Underground parking spots cost $30000-$50000 to build in D.C. This smells like a systems problem.


The Washington City Paper has a (rather lengthy) post on parking in D.C. Fair warning: it’s pretty wonky with zoning rules, ordinances, etc. However, the numbers caught my eye:

An underground parking spot costs between $30,000 and $50,000 to build, and residents pay for it one way or another.

“Let’s say it’s $100 per month. If you built the parking space, and it cost you $40,000, $1,200 per year doesn’t cover it,” says Four Points Development’s Stan Voudrie.”


On the supply side of the parking problem, costs are fixed: You can’t dig a hole and line it with concrete on the cheap. Demand is more dynamic, and to some extent, it responds to price. Unlike in suburban areas, most District landlords don’t pair spaces with the unit, which means that tenants pay between $100 and $300 more per month for their cars.

As these numbers appear, I suppose it’s more of a simple equation problem, but I feel like we could easily transform this into a nice, in-depth linear systems problem. For example you could contrive or find the cost of a garage parking spot and monthly fees, and figure out which one will pay itself off faster.

Basically, I’m just excited to finally find a potential systems problem that doesn’t involve cell phone plans.

Guiding Questions

  • How much do underground parking spots cost in our area?
  • How much do parking garage spots cost?
  • And what are the monthly costs of renting a spot?
  • How long would it take to pay off each spot?

Suggested Activities

  • Depending on how in-depth you want to get into this, you could turn this into some sort of apartment building project, or you could restrict it to a few-day investigation, having student find when each type of spot will pay for itself.
  • Again, if you really want to dig into this, you could ask student to develop a pricing system for their parking spots (i.e. would underground spots be worth more? what about spots closer in?).

Aside: this is like my fourth post on traffic or parking and mathematics. I’m starting to wonder if PBL could stand for “Parking Based Learning”.

The FAA wants to “take a fresh look” at rewriting the rules on electronic gadget usage on planes. How many flights equals “a fresh look?”


Check out this NYTimes article. Apparently there’s some encouraging news for those of us with e-devices, which is everyone: the F.A.A. is going to review the rules for takeoff and landing whilst using particular electronic devices. Surprisingly it appears as if airlines could start allowing electronic devices right away but would have to test the devices themselves. But not only the devices, but, well, I’ll let you read:

Abby Lunardini, vice president of corporate communications at Virgin America, explained that the current guidelines require that an airline must test each version of a single device before it can be approved by the F.A.A. For example, if the airline wanted to get approval for the iPad, it would have to test the first iPad, iPad 2 and the new iPad, each on a separate flight, with no passengers on the plane.

It would have to do the same for every version of the Kindle. It would have to do it for every different model of plane in its fleet. And American, JetBlue, United, Air Wisconsin, etc., would have to do the same thing. (No wonder the F.A.A. is keeping smartphones off the table since there are easily several hundred different models on the market.)

Emphasis mine. That sounds like a lot of combinatorics and permutations to me.

Guiding Questions

  • Which kind of device would require the least/most testing?
  • Which airlines could conceivably do this in the least amount of time, with their fleet size?

A bit of research on airline fleets, a bit of googling on the different types of electronic devices (e-readers, MP3 players, etc), and you’ve got a nice permutations problem.

Seven (Sneaky) Activities To Get Your Students Talking Mathematically

So we all know it’s importance. We all understand that students learn more when they do it. The problem is how to facilitate it.

“It” is getting students to talk mathematically. As a teacher, you can’t really say “ok kiddos, work on this thing together and I WANT TO HEAR THAT MATH TALK.” Well, maybe you can, but I’m not able to. I have to be more insidious, more conniving, more sneaky to get my students using mathematical vocabulary and verbalizing mathematical arguments.

Over the past couple years I’ve tried to aggregate some of the more successful teaching activities into my toolbox. This post will introduce seven of these sneaky, math-talk-producing activities. Not only that, but they are a) easily translatable to many content areas and b) quickly producible. In many cases you could create these activities with a word document and maybe 30 minutes time. You could be doing this tomorrow.

I’ll credit each activity individually, but a bunch are cribbed straight from Malcolm Swan’s work here and (related) the Shell Centre here. Go to those places and learn stuff.


Students match cards to various similar or equivalent things. I have a sneaking suspicion that it’s simple the tactile nature of using cards (or shoot, just paper cuttouts), that makes it feel like a game.

What I also like about it is that as students are matching, they naturally go back and revise their work. It’s a wonderful thing when students come to the last grouping and they realize it’s incorrect, then they go back through and find where they erred.

This is perfect for almost anything that utilizes multiple representations, which is pretty much all of Algebra. In fact, I’d be willing to bet that there isn’t a mathematical concept where you could use some sort of Matching activity.

(From Shell Centre MAP Project: Interpreting Distance-Time Graphs)

(Informally) Evaluating Student Work Samples

Being doubly sneaky, I love having students analyzing and evaluating samples of mathematical work because it has a nice side-effect. Not only are the students deciphering mathematical information, but they’re also inadvertently figuring out for what makes good math work. (Sneaky, right??)

As students are asked to follow some protocol (see below), they’ll no doubt verbalize things like “I like this work because it’s clear” or “I don’t understand this work because it’s so messy.” And I’m all BOOM GOES THE DYNAMITE, I KNOW!

I like the protocol shown below. I would NOT have students give it a grade or evaluate it against a rubric or something like that. That tends to make the goal of the assignment to check things off from a grocery list (on the other hand, I DO like having students evaluate themselves against a scoring guide).

(from Shell Centre MAP Lesson Units: Geometry Problems)

Determining “Truthiness”

With a nod to Stephen Colbert, “truthiness” in this case refers to students deciding whether a statement is always, sometimes, or never true. This always seemed natural for Geometry because of activities like this:

(from Shell Centre MAP Lesson Units: Evaluating Statements about Length and Area)

However, recently I came across things like this from the Shell Centre, using the Truthiness activity for Algebra and equations:

(from Shell Centre MAP Project: Sorting Equations and Identities)

You could certainly also utilize the Truthiness activity for when you have a variable in the denominator. Truthiness (I’d imagine) works great with things like asymptotes and discontinuities.

Ordering mathematical artifacts

Basically, here’s some stuff, put it in order from “least [something]” to “most [something].” In this case, [something] = “Square-ness”.

Fun story: I was in the middle of some PD and I showed a teacher this activity and he said he hated it because it was “watering down” math language. And that this was just one in the long line of “new math” instruction that was more about creating “warm and fuzzy feelings” than delivering concrete rules to follow. He also said that we were doing a disservice to the creators of mathematical definitions by doing activities like this and it was pretty much the problem with society.

That was a fun conversation!

Anyway, I love stuff like this. It, again, “forces” the mathematical dialogue and arguments. I stole this from Jason (@jybuell) at alwaysformative.blogspot.com (even though the source materials specifically state that these are not intended to be classroom instructional materials – OOPS!).

Similar: EmergentMath Problem of the Year 2011 Winner, Mr. Honner’s Equilateral Triangle poser.

“Odd one Out”

Simple instructions: pick the one thing that doesn’t below.

Once again, you don’t even need to prompt students to explain their reasoning, because they’ll explain it to each other. Especially if there’s a disagreement about which one doesn’t belong

More complex perhaps: finding three mathematical things that all could potentially be the “odd one out.” Or, “find a reason that any of these three don’t belong.”

(from Swan, Malcom, 2005. Improving learning in mathematics: challenges and strategies. Link)

Classifying Mathematical Artifacts

Much like the Matching activities, I think it’s just the tactile nature and the natural work-revision that comes along with an activity such as this that makes it really valuable and conversation-producing.

(from Swan, Malcom, 2005. Improving learning in mathematics: challenges and strategies. Link)

Any Questions?

Act 1 – Pop Box Design from Timon Piccini on Vimeo.

Of course, we all know Dan Meyer as the Moses of “Any Questions”, but if Meyer is Moses, then Timon Piccini (@MrPicc112) is Joshua, because he’s been crushing it lately.

The “Any Questions?” activity is a sort of stripped down (and cleaner) version of the “Need-to-Know” process (read more about that here). The twitter hashtag #anyqs is a great place to find stuff. However, on twitter everything is assumed to be in rough draft form. A great Any Questions? will launch students directly into the content you are intending them to learn about. That differs a bit from stuff like this, that may solicit five different content-related questions. Those have a place too, but probably belong in a different category.


What are some of the activities you conduct to promote mathematical discussion in your classrooms? Please share links and ideas in the comments!