Monthly Archives: June 2013

Problem Makeover – Carpeting

This week’s problem makeover from Dan Meyer features a room in need of carpeting:


There’s a lot of stuff going on in this problem:

  • Overlaying rectangular carpet over an irregular shape.  We’re going to need to worry about waste at some point, perhaps.
  • The double-sided tape.  We’ll have to look at the perimeters of the individual carpet pieces and add them up later.
  • Costs.  We definitely don’t want to waste carpet or tape, or at least minimize the amount of waste.
  • Meters vs Yards vs Feet.  Yeah, this aspect of the problem is just mean  Maybe if Lesa had her fill of coffee before measuring the room, she would have used feet.

I think the piece of this problem which may be easy for kids to overlook, and least comprehend, are those seams.  Seams are bad, and add to cost.  This problem needs something tactile to get kids thinking about the seams and coming up with their own ways to  think about them and minimize them.


Groups of students will be given a bunch of 4×4 inch sqaures and a bigger shape to tile. Tiles1

The instructions are simple:

Use the squares to cover the triangle in the most efficient manner possible.

As groups complete the task, they will then meet with other groups and defend why they feel their coverage method is the most efficient.

Tiles2I’ve left the phrase “most efficient” open for interpretation here.  I’m hoping that a strong definition of efficiency will emerge from group discussions.  As groups continue discussions and start to reach agreement on the characteristics of efficiency, have members of teams approach the front board, and start a class-wide list of these characteristics.  Here’s what I am hoping will emerge:

  • Using the least number of squares as possible is important.
  • Having squares overlap, if it can be avoided, is preferable.
  • Seams are ugly, but necessary.  When we are able, we want to minimize these seams.

This last bullet may be the toughest to coax from kids.  But this is a good time to bring in carpeting as a nice analogy.  Does the carpet at your house have seams?  Where are they?  If you were laying carpet, where would you try to put them?

Once we have a stronger definition of efficiency, it’s time to try a second tiling:


This time have students write up their solution and defend their efficiency.  There’s a nice opportunity for differentiation here.  Students with a strong geometry background could use rules regarding secants and chords to generalize the optimal solution, or even use software to model the scenario.  Middle school students, or those with less exposure to geometry, could simply measure the seams.

Now for the main event.  Students will be asked to tile one last shape:


This time, we add a new wrinkle to the problem.  To get students thinking harder about efficiency, we add some costs to the mix.

  • Each square used costs $10.
  • Every inch of seam in the tiling will cost $2 per inch.

It’s Saturday morning during the summer, so I’m not sure if these are “good” costs for the task, but we can adjust them.  Teams will calculate their costs, defend their solutions, and be assessed on how well they minimized the costs.

Geometry isn’t usually my “thing”, but this challenge appealed to me, because the original task was so filled with stuff that I know many students would shut down.  And I feel there is a natural opporunity here for students to assess their solutions and communicate.  In the end, I want my students to:

  1. Communicate their problem-solving process
  2. Defend their process, and evaluate the process of other teams
  3. Utilize their findings in different situations.

What’s My Role in the #MTBoS?

If I could change the MTBoS to make it better, I would make it less of an “underground”, almost secret, society and work to make our ideas more mainstream.  Let’s co-write articles for NCTM.  Let’s share beyond the 140 characters.  Let’s begin to become a force for change beyond our followers.  I look forward to the day where my colleagues don’t look at me with strange glances when I mention a great activity I found on a blog.

In the last few week’s, there has been a lot of back and forth discussion regarding the present and future of the “Math-Twitter-Blog-o-Sphere”.  The MTBoS is the community of math educators who share ideas, stories and friendships through Twitter and blogs.  Its a wonderful and growing community of diverse educators, many of whom have formed real relationships through the love on online math sharing.  But it’s also a place which can be intimidating to new tweeters and bloggers.  To be honest, until about a week ago, I had to keep looking up “MTBoS” to remind myself what it stood for.

Last week, discussion of the MTBoS was featured at the weekly online conference at the Global Math Department. The “If I could change” prompt I completed above was one of the closing activities from the hour of sharing.  Some quotes which struck me from the discussion appear below.  I apologize if I don’t cite names here, as it was hard to follow who was speaking all of the time on the playback.

“I feel very isolated in my own department” – I could not agree more with this.  More than anything else this community not only makes it safe for me to share new, perhaps game-changing, pedagogical ideas, but lets me hear from educators I respect and admire on a daily basis.  There have been times when I felt   uncomfortable with sharing ideas locally, for a number of reasons, and the MTBoS makes it safe to be creative and different.

“I know stuff, and I am obligated to share it” – this sums up nicely my rationale for the blog.  I’m often surprised when I look back on the lessons I have developed over 16 years, and more surprised when other teachers find them unique, when it never really occurred to me I was doing anything special.  There’s such a great feeling when I read someone else’s blog, see a lesson and think “man, why didn’t I think of that?”, and immediately share it with the 40 math teachers in my department.

“What’s relevant is that it is for the kids” – perfect!  There are a lot of bells and whistles is teaching ideas, including an avalanche of tech tools.  Sometimes it helps to take a step back and think “how does this improve anything?”  If there is a MTBoS mission statement to be written, it must be written around the idea that we all want to help kids learn math better.

It seems like a good time to evaluate my personal mission as part of the MTBoS. I can’t state that I am a “primary” member; rather, I tend to hover and grab ideas or join discussions when time allows or interests dictate.  So, who am I, what am I doing here, and how am I contribtuing to the good of the cause?


I started blogging about 2 years ago because I felt like I had a lot of math stuff worth sharing.  I had always enjoyed sharing teaching ideas and lessons with colleagues in my building, and blogging just brought it to a whole new level.  There really isn’t a rhyme or reason to my posting schedule.  When I come across something neat, or a great experience occurs, I blog about it.  I also have a backlog of a lot of drafts of incomplete ideas, which I hope to get to…someday.

The blog has been helpful in that it is now a warehouse of some of my teaching experiences.  When a collegue now comes to me looking for an idea, or wanting more info on something, I can now send out blog links.  I am sometimes disappointed when I don’t get feedback on ideas, but then I can look at my blog stats and see which ideas are being “pinned” on Pinterest, or linked to from other places.  It’s often suprising to me some of the activities, which I never thought to be special, get picked up and shared by new teachers.  It’s a good feeling to help out new educators in building their filing cabinet of teaching ideas.


I now get my best classroom ideas primarily from items I see on Twitter.  From articles to videos to classroom lesson ideas, I am constantly looking for something new to share with colleagues.  My advice to my local math teacher friends is to join twitter and follow just a few primary folks at first.  You don’t need to check in every day or every hour.  The beauty of twitter is that will all be there when you have time to look.  There seems to be a fallacy out there that twitter is a time-consuming intrusion.  And it can be, if you want it to be.  It can also be a wonderful, low-pressure way to think outside of your building.

I use twitter to participate in chats.  During the school year, my favorites are #sbgchat (standards-based grading) and #statschat, along with #mathchat.  Where else can you rub virtual shoulders with authors and national experts?


I actually have met very few of the people I follow on twitter.  At the AP Stats reading a few weeks ago, it was my pleasure to meet Shelly (@druinok), and touch base, and I look forward to meeting many more at TMC13 in Philly later next month.  Twitter’s great, but there is no replacement for a real, face to face, argument over how to teach complex numbers.  I look forward to it over a few beers.


The end of the school year bring with it the end of my tenure as an instructional coach in my district.  By choice, I am heading back to my high school classroom.    Part of my decision is based on all of the great ideas I have acculumulated and hope to bring to to my classes.  It’s a bit overwhelming really.  The blog will continue, but maybe with some more classroom focus.  I doubt I will change the name of the blog, as many people have it linked.  But a new tagline should be coming.  Suggestions encouraged!

Classroom Resources from the AP Stats Reading

Last week, I returned from 7 days in Kansas City where 650 of my closest friends and I successfully completed the “Million Question Challenge”.  This year, over 170,000 students took part in the AP Stats exam.  With each exam having 6 questions, this was the first year the readers had the million-question task.

In the evenings, there are lots of great learning and networking opportunities for teachers.  One of the highlights is “Best Practices” night, where this year 17 teachers shared their classroom action.  I presented my ideas and experiences with co-teaching in AP courses beyond stats (which I have featured on the blog before): what a fun and intimidating experience to present in front of so many people whose work I admire!  My slides, below, outline my experiences working with teachers an AP Psych and Chemistry, and some examples problems from Biology.

You can check out presentations and support materials for many of the Best Practice speakers at Jason Molesky’s Statsmonkey site.  Some of my favorites, things I am definitely looking forward to trying or exploring, include:

Kevin DiVizia’s “Scatterbrained Fathom”: to collect data to later use as an opening to the meaning of r-squared in scatterplots, Kevin takes his classes down to his football stadium’s turf field, “the world’s biggest ruler”, and has students launch stomp rockets.  A very cool data collection idea.  Does vertical height or participant weight make a difference ?  If the reaction of the room is any indication, the Stomp Rocket people will have many new customers this coming year!

Robin Lock – StatsKey.  This free online site has pre-loaded data you can use to explore topics from the AP Stats curriculum.  I enjoy the sampling distributions area, where samples can be drawn repeatedly from a large population and their means analyzed.  Check out this sampling distribution of samples of size 30 from 2011 movie budgets, a population which is skewed right.


ConclusionLuke Wilcox’s “Understanding and Visualizing Significance Level”. Do your students REALLY understand what they are saying when the write hypothesis test conclusions?  Stop using flippant phrases like “If the P is low, reject the Ho”.   and insist that your students write out specific language from day 1.

Check out the many great ideas on the Statsmonkey page.  Thanks to Adam Shrager and Jason for organizing the evening.  Looking forward to next year already!