Monthly Archives: April 2014

Hitting the Home Stretch: Exponents, GCF’s and LCM’s

This is a busy stretch in my school year.  My 2 Prob/Stat classes are nearing the end of new material with PA Keystone Exams in Algebra 1 looming. For my College Prep class, about half have not yet taken the Keystone while the rest took it last year as 8th graders. In Academic, all students will take the Keystone in May.  Combine this with my AP Stats class taking their final exam this week, with the AP Exam next week, and my track and field meet responsibilities building as the season reaches its peak; it’s a hectic time of year.

In both of my Prob/Stat classes, we are beginning unit on polynomials.  The Prob/Stat class is a course we offer between Algebra 1 and 2. While the course contains much Prob/Stat material, we also clean up some concepts from algebra.  Unlike other algebraic units like systems of equations where there are many rich examples and opportunities to differentiate, the start of a polynomials unit often feels static. Here are two activities I have used this week for Laws of Exponents and GCFs/LCM’s.


This activity worked equally well in my college-prep group (for whom this was review material), and my academic group (where this was mostly new).  The file below contains 16 cards with numeric statements.  Break your class into teams of 2, 3 or 4.  The job of the group is to identify the true statements and the false statements.  For this activity I banned all calculators.

The letters on the cards are not anything the kids need to worry about, but allow me to easily check progress. The cards with vowels are all the false statements.

I was surprised at how much trouble my college-prep group had with building the correct false pile.  To reach some consensus on the false pile, I asked every group to hold up one card they knew to be absolutely, positively false.  Many groups mistakenly agreed that any number raised to the zero power was worth zero, which led to a class argument on who was right.

Tomorrow, we will look more closely at the falses.  In the file above, note that the cards are arranged in groups of 4. In the first group, we will review the addition rules for exponents; then the subtraction rule in the next group of 4; then the multiplication rule for the next group.  In the end, this felt much more satisfying, with increased engagement and peer discussion than simply listing rules on the board.


The speed dating concept is one many math teachers have stolen from the great Kate Nowack, and it worked perfectly in my Academic class to work through greatest common factors and least common multiples.  After doing just one example on the board, desks were arranged  into a pairs facing each other, down one long row.

Speed Dating

All students were given a card with a monomial.  They then worked with their partner facing them, and found the GCF of the two monomials.  The first time around, my co-teacher and I provided help to just about all groups.  After teams found their GCF’s, all students on the right-hand side stood and moved down one seat and worked with their new partner.  There were so many plusses to this activity:

  • all students were repsonsible for their own monomial
  • all students were engaged: no hiding behind a worksheet
  • students worked together, and with different partners each time

Some of the cards I handed out are shown here.  I tried to have a variety of cards which clearly shared factors, with different powers of x and y.


I was very impressed with how my class performed on this activity, and we moved onto a second round where LCM’s were found. This time I had students trade cards, and the left-hand side shifted down each time.

Let your kids work together, discuss and find patterns – the notes then write themselves.

Professional Development: Who Owns the Responsibility?

Last weekend, I had the oppotunity to speak at the Association of Mathematics Teachers of New Jersey Technology Conference on my experiences with Desmos, where I shared a number of the Desmos activities and ideas I have posted here on the blog.  Conference speaking and classroom direct instruction are totally different experiences.  There a few things I need to think about in preparing for a conference which aren’t part of my daily routine:

  • How many people will there be?  Conference sessions are mini-popularity contests…I may have 5 people in the room, I may have 50.  You just never quite know what to prepare for.  Small groups are great for encouraging discussions.  Larger groups have a difference set of engagement challenges.
  • What’s the overall background of the room? People who will come to see me could be a mix of veteran and new teachers, with different stories to share, and levels of comfort with my topic.  Taking the temperature of a room quickly and finding a baseline comfort zone is key.
  • Do I really know what the heck I am talking about? I’ve offered to present because I feel I have something to share; but is my message unique or helpful?  Could there be someone in the room who knows far more about this topic than I do?

It can be a stressful experience presenting in front of peers, but also highly rewarding and a great way to make professional contacts!

Overall, I feel my session went quite well.  After all of the stragglers found their way inside, there was a full house (or lab) of about 35 folks, with a positive vibe in the room.  I think I met all of my goals, and I believe many left the session with tangible ideas for their classroom, and ideas to share with colleagues.  If I accomplished all that, then I feel successful.

For this speaking opportunity, I borrowed some equipment and recorded my talk, something I had never done before (see video below).  Upon review, there is a common theme running through the talk which has cause some post-talk reflection for me.  A few times I ask the group about their experiences with online tools, with some (to me) surprising results from the room of 30-35:

  • Only 1 or 2 had used Desmos before.
  • Only one had used Geogebra before.
  • Only one had heard of Edmodo before.

How could this be?  Here was a group of enthusiastic educators, concerned enough about their craft that they sought out professional development on a Saturday, and very few knew about these tools.  Is this small sample group indicative of all math teachers?  Should I be as surprised as I seemed in the video (really…watch my expression when I ask about Edmodo)? The good news is that hopefully some exponential growth occurs, and these teachers tell their colleagues, who then share with their colleagues…and so on….

But what of those teachers who do not seek out conferneces?  How do they find new resources?  Or are they even looking?  Do teachers have a professional responsibility to seek ro revise their ideas and practices?  I won’t pretend to have any answers in this blog post; rather I’d be eager to hear some thoughts on these questions from my readers.

And while my session has a clear technology slant, does the variation in learning experiences extend to math pedagogy in general?  Can teachers defend their classroom practices, and seek our resources for revision if needed?  How many teachers have considered how Common Core shifts will effect their classroom structures? Have teachers considered the Standards for Mathematical Practices and how they apply to their classrooms? Where do teachers go to find professional development opportunities which meet their unique needs?

And, most importantly, what are the responsibilities of classroom teachers, curriculum specialists and administrators in facilitating these reflections?  It’s a lot to chew on.

Below is video of my Desmos session.  Seeing myself on camera is at the same time cringe-worthy and thrilling…so much to learn from.  Man, do I gesture with my hands…. a lot!  Feel free to comment, share or heckle!


Chi-Squared Tests: Rock, Paper, Scissors.

At the beginning of the school year, I shared a post about a fun Rock, Paper, Scissors applet on the New York Times website.  Back then, my class used the applet to collect data for 2-way tables, and considered appropriate methods for displaying the data set.

Fast-forward 6 months: my AP Statistics class is knee-deep into hypothesis testing, and we’re now up to Chi-Squared tests.  These are some of my favorite tests, as the data is often richer than what we find in tests for means or proportions.  Here’s how we used the Rock-Paper-Scissors applet to produce data:

  • Teams played the game in veteran mode.
  • In round 1, teams were given 3 minutes to play the game normally, which I’ve labeled the “guts” method.
  • In round 2, teams were given 3 minutes to play the game randomly, using “randint” on their calculator to generate a digit from 1 to 3, which corresponds to a move.

We then considered an appropriate test for assessing the data.  This comes on the heels of Chi-Squared Goodness of Fit tests.  But here we have two samples, and we want to determine if the proportions are similar in both samples: this was our first test for homogeneity, and it was easy to move through the mechanics of the test.


Doug Page also shared a worksheet he has developed for using the Rock, Paper, Scissors applet.  I do not have contact info for Doug, but I hope he provides some details on his success in the comments.

This activity will now become a yearly staple in my AP Stats arsenal!  Enjoy.