When explaining proportions one example that always captures the students attention is to find the distance to a city on the map (I use New York). On the map in my room, NYC is 37.5 inches away with a scale of 1.5 inches for every 100 miles. We then set up the following proportion:
I make sure to assert the concept of keeping the units consistent (inches on top on both sides etc), and I show the differing ways it can also be set up.
After we attain the answer–which comes out to 2500 miles–I then allow the student with smart phones to use the map function to find out what the distance comes out to (about 2800 miles). I challenge the students to think why it’s farther using this map than a real map (The zig zaggy-ness of the roads). Which makes flying so much more efficient–not only do airplanes go faster, they create their own straight-shot path.
(And while I’m on the topic, I like to point out why planes are able to go 650 mph at 30,000 feet but not nearly as fast when they first take off (thinner air). The same reason more baseball players hit more homeruns at Coors Field in Denver and why my golf swing mysteriously had 15% more power when I played in Denver.)