Triangle Congruence Lesson 2
Students grapple with congruence through rigid transformations, then conjecture a "shortcut" set of conditions that also ensure congruence.
Activate prior knowledge, from the Congruent Halves activity in Lesson 1 (suggested discussion format: think-pair-share with cold-calling):
Provide students with the mathematical definition of congruence: "Two shapes are congruent if and only if there exists a sequence of rigid transformations that map one shape onto the other."
OPTIONAL EXTENSION for HIGH SCHOOL STUDENTS: Focus on the "If" and "Only If" parts of the definition (SMP 6 again)
Come back to the idea that students might think, "If two shapes have all the same side lengths, then they are congruent." Support students to Attend to Precision and put this into a conjecture OR dispute it with a counter argument (if they haven't already). Present this either as a conjecture students made earlier in the lesson, if it came up in discussion, or as a common conjecture if it hasn't come up.
Give students the Lesson 2 Student Activity Sheet "These Triangles Aren't Congruent, Are They?" (download from Materials section above).
You might have students find all the possible permutations of the three side lengths as a class.
For accountability purposes, you might assign different pairs to construct two of these triangles on the same piece of paper with a ruler and a non-collapsing compass, or on the same page using dynamic geometry software. Student 1 in each pair should construct the first triangle with their partner supporting them, and Student 2 should construct the second triangle.
Ask students, "Is your triangle congruent to your partner's triangle? Decide together. Can you use the definition of congruence to convince me that your triangles are or are not congruent?"
Put pairs together in groups of four so that they have four different triangles:
"Based on your work with the triangles, let's make a conjecture about criteria for triangle congruence."
Ask students, "Are you convinced that this conjecture is true for triangles?" and "Have we proved this conjecture is true for any pair of triangles with corresponding sides?"
Exit Ticket: Ask students to address the lesson's essential questions, in writing.
Use this to explain why builders use designs like these:
and not like these:
...to make rigid structures that don't end up leaning over like this:
Facilitate meaningful mathematical discourse:
Support productive struggle in learning mathematics:
Leave your thoughts in the comments below.
CCSS, Content Standards to specific grade/standard
CCSS, Standards for Mathematical Practices
PtA, highlighted Effective Teaching Practice and/or Guiding Principle CCSS