# A Plane with a View

I travel a lot for work and therefore spend quite a bit of time looking out airplane windows.  I always think of model trains with the little miniature villages, people, cars, etc. as the planes are taking off or landing.  I also find it fascinating to see all the geometric shapes that are so apparent flying over the different landscapes – from farmland to cityscapes, geometry is everywhere.

As I stared out the window the other day,  trying to come up with something ‘mathy’ to write about, it dawned on me that I was looking at the perfect ‘mathy’ thing– the farmland with its various geometric shapes such as circles, rectangles, triangles and trapezoids.

Technology allows us to take planes-eye view pictures of these real-world images and bring them to our students.  Mathematical modeling comes to mind, from the Common Core Standards for Mathematical Practice. Real-world modeling leads to amazing student engagement and leads fluidly into other standards, such as reasoning abstractly and constructing viable arguments when testing conjectures.

Let’s think about something as simple as geometric shapes.  When we introduce students to shapes, we tend to draw pictures or show them shapes in various colors, sizes and orientations.  We might even use dynamic geometry software such as Sketchpad.  It would be more interesting to incorporate real-world instances of those shapes, such as satellite images from Google Earth, and use these shapes to help students think and reason about why certain shapes are being used.  For example, why are some farm crop fields in the shape of circles and others in the shape of rectangles?  Does it have to do with the crop itself or the watering tools the farmer has?  If you are going to use a circular field, what’s the maximum radius for that field?  What’s the area of that field?  How many seeds could be planted?  What’s the volume of the crop produced?

If we go in a different direction, and think about the teaching of parallel lines and transversals, where we often use a static picture of two lines with a transversal (our typical way to introduce this concept). Perhaps we are a little more dynamic and use Sketchpad, but why not step it up a notch and incorporate a satellite image of a city in conjunction with Sketchpad and introduce those concepts in a real-world context? The learning becomes both dynamic and relevant.

Almost every city has parallel streets, right angles and transversals, and a variety of geometric shapes, so using real-world images and dynamic geometry software, such as Sketchpad, to introduce the concepts helps make the connections and promotes critical thinking.   Students’ questions about mathematics will have more purpose and lead to geometric properties and definitions that have meaning.  Why are the streets parallel? How do you know? What’s the shortest distance from point A to point B mathematically and realistically?  What might be the reason for the circle round-a-bout in the park? What’s the predominant building shape and why do you think that might be?  Math becomes cross-curricular, relevant and challenging; no matter the level of student.

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