Monthly Archives: September 2013

Visualizing Shared Work Problems

Fred can paint a room in 5 hours, working alone.  His friend, Joe, can paint the same room in 7 hours.  How long will it take for them to paint the room, working together?

It’s a shared-work party, people!  Get your party hats on and let’s look at a visual method for exploring these often mundane problems.  This past summer at Twitter Math Camp, I participated in an algebra 2 group where part of our time was spend considering methods to re-think the traditionl approach to rational functions and their applications.  Thanks to John Berray for the great conversations, which led to some changes in how I appoached shared work problems this year.

My approach this year started similarly to previous years: guiding a dicussion with the class, with the goal of developing models for the amount of work done by each painter.  I find that quesitons like “How much of the job will fred have complete after 1 hour? 2 hours…etc” will usually lead to the models we seek.  What I did differently this year was graph the two work functions.  Using the Desmos calculator works nicely, and allowed for a discussion of the problem much richer than if the expressions had been just jotted down on the board.  Many students followed along on their TI calculators.


From here, we can make connections betweem the functions, their graphs, and make conjectures about the sum of these functions.


In my class, students certainly completed similar problems (including distance / rate / time), with the graphs serving as a check and visual affirmation.  With the graphs, we could also look at adaptations to the theme, such as “what happens if one of the painters shows up 2 hours late?”


Also, problems where the combined time was given, with the goal of finding a missing individual rate, were explored and discussed.


Click the icon below to play with this model on your own.  This is a great opportunity to let students observe function behavior and communicate results from a graph.

UPDATE: The Desmos folks flew with this one, and added a whole bunch of bells and whistles.  Click the graph below to experience their shared-work extravaganza.

Rock, Paper, Scissors and 2-Way Tables

Last weekend, the evil Michael Fenton posted a link to an online applet which will now occupy you for the next 2 hours.  It’s not too late to run away now…

Still with me? An adventurous soul, you are.  Anyway, the NY Times online Science section has shared an online game of “Rock, Paper, Scissors”, where you can play against a choice of computer opponents.  The “Novice” opponent has no understanding of your previous moves or stratgey.  But, the “Veteran” option has gathered data on over 200,000 moves, and will try to use its database to crush your spirit.


My Advanced Placement Statistics class today was preparing for their first chapter test, where topics include 2-way tables and marginal distributions.  Time to abandon my planned review and play!  Here’s what we did:

Each group (I have 6 groups of 4) was given a netbook computer and the NY Times site.  Half of the groups were told to play against the “Novice” player, while the other half challenged the “Veterarn”.  Each group played 20 times, and pride was on the line as groups considered their moves carefully.  Class data was gathered and compiled into a 2-way table.


But just how good are we at outsmarting the computer opponent?  In round 2 of this activity, groups again played 20 games, switching their opponent.  This time, however, I directed groups to choose their moves RANDOMLY.  Groups used their graphing calculator to generate a random number from 1 to 3, which determined their move.  The NY Times site provides some info regarding randomization:

A truly random game of rock-paper-scissors would result in a statistical tie with each player winning, tying and losing one-third of the time. However, people are not truly random and thus can be studied and analyzed. While this computer won’t win all rounds, over time it can exploit a person’s tendencies and patterns to gain an advantage over its opponent.

Groups played 20 more times, and a new table was created for this “random” round.  Last round strategy was labeled the by “guts” round.


With the data now on the board, groups were given a few minutes to summarize their findings.  Did we improve by being random?  Did we improve in any particular area?  This turned out to be an engaging review of marginal distributions, and a good opportunity to discuss ribbon graphs, which come up in AP Stats as a useful graphical display.  Below, Excel can be used to compare the “Veteran” opponent results.

Ribbon Graph

Thanks Mike, for sharing such a cool link!

I’ve Joined the Flipping Revolution

Two weeks into teaching Algebra 2 for the first time in a long time, and things are going great so far, but time to start one of the more potetnially tedius chapters in Algebra 2: Rational Expressions and Equations.  Taught “by the book” this can become a 2-week journey of nasty-looking expressions, scary worksheets and those dreaded “shared work” problems.  A perfect time for me to take my first real dive into “flipping” my classroom.  Here’s what I have done so far:

VIDEOS: I used Doceri to create videos for each of the sections in the chapter.  While I love Doceri (since I can do videos from my couch AND they upload easily to YouTube), using it for this chapter has not been ideal, since the problems get nasty and long quite quickly, and you can’t scroll the screen.  May use SMART Notebook for some down the road.  And a few “takes” were made, as it’s easy to screw up making a video when you have Jeopardy on mute in your living room.  Even my final version of the first video below has, for me, a “cringeworthy” error in vocabulary.  Here are my first two unit videos:

HOMEWORK:  Students have been given notesheets, with the problems in the videos provided.  Their job is to watch the video, take notes, and then complete just a handful of problems related to the idea.  My intent with these problems is not to provide anything tricky: just enough to demonstrate some mastery of the material.  We’ll save challenging problems for class.  All of this material is posted on Edmodo for my students, so they can go back if they need.

Just a few days in, and the reaction of the students has been quite positive.  They appreciate that they will not be given homework designed to keep them up for hours, and that the communication during class time is more whole-class, rather than lecture.  Some more classroom observations:

  • Homework is no longer an ending of a lesson, it’s the beginning of a journey.  Students come into class ready to apply what they experienced.  I have the ability to raise the difficulty of problems based on what I am sensing from the class.  I don’t need to wait until the day after a night of homework struggle to measure my students’ progress.
  • I am not spending a dis-proportionate amount of time at the beginning of class dealing with homework issues.  In earlier years, I assigned homework in the same manner I suspect many teachers do: give an assortment of problems..enough for students to feel successful, but with a few to provide challenge to those students who need it.  The next day, this approach often yielded well-intended, yet essentially wasteful, conversations where I went over problems in front of the class.  From my eyes, this seemed like “help” to the class, when from a wider view it is easy to see these discussions are only absorbed by small pockets of students.  And since the daily “let’s go over the tricky HW problems” portion of the day has been removed….
  • I am planning more investigative experiences into my routine.  Today, for the last 30 minutes of class, students borrowed netbooks from down the hall, and used Desmos to explore possible dimensions of rectangles with fixed area…a set-up for graphs of rational functions in a few days.  Part of this exploration turned into “play with Desmos, and do some stuff”.  Good!  Tomorrow, we will check out a shared-work video, and start making some connections.
  • Students are accountable for their learning.  They are welcome to view the videos multiple times, pause, or skip if they desire.  But, full disclosure….I am doing this now with an honors class.  Looking forward to trying his with my academic algebra 2 in the spring, and reporting out.
  • While I am using class time to tackle the most difficult problems, that is not to say my students to not have rigorous assignments.  Besides the “flipped” homework, I am also assigning more complex tasks, with a two-week window for students to choose, write-up, and turn in problems.  More on this in a later post.
  • I am also “flipping” lectures in AP Stats, through videos my colleague produced last year.  For these, I have created short Google Form quizzes which assess the main points of the video.  The data from these forms have been helpful in clearing up misconceptions during class meetings.
  • I will ALWAYS produce my own videos, or rely on those of my colleague for stats.  My students rely on me to be their guide, and I will always meet that expectation.  I will not let an anonymous guide be the primary source for my class.

Would appraicte your constructive feedback, suggestions, and classroom stories!  Now back to the iPad.

Use Appropriate Tools Strategically

This semester, my Algebra 2 students will be exposed to a wealth of math tech tools.  Graphing calculators will be a big part of what happens in my classroom; not only because they are great tools for discovery, but also because I feel some responsibility to have students understand the appropriate use of these tools as they head towards AP classes.  Forcing a tool upon students because it will help them on a test is weak, I know…I cry myself to sleep sometimes…though I do rely on the technology to craft discovery moments in my class.

But I also want my students to experience other tools, like the Desmos calculator (which we will use later for the world-famous Conic Sections project), Geogebra and Wolfram|Alpha (reviewed earlier here on the blog).  So, how do I get my students to experience all of these tools, and start to make measured decisions about how and when to use them?  Hey, we have a Standard for Mathemaical Practice for that!

CCSS.Math.Practice.MP5 Use appropriate tools strategically.

Lost in the great stuff on precision, modeling and reasoning is this awesome nugget, with a specific focus on tech tools:

Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

Nice!  Exactly what I am looking for!  So, how do you do that?  How do you get students to start comparing and assessing tools?

Here is the day 1 assignment I gave to my students in Algebra 2, as posted on Edmodo:

Write a product review for the Desmos online calculator. Consider its pros and cons, and whether you feel this is a site you would recommend to others. One side of a piece of paper MAX. Screen shots allowed and encoraged. Can be turned in via hard copy, or electronically.

Thats it.  Go ahead, kick the tires, and tell me what you find.  I didn’t make students aware of my on-going man-crush with Desmos or that I had done a webinar for them.  I had no idea what I was going to get back.  And since I stress written communication in my classes (moreso, I suppose, than many math teachers), this gave me a first writing sample to analyze.

The results were largely encouraging.  While many students focused solely on the graphing of functions, some students demonstrated evidence of digging deeper, looking for characteristics which make Desmos unique.  Some snippets:

Unlike the normal graphing calculator, it graphs your equation as you are typing it and allows you to delete parts of the equation if your graph isn’t what you wanted it to be. Desmos also provides the general equations for many different lines, parabolas, and other more advanced graphs.

The example graph list on the left side of the screen acts as a jump start and learning tool to give confused students a boost in the right direction.

It is quick, simple, and efficient to use and is recommended to all users that seek a tool for graphing. The designs are not distracting but sleek and a simple white to emphasize the purpose of the tool, for math and nothing else.

The Desmos is different, its not complicated at all, it can do so many things that most calculators can’t, and it’s free. The fact that Desmos is free is really what makes it so much better than all of the other because you don’t have to shell $120 out of your pocket for a calculator that has all the same capabilities that Desmos has.

But not all is sunny, as some students noted some “Cons”:

With the internet calculator, there is the obvious issue of no internet, no calculator. Also, I found some buttons were tough to get to such as the “pi” key which required me to press several buttons in order to get that one.

One last thing about the calculator is the fact that it can be downloaded as an app, but only on apple products. For android users, like myself, you would have to use the calculator through the internet which isn’t as easy to use as through an app.  Also, the app is accessible without wireless internet connection, but android users need the wireless connection to use the Desmos calculator.

All told, a good first writing assignment for my students, followed by some discussions of tools and their appropriate use.  As we travel through Algebra 2, many chances to compare tools, and discuss the best tool for the job.  Looking forward to doing another product review, using Wolfram|Alpha.  Stay tuned.

The More Things Change…the More They Change

This fall, I am coming back to the classroom after 2 years serving as an instructional coach for my school district.  It’s been an exciting few weeks, setting up a classroom for the first time in a long time, and it occurred to me that I am using many, many digital tools in my classroom which simply did not exist when I last taught….just 2 years ago!!!  This reinforces the need to stay on top of new technologies and look for what fits in your class, and improved student engagement …it’s easy to fall a few generations behind.  Here’s what I am featuring in my classes this year:

Edmodo – I am already loving Edmodo for my classes, after just one week.  Best described as “Facebook for education”, Edmodo allows you to have a course-specific area online, with students given a code to join your class.  Post links, files and resources, and let students participate in online quizzes and polls.


There’s really no reason for students to ask “what did I miss” when they are absent, as Edmodo moves your class beyond the regular class period.  Has an ipad app, and many students here have installed the iPhone app.  And it’s free!  My first day student inventory moved from the standard “get this signed and fill out the questions” paper form to an online survey posted on Edmodo, which brings me to my next tool…..

Google Forms – OK, Google Forms are not new, but the entire concept of Google Drive and sharing content with students and colleagues has changed greatly in 2 years.  I am using a Google Doc calendar to plan my classes, and a shared doc is being used to team-plan Algebra 2 classes across phases.  And the first-day survey has moved online, check out my survey.  The responses served as great discussion starters for the next day.

Remind101 – this is my first time using Remind 101, and many students in our school are already accustomed to its concept.  Create an account and a class, which generates a “join” code for students and/or parents.  Then, log onto their site and send reminders of assignments, tests, quizzes, or anything you want to communicate to your classes.  It’s 1-way communication; quick, easy to use, and free.

Doc CamDocument Camera – When my department head discovered I was coming back to the classroom, he asked if I had any supply requests.  I only had one: document camera, and already this has been a great tool for my classes.  I had seen a number of teachers using document cameras in their classes, in many engaging ways.  My interest was further piqued at this year’s Best Practices Night at the AP Statstics reading.  Daren Starens, author of The Practice of Statistics, spoke on “Making Homework Count”, and a method for assessing homework he calls the “visiting artist”.  Randomly select a student, place their work under the document camera, and lead a discussion of the work.  Check “critique the reasoning of others” off your Standards of Mathematical Practice list every day!  The model I am using costs about $60, from IPEVO, and can be found on Amazon.

PollEverywhere – I do not have a set of classroom clickers, but no worries here! On day 1 of my Statistics class, student teams were asked to analyze an “unusual” data set.  To generate ideas, groups shared their observations using PollEverywhere.  Another free tool, it’s easy to set up an on-the-fly assessment.  As student ideas were generated, they appear on the wall, and the discussion flies!

Poll Everywhere

Digital Textbook Editions – I taught AP Statistics for 8 years before leaving the classroom, and used The Practice of Statistics, 2nd edition, for all of those years.  This year, we have moved to the 4th edition, with a digital version for the instructors.  I have told my students to leave their texts at home this year, unless they hear otherwise.  This is not only due to the digital textbook, but also….

Livebinders and “Flipping” Videos – For the first time, my Stats colleague and I are working to place a large amount of course content online, using Livebinders.  This is a content curating tool, and you can check out many public binders teachers have shared on many subjects. We hope that students will use this as an ongoing resource, going back to review before assessments, and creating the culture of Stats as a whole-course, rather than disconncected units.  In addition, my colleague Joel Evans has taken the notesheets to the next level!  CornellFirst, he converted many of my chapter guides to a Cornell Notes format.  But, even better, we are now “flipping” many of our lectures, through section-specific notes.  We hope this will allow for more activity-based instructional time, with a high payoff in student engagement.  In Joel’s first year of trying some flipping videos, his AP pass rate was 100%.  Does correlation imply causation?  Maybe, maybe not….but no matter how successful you feel your classroom has been, it is always a good time to review methods and take them to the next level.  Check out some of the Stats flipping videos, and our many class resources, on our AP Stats Livebinder.

So many great tools to transform your class culture.  Don’t be afraid to try something new!

Breaking Apart Sums and Differences of Perfect Cubes

The first few days of math class…an awkward time for both students and teacher.  The kids haven’t picked up on my mannerisms yet, aren’t sure why I fly around the room like a maniac, and worse, they aren’t laughing at the jokes.  I tend to use the first few days of any class seeing how far I can put my foot on the gas…what do my students understand?  Where are there gaps?  Who will I need to sit on during year?  Without exception, I dedicate at least part of the 1st day with students at boards, shouting out review problems.  The problems are strategically chosen to allow for initial success, dust off some cobwebs, provide for discussion when we hit some road blocks, and most importantly let students know that it will be perfectly acceptable to struggle in my class….as long as you keep trying.

I’m trying a differenmt apporach with an Algebra 2 course I am teaching this semester, and hoping to build some discovery and communication moments right in the first few days.  As their first day assignment, students fill out a Google form with information about themselves: hobbies, goals, clubs, etc.  As part of the form, I am adding this task: “Tell me everything you know about this graph:”


For day 2, I’m hoping the responses will provide a review of vocabulary (intercepts, roots, solutions, even rotational symmetry perhaps?), and some table-setting for what’s to come.  In our district, I can expect that Algebra 2 students will have a solid background in linear functions and basic polynomial operations, mostly limited to quadratics.  Cubics for the most past have not been explored yet.  And while polynomial multiplication and factoring are not new, rarely do students see polynomial division before algebra 2, so I will bring this into the discussion as a new idea.  NOTE: our district uses the Everyday Math program in the early grades, which stresses partial quotients.  Wondering how this will play when I attempt polynomial division…update may be coming!


Starting algebra 2, students should be able to “read and recognize” the following polynomial patterns: difference of squares and perfect-square trinomials.  But beyond this, I want students to be able to relate factored form to graphs, which often seems to be marginalized in the drive to practice process.  So, one of my first lesson openers will be a short and sweet challenge.  Does the following polynomial factor?

In their teams (my students always sit in groups), I will provide some time for students to consider this problem, and observe their trials.  I expect that will have a few groups attempt (x-2)^3, which will end badly, but hopefully lead to more trials.

CubicSo, how do we cross the bridge to the formulas for differences, and sums, of perfect cubes.  Time to start looking at some graphs, in particular the functional form of the given expression:

What do we notice with this graph?  And what characteristics will be helpful with the factoring problem at hand?  Here is where I hope students drive the discussion:

  • This graph has an x-intercept of 2.
  • This means that x-2 is a factor.
  • There are no other obvious intercepts, but we can employ long division here.

So, x^3-8 DOES factor.  Do other cubics factor?  How?

With their teams, students will now be given a few more cubics to factor:


What patterns do we notice?  Can we develop a general rule for factoring difference of cubes, and even sums of cubes?

Guiding the discussion towards a generalization, without students feeling forced-fed, is part of the art of teaching.  Hoping these first day discussions tie together lots of previous knowledge with a discovery moment.  I am not sure how it will go, but I hope to set the table that nothing is given for free.  Show me what you know!