We’ve been exploring the ideas of similarity and congruence in my freshmen classes. Today the students developed congruence by SSS – with straws.
Supplies needed: drinking Straws (1-2 per student), string (dental floss was too slippery), and scissors.
Question 1: Are all quadrilaterals with congruent corresponding sides necessarily congruent?1
Students began by cutting a drinking straw into four pieces, threading the string through, and tying it off to form a quadrilateral. They quickly realized that they could “move” the angles, so nope, this is not true. Unexpected bonus: discussion of concave/convex.
Question 2: Are all polygons with more than four sides that have congruent corresponding sides necessarily congruent?
Repeat the bit with the straws. Some of them realized they could easily create this by carefully cutting one of the straws from their quadrilaterals. Conclusion, nope, more “moving” angles.
Question 3: Are all triangles with corresponding sides congruent necessarily congruent?
Make the triangle with straws… conclusion, yep, this does guarantee congruent triangles as the angles are “locked”.
We spent the rest of the period trying to explain why this is the case (and formalize the language after a bit). We also discussed whether of not this constituted a proof.
On the way out, one student thanked me for the lesson. He said it was fun to really understand why it worked. Wow. I was thanked. For a math lesson. Wow.
Up next: SAS congruence with straws and pipe cleaners.2
1We’ve spent the past few days on “if-then” statements and counterexamples as a lead in to this.
2I thought it was time to start sharing.