Looking for Congruence Shortcuts
Lesson 3 of 4
8th or HS Geometry
90-120 minutes
Description
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!
Materials
OR
Introduce
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?"
[Answer: no]
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:
- Students who aren't sure how to use a ruler (especially for non-integer lengths)
- Students who aren't sure how to use a protractor
- Students who are having trouble getting started
- Students who are stuck after drawing one line
- Students who are trying to use ALL of the measurements given
- Students who are using "guess and check" with some of the information and checking using the rest of the information
- Students who are just using a subset of the measurements given, e.g. 2 sides and the included angle, or 2 angles and the included side
- Students who are trying to calculate the third side analytically (maybe by misapplying the Pythagorean theorem) in order to use the previous lesson's methods
- Students who are using measurements in the wrong place -- e.g. putting the measurement of Angle A at the intersection of AB and BC.
If needed, stop the activity to provide (or invite a student to provide):
- A tutorial on using rulers to create a line with a certain length
- A tutorial on using protractors to create a ray at a certain angle
- Some hints from students who are having success getting started
When most students have drawn a triangle that matches yours, invite 2-3 students to share their strategies. You might invite:
- A student who has put a measurement in the wrong place so that all their measurements are "correct" but the triangles aren't congruent
- A student who used all the information, e.g. two segments with a given angle between them (Side-Angle-Side) and also drew the other two given angles at the ends of the rays (Side-Angle-Side-Angle-Angle), or a student who drew rays on either side of a given side (Angle-Side-Angle) and then measured the given side length along one ray, and added the third angle at the end of that side (Angle-Side-Angle-Side-Angle)
- A student who used only three pieces of information (Angle-Side-Angle or Side-Angle-Side, most likely).
Explore
If you are using
Illuminations:
"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"
- "If all three pairs of corresponding sides are congruent, the triangles are congruent."
- Ask, "What pieces of information did you use to draw your triangle that was congruent to mine? Did anyone have a strategy where they only needed some of that information?"
- It's fine if students just say, "Two sides and one angle." The activity will show their conjecture either is or is not true.
- Ask, "What other sets of information might be enough to guarantee that three triangles are congruent? Any guesses?"
- Record any conjectures.
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.
Synthesize
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.
Exit ticket:
Have students write about the following:
- Given Triangle ABC, make a list of every possible set of measurements you could be given that would result in definitely drawing a congruent triangle.
- Dory says that if you have two triangles and you know any two pairs of corresponding sides are congruent and any pair of corresponding angles are congruent, then the triangles have to be congruent. Nemo says he doesn't think that's true, but he's having a hard time explaining just when it is true. Please help Nemo, using what you explored today.
- Agree or Disagree: Looking at examples in the app or the game proves that these shortcut sets of conditions guarantee triangle congruence. Why do you agree or disagree?
Teacher Reflection
Tools and Technology:
Leave your thoughts in the comments below.