Triangle Congruence Lesson 4
Through a "Simon Says" style game, students prove triangle congruence theorems SSS, SAS and ASA using transformations.
Review the three proposed criteria for triangle congruence. Make sure the statements are clear and precise (SMP 6). Remind students that we haven't proved them yet, just conjectured.
Listen to students' ideas about what it would take to prove they work for any triangle. Summarize student statements with something like the following: "We'd have to prove that no matter where the triangles are or what the sides and angles are, as long as one of the sets of criteria is true, we can definitely always translate, rotate, and reflect to get one triangle onto the other."
Introduce students to the Simon Says Proofs directions (slide 3 of the
PowerPoint slides), where the teacher is Simon and the students are the players.
Encourage students to try each of the starting conditions several times. Each time a new student takes the role of Simon.
Bring students back together. Ask them the following questions:
Emphasize again that, "We want to prove that no matter where the triangles are or what the sides and angles are, as long as one of the sets of criteria is true, we can definitely always translate, rotate, and reflect to get one triangle onto the other."
Ask, "How does getting good at the Simon Says game help us show that, no matter where the triangles are or what the sides and angles are, as long as one of the sets of criteria is true, we can definitely always translate, rotate, and reflect to get one triangle onto the other?"
Have each group of four play the game until they are confident that they could come up with a set of transformations that map one triangle onto the other no matter where the triangles begin. Have students fill out a flowchart for one or more starting conditions. Suggestion: Students might do this as homework.
Suggestion for Differentiation:
Collect students' flowchart descriptions of how they would transform an arbitrary triangle satisfying the given conditions onto another arbitrary triangle.
Choose one or two steps of the flowchart to ask students to use the definitions of rigid motions to prove how they know that Point E will end up mapped onto Point B, and/or Point F will end up mapped onto Point C. Most students will use a translation, followed by rotation, followed by reflection (if needed). In that case:
This should result in a somewhat formal proof by transformations for each of the three triangle congruence cases.
Here are fully justified responses for the SSS, ASA, and SAS cases. Note that for the AAS case we can calculate that the 3rd pair of angles must be congruent and turn the problem into an AASA case which works just like ASA.
The authors wish to acknowledge that students writing these transformation proofs with some degree of rigor may feel that they should be treated to a parade led by the superintendent, and get a key to the city from the mayor or something. These proofs are HARD in that they stretch students (and their teachers) to think about Geometry differently. But, these proofs are doable and a sign that students have mastered productive struggle! What can you do with your students to celebrate their success?
One important aspect of summarizing is to debrief about what helped kids persevere with the Simon Says game and the proofs that their flowchart steps work as well as how awesome it was that they did persevere and how proud they can be of what they came up with.
The other point to emphasize in this Lesson ARC is how these proofs help them understand which triangles are good candidates for being congruent and how much simpler it is to use these proven shortcuts (SSS, ASA, SAS) vs. having to come up with a general mapping rule for every new situation they encounter. So, for example, instead of coming up with transformation proofs for each new triangle situation they encounter, they can just figure out how to "turn these into a problem I've solved before" by figuring out which congruence theorem shortcuts are relevant in each case (SMP 7).
Build procedural fluency from conceptual understanding:
Pose purposeful questions:
Support productive struggle in learning mathematics:
Leave your thoughts in the comments below.
Thanks for putting all of these great resources in one place! Having multiple visual representations of transformations can be so beneficial for all learners!
CCSS, Content Standards to specific grade/standard
CCSS, Standards for Mathematical Practices
PtA, highlighted Effective Teaching Practice and/or Guiding Principle CCSS