, Drew! Anything else?

I don’t know how I find time to plan. Between getting tests ready and grading the homework I collect every night, there’s just not enough time.

Wait…you collect and grade homework every night?  Why the hell are you doing that?  Let’s have a chat about the intent and value of homework.

How did college prepare me for any of this?

They didn’t. You need another beer Drew?  The lack of classroom realism education colleges provide seems to be a common tale, and for my money is one of the big problems we need to tackle if we seek to improve the profession. So here comes the big reveal I’d like to share with all new teachers and/or those considering the profession.  Ready?  This is free advice, though putting a buck in the tip jar is always welcome.

The day-to-day profession of teaching is nothing like the tired stereotypes, nor does it resemble the vision you have of teaching when you think back to your own schooling. It’s likely that for the first few years of your profession you will adhere to stereotype: neat rows, textbook pages, planned lectures. The path of your career, your effectiveness as a teacher, and your own happiness all rest on how quickly you challenge your methods. Find colleagues doing great things, learn from them, tweak them, and make them your own.

PodcastThis past week I caught up on past episodes on the Freakonomics Podcast while working out. One episode from November was of particular interest, titled “Is America’s Education Problem Just a Teacher Problem?“. When I talk about the teaching profession with non-educators, my prescription for “fixing” the system often comes down to 3 easy-to-understand steps:

  1. Hire the best people
  2. Train them well
  3. Keep the best and let them lead

Isn’t that how all professions should operate? Where are we faling short? And while this podcast started off slow, with a narrow focus on where teachers tend to fall in their graduating classes, it was a later conversarion regarding teacher training which moved me. The quotes below come from David Levin, co-founder of KIPP schools, which you can find on the podcast transcript.

The way we train teachers is fundamentally broken in this country.

Yep….My bachelors degree is in mathematics. Only after graduation did to I go back to Drexel University to obtain teaching certification. I clearly recall a conversation with my advisor back then, who touted the Drexel program with “we believe anyone can teach”, and assurance that I could teach with an intern certificate after taking just 1 class.  Somehow I made it through, but I can’t recall much from my training which was helpful to me as a young educator. I’ve also discovered that not everyone with a bachelor’s degree can teach…not surprising.

It’s disproportionately theory-based. …I’m not sure of what good the theory of math instruction is if you don’t actually know how to deliver a lesson on math as well.

I would go even further here. It’s not just about planning and delivering a lesson – it’s about having the expectation that lesson study is a critical element in becoming an effective educator. How do we know if a lesson is effective? How do we adjust for student needs? What do we do to improve even our best lessons?

Sometimes the best math teachers weren’t necessarily the best math students, because you know you often teach better what you weren’t so good at, because you actually had to work to learn it.

Whoa….so, should we be hiring all of the top students to teach, or no? I immediately had a flashback to college calculus. I fully admit to not being a top math student, mainly because my work habits were supect. But I could learn new things quickly on my own if I had the inclination. I have to say that I didn’t really understand calculus until I started tutoring it. Having to explain the chain rule to a peer caused me to think about how it worked, to prepare for snags, to consider how ideas fit within the big picture. There’s an assumption that math teachers simply transmit information, with little regard for how we facilitate learning. This was quite a refreshing statement.

Back to my conversation with Drew, where it was also a great time to confess many of my teaching sins from over the past years, and measure the reaction. So, over some beverages, I had to confess something to Drew about the AP Stats class he attended, which was in my 3rd year of teaching the course.

Drew, I didn’t know what the hell I was doing back then.

It’s true. Sorry. I mean, everyone passed the AP exam and everything, but I look back on some of my old techniques and can confidently place them in the “stuff I would never advise people to do” file.  Drew gave me look on par with the wide-eyed glare a 9-year-old gives you when you finally tell him Santa isn’t real, that awkward, incredulous stare which is followed by knowing head nods.

So what changed? I used the same textbook, provided many of the same materials, revised my tests a bit. Tweaking problems and getting better at identifying the tricky part of problem 45 on page 312 just isn’t enough.  My development as a teacher comes from finding and utilizing a support system to help me improve. This goes beyond conversations with building colleagues, though having an enthusiastic department here at my home school has been valuable. Local groups like the Philadelphia Area Stats Teacher Association and ATMOPAV have helped me develop a local network of master teachers with whom I can discuss lessons. Recently, reaching out through twitter and the AP Stats message boards grew my learning community, and help me continually improve.

I wonder how many teacher prep programs are helping teachers find and build their own professional networks? In this age of connectivity, it’s simply silly to leave out this crucial piece.

Class Opener – Day 74 – PolyGraphs

My 9th graders have only about 2 weeks left with me before their final exam. Most of them will move on to Algebra 2 next semester, so my strategy with them has been 2-pronged: ensure we are produtive with new material and put them in a “happy place” to make a seamless tranisiton to Algebra 2. With a unit review today, and a pre-holiday-break quiz Monday, this was a perfect time to test-drive the new FREE PolyGraph activity from my friends at Desmos, along with the awesome work of Dan Meyer and Christopher Danielson. The Parabola activity sounded perfect for my class, though there are also activities featuring linear functions, rational functions and hexagons.

My freshmen have limited understanding of quadratic functions. While we have encountered some useful vocabulary regarding parabolas in my class (intercepts, vertex, domain and range), these students have not had a formal unit in graphing them yet. I was curious if students could transfer what they already knew to a new scenario. I was tempted to do a quick review of vocabulary before sending kids to the lab, but thought better of it. I want gut reactions.

In the activity, one student acts as the “picker” and chooses 1 parabola from a set of 16. The “guesser” then asks yes or no questions to help narrow down which parabola was chosen. “I don’t know” is also available as an option, if the question is not clear.

polygraphBetween games, students are given challenges to help guide their understanding of vocabualry and “good” leading questions.  I found these “intermission” questions to be extremely helpful, and noticed that the quality of the questions students asked improved after participating in them.

Some obeservations about my students in this activity, which we did for about 30 minutes.

  • Students didn’t have vocabulary to describe parabolas which “open up” (a>0) versus those which “open down”. The question “is it a smiley face or a frowny face?” was used by more than one student and led to some side discussions of what this meant.
  • Students also recognized that parabolas could have different widths, and describing the differences between these was more challenging. Questions like “Is it wide?” or “Is it narrow?” are helpful for identifying some extreme curves, but without a baseline for what a “regular” curve looks like, this leads to some confusion over which parabolas should be eliminated.
  • In the first round, few student mentioned the x or y-axis in their questions. Later, I noticed these became valuable tools for elimination.
  • Questions which attempted to use the vertex showed mixed success. “Is the vertex positive?” is unclear, but these attempts improved with more game plays. Similarly, attempts to describe domain or range often needed more work.
  • Students can be sneaky, and mine are no exception. Some students attempted to bypass mathematical conversation by asking “Is it in the top row?”.  Nice try – until they realize the parabolas are mixed up. Also, sometimes students were assigned to play against students sitting right next to them. Not ideal, but workable.

Here’s where I would go with this, if my next unit was on a formal discussion of quadratics: copy the student-developed phrases like “smiley face” nto a document. As we encounter those ideas formally in graphs, develop more math-specific language, match them up with the student descriptors, and improve the document. I want students to take ownership of their descriptions, and allow for their self-generated language. Hopefully, this builds richer connections to the vocabulary.

At the end of each class, I had students complete a Google Form evaluation. I appreciate the feedback from students who took this task seriously!

What did you enjoy most about this activity?

  • This activity was really really FUN!! I liked it because I was able to interact with my classmates. I had fun as well as learn.
  • It was interesting to see the language people used to describe the graphs.
  • i really liked that it was an interactive activity that we could do with our class mates. It really allowed us to think about math in a fun way!

How did this activity increase your understanding of parabolas?

  • I learned that a “smiley face” is positive and a “frowny face” is negative. these math terms will probably be useful.
  • I had to really think about the vocabulary and what I was saying while asking and answering questions
  • It forced me to think mathematically and use many math terms to figure out the answer.

Down the road, I think it could be fun to have one class code and invite students from a number of schools to join in. Knowing that their partner was somewhere in the room caused some goofy behavior, and I wonder how much more focus they would have if their partner was from a different school. In the end, I appreciate this activity because it is fun, forces students to think mathematically, and has clear entry points for class discussion after leaving the computers. Finally, can we have students use their already-existing Desmos accounts for logging in? I like that the data from all students is shared with me, and would be even better if their activity data is all in one place.  Awesome job team!

Class Opener – Day 73 – You Can’t…Because You Can’t

In the past few days, my 9th graders have worked through a chapter on polynomials: multiplying, factoring, solving, simplifying. There’s a lot of process here, and often my fear is that students attempt to memorize short-cuts (such as the old stand-by…FOIL) without fully understanding the reasons WHY procedures are valid. It’s an easy “out” to tell students they will need procedures for their next class – I drink from this well sometimes – but I need something more my students. I want them to be able to clearly articulate and verify, using precise vocabulary, the rationale for all steps they take in math class.

In today’s opener students were presented with two problems on the board I had “solved”, and were asked to comment on my procedures:


The problem on the left is one we had completed yesterday in class, and a number of students noticed that one of the solutions, zero, was missing. I asked students to identify reasons why my solution gave a different solution set:

Because you don’t get the right answer.

Because zero is supposed to be an answer.

We’re not quite getting to the heart of the matter….I asked students to look over my solution to the problem on the right and comment.

You have to subtract the 5 over first.

You need to set it equal to zero.

In each case, students were fixated more on what I should have done, rather than what was presented in front of them as a solution.  Time to re-direct the conversation – I asked students to think about each step I had done in the problems, and tell me specifically which step was in error. This is a much more uncomfortable experience.  For each problem, the steps “feel” right. In both of my classes, the breakthrough eventually came, with some coaxing:

Students: You set both factors equal to 5. You need to set them equal to zero.

Me: Why can’t I set them equal to 5? The equation equals five.

Students (eventually): Because if two things multiply to make zero, one of them must equal zero.

Now we are getting someplace. The zero-product property is often taken for granted in this unit, but it is a powerful little engine. Name two numbers which multiply to make a product of 5….is it guaranteed that one of the two numbers MUST be 5? Nope. Zero is the hero. Hoepfully, some new conenctions were made regarding the nature of zeroes here.

The problem on the left was a much tougher nut to crack. The conversation eventually focused on the “other” solution – zero – and the perils of dividing by zero. Definitely look for more “devil’s advocate” moments as we explore rational expressions further.

Class Opener – Day 72 – Fermi Questions

Today’s class opener caused a student to string together a sequence of words I don’t think I’ve ever heard in this history of sentences:

Can we please just factor now?

Today I exposed my freshmen classes to Fermi Questions, a series of unusual estimation experiences, like the one we started with below:


The Fermi site here provides a slider where you can change the power of 10 to integer values. Some students had trouble wrapping their heads around the expectation, until some students summarized the ideas quite nicely:

  • It has to be in the thousdands.
  • I don’t think it could be in the millions.
  • Think about how many it would take to go along the side and multiply.

Students really got into the estimates, and I enjoyed listening to them argue their position with neighbors while attempting to estimate unknown quantities. I facilitated the group-think by moving the slider based on loud “higher” or “lower” from the group, until it seemed we were satisfied.  The site then gives you a result and a score based on how close you were.  There are a few thousand questions on the site, and we got through about a dozen today before settling into class.

Some of my favorite questions are those which demand a negative exponent, such as this one:

Determine the diameter of a 22 caliber bullet divided by the length of the Nile river.

Do we think it is one-tenth the length? One hundredth? One millionth? This was a fun way to re-visit laws of exponents, especially negative exponents.

While most of the class was engaged in the discussion, a few shyed away, which led to the quote at the start of this post. Are this questions really so threatening to students that they would RATHER factor? It also plants the seeds for some potential stats data collection, down the road.