One way to discover or apply the formula for the area of a triangle is to explore area on the geoboard. The initial activities should be along the lines suggested in my Geometry Labs Lab 8.4. (Free download.)
After that, one can zero in on triangles, by asking a question like: “Find triangles of area 15.” After some right triangles have been found, challenge students to find some other examples, in fact, as many examples as possible. If you keep pushing them to find more, these are likely among the solutions:
Such a set, or another one of the same type, strongly suggests that the height is involved, and with appropriate bookkeeping, the formula is within reach. (Of course, the most straightforward explanation / proof of the formula is by splitting the triangle into two right triangles by dropping an altitude.)
If many or most of your students already know the formula, or if you prefer to introduce it some other way, you can still use this activity as a domain to apply it. This is certainly more interesting than finding the areas of given triangles. However, if application is the goal, I would ask students to “Find all the triangles that have one horizontal side, and area 15“. I count 44 on a standard 11 by 11 geoboard. Am I right?
While planning one of my summer workshops yesterday, I came up with a related challenge: “Find triangles with area 15, such that none of their sides are horizontal or vertical“. This is not easy, and the usual formula does not help. In fact, it may be best to use this as a bonus activity for students who enjoy a tough problem. The reason is that random trial and error may yield an answer, but the search is most likely to succeed if it is systematic. One way to help students organize their exploration is encapsulated in this GeoGebra file. Students can see the coordinates of the vertices (which may lead them to an interesting discovery,) as well as the triangle’s area. One vertex is fixed at the origin, the other two can move on their respective blue segments. The segments can be moved by moving their visible endpoint. Here is one solution:
I did not do that systematic search on a geoboard, or on GeoGebra. Instead I used a number pattern that emerged from my early experimentation with the latter. Well, to be honest, not only from that experimentation: it is related to something I knew but failed to apply to this problem early on.
In any case, using the number pattern, I concluded that there are 20 such triangles on a standard 11 by 11 geoboard. Am I right? I hope so, but I can’t say I’m sure!
(The next day:) Alas, I was wrong. The figure above is of one possibility I had missed yesterday. I had to make some changes to the GeoGebra file and this post! I don’t know how many solutions there are, but the number is definitely greater than 20.
For more on geoboard triangles:
- Visit the Geoboard home page on my Web site
- This post follows up on the “triangles of area 15” question.
- I made the geoboard figure above on one of the several online geoboards, the one on the Math Learning Center site.
- Matt Enlow and Paul Jorgens helped me think about the third problem.