Category Archives: Technology

Seeing Stars with Random Sampling

Adapted from Introduction to Statistical Investigations, AP Version, by Tintle, Chance, Cobb, Rossman, Roy, Swanson and VanderStoep

Before the Thanksgiving break, I started the sampling chapter in AP Statistics.  This is a unit filled with new vocabulary and many, many class activities.  To get students thinking about random sampling, I have used the “famous” Random Rectangles activity (Google it…you’ll find it) and it’s cousin – Jelly Blubbers. These activities are effective in causing students to think about the importance of choosing a random sample from a population, and considering communication of procedures. But a new activity I first heard about at a summer session on simulation-based inference, and later explained by Ruth Carver at a recent PASTA meeting, has added some welcome wrinkles to this unit.  The unit uses the one-variable sampling applet from the Rossman-Chance applet collection, and is ideal for 1-1 classrooms, or even students working in tech teams.  Also, Beth Chance is wonderful…and you should all know that!

starsIn my classroom notes, students first encounter the “sky”, which has been broken into 100 squares. To start, teams work to define procedures for selecting a random sample of 10 squares, using both the “hat” (non-technology) method, and a method using technology (usually a graphing calculator). Before we draw the samples however, I want students to think about the population – specifically, will a random sample do a “good job” with providing estimates? Groups were asked to discuss what they notice about the sky.  My classes immediately sensed something worth noting:

There are some squares where there are many stars (we end up calling these “dense” squares) and some where there are not so many.

Before we even drew our first sample, we are talking about the need to consider both dense and non-dense areas in our sample, and the possibility that our sample will overestimate or underestimate the population, even in random sampling.  There’s a lot of stats goodness in all of this, and the conversation felt natural and accessible to the students.

Studestars1nts then used their technology-based procedure to actually draw a random sample of 10 squares, marking off the squares.  But counting the actual stars is not reasonable, given their quantity – so it’s Beth Chance to the rescue!  Make sure you click the “stars” population to get started.  Beth has provided the number of stars in each square, and information regarding density, row and column to think about later.

But before we start clicking blindly, let’s describe that population.   The class quickly agrees that we have a skewed-right distribution, and take note of the population mean – we’ll need it to discuss bias later.

Click “show sampling options” on the top of the screen and we can now simulate random samples.  First, students each drew a sample of size 10 – the bottom of the screen shows the sample, summary statistics, and a visual of the 10 squares chosen from the population.


Groups were asked to look at their sample means, share them with neighbors, and think about how close these samples generally come to hitting their target.  Find a neighbor where few “dense” area were selected , or where many “dense” squares made the cut, how much confidence do we have in using this procedure to estimate the population mean?

Eventually I unleashed the sampling power of the applet and let students draw more and more samples.  And while a formal discussion of sampling distributions is a few chapters away, we can make observations about the distributions of these sample means.


And I knew the discussion was heading in the right direction when a student observed:

Hey, the population is definitely skewed, but the means are approximately normal.  That’s odd…

Yep, it sure is…and more seeds have been planted for later sampling distribution discussions. But what about those dense and non-dense areas the students noticed earlier?  Sure, our random samples seem to provide an unbiased estimator of the population mean, but can we do better?  This is where Beth’s applet is so wonderful, and where this activity separates itself from Random Rectangles.  On the top of the applet, we can stratify our sample by density, ensuring that an appropriate ratio of dense / non-dense areas (here, 20%) is maintained in the sample.  The applet then uses color to make this distinction clear: here, green dots represent dense-area squares.


Finally, note the reduced variability in the distribution from stratified samples, as opposed to random samples. The payoff is here!

Later, we will look at samples stratified by row and/or column.  And cluster samples by row or column will also make an appearance.  There’s so much to talk about with this one activity, and I appreciate Ruth and Beth for sharing!

Pulling In To the Station

My school isn’t 1-1 with technology yet, though there are rumblings we will get there next year….or the year after….or 2031…anyway, it’s time to get techy!  My new classroom features 4 computer stations in the back – nice to have, but not super-helpful with classes of about 24 each. Station-model classroom structure has been super-helpful in my pre-calculus class in the first month. Besides the chance for all students to participate in rich technology-based activities, I’ve had the opportunity to carve out valuable small-group time with students.  Here’s an example:

In our first pre-calc unit, we review functions and their shirts, folding in new ideas like the step function, piecewise and even/odd functions.  My objective for the class was for students to consider functions in varied forms.  As students entered class, playing cards were drawn to establish their groupings, so there were 3 groups of 7 or 8.  With 15 minutes on the classroom clock, students started on their first station:

  1. Group 1 gathered in a small group with me in a circle of desks, where we worked through proving functions even or odd, and sketching their graphs.
  2. marbleslideGroup 2 worked at the computer stations on a Desmos Marbleslides featuring quadratic functions, with many students pairing up to work together. If you have never tried a Marbleslides, run and play now – we’ll wait for you to come back…
  3. Group 3 worked out in the courtyard (hey, my new classroom leads outside – which is nice) on a group task involving a piecewise function.

After groups had rotated through all 3 activities, we had time to recap / share and assess our learning over the hour.  Here’s why I need to do this more:

  • The small group station let me touch base with every student, assess strengths, find out what we need to work on, and provide feedback to everyone.
  • Marbleslides is sneaky awesome! When students begin to obsess over function shifts and how to restrict domains and don’t want to peel away from their computer, you know something is going right.
  • Class went fast! It felt like the mixed practice from Let It Stick was now becoming part of my classroom culture.
  • My pre-calc is mostly 11th and 12th graders, who have had a pretty traditional classroom experience in their math lives.  I can sense they appreciate that something difference is happening.
  • All students are responsible for their learning.  Even the least-active task, the piecewise function, was used the next class for sharing out and a jumping-off point.


Activity Builder Reflections

We’re now about 9 months into the Desmos Activity Builder Era (9 AAB – after activity-builder). It’s an exciting time to be a math teacher, and I have learned a great deal from peeling apart activities and conversing with my #MTBoS friends (run to to start peeling on your own – we’ll wait…). In the last few weeks, I have used Activities multiple times with my 9th graders.  To assess the “success” of these activities, I want to go back to 2 questions I posed in my previous post on classroom design considerations, specifically:

  • What path do I want them (students) to take to get there?
  • How does this improve upon my usual delivery?


AN INTRODUCTION TO ARITHMETIC SERIES (click here to check out the activity)

My unit or arithmetic sequences and series often became buried near the end of the year, at the mercy of “do we have time for this” and featuring weird notation and formulas which confused the kids. I never felt quite satisfied by what I was doing here.  I ripped apart my approach this year, hoping to leverage what students knew about linear functions to develop an experience which made sense. After a draft activity which still left me cold, awesome advice by Bowen Kerins and Nathan Kraft inspired some positive edits.

seatsIn the activity, students first consider seats in a theater, which leads to a review of linear function ideas. Vocabulary for arithmetic sequences is introduced, followed by a formal function for finding terms in a sequence. It’s this last piece, moving to a general rule, which worried me the most.  Was this too fast?  Was I beating kids over the head with a formula they weren’t ready for? Would the notation scare them off?

plotsThe path – having students move from a context, to prediction, to generalization, to application – was navigated cleanly by most of my students.  The important role of the common difference in building equations was evident in the conversations, and many were able to complete my final application challenge.  The next day, students were able to quickly generate functions which represent arithmetic sequences, and with less notational confusion than the past.  It certainly wasn’t all a smooth ride, but the improvement, and lack of tooth-pulling, made this a vast improvement over my previous delivery.

DID IT HIT THE HOOP? (check out the activity)

DAN.PNGDan Meyer’s “Did It Hit the Hoop” 3-act Activity probably sits on the Mount Rushmore of math goodness, and Dan’s recent share of an Activity Builder makes it all the more easy to engage your classes with this premise. In class, we are working through polynomial operations, with factoring looming large on the horizon.  My 9th graders have little experience with anything non-linear, so this seemed a perfect time to toss them into the deep end of the pool.  The students worked in partnerships, and kept track of their shot predictions with dry-erase markers on their desks. Conversations regarding parabola behavior were abundant, and I kept mental notes to work their ideas into our formal conversations the next day.  What I appreciate most about this activity is that students explore quadratic functions, but don’t need to know a lick about them to have fun with it – nor do we scare them off by demanding high-level language or intimidating equations right away.

The next day, we explored parabolas more before factoring, and developed links between standard form of a quadratic and its factored form. Specifically, what information does one form provide which the other doesn’t, and why do we care?  The path here feels less intimidating, and we always have the chance to circle back to Dan’s shots if we need to re-center discussion.  And while the jury is out on whether this improves my unit as a whole, not one person has complained about “why”…yet.


Last night, the Global Math Department hosted a well-attended webinar featuring Shelley Carranza, who is the newest Desmos Teaching Faculty member (congrats Shelley!).  It was an exciting night of sharing – if you missed it, you can replay the session on the Bigmarker GMD site.


Activity Builder – Classroom Design Considerations

This past summer, our forward-thinking math-teacher-centric friends at Desmos released Activity Builder into the wild, and the collective creativity of the math world has been evident as teachers work to find exciting classroom uses for the new interface. Many of these activities are now searchable at – you’re welcome to leave now and check them out – but come back…please?

Its easy to get sucked in to a new, shiny tech tool and want to jump in headfirst with a class. I’ve now created a few lessons and tried them with classes which range from the “top” in achievement, to my freshmen Algebra 1 students. In both cases, I’ve settled upon a set of guiding principles which drive how I build a lesson.

  • What do I want students to know?
  • What path do I want them to take to get there?
  • How will my lesson encourage proper usage of math vocabulary?
  • What will I do with the data I collect?
  • How does this improve upon my usual delivery?

It’s the last question which I often come back to. If making a lesson using Activity Builder (or incorporating any technology, for that matter) doesn’t improve my existing lesson, then why am I doing it?

One recent lesson I built for my algebra 1 class asked students to make discoveries regarding slopes and equations of parallel and perpendicular lines. Before I used it with my class, a quick tweet 2 days before the lesson provided a valuable peer-review from my online PLC.  It’s easy to miss the small things, and some valuable advice regarding order of slides came through, along with some mis-types. The link is provided here in the tweet if you want to play along:

The class I tried this with is not always the most persistent when it comes to math tasks, but I was mostly pleased with their effort. Certainly, the active nature of the activity trumped my usual “here are bunch of lines to draw – I sure hope they find some parallel ones” lesson.

As the class finished, I called them into a small huddle to recap what we did. This is the second lesson using Activity Builder we have done together.  In the first, the students didn’t know I can see their responses, nor understand why it might be valuable.  In this second go-round, the conversation was much deeper, and with more participation than usual. In one slide, the overlay feature allowed us to view all of our equations for lines parallel to the red line:


We could clearly see not only our class successes, but examine deeper some misunderstandings.  What’s happening with some of those non-parallel lines?  Let’s take a closer look at Kim’s work:

Parallel 2

What’s going on here? A mis-type of the slope? The students were quite helpful towards each other, and if nothing else I’m thrilled the small group conversation yielded productive ideas in a non-threatening manner – it’s OK to make errors, we just strive to move on and be great next time.  The mantra “parallel lines have the same slope” quickly became embedded.

The second half of the lesson was a little bumpier, but that’s OK.  Before questions regarding slope presented themselves in the lesson, storm clouds were evident when the activity asked students to drag a slider to build a sequence of lines perpendicular to the blue line.  Observe the collective responses:


So, before we even talk about opposite reciprocal slopes, we seem to have a conceptual misunderstanding of perpendicular lines.  I’m glad this came up during the activity and not later after much disconnected practice had taken place.  In retrospect, I wish I had put this discussion away for the day and come up with a good activity for the next day to make sure were all on board with what perpendicular lines even look like, but I pressed ahead.  We did find one student who could successfully generate a pair of perpendicular lines, and I know Alexys enjoyed her moment in the sun.


What guiding principles guide you as you build activities using technology? How do they shape what you do?  I’m eager to hear your ideas!



Activity Builder Reflections

The super-awesome Desmos folks set Activity Builder into the wild this past summer, and it’s been exciting to see the creativity gushing from my math teaching colleagues as they build activities.  So far, I have used Activities with 2 of my classes, with mixed success.

In my 9th grade Prob/Stat class, I built an Activity to assess student understanding of scatterplots and lines of best fit.  You can play along with the activity if you like: go to, and enter the code T7TP.  I am most excitied by the formative assessment opportunities an activity can provide – here are 3 places where I was able to assess class understanding.

In one slide, students were shown a scatterplot, and asked to slide a point along a number line to a “reasonable” value for the correlation coefficeient, r.  The overlay feature on the teacher dashboard allowed me to review responses with the class and consider the collective class wisdom.

r overlay

In another slide, students were again given a scatterplot and asked to set sliders for slope and y-intercept to build a best-fit line.  Again, the overlay feature was helpful, though it was also great to look at individual responses.  This led to a discussion of that pesky outlier on the right – just how much could it influence the line?

LSRL overlay

Finally, question slides were perfect for allowing students to communicate their ideas, and focus on vocabulary. In our class debrief, we discussed the meaning of slope in a best-fit line, and its role in making predictions about the overall pattern.

 question roll

But all has not been totally sunny with Activity Builder.  In my Algebra 1 class, I built an Activity to use as a station during class.  Splitting the class in half, one group worked with me on problems, while the rest worked through the activity, then flipping roles halfway through class.  You can try this activity at, code 3FGM.

Storm clouds approached early, when a student complained that they didn’t know what to do – even though the first slide offered instructions to “Drag the points…”.  Quickly my “I’m an awesome teacher who uses stations” fuzziness turned into saltiness as students clearly were not following the activity faithfully.  Here’s what I learned:

Leading class through an activity beforehand would have been helpful. In the future, I’m going to make a vanilla lesson which walks students through simple tasks – dragging points, answering questions, entering equations, adjusting sliders – and let them see how I can view and use their responses.  Just setting a class into the wild, especially a class which often struggles with instructions, didn’t work so hot.

Last Saturday, I led a group of about 20 teachers in an Activity Builder workshp at the ATMOPAV Fall Conference at Ursinus. I had 3 goals for the assembled teachers for the hour:

  • Experience activities through a student perspective.
  • Experience the teacher dashboard.
  • Start building their own activities.

Some have asked for my materials, and I can’t say I have too much to share.  Check out my Slides and feel free to contact me with questions about the hour. Some highlights of the group discussions:

  • When is the best time in a unit to use an Activity?  So far, I have used it as an intro to a unit, and also as a summary of a unit.  The difference is in the approach to task.  An intro activity should invite students to explore and play, and think about generalizations – include lots of “what do you think?” opportunities.  In my summary activity, I asked specific questions to see if students could communicate ideas based on what we had learned.
  • Think about how you will leverage to teacher dashboard to collect and view ideas.  How does the overlay feature let all students contribute and build class generalizations in a new way?  How will you highlight individual student responses to generate class conversation?
  • Ask efficient questions.  There’s really not a lot of room in the text for long-ish tasks.  Keep things short, sweet, and focused.
  • Many teachers wanted to know more about building draggable points.  The way I do this is to create a table, enter some points, and use the Edit feature to make the points draggable.  Your best bet may be to take an already existing activity and pore through its engine, which reminds me….

Desmos is now assembling an searchable archive of vetted activities.  Go to, and use the search bar at the top-left.  I highly recommend any creations by Jon Orr, Michael Fenton and Christopher Danielson.

copyAnd finally, an exciting new feature to Activity Builder just appeared today – you can now copy slides within an Activity.  Click the 3 dots to duplicate a slide and use it again, or edit a graph to use later.

ISTE 2015 – Keep the Learning in Focus

Anytime I do a blog post which is a list, my traffic shoots up.

– A friend / tech-blogger

cuethinkThis post has been rattling around in my head since the end of the ISTE (International Society for Technology in Education) Annual Conference last week. I most appreciated the chance to meet and discuss technology war stories with folks I had only “met” before through twitter, make new connections in the math world, and think about how new ideas and products will change my classroom culture.  But trying to summarize the experience in a blog post is difficult.  There’s just too much stuff – new tools, inspiring people, great school action – how can I fit it all in?

I got it — I’ll make a list!!!

Often, the most popular presentations are those which feature lists – it’s a great device for getting foot traffic to your session.

  • Amazing Chrome Apps and Extensions to Enhance Teaching and Learning!
  • The Magic Bag of New Presentation Tools for Teachers
  • 60 in 60 – App Attack

I confess I enjoyed Steve Dembo’s list session – “Something Old, Something New” – which challenged participants to share tools on Twitter and think about how “old” tools still could be thoroughly effective in the classroom, and not to toss them for new things, while also embracing the new.  Sessions featuring lists can be intoxicating hours of “wow”.

After my session featuring Desmos on day 1, I was energized to think about a session for next year.  There were few math-specific sessions at ISTE, and the group I worked with seemed appreciatice and eager for more.  There are many, many math tools I use in my classroom, and I’d love to share them…I even have a title:

The Math Tech Tool SmackDown!

60 minutes – 10 to 15 math tools, lots of oohs and aahs over their wonderfulness, some quick examples, a few cute anecdotes, and everyone leaves happy.

Teachers LOVE lists!

The list is also a cop-out.

List sessions are often one-sided affairs.  The presenter moves rapid-fire through tools with examples, and the time crunch to get to everything means little time for discussion.  The application and personalization are left for the user to figure out later.  They aren’t BAD sessions at all (heck, my last post on this blog is a list…and you’ll find many other lists buried here on the blog), just know going in that discussion of pedagogy will not be the order of the day.  Follow up that list session with a smaller group opportunity and syntthesize your new learning immediately.

I’m suddenly feeling less excited (and a little guilty) about my Math Tool Smack-Down.  Some twitter sharing from a colleague helps lend some clarity to my thoughts:

Yes!  It’s about best teaching practice – not the tool (duh!).  It’s easy to forget that in the tsunami of stuff (and swag) at a big tech conference.

Jed Butler is such a great math resource, and an awesome friend.  He came as a participant to my Desmos session, and ended up being a vital resource when the tech went south.  He also acted as my button-pusher, and general problem-solver.  On the last day of the conference, a lunch conversation of math tools developed into a potential ISTE talk for next year, featuring problem-posing as a framework for making use of apps and tools.  Such exciting conversation, and there will be a lot more to come this month when Jed and I (along with Mike Fenton and Glenn Waddell) share Desmos morning sessions at Twitter Math Camp.

Extending conversations beyond conferences – one of the most powerful aspects of my participation in the Math-Twitter-Blog-OSpehere.  Keep a lookout here on the blog as we get deeper into July as the group shares out classroom ideas.

Thanks to Priness Choi for sharing out her experience in my session.  Yes, I move around a lot!

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!