Mapping the U.S.A. (with Oblique Triangles)

I have seniors these days and we're doing Precal. The students learned in Geometry that any three side lengths will determine a unique triangle (as long as the sides obey the Triangle Inequality): lengths of 24, 9, and 30, for example, can only be fit together in one way to create a triangle. Only later, however, do they learn about the Law of Cosines, which allows you to find the angles of that particular triangle. So starting from “side side side” information, they can calculate an angle, then use a protractor to draw the triangle accurately. It occurred to me that we could create a map by drawing triangles in this way. If the students looked up their own three cities and/or towns, and the distances among those places, they could make their own triangular scale map- and if they all used the same scale, we could fit their maps together.

The restrictions I gave them were:

-the scale is 1 mm = 1 mile.

-the distances cannot exceed 300 miles on any side of a triangle.

-the places must be within the 48 states.

Here’s how it’s looking overall right now. You can actually see the country taking shape.

The Angle Game

Here’s something I’d like to write as a free internet game (because what I really need right now is another project).

So far it’s been a game that my students play in pairs, on paper. The paper between them has a diagram like the one below:

Each student picks a marker to be his or her color. The first student begins by choosing any angle on the diagram and estimating its measure, then writes that measure on the diagram.

The second student can then either do likewise, making up a measure for any angle on the diagram, or writing in the measure of an angle that has already been determined by what the first student wrote. For example, if the first student wrote in 75 degrees, the second student might write “75” in the angle vertical to the first one, or just go elsewhere and make up another angle measure.

The students proceed like this until all angles are determined, and everything agrees. The trouble, of course, is that often the angles end up in disagreement, especially on the more complex diagrams:

 

(See the conflicts here?)

The student whose turn it is has the opportunity to perform corrections, crossing out as many angles as necessary and writing in new measures, in his or her color, until the angles are brought to agreement. The student then completes the turn by writing in a new angle.

At the end, the student who claimed the most angles wins.

Here are the blank diagrams I have so far for playing the game.

Back to the computer game idea: it would be a lot of fun to have a.i. opponents that behaved in distinctly different ways (like in the game “Scorched Earth,” in which you could play against a “Pool Shark,” a “Cyborg,” etc.). For example, you could have an opponent that notices your mistakes, but doesn’t correct them at first, waiting to see how long it takes you to pick up on them. If you seem like a vulnerable target, the opponent will lay traps and wait until there are lots of discrepancies on the board, and then pounce to correct them all. Or you could have a crazy player who constantly does lots of corrections that are actually incorrect.

Steal my idea, somebody- write this game! I am never going to get around to this.