Triangle Congruence Lesson 3
What is the minimum info needed to prove two triangles are congruent? Play a game where the less info you give, the more points you get!
Lesson 3 Student Activity Sheets "Match My Triangle"
Teacher Note: There are 4 different versions (A - D). The only difference is what triangle students are given to start with. Depending on whether your students work in pairs, groups of 3, or groups of 4, you'll either need A and B, A, B, and C, or all of A - D.
Activate prior knowledge from Lesson 2:
"Please remind me about the triangles we were drawing yesterday. What information did we have about those triangles?"
[Answer: "We knew all three side lengths."]
"With yesterday's triangle, if we correctly matched all of the information, could we draw two triangles that were NOT congruent?"
On a piece of paper, draw a triangle and measure the side lengths and angles. For formative assessment purposes you might want to include a side length that is not a whole number.
Give students a challenge, along the lines of: "I've drawn a secret triangle, and measured 2 of the sides and all 3 angles. Do you think, with the given information, that you will be able to draw a triangle that exactly matches mine? Or will some of our triangles be different?"
Circulate among students and attend to their strategies for recreating your triangle. Take note of:
If needed, stop the activity to provide (or invite a student to provide):
When most students have drawn a triangle that matches yours, invite 2-3 students to share their strategies. You might invite:
If you are using
"Mathematicians are always looking for generalizations and shortcuts. Instead of having to check all possible measurements to show that two triangles are congruent, they like to say, "Hey, we know the same basic facts about these triangles. Is there a shortcut that means they ALWAYS have to be congruent, like how any two triangles with all congruent corresponding sides HAVE TO be congruent?"
Write "Conjectures about Triangle Congruence"
Show students the
Congruence Theorems Illuminations App. Ask them, "How can we use this app to confirm or disprove our conjectures?"
If students need more prompting, ask, "What would happen if the information was NOT enough to guarantee that the triangles are congruent?" "What would happen if the information WAS enough to guarantee that the triangles are congruent?"
Instruct students to use the app and to test and record as many conjectures as they can, using the Looking for Shortcuts student activity sheet (download from Materials section above).
If you are using the "Match My Triangle" Game:
Introduce Round 1 of the Match My Triangle Game. Explain to students that if you were playing the game, each student who had drawn a triangle that matched yours would get 1 point, and you would get 1 point for each triangle that matched yours.
Put students in groups of 4 and have them play Round 1. If you have clear plastic sleeves of whiteboards, students can play Round 1 several times. After the first time, they can draw and measure their own triangles for their groupmates to try to match.
Differentiate instruction by letting some groups of students play Round 1 until they are confident that they have strategies to draw congruent triangles based on plenty of given information. Introduce other groups to Round 2 (more points if less information is given; partners try to draw different triangles). Round 2 can be made even more challenging by adding the rule that each player has to give a different set of measurements.
As groups become proficient at Round 2, introduce Round 3 to small groups or the class. If you don't get to Round 3 in most or any groups, that's okay! You can play Round 3 as a class as part of the Summarize/Synthesize time.
Ask students to share sets of measurements that seemed to always produce congruent pairs of triangles.
Invite students to name and describe the conditions for each of these conjectures in their own words, also sharing the way math textbooks usually refer to them. Sometimes students remember the difference between SAS and SSA better if they've made a connection like "SAS looks like an alligator mouth when you label the congruent parts."
Ask students to share conjectures that do NOT always seem to produce congruent pairs of triangles.
Highlight the difference between saying "you need all three pairs of sides to be congruent" and "you need two sides and one angle" -- which angle it is matters! Encourage students to "attend to precision" (SMP 6) in telling which angle or which side was needed.
NOTE: The ambiguity of SSA may not have come up if students were playing Match My Triangle because either they may have all made the same choices, or may not have picked ambiguous cases, or may never have given that set of instructions. If that's the case, you might problematize the "you need two sides and one angle" assertion by having the students play Round 2 against you, telling you 2 sides and 1 angle, using the original triangles on their sheets. Show them that, in some cases, you can win the round by not matching their triangle, and in other cases you can't. Note that the ambiguity of SSA will be explored in more detail in Lesson 4.
Have students write about the following:
Tools and Technology:
How do you see either one of those roles being important in your classroom with this lesson?
Leave your thoughts in the comments below.
What is the minimum amount of information needed to prove two triangles are congruent?
CCSS, Content Standards to specific grade/standard
CCSS, Standards for Mathematical Practices
PtA, highlighted Effective Teaching Practice and/or Guiding Principle