Eglinton-Scarborough Crosstown

As a teacher, it pays to be opportunistic.  At least in terms of student engagement.  Take Toronto transit planning for example (or more like a ridiculously embarrassing lack of transit planning in the GTA).  Some students are bound to know about the most recent craziness surrounding a subway line in Scarborough.  Bringing it up may get other students interested in current events.  Nothin' like a good ol' subway debate to get a class riled up and excited for some learnin'.

Anyways this week I came across this map:

When a construction company is tendering a job, an estimator (or intern) may need to do take-offs: a fancy way of saying figuring out how much stuff they need to buy so they can put a cost to a job.  Here's a good math question:

Using the map above, how much material will they need to remove to construct the tunnels in the underground section of the LRT?

You may say there's not enough information there, which is probably true unless you make some big assumptions.  You can find the length of the tunnel using Google Earth.  If you explore the Eglinton Crosstown website you can find this rendering of the tunnel boring machine launch site:

From the picture, it's apparent that there are 2 adjacent tunnels.  Based on the size of a person (maybe 1.6 or 1.7 metres), you can estimate the diameter of the tunnels.  Using the length of the tunnel and the diameter, you can calculate a volume (after some unit conversions most likely).

Real-World Problem Solving

There are many other ways to find out the size and lengths of the tunnels from reports and construction drawings.  That's the awesome thing about these kinds of questions.  Plenty of ways to an answer.  Also, the answers may vary depending on what assumptions are made!  You can have a good class debate to try to figure out who is closest and why.  

You could also put some numbers to it.  How much would it cost to remove the rock/dirt?  Where should it go?  How big would a pile of it be?  How do I make sure I cover my assumptions?  These are all questions an Estimator has to deal with when pricing a job.  And it all comes from some simple geometry!  Great for any math class.  A lot of Construction and Engineering problems can be boiled down to simple math.  

This is also a perfect opportunity for some Problem-Based Learning.  The Eglinton LRT could be a theme for a unit.  The problem would be to estimate the cost of the whole rail line.  Several math concepts could be brought in to help solve the problem: areas and volumes, slopes of lines, finance, scale, understanding and creating graphs.  It's a way to give a bunch of disparate and lonely math concepts some common context and interest.  There is also the human factor to consider: the impact on the communities and the environment.  Definitely some interdisciplinary potential there.

I do foresee a problem however.  If I was ever to do this activity with a class I'd probably get so excited I'd end up talking about tunnel boring machines for an hour and put everyone to sleep.  The woes of being a Civil Engineer... not everyone cares about dirt as much as you do.  Vince understands.