Why isn't ASS a member of the "Fab Five"?

I have taught Geometry for many years, and a big part of the curriculum has been proving congruent triangles. If you teach this, you know about what I refer to as the "Fab Five":

  • Angle-Side-Angle (ASA)
  • Side-Angle-Side (SAS)
  • Side-Side-Side (SSS)
  • Angle-Angle-Side (AAS)
  • Hypotenuse-Leg of a Right Triangle (HL Rt. △)
I introduce this list with accompanying diagrams illustrating each method. Here is a sample:


After generating the list of the "Fab Five", I emphasize to the students that each method requires three parts from each triangle. This, in turn, always results with students asking: "Hey Mr. K. What about AAA and ASS?", the latter of which usually produces a number of giggles.

So I ask them: "If you know that three angles in one triangle are congruent to the three corresponding angles in another triangle, why are the triangles not necessarily congruent?" After a little guidance, students will recall our work from earlier in the year with dilations and scale factors and the fact that a dilation preserves angle measures. This becomes our first introduction into similarity. So, AAA is not a member of the "Fab Five".

This brings us to Angle-Side-Side, aka "ASS". Now I understand the teenage mind pretty well, even if it's been quite a while since I had one. The mere mention of the term and writing it on the board is usually an opportunity to have a few laughs - always an important part of my class culture. Some teachers explain that "It's a bad word, so you can't use it as a proof reason." That's cute, but I'm not sure if teachers really explain why it doesn't work.

So I present the students with this:



In the past, students used their compasses, protractors, and straightedges to draw triangles with these measurements. Then I would display the triangles on the board and ask the students for their observations. Some cases produced congruent triangles, some did not. Of course, this depended on their construction & measurement abilities.

This year, I had the students construct the triangles using the Desmos Geometry app. I had them first make a triangle and show the side and angle measures. Then, the students manipulated the triangle to match the given parameters:


This brought up issues of accuracy (how close should it be?) and placement of measurements (does m∠ABC have to equal 60° or can m∠CAB = 60°?).

After they made their triangles, I gave the students patty paper and asked them to trace their triangles. I then collected their papers and laid/stacked them under the document camera to see which parameters produced congruent triangles.




The results and conversations were really powerful. Many "ah-ha" moments were made. And most importantly, the students realized why ASS is not a member of the "Fab Five"!

 

Finally, here is a LINK to a presentation which includes screenshots and links to each of the Desmos Geometry constructions.

Thanks for reading!







Comments

Post a Comment

Popular posts from this blog

Paper Plates & Fuzzy Sticks

Image
As the school year enters fourth quarter, teachers are blessed/cursed by the arrival of nice weather. For those of us in the northeast, we rejoice the end of winter and celebrate the boost to the spirit from the new spring. But along with that comes the battle to keep the students on task and motivated as they can sense that the end of the school year can't be that far away. I have an extra challenge as my classroom looks out on the tennis court and track, and I am always competing with teens enjoying gym class for the attention of my students. One of the last units that we cover in Geometry is our unit on Circles. Having taught Geometry for a number of years, I have always been on the lookout for new ways to engage students with the many terms, concepts, and relationships that come with the Circle unit. While it is always easy for me to immediately go to the Chromebooks and have the students use Desmos and/or GeoGebra to investigate, I decided to go low-tech (again) this ye
Read more

Geometry Modeling Set From the Thrift Store

Image
A couple of weeks ago, I had some time to kill between dropping my daughter off at a friend's home and picking up my son from musical rehearsal. So I decided to visit the local thrift store. I've made out well there over the years, picking up toys for the kids, a record player for my parents, and a variety of other items for the home. But on this day, I found myself drawn to the toy section. This is funny because my own children are (sadly) of an age where they have outgrown toys. Scanning the shelves, there were a number of jigsaw puzzles, "well-loved" toys, and many games. Among the many boxes, I found this: Over the past few months, I have spent many hours constructing polyhedra structures using straws and wire. Here are a few of my creations: So when I came across the Discovery Kids 77-Piece Build & Play Construction Fort for the low-low price of $3.99, I just had to have it. Little did I know, it would quickly become a favorite source of math
Read more