Friday, August 24, 2012

My Favorite Lesson - "How Big is Chicago?"

This is one of the first activities I developed entirely on my own, and the first one I ever taught to an actual class of students.  I've now done it four times, and it's been met with mostly success.  It's an activity that I really like and plan to continue improving.  It's worked very well with classes of 5th-7th graders, but can be adapted for older students.  I've fit it into a 50-minute block, and while time is tight it's definitely doable in that timeframe.

How Big is [Your City]? 

What you'll need:
  • Rulers
  • Markers/drawing supplies
  • A class set of maps (at least one per student or group, but make extras in case they get really into it!).  I use maps of the City of Chicago because that's where I've taught the lesson.  You can [should] use your own city/county/state/ward.  Interesting shapes are better than squarey ones.  More on the maps:
    • The maps need to have a scale.
    • You might be tempted to use large paper for your maps.  It's more trouble than its worth!  Maps on 8.5"x11" paper are just the right size, and are perfectly legible.  
    • Try to get as "clean" a map as possible.  Students will get distracted by whatever miscellaneous printing is on the map (roads, El lines, district numbers, anything!)
    • This is the map I use  [PDF alert!]. I have a paper copy that has the city border outlined that I use for making student copies.  It's not perfect, but it's the cleanest one I could find of the city that also had a scale.  
Start by asking students some various ways to measure the "size" of a city.  Answers here can vary, it's more to get them thinking.  Answers have ranged from the standards of population and land area to thoughts about economic power, tourism, or a city's fame. (this part can be cut for time.)

Now, introduce the task - use the map to estimate the land area of your city.  The problem solving skills of the students will probably influence the amount of guidance that you have to give from this point.  If they're pros at compound shapes and calculating areas, then just hand out the maps and let them roll.  Otherwise, this might be the time to introduce some vocabulary like partition, dissection, approximation, or maybe work a quick example with a simpler map (this part is where I need to do some work before teaching it again.)

Hand out the maps and supplies.  The goal here is that they'll start drawing rectangles / triangles / parallelograms that approximate the edges of the mapped area.  This is where the spare maps come in handy, because they will want to re-start their drawing, either because of a mess-up or because they thought of a better way halfway through.

Once they've got a partition that resembles the map outline (you can have them check with you before proceeding,) they should start measuring the shapes and calculating areas.  I find it's easiest to get a total area in square inches or centimeters, then do one unit conversion at the very end.  Encourage students to write their measured dimensions directly on the map - it'll be easier for them and you to find errors if all the dimensions are in one spot rather than scattered around (calculations should be on separate paper.)

The scale conversion has presented a problem for some of my students.  Most of them get that on the map (my map at least) 1 inch = 3 miles, but some then try to use this same factor to convert square inches to square miles.  One way that helps explain the conversion without algebra is to draw a picture like this:
I think a lot of students struggle with the idea of what exactly square/cubic/etc units are, and how they compare to linear units.  This activity might help them figure out the difference between them.

If you've done it right, the kids will be burning to know what is the actual area of the city?  I like to show them this list of large US cities.  It's particularly nice because you can sort by population, land area, and pop density.  Beware that this list doesn't include small-but-large-area cities (the top 4 US cities by land area alone are actually all in Alaska [as per this list].)

I generally conclude at this point, but here are some extension questions I've pondered:
  • Using your method, how close do you think you could get to the actual area?  How many shapes would this take?  
  • How would you design a computer program/algorithm to calculate areas using this method?
If you're tech savvy and have the resources in your class you might try to import the map file into Sketchpad or Geogebra so that students can dynamically adjust and recalculate the areas of their shapes.  In this case, the activity becomes less about measuring and precision and more about getting the partition just right.  You lose the kinesthetic aspect of physically drawing and measuring, but gain the ability to experiment with many more types of shapes.

Let me conclude with two examples of student work.  This is from the first time I taught the lesson, and most of the class didn't actually make it to the measuring phase.  One pair made it all the way through and got an extremely close estimate for the land area.  This is their map:
These students even went back and refined their original number by subtracting the area of the small triangle in the bottom left.  The map tells you that they had a problem-solving plan when they started drawing.  

This second map is from a group that didn't make it through to the end calculation, but it illustrates some of the pitfalls of doing this activity:
Particularly in the top left you can see how they started to make shapes fit willy-nilly rather than focusing on the end calculation.  These students could have used more support and guidance as they worked on the project.  

Thanks for reading if you've made it this far.  I hope you liked this activity as much as I do.  Please let me know your thoughts/suggestions in the comments.  

1 comment:

  1. Hi, Kyle,

    I love this lesson idea -- if I was still in the classroom I would find a way to incorporate it!

    I came across your blog via David Wees, and as a fellow mathematics educator I thought you might be able to help in spreading the word about an educational TV show for preteens about math that we're putting together. "The Number Hunter" is a cross between Bill Nye The Science Guy and The Crocodile Hunter -- bringing math to children in an innovative, adventurous way. I’d really appreciate your help in getting the word out about the project.

    I studied math education at Jacksonville University and the University of Florida. It became clear to me during my studies why we’re failing at teaching kids math. We're teaching it all wrong! Bill Nye taught kids that science is FUN. He showed them the EXPLOSIONS first and then the kids went to school to learn WHY things exploded. Kids learn about dinosaurs and amoeba and weird ocean life to make them go “wow”. But what about math? You probably remember the dreaded worksheets. Ugh.

    I’m sure you know math is much more exciting than people think. Fractal Geometry was used to create “Star Wars” backdrops, binary code was invented in Africa, The Great Pyramids and The Mona Lisa, wouldn’t exist without geometry.
    Our concept is to create an exciting, web-based TV show that’s both fun and educational.

    If you could consider posting about the project on your blog, I’d very much appreciate it. Also, if you'd be interested in link exchanging (either on The Number Hunter site, which is in development, or on which is a well-established site with 300,000 page views a month) please shoot me an email. We're also always looking for input and ideas from other math educators!

    Thanks in advance for your help,