Monthly Archives: March 2013

“Wow” Moments with Wolfram|Alpha

The Siemens STEM Academy offers great resources for teachers, from lesson plans, to blog posts from teachers, to fantastic free webinars.  Full disclosure: I have written for the STEM Academy blog, and been a part of the Academy summer program…but I am but a small fish in a cool ocean of resources!

This week, the Academy hosted a free webinar featuring a demonstration of the dynamic knowledge provided by Wolfram|Alpha.  Having used Wolfram Demonstrations before in my classroom, I was looking forward to learning more about this search tool.  Crystal Fantry provided an hour-long overview of this exciting resource, and ideas for classroom uses.  It’s amazing how many “wow” moments I have these days with the new tech tools our students can have in their hands, but this one goes beyond that.  Knowing that students have access to resources like this should cause us all to think about our roles as math teachers / facilitators….this is a game-changer!

So, just what is Wolfram|Alpha?  The site is simple, just enter what you want to search for, and off you go…but this tool is so much more than that.  The “about” from their website provides some insight:

Wolfram|Alpha introduces a fundamentally new way to get knowledge and answers—not by searching the web, but by doing dynamic computations based on a vast collection of built-in data, algorithms, and methods.

So, what the heck does that mean exactly?  Let’s learn by diving in.  And while you can use Wolfram|Alpha for far more than math, this is a math blog so let’s focus in on some math….

Try this: “y=2x +3”.  Let’s start with something simple…what does Wolfram|Alpha give us?


Fun stuff.  A nice graph, the domain, and alternate form.

How about this: “3x+5=2x-9”

Also nice, a plot of the the functions.  And the equations’s solution..but what’s this…a “step-by-step solution”?  If you are logged in (free accounts) you can step through the solution:


So, what happens now when you give that worksheet of equations to solve for homework?

There are a lot of other neat computations to explore, try some of these as starters:

  • “y=(x+2)(x-3)”
  • “inverse y=x^2+3x+1”
  • “sin(x)+cos(x)=1”
  • “Integrate x^2 dx from 0 to 5”

WA3But Wolfram|Alpha goes beyond quick lists and computation.  How about “Pascal’s Triangle mod 5”. Or “triangle sides 3, 6, 8”, or try the elusive 17-gon, and see the many facts to check out.


I have only scratached the surface of the many features, and there are also lots of nooks, crannies and links for you to explore.  I’m eager to use this tool with students as a means to research new ideas, and make some sense of their characteristics.  For example, let’s think about domain and range, as I ranted about in a previous post.  I like that Wolfram|Alpha expresses domains using set notation, and this is a great opportunity to have students research new functions.  Most of what we do in Algebra 1 deals with linear functions, so we get a lot of “all real numbers” domains.  Expose your students to non-linear functions, once they know how to make their x,y tables.  Try these:

  • y = 5 / x
  • y = rt (x-2)
  • y = 1 / (x^2 – 9 )
  • y = 2^x
  • y = x^2 – 4

And what to do with these new functions?  Let’s place them into categories, share our findings, and communicate our ideas.  Give each group 2 or 3 new functions to look at and share their findings on  This site, formerly called WallWisher, allows everyone to contribute their ideasd and move them around the canvas.  Here’s a sample of my function domain wall, click the link to contribute your own, play around the wall, and double-click in any empty space on the canvas to contribute.  Or sign up for a free account and create your own wall.


Thanks to Kyle Schutt (@ktschutt) and the gang at Discovery Education for providing these great webinars.  Be sure to check out the Siemen’s STEM Academy blog for more great resources, blog posts, and archived webinars.

Probability Openers – Separating the Possible from the Plausible

A recent problem I reviewed from the Mathematics Assessment Project (MAP) caused me to refelct upon the coverage of probability in our classrooms.  The website provides sample tasks and assessment tools for schools and districts as they adjust curriculums to match the Common Core.  In the standards, probability begins to take center stage in grade 7:

Investigate chance processes and develop, use, and evaluate probability models.

Probability is treated like the ugly step-sister in many math courses: ignored, shoved to the side.  Look at standardized testing results, including AP Statistics, and you will find that probability standards often produce weak results.  The isolated fashion with which we treat probability is certainly not helping.  Let’s develop strategies to not only re-think probability, but to encourage communication of ideas and develop understanding.

I have taught probability at many levels: as an 8th grade teacher, as an AP Statistics teacher (and reader) and as the author of a Prob/Stat course delivered to 9th grade students.  This year, I taught my first college course in Statistics, where the problems with probability persist.  The picture below is from my college Stat 1 class, which you can also see on the great site Math Mistakes, by Mike Pershan.  Visit and provide your input:


Here are two activities I hope you can use in your classrooms to help fight the probability battle.


The Core Standards provide a framework for our students base knowledge of probability:

Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

Before we can start diving into formulas and fractions with probability, we need students to understand, and be able to express, whether an event is likely or not likely.  For simple events, like flipping a coin or drawing a card, this can be an easy discussion, but what happens when we start talking about complex events, like flipping multiple coins, tossing 2 dice, or choosing a random car from a parking lot?  Download my Probability Opener, which asks students to assess events and compare their probabilities.  Answers are provided.  The events start off innocently enough, but soon meander into more complex tasks, where students are asked to identify the more likely outcome:

You roll the dice and move your piece in the game of Monopoly…

A:  You end up on Boardwalk

B:  You end up in Jail

This is also a great activity if you have a classroom clicker system, or for Poll Everywhere if cell phones are allowed.


This activity is an adaptation of a Spinner Bingo problem from the Mathematics Assessment Project site I mentioned at the top of the post.  Visit the MAP site for not only the task, but a rubric and samples of student work.

The problem presents a scenario where students are asked to assess a spinner game, and bingo cards created for the game.  Read the files given on your own, but here is a quick summary:

  • 3×3 bingo cards are filled out, using non-repeating numbers from 1-16.
  • Two spinners with 8 equal spaces numbered 1-8, are spun, and the sum computed.
  • Players mark off their bingo cards if the sum appears on their card.

This is a nice scenario, so let’s adapt it and use it as our unit opener.  Having students play a game, and develop and justify a strategy, is a great way to get started.  Here is a lesson guide I have written to help you get started.  Also, the site Unpractical Math provides a virtual spinner applet you can use.  It’s often easier to just dive in and play, so here is a brief video demo of the game and how you can play it in your classroom:

Please let me know if you use either of these activities, and would appreciate your feedback. Thanks.

Dive Into Screencasting

Looking forward to an afterschool session tomorrow on screencasting.  I have done a number of screencasts, using different (mostly free) products, for many educational purposes.  Sometimes, I use screencasts to explain class problems, so students can review procedures.  Other times, I have made screecasts to demonstrate ideas for colleagues, like Flubaroo, Google Drive and Sketchpad.  Here are some resources for both my district colleagues, and my blog friends.


Screencast-O-Matic:  Free sign-up, requires java, and can upload directly to YouTube.  15-minute screencast limit.

Jing:  Free download and sign-up.  5 minute screencast limit.  Share to twitter and Facebook.

SMART Notebook:  I’m always surprised when colleagues don’t realize a recorder comes with this software.  Create wmv files, which upload to YouTube.

doceriDoceri:  Free for the iPad.  I love sitting on my couch on a Sunday morning and crafting a lesson at the touch of a finger.  Program “stops” into your playback, and record your voice over the playback.  Uploads to YouTube.


Making a video which will be useful to the viewer can take practice and patience.  Here are a few tips I have for somebody new to the screencasting game:

  1. Keep it short and snappy.  Think about your message, and deliver it succinctly.  I have not run across many people looking to rewind and replay my mad ramblings.  Try not to improvise or go off-message…which brings me to…
  2. Write a script, or at least an outline.  Think about the bullet points you want to cover in your screencast, jot them down, and stick to the script.
  3. Rehearse the script.  For me, this means simply going through my bullet points.  What are you going to say, in what order, using what resources?
  4. Rehearse the timings.  When I first got started making screencasts, this was something I often did not think out enough, and now obsess over.  What do you plan to show on your screen?  What pictures, website, programs or applets should be open?  Don’t wait to hit record before opening a program, which you know will choose that moment to act up.  Have everything minimized and ready to go, and do a run-through.
  5. Be prepared to start over.  I need to start making a director’s cut version of my mistakes in screencasts.  Quality matters.  Fortunately, with a screencast, you can start over and make it better.

Here are some sites and articles which are also helpful to the new screencaster:

Teacher Training Videos: a wealth of videos which walk teachers through the basics of many tech tools.  Find screen capture tools near the bottom of the page, on the left.

Educause – Why do screencasts?  Mike Ruffini promotes the benefits of creating screencasts, along with strategies for implementation and evaluation.

Turn To Your Neighbor blog – a quick-start guide to getting started with screencasting.  Includes a pdf of the quick-start guide.  Great suggestions for the new user.

TeachThought – How to Screencast Like the Khan Academy.  Has overviews of products for the advanced user (read: not-free).  More insights into the benefits of screencasts.

STEM Fizz: The Friday Institute for Educational Innovation.  Not necessarily dedicated to screencating, but good ideas for flipping the classroom and the rationale behind this practice.

When you have recorded your first video, and you start populating the YouTube channel, let’s embed those videos in blogs and webpages to share with the world.  Here is one on the plug-in for Google Drive, called Flubaroo, which has proven popular as an easy tool for educators.  I like to think I have gotten better since this video, one of my first.  And I have also learned the value of a quality microphone.

Exciting Action at T^3 – Saturday

Back for a second day at the Texas Instruments Teachers Teaching with Technology Conference, and hearing some great ideas from the sessions I have attended.  The highlight of my day today was a panel discussion of  the Nspire app for iPad, which I have just loaded.  The panel was faciltated by TI brand ambassador Dr. Mayim Bialik, who has been a presence throughout the conference on sessions regarding the new app.  Dr. Bialik was joined on the panel by teachers Sheri Abel and Stephanie Ogden, who piloted the Nspire app in their classrooms.

Stephanie summarized her feelings of the ipad app:  “This is what other instructional tools aspire to be”, while Sheri shared a level of engagement from her students she had not before experienced.  As one example, Sheri had students drag and re-drag axes and functions in order to develop a conceptual understanding of domain and range.  The ability to drag and discover has been the greatest source of positive discussion from the app this weekend, and is seen as a gamechanger.

ipad appWhile the app is an exciting tool for discovery, the panel also noted that ipads certainly cannot be used for standardized tests.  To me, it will be a tough decision for schools to make a decision between ipads and handheld: the expense is not insignificant.  Also, schools may need to make a choice between continuing a TI 84 culture, or transitioning to an Nspire culture.
Coming soon, I will take the app back to my schools and hope to work through a lesson or two with classes.  I will share my reflections in a later blog post, so stay tuned….
Earlier in the day, I was excited to experience first-hand the new color 84, which is due to be available in stores for back-to-school time.  The session, 84 Plus C in Secondary  Math in Preparation for AP Calculus, was facilitated by Fan Disher
Fan walked the group through a number of problems from AP Calculus which require graphs.  This was a great opportunity to test-drive the new features.  If you are a veteran 84 user, you will identify the commands and their locations, with some new bells and whistles.  Here are my favorite new features:


The opportunity to graph functions in color, and have a color background, makes it so much easier to identify and compare graphs.  Pictures can be added as backgrounds.  The calculator comes with 5 pictures, just begging for function-modeling, and your own photos can be added by using TI Connect.

No more AAA batteries!  water fountainIf your math department spends crazy money each year just on batteries, this is a huge improvement. Calculators can be recharged using cables, or a docking station.  In chatting with TI trainers, they tell me that a calculator will hold a change for about 20-30 hours, or about 2 weeks of occasional use by a student.

Me calc

Besides utilizing the new features of the 84C, this session also modeled composition of functions, and how to use the 84 to facilitate class discussion of domain and range of composites.  Problems from AP Calculus were used to look at graphs, shaded areas, function tables, and TI’s real-type features.  Great stuff I will take back to my colleagues.

On Friday, I had the opportunity to sit down with Mari and Dale from TI to discuss some of my concerns about the latest wave of TI products, in particular that there is so much new stuff coming out at the same time….so much so that it becomes tricky for teachers to select a product and standardize their building technologies.  Dale, who has been involved with TI for 16 years, did an excellent job of walking me through TI’s thought process.  With the 84 color, TI has acknowledged that, while the Nspire and its iPad app have improved educational capabilities, the is a large core audience which is attached to the 84.  Ignoring them or forcing them into the Nspire would not be wise, or very nice.  Also, the overall look of the 84 had not changed for some time, even with vast improvements, and decreased cost, of memory and color screens.  So, now we have the 84 C!  And its improved interface is worth considering for your classroom.

If you read my blog post a few weeks ago, where I broke up with TI, I need to give you a status update on our relationship.  I’m happy to say that we talked things out, and I will be happy to be in a relationship with TI for quite some time!

T^3 Conference – My First Look at the iPad App

The 25th annual Texas Instruments Annual conference started today, and after an inspirational opening by Leland Melvin, I was eager to get to my first session on the new Nspire iPad app.  I think I am currently in my “stubborn” stage…refusing to pay to buy this until I understand its worth.  So, for now, I will sit back, try to learn, and gauge the reaction of the room.  The app is currently “on sale” at a price of $4.99, which is $25 off the regular price.  Whether the regular price is way to high, or the sale price is too low….I’m not sure yet.

The icon for various page types (calculator, graphs, geometry, etc) will be familiar to those who have used the Nspire computer software.  Also, a short introduction / tutorial is provided, but what fun is that?  Today’s presenter was Andy Kemp, who led us on an hour-long tour of the features of the new app.

A math keyboard is provided, and seems to be designed to have a different look and feel than the standard iPad keyboard.  Templates for things like fractions, matrices and derivatives seem to be intuitive.  Also, some buttons can be held and pressed to give extra functions.  Holding down the cos key will give a menu for sec, and their inverses.

Ipad app

“The ability to move the graph around with your fingers and manipulate it is what is fundamentally different”.  The function rule will change as the graph changes.  This is nothing much different than the handheld, but the transformations are much smoother with the ability to drag with fingers, rather than using the closed hand tool.   A variable button allows us to recall previous functions, and use them to create new functions.

I do appreciate and like how the sliders work on this app.  The slider box can moved and resized much more easily, and the settings can be changed by holding down on the slider box.  Also, the animation feature is much more obvious and easier to implement.

Here’s what I don’t quite understand.  To find roots of a function, TI still wants me to identify a lower and upper bound, after which it will then search for roots. This, to me, is not as convenient as the Desmos calculator, where landmark points are identified easily, and can be turned off if I like.  Not sure I understand TI’s obsession with requiring boundary selections.

My first impression is that I would use this more than the teacher software, which to me is often clunky and slow.  I like that I can create files which can feature graphs, data, and functions.  And TI continues to update its arsenal of activities and files.  So, eventually, I will probably relent and purchase the app for my ipad.

Follow-Up on Math Term Expungement

In a comment from my recent post “3 Phrases From Math Class we Need to Expunge“, Tina from the blog Productive Struggle shared a Google Doc she has been assembling of terms and “tricks” we all could evaluate in our math courses.

Tina is requesting 3 categories of entries:

What tricks do you hate when students shout out?
What words do your students use without understanding?
What notation do you wish students started using earlier?

Here is the link to the document: Tina’s Google Doc

My favorite so far is Tina’s idea to introduce subscript notation for sequences and series earlier in math courses.  It always surprised me how much trouble that was for my 9th graders….silly almost.

3 Phrases from Math Class we Need to Expunge.

A brief twitter exchange last night between myself and the great NY math educator Mike Pershan caused me to get off my rear to assemble a post which I had kicking around my head for some time now, a list of terms and shortcuts we use in math class which, while well-intentioned and used everyday by many math teachers, aren’t necessaily helpful in causing kids to understand their underlying math concepts.


In a recent in-service with middle-school math teachers, I used a video by Phil Daro (one of the authors of the Common Core math standards) to have colleagues reflect upon the practice of “answer getting”, short-term strategies employed by teachers to get students through their immediate math assessment, but with little long-term value in math understanding.  Click on the “Against Answer-Getting” tab for the video.

So, here is my first list of nominees for elimination, and some strategies for helping students develop underlying algebraic ideas.  It probably won’t be my only list, and I welcome your candidates and thoughts.


This is a device I often see in pre-algebra classrooms, often times as a poster for easy reference, other times as a mantra for the students to help complete worksheet problems.  From the site

TIP: For subtracting integers only, remember the phrase

“Keep – change – change
So, we have a short and snappy device which helps us with just one specific type of integer problem.  It’s not wrong, just too specific, and do students understand why it works?
What to do instead:
Let students develop their own summaries of integer problems, and create their own posters which describe their findings.  Use integer zero-pair chips or online applets, like from the National Library of Virtual Manipulatives (search for “chips”).  Number line applets can also help students visualize addition and sibtraction problems.  Have students write stories about given integer and subtraction problems, and have students peer-assess work for proper use of math terms.  Eventually, have students debate the possible equivalence of integer pairs:
  • 5 – (-2) and 5 + 2
  • a + (-b) and a – b
  • a – b and b – a


The ad-laden math site Coolmath gives its own snazzy description of foil:

We’ve got a cool little trick called “FOIL” for multiplying binomials….it’s really just an easy way to do the distributive property twice, which would be really messy and confusing to do.

YEY!  You mean I can multiply stuff without that nasty and scary distributive property, without actually talking about the distributive property!  Yey shortcuts!  I’m in! {insert sad face}

Folks, ditch FOIL, and use the opportunity to talk about the double-distributive property.  Re-write the binomials as an equivalent expression and multiply.  Set the stage for factoring and note how much more understanding factoring by parts takes on.  And, now we can tackle those “messy” trinomials too.


Try this exercise tomorrow: take a class tht has been through Algebra 1, and as an opener tomorrow ask them to explain what the phrase “Cancel Like Terms” means when dealing with a rational expression.  Or, if that is a bit too scary, simply ask your students what it means to recude a fraction.  This is a nice activity to do as a Google form, and have students assess the explanations.  Many students will give an example as a definition, which is not what we are looking for here.  How many students discuss factors, GCF’s, numerators or denominators?

Reducing a rational expression means to divide both the numerator and the denominator by the greatest common factor of both numerator and denominator.  (Incidentally, also try having your students provide steps for finding a GCF.  This one also reveals what your students understand.)  The great part about this procedure for reducing is that it works equally well for each of the following expressions:

To many of our students, cancel is digested as “cross-out stuff”.  We have better vocabulary for it, so let’s encourage its use.