How Do We Assess Efficiency? Or Do We?

A problem on a recent assessment I gave to my 9th graders caused me to reflect upon the role of efficiency in mathematical problem solving. In particular, how much value is there in asking students to be efficient with their approaches, if all paths lead to a similar solution?  And should / could we assess efficiency?

The scene: this particular 9th grade class took algebra 1 in 7th grade, then geometry in 8th.  As such, I find I need to embed some algebra refreshing through the semester to dust off cobwebs and set expectations for honors high school work. For this assessment, we reviewed linear functions from soup to nuts. My observation is that these students often have had slope-intercept form burned into their memory, but that the link between this and standard form is weak or non-existant.  Eventually, the link between standard form and slope ( -A/B ) is developed in class, and we extend this to understanding to think about parallel and perpendicular lines.  It’s often refreshing to see the class see something new in the standard form structure which they hadn’t considered before.

The problem: on the unit quiz, I gave a problem which asked students to find the equation of a line parallel to a given line, passing through a point.  Both problem and solution are given in standard form.  Here is an example of student work (actually, it’s my re-creation of their work)….

linear problem

So, what’s wrong with the solution?  Nothing, nothing at all.

Everything here answers the problem as stated, and there are no errors in the work. But am I worried that a student took 5 minutes to complete a problem which takes 30 seconds if standard dorm structure is understood?…just a little bit.  Sharing this work with the class, many agreed that the only required “work” here is the answer…maybe just a “plug in the point” line.

My twitter friends provided some awesome feedback….

Yep, we would all prefer efficiency (maybe except Jason). Thinking that I am headed towards an important math practice here:

CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated reasoning.

It may be unreasonable for me to expect absolute efficiency after one assessment, but let’s see what happens if I ask a similar question down the road.

Confession, I really had no idea what #CThenC was before this tweet.  Some digging found the “Contemplate then Calculate” framework from Amy Lucenta and Grace Kelemanik, which at first glance seems perfect for encoruaging the appreciation for structure I was looking for here.  Thanks for the share Andrew!

Yes, yes!  Love this idea.  The beauty of sticking to standard form in the originial problem is that it avoids all of the fraction messiness of finding the y-intercept, which is really not germane to the problem anyway. Enjoy having students share out their methods and make them their own.

What do you do to encrouage efficiency in mathematical reasoning?  Share your ideas or war stories.

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.

Residuals from the Past Month

It’s been a busy month of October. I don’t have a new lesson or resource to share this time – instead, here are some things which have been rattling around my brain.

Last night at the weekly Global Math Department online meet, NCTM President-Elect Matt Larson shared new and “in the works” resources for members, and a revised vision for PD in the coming years.  In the past few months, I have been fortunate to speak personally with both Matt and current President Diane Briars, and I am happy to hear that NCTM not only values the work of the Math-Twitter-Blog O-Sphere, but are now beginning to take lessons from the successes of ourline community and bring it to the national organization.

With regional conferences starting this week, I am most excited to see a new website NCTM has established to encourage ongoing dialogue:  I won’t be able to make any of the regionals, but will be checking in from afar on this new site. I’m already enjoying the sharing from presenters, and the sense of ongoing discussion.

I re-arranged my bedroom furniture this summer, and I’m not sure I like it better.

This afternoon, I took one of my daily walks through the neighborhood, with the Bill Simmons podcast as my companion. His guest was Judd Apatow, and the conversation turned towards the negative aspects of celebrity.  Judd mentioned Eddie Murphy who started as observational comic, then became hugely famous, as someone whose work was altered by the seclusion of celebrity status. No longer able to make the every-day connection to his audience, the observational aspect of Eddie’s career withered away, and was replaced by other things.  Judd Apatow, sensing a need to re-visit his comedic roots for inspiration, dusted himself off to do stand-up and has caught his comedic second wind.

Is there a lesson here for teachers who leave the classroom to become administrators? How long does it take for separation from the classroom to take root – and can (and should) it be re-visited now and again?

Sometimes I wonder why nobody has been arrested yet for assaulting one of the Impractical Jokers

I have 3 quite different preps this semester, and I am professionally miserable because of it.  With block scheduling here, teachers have 3 courses each semester.  Now and then, 3 preps is not a big deal.  But I teach each course with someone different (or a different group) from the department, and I rarely share prep with any of them.  I’m also the only member of my department to have 3 preps, and this is the second semester in a row this has happened.  OK…I’m getting real close to my whining quota here, but I don’t think I am doing a good job right now.  Instead of having laser-focus on my courses, I find myself all over the place.  This is not helping my students and I am worried.

Some of my AP students report that they will go trick-or-treating next week.  For me, high school age is when you are out of the candy loop.  Am I right?

My Math Club kids are the most enthusiastic bunch I have “coached” in recent memory.  And the weekly Math Madness contests have been great for getting kids to talk about problem solving approaches.  I don’t usually enjoy doing math for competition’s sake, but we have been holding weekly de-briefs after each contest and the conversations have been informal, spirited and genuine.  I’m lucky to work with such a great group of kids!

Today is “Back to the Future” day – October 21, 2015.  The day Marty McFly visited the future on the big screen.  And I passed a DeLorean on the way home from work (no lie, this really happened!)

My new local hero is a colleague of mine at my school who teachers Anatomy and Physiology, Chris Baker.  In addition to being an awesome role model for kids, and someone passionate about his craft, he has jumped deeper into the Twitter pool and has embraced 20% time as part of his classroom culture.  Consider giving him a follow – he’s a good egg!

Making It Stick…With Beanbags

The book Making It Stick – the Science of Succesful Learning has caused me to consider how I approach practice and assessment in my math classroom. The section “Mix Up Your Practice”, in particular, provides ideas for considering why spaced practice, rather than massed practice, should be considered in all courses.

But it was an anecdote which began the chapter on spaced practice which led to an interesting experiment for stats class.  The author presents a scenario where eight-year-olds practiced tossing bean bags at a bucket.  One group practiced by tossing from 3 feet away; in the other group, tosses were made at two buckets located two feet and four feet away.  Later, all students were tested on their ability to toss at a three-foot bucket.  Surprisingly, “the kids who did best by far were those who’d practiced on two and four-foot buckets, but never on three foot buckets.”


Let’s do it.

My colleague and I teach the same course, but on different floors of the building during different periods. Each class was given bean bags to toss, but with different practice targets to attempt to reach.

  • In my class, lines were taped on the floor 10 and 20 feet from the toss line.
  • For Mr. Kurek’s class, one target was placed 15 feet from the toss line.

Photo Oct 05, 9 33 54 AMAfter every student had a chance to practice (and some juggling of beanbags was demonstrated by the goofy….), I picked up my tape lines, and placed a new, single line 15 feet from the toss line.  Each student then took two tosses at the target, and distances were recorded (in cms).

We then analyzed the data, and compared the two groups (the green lines are the means):

bean bags

I love when a plan comes together!  The students, who did not know they were part of a secret experiment, were surprised by the results – and this led to a fun class discussion of mixed practice.  Here, the mixed practice group was associated with better performance on the tossing task. Totally a “wow” moment for the class, and a teachable moment on experimental design.

An Egg-Cellent Simulation

The scenario I used for a fun lesson with my 9th graders this week comes from a talk by James Bush from Waynesburg College, which I attended at the US Conference on Teaching Statistics in May.  James is a master of finding clips from TV and movies to use in his class to encourage discussion, and this clip from the Jimmy Fallon Late Show features a game called “Egg Russian Roulette”. I have embedded a clip here, but you can search for many times Jimmy played this game on his show.

For James’ college statistics courses, this clip is a helpful opener to the hypergeometric distribution, where we are interested in multiple successes from draws done without replacement.  While this setting could eventually be presented to my AP Stats students, it lives a bit outside of the scope of what we do at the college level. But there are some strong entry points for discussion with my 9th graders, including probability trees, conditional probability, and simulation.


Before showing the video, two volunteers were called to participate in a mystery game.  The two student volunteers became a bit nervous over their decision when a carton of eggs was produced, eventually shown to be filled with plastic eggs (awesome idea by James!). My first chance to try this with volunteers on my own was at Twitter Math Camp in June, and lots of fun tweets followed.

Thanks to Richard Villanueva, who recorded many of the My Favorites from Twitter Math Camp, we have coverage of the ganeplay.  Check out all of the videos of My Favorites from TMC15 on his YouTube Channel.

Next, I asked the class to think of questions they have about the game they saw, or in general about Egg Roulette.  A good starting list developed:

  • How likely is it that Tom Cruise would lose that quickly?
  • Once Tom picks a raw egg, how likely is it that Jimmy is safe on his draw?
  • Is it better to go first or second?

Photo Sep 18, 8 06 14 AM I then challenged my student groups to sketch out the first three rounds of egg roulette, and find the probability of Tom losing in 3 rounds. We had worked on trees the day before, and this game presented a good chance to apply what we had discussed earlier.

Photo Sep 18, 8 13 52 AM


After our analysis of the first three rounds, the conversation then moved to strategy: is it better to go first or second, or does it not matter?  Our “gut reaction” poll revealed that “it doesn’t matter” was the most common response, with “go first” was in second place.  The thought behind going first is that you could easily draw a hard-boiled egg, and thus put pressure on the other player.

To simulate the game, pairs of students were given one suit from a deck of cards.  The ace was moved aside, leaving 12 cards (representing the 12 eggs).  The 10, jack, queen and king then  represent raw eggs.  After shuffling the cards, students dealt cards into two piles, Tim and Jimmy. When a player was dealt two raw eggs, the game ends and the result recorded.  We were quickly able simulate over 50 plays of Egg Roulette, and the class results were recorded.

Egg Simulation

One student quickly identified, and then defended, that the game can NEVER go the full 12 rounds.  Also, some students noted that the second player (here, Tom) has one extra opportunity to lose the game.  A second straw poll revealed that student perceptions on the game had shifted – few thought it was a 50/50 game, and many saw that the first player held a disadvantage.


In teams, students are now challenged to explore a similar (yet shorter) Egg Roulette game, compute theoretical probabilities, conduct a simulation, and analyze the results.  I’m looking forward to some interesting-looking trees.  The document here shows guidelines for this activity, some assignment ideas, and a full tree for the first 3 rounds of Egg Roulette.

A Sneak Preview of My 2015-2016 Classroom

Today is the last day of summer vacation. In the past week, boxes have been unpacked, t-shirts and class decorations have been hung, and my awesome school custodians have provided me with even more whiteboard space – all the better for getting students up and moving


But beyond the physical layout for this year, here are some ideas I’ll focus on this coming year, many provided by my friends in the Math-Twitter-Blog-O-Sphere, the #mtbos for short.

GREETING STUDENTS WITH HIGH FIVES – Intertwined with all of the mathy goodness of Twitter Math Camp this past July was a simple and powerful device for student engagement from my friend Glenn Waddell – the High Five.

Each day last year, Glenn met his students at the door to give them a high five – a simple, caring gesture to establish a positive tone for class.  I often meet students at the door before class or linger in the hallway for informal chat, but I love the tradition and rapport Glenn establishes here and hope to emulate it.

ESTABLISH SEMI-REGULAR RANDOM GROUPINGS – this gem comes from Alex Overwijk, who is the king of Vertical Non-Permanent Surfaces and Visible Random Groupings. This year, I plan to randomly change my seating chart once each week, or at the start of a new unit – whichever seems to make the most sense at the time.  Traditionally, I’ll assign groups on my own and change them once or twice in a semester.  With some classes, I’ll allow students to choose their own groups.  But I have found that these practices often foster group-think, where a group will together develop the same bad habits through their work together.  I want more interaction, more sharing of ideas, especially in cases where students otherwise would not have encountered each other. I’m planning to assign each student a playing card on the first day, and set the new groups by dealing cards on the desks on days when it’s time to change.  I also confess here that a static seating chart was a huge crutch for me, as I would print out student names for me to glance down at when I needed.  Which leads into another goal for the new year…

better jobI MUST LEARN NAMES DAMMIT! – I confess this could be one of my weakest areas as a teacher. I could make all kinds of excuses for it, but it comes down to this – I drop the ball when it comes to learning and recalling my students’ names. We start school next Tuesday with a 4-day school week, and my goal is to know all names as they walk in the door by the first Friday.  I have already gone through my class rosters (which conveniently provide photos). How awesome would it be to know student names before they even walk in the door?

And beyond my current students, I am brushing up on names from students I taught last year. I’ve missed out on these connections for too long, and it’s my fault – time to work harder at it.

IMPROVING MY HOMEWORK PRACTICES – I don’t grade homework anymore, and in many cases have changed the nature of assignments. I’ve settled into the philosophy that I would rather have students think about a handful of meaningful, discussable problems rather than complete a laundry list. This year, I am looking to include more articles and video clips for students to observe and discuss in lieu of traditional assignments.

To go over homework, I often employ random methods to share works on my document camera, with mixed success. I’m finding that since I don’t directly look at assignments anymore, the completion is spotty at all levels. I may need to go back to a few minutes of checking and informal greeting at the start of a period to improve assignment fidelity.

grabUSING REFLECTOR TO ENCOURAGE PARTICIPATION – It can’t be the new school year without a new tech tool to try out. This year, I am looking forward to using the Reflector 2 program from the folks at Squirrels. This inexpensive software, loaded onto my laptop, allows me to relfect the screen from my ipad or iphone onto the laptop. I’m hoping this will allow me to be more hands-free for presentations, and hand over the ipad to students to take control – using Desmos or Deoceri to create works and share in front of the class. Also, I’m wondering what a class would look like where students could reflect their own phones onto the screen and share works. Day 1 of class could feature a “load test” – what happens when many, many students all try to reflect their graphs at the same time?

Now, out to the craft store to buy some last-minute stuff!

Desmos Lessons for AP Statistics

In the past year-plus, Desmos has added useful features to help those of us in the statistics world. The elegant addition of regressions (check out my tutorial video) has been a welcome new feature, and simple stats commands have also been added for lists.  Here are 3 Desmos creations which will become part of my classroom lessons for the 2015-16 AP Stats year.


That dreaded r-squared sentence…..yep, the kids need to memorize, but let’s add some meaning behind the “percent of variation due to the linear relationship….blah blah blah…” mantra.  Here’s an activity I do with my classes which has helped flesh out this regression idea.  To start, every student is handed a card face down with a prompt.  On my signal, the students turn over the card and respond to the prompt, with specific instructions not to discuss their response with classmates.  Here’s the prmopt:

An adult male enters the room. Estimate his weight.

After some nervous mumbling, I now hand out a second prompt card, and we will repeat the process.  But this time the card looks a little different.

An adult male who is {*see below} tall enters the room. Estimate his weight.

This time, I have 6 different versions of cards, and they are randomly scattered about the room.  Some cards say “5 feet, 6 inches” for the height, with other cards for 5’9″, 6’0″, 6’3″, 6’6″ and 6’9″.

After responses for both cards have been given, the responses are written on the board, along with the associated heights for the 2nd round of cards.  How did the background information given in the 2nd set of cards influence our responses?  Now the bait has been set to look at the Coefficient of Determination on Desmos.

rsqr1In this Desmos, heights and weights of adult males are given in a scatterplot. Activating the first folder – “using the mean of y1 for prediction” shows us the mean of all weights, and associated errors if the mean weight were used to predict for all men. The folder is activated by clicking the open circle to the left.

rsqr2Next, we can explore how the regression line helps improve predictions. Click the “LSRL and explained variation” folder and note the reduction of error.  The calculation for r-squared as the reduction of error is given, and can be compared to the calculated r-squared value from the regression.  Also, points in the scatterplot are draggable – so play away!


I have done this exploration of regression facts for many years, using worksheets from Daren Starnes along with Fathom. I find this Desmos version to be much easier for kids to handle, and it can be saved for future discussion.  And while in this demonstration I have all of the commands prepared for you, I would walk students through entering the commands themselves in class.

lsrl1First, we have a scatterplot with its LSRL included.  Activate the “mean of x and y” folder” and notice the intersection of these value lines. Here, the points are all draggable, so we can easily generalize that all LSRL’s pass through the point x-bar, y-bar.

lsrl2The second discovery is a bit more subtle.  Click the next folder, and now we have new lines 1 standard deviation in each direction for x and y.  Clearly, our intersection point is no longer on the LSRL, but what is its significance?  How far do we rise and run to get to this new point on the LSRL?  Some calculation and discussion may help students discovery this fact about the slope of an LSRL:


This is not a fact students need to memorize in AP Stats, but certainly the discussion builds understanding of regression beyond what our calculator provides.


binomialLists on Desmos have strong potential for investigating a distribution by using a formula repeatedly.  In this Desmos demonstration, students investigate the behavior of the binomial distribution, using sliders to define values for n and p in the distribution.  Activating the normal curve folder allows us to assess the “fit” of the binomial distribution against a normal curve.  I added the purple dots near the top to make it easier to investigate where the normal approximation is strong/weak in approximating its binomial cousin.

While Desmos has a while to go before it will replace graphing calculators in my AP Stats class, these activities will be part of my classroom this year.  Looking forward to creating and sharing more!