Summary
Students investigate triangle congruence and complete proofs using the theorems they verified.
Essential Question(s)
Are there congruence shortcuts for triangles?
Snapshot
Engage
Students respond to a prompt asking the essential question.
Explore
In groups, students test possible congruence theorems.
Explain
Students present the possible shortcuts, explaining why they did or did not work.
Extend
Students create a construction project using congruence theorems.
Evaluate
Students complete proof puzzles to solidify their knowledge of the congruence theorems.
Materials
Triangle Congruence Possibilities (attached)
Triangle Congruence Investigations 1 and 2 (attached)
Proof Puzzles (attached)
Access to dynamic geometry software, like Geogebra for each group (A physical manipulative, like AngLegs, or ability to construct segments and angles would work too.)
Engage
Provide the following prompt for students: "A contractor has just assembled two massive triangular trusses to support the roof of a recreation hall. Before they are hoisted into place, the contractor needs to verify that the two triangles are identical.How should he do this?" Allow students time to think and take five student responses.
Pose the question: "Might there be a shortcut?" Students will probably readily agree that there should be. Ask, "What do you think is the minimum number of parts we can test to reach a conclusion?"
Ask students to provide all combinations of three adjacent parts they can think of (you may suggest S for side and A for angle). Allow them to work in pairs to create a list. Choose two groups to write their lists on the board, then other groups can add to what they have. Discuss the possibilities together.
Come to a consensus: "Are these are the only six ways that the same three parts of two triangles may be compared?" (Yes, as any additional ones would be equivalent.) Then we will investigate to see which of these work.
Explore
Day 1: Split students into groups of two or three. Provide each group with one investigation (either 1A, 1B, or 1C from the "Triangle Congruence Possibilities Investigation 1" handout) and a device on which to access Geogebra. Allow students time to work through their investigations and reach a conclusion. Walk around as students are working, helping students who struggle, but be careful not to direct them outright.
Some good questions to ask might be:
Are you sure?
Does your group agree?
What have you tried already?
What do you think you should try next?
Does that work in any case?
Have you tried another case? This could be for students who draw conclusions very quickly; encourage them to create another case and try it too.
Because it is likely that more than one group will have completed each A, B, and C, have the A groups get together to discuss their findings and do the same for the B groups and the C groups. Have each new group reach a consensus about the results of their investigation.
Day 2: Similar to Day 1, split students into groups of two or three (it's not necessary to keep the same groups from Day 1, so keep or change them as you wish). Provide each group with either investigation 2A, 2B, or 2C from the "Triangle Congruence Possibilities Investigation 2" handout and a device on which to access Geogebra. Repeat the process from Day 1 with this second investigation.
Explain
Day 1: Have a representative (or two) from group 1A present their findings and then do the same with the groups that went over 1B and 1C. Visual aids are suggested during these presentations, since the other groups have not had a chance to think about the conjecture being presented each time.
After the presentations, discuss the results as a class and come to a conclusion. Record the SSS, SAS, and SSA conjectures for future reference.
Day 2: Similarly to Day 1, have a representative (or two) from group 2A present their findings and do the same with 2B and 2C. Discuss as a class and reach consensus. Record the ASA, SAA, and AAA conjectures for future reference.
Extend
Student groups of two or three should create a presentation in response to the following prompt:
The PTA from our school is looking to sponsor a student project for the beautification of our grounds. Create a proposal for this project that is made of congruent triangles. The proposal should include a detailed sketch of the item, with all congruencies clearly labeled.
Evaluate
Split students into groups of two or three and supply them with the "Proof Puzzles" handout (only proofs one through four). Give them time to piece together the puzzles in a logical order, encouraging them to think about how they constructed their triangles in their previous investigation. After they have completed the puzzles, they should write them up with a justification of their sequence.
Student groups should compare their solutions and come to consensus.
Resources
Serra, M. (2003). Discovering geometry: An investigative approach (Vol. 4). Emeryville, CA: Key Curriculum Press.
K20 Center. (n.d.). Geogebra. Tech Tools. https://learn.k20center.ou.edu/tech-tool/2352