Shadows, Mirrors, Scale Models, & School Measurement

I’ve been itching to do this for days, but clouds and rain got in the way.

The past two weeks have been overcast and have had rain in the forecast nearly every day, so I was excited yesterday when at lunchtime I saw a break in the clouds, sun in the sky, and shadows on the ground! We would take a mini field trip outdoors and find the height of our school! This would be done by measuring shadows, drawing similar triangles (one from the school, its shadow, and the line of the sun’s rays connecting the top of the school to the furthest reach of its shadow; the second triangle from a pole, its shadow, and the sun’s rays, the pole being an item we could measure both height and shadow length directly), setting up a proportion, and solving for the unknown school height.

After lunch we began class with a warm-up challenge, then I posed the question all WCYDWT style to pique student interest: How tall is Patterson High School? I tried to ‘be less helpful’ as students threw out a variety of ideas of how we could figure it out (measure the height of my classroom and multiply times three for three floors, climb up to the top of the school with a ladder and tape measure, etc.). After some discussion, I turned out the lights and used a flashlight to cast a shadow on the floor. We talked about how shadows are formed and drew pictures on the board.

Unfortunately, as we stepped outdoors the sun was disappearing behind a cloud! Disheartened, we went back indoors and wrapped up construction of scale models of various classrooms in the school.

Last night, saddened by my shadowless plight, a teacher on Twitter suggested a cloudy-day alternative (or any-day comparative addition) of using a mirror to create similar triangles. When you can see that top edge of the building in a mirror laid on the ground, your line of sight to the mirror is reflected at an equal angle up to the top of the school. Add in yourself, the building, and the distances from the mirror out to both, and you’ve got similar triangles! The funny thing is that, even though I have practice questions in the project about mirrors and similar triangles, it never occurred to me to actually get a mirror and do this. Perhaps because my main experience with mirrors is as vertically affixed to a wall instead of portable and horizontal? So, in any case, this morning I bought a \$2 compact mirror at the grocery store.

There’s a happy ending though! Today the weather was beautiful, so we made the trek outdoors again. Students and faculty stared as we walked through the halls, tape measure, pole, yardsticks, mirror, and notebook in hand. We got outside and set to measuring/recording. A teacher on the second floor leaned out of his window, said hello to the students, and asked one of them to toss up a chalkboard-eraser that had fallen out. We measured the school’s shadow, the height and shadow length of the pole (students knew how this would yield similar triangles due to our talk yesterday). Then students took turns moving back and forth until they could see the top edge of our school in the mirror on the ground, and then we measured one student’s eye height, the distance from that student to the mirror, & the distance from the mirror to the school. Since we hadn’t discussed this in advance, they weren’t sure exactly how these mirror measurements would fit in to help figure out the school’s height.

We raced back indoors since the end-of-day-announcements would be coming on in only a few minutes. The students drew similar triangles all over the board, figuring out how the mirror would be used as well as the shadows.

Mirror-Caused Similar Triangles and Calculation

Using proportions and coming up with different answers for the school’s height, we discussed sources of error in measurements, and brought the lesson to a conclusion just as the loudspeaker came on!

All in all, an exciting day and a reinforcement for why I believe in project-based learning. Perhaps the ‘revolutionary’ new Baltimore teachers’ contract will tie our evaluations and pay to how many of our students can determine the heights of unknown buildings using only a few tools and the power of math, rather than just looking at standardized test scores? Well, I can dream, can’t I? 😉

Filed under math, teaching

3 responses to “Shadows, Mirrors, Scale Models, & School Measurement”

1. Cool! That (mirror activity) was totally one of the things we did last year for the measurement unit. I found out the hard way that putting mirrors at the base of trees is a bad idea, because the ground is uneven and your results come out crazy. Works much better with buildings and light poles.

2. You might like the Noon Day (Measuring the circumference of the earth) collaborative project (http://ciese.org/noonday) I started it over a decade ago at CIESE and a colleague of mine continues to run it. Here’s an article about it.

http://web.mac.com/ihor12/home/Articles/SpiritEratArticle.ic.pdf

-Ihor