My comments are mostly about SAS, ASS, etc. Since my brain is almost hard-wired to my computer these days, some of the activities I suggest have a dynamic geometry flavor--Geometer's Sketchpad and Cabri can be used.
General question: Given three numbers (which can represent lengths or angles, depending on the specific question you're asking), does there exist a triangle with these measurements? If yes, how many (non-congruent) triangles?
If the 3 numbers represent lengths, the existence question leads to the triangle inequality, the Pythagorean theorem, and (if you're teaching trigonometry) the law of cosines. The uniqueness question leads to the SSS criterion for congruent triangles. Some interesting triples: 2 3 4 3 4 5 2 3 5 2 3 7
Hands-on explorations: use linear things connected with hinges at the ends: pieces of drinking straws and pipe cleaners?
Ruler and compass/dynamic geometry explorations: Let a, b, c be the numbers of a triple. Construct a segment of length a. Construct a circle of radius b at one endpoint of the first segment (so the endpoint of the side of length b will be on this circle) Construct a circle of radius c at the other endpoint of the first segment (so the endpoint of the side of length c will be on this circle) But the endpoints of the sides of length b and c coincide, so this is where the circles intersect. There are, in general, 2 such points. So you should get 2 triangles. When are they congruent? Might you only have one intersection point of the circles? How? Might you have no intersection points? How?
[This is really cool in Geometer's Sketchpad, and makes you think about the situation in a very different way. Maybe I'll put a sketch on my Web page.]
You can do similar things with SAS, ASS (that's what you are if you think it's a theorem) and AAA. Note that ASS is _almost_ a theorem: two triangles with the same A, S, and S are not necessarily congruent, but there are only two choices for the congruence class of the triangle. With AAA, there are infinitely many choices.
Real life application: Rigidity. One of the many bridges I can see from my window at the moment is made entirely from triangles. Why? Why not just make a framework of squares? [Partial answer: try the drinking straws and pipe cleaner experiment with a square. Will it hold its shape? NO! It squashes to a rhombus at the slightes nudge. You wouldn't want this happening on a bridge.]
Extensions: analogs of SSS, etc., for quadrilaterals? What additional elements would you have to add to a given quadrilateral to make it rigid (e.g., a diagonal)?
p.s. I vote for trapezoids having at least one pair of parallel sides. Then a trapezoid is a special kind of quadrilateral, a parallelogram is a special kind of trapezoid, etc.
Susan Addington until June 15, 1996: After June 15, 1996: firstname.lastname@example.org email@example.com The Geometry Center Math Department 1300 South 2nd St., Suite 500 California State University Minneapolis, MN 55454 San Bernardino, CA 92407 (612) 624-5058 (909) 880-5362 fax: (612) 626-7131 (fax) (909) 880-7119 WWW at Geometry Center: http://www.geom.umn.edu/~addingto/ WWW at CSUSB: http://www.math.csusb.edu/faculty/susan/home.html ***** NEW MATH GAME! ****** Check my Web pages for The Number Bracelets Game.
On Thu, 30 May 1996, Rex Boggs wrote:
> I am just starting our unit on 'plane shapes' in year 9 maths. One of the topics is the > usual stuff on congruent triangles - SSS, SAS, etc. I haven't found a particularly good > way to teach this topic. > > The best I can do is to ask students to construct a triangle with a given set of sides > and angles - eg A = 45 degrees, b = 4 cm, c = 5 cm, and show that everyone _has_ to > construct the same triangle, while with A = 60 degrees, b = 5 cm and a = 4.6, there are > two triangles that meet those conditions. > > I reckon kids find this topic a confusing, especially if in the triangles in the > diagrams look congruent but don't meet a congruency test, or don't look congruent but > do meet one of the tests for congruency. > > While I'm on the topic of geometry, here are a few other discussion-starters: > > * do you have any great protractor and ruler activities for teaching or revising how to > use these objects? > > * does a trapezium (trapezoid to US citizens) have _exactly_ one pair of parallel > sides, or _at least_ one pair of parallel sides? > > * does the 'kite' really deserve its own name? It really doesn't have much going for > it. > > * I have had great difficulty finding any life-related applications of the parallel > lines theorems, other than in the trivial case where the the transversal meets the > parallel lines at a right angle. For that matter, I haven't found much in the way of > life-related applications for the angle sum of a triangle, quadrilateral, etc. Any grand > ideas? > > Thanks for any assistance. > > Cheers > > Rex > >