Those Triangles Aren't Congruent...Are They?
Lesson 2 of 4
8th or HS Geometry
60-90 minutes
Description
Students grapple with congruence through rigid transformations, then conjecture a "shortcut" set of conditions that also ensure congruence.
Materials
Introduce
Activate prior knowledge, from the Congruent Halves activity in Lesson 1 (suggested discussion format: think-pair-share with cold-calling):
- "What strategies did you use to find congruent halves? What did you look for to give you hints about where to draw the dividing line?" (SMP 7)
- "When you were confident that you had congruent halves, what strategies did you use to decide how to translate, rotate, and reflect one half to show that it exactly matched up to the other half?" (SMP 7)
- "How do you know two shapes are congruent? How did you convince skeptics that your two halves were congruent?" (SMP 6 -- Attempt to elicit from students a range of precision, from "They looked the same" to "They matched up exactly" to "All the sides were the same" to "I could reflect/rotate/translate one to line up with the other.")
- Make a list of these.
- If "All the sides are the same" isn't on the list, add it, perhaps saying, "Here's another common explanation." If students argue with it, remove it, if they convince you.
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."
- "Look back at the list of ways we convinced ourselves and skeptics that shapes were congruent. How does this match up with what you were explaining about how you knew the two halves of your shapes were congruent?"
- "How is this different from what you were thinking?" (SMP 6 again -- It is not exactly the same to say, "the two shapes match up exactly" or "The two shapes are the same exact size and shape" or "They are the same" and to say there is a sequence of rigid transformations that map one onto the other. It is essentially the same to say "If you flip or rotate the shape you can set one on top of the other and they match.")
OPTIONAL EXTENSION for HIGH SCHOOL STUDENTS: Focus on the "If" and "Only If" parts of the definition (SMP 6 again)
- Explain/remind students about the meaning of "if and only if"
- Write, "IF there exists a sequence of rigid transformations that map one shape onto the other, THEN the two shapes are congruent."
- Write, "IF two shapes are congruent, THEN there exists a sequence of rigid transformations that map one shape onto the other."
- Ask, "Which would you use to justify why two congruent shapes have all of the corresponding sides congruent?"
- Ask, "Which would you use to justify how you know that these two shapes are congruent?"
- Ask, "Can you explain in your own words what the difference is between the two halves of the definition?"
Explore
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.
- Support homing in on the precise conjecture "Two figures are congruent if every pair of corresponding sides is congruent." (SMP 6)
- "When you say, "All the sides are the same, do you mean they both have to be equilateral?"
- "When you say, "All the sides match up," can you put that in more mathematical terms?"
- "Do you remember the mathematical term for the matching sides?"
- "Does the order of the side lengths matter? Like, is a quadrilateral with sides 4, 5, 6, 7 congruent to one with sides 4, 5, 7, 6? What about a triangle with sides 5, 7, 9 and one with sides 5, 9, 7?" (SMP 7?)
- Ask students, "Do you think this is true?"
- The quadrilateral question above is a pretty dead give-away; use it if students seem convinced that this will work. If they think it's still true have, them draw a picture to show you the transformations that map a quadrilateral with sides 4, 5, 6 and 7 onto one with sides 4, 5, 7, and 6.
- Encourage them to draw a counter example and show that it doesn't work.
- Bring back the idea of the triangle. Have them also try to convince you that the order of the side lengths matters in a triangle. Whether they say, "It doesn't matter," or "It does matter," convince them that you're skeptical. Tell them, "I won't believe you until you show me that every possible triangle with side lengths 5 cm, 7 cm, and 9 cm is congruent to every other triangle with those side lengths." (SMP 3, SMP 6)
Give students the Lesson 2 Student Activity Sheet "These Triangles Aren't Congruent, Are They?" (download from Materials section above).
TEACHER NOTE:
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?"
- Before they get too far in, ask students how tolerant they should be of slight differences, and why they think this is the case (SMP 6)
- If students are very familiar with the definitions of the rigid transformations and constructing them, have each pair construct with compass and straightedge or dynamic software the transformations that map their triangles onto one another.
- If students' knowledge of transformations is more informal, have them use patty paper to convince each other whether or not their triangles are congruent.
Put pairs together in groups of four so that they have four different triangles:
- Check each other's work. Do you agree with the other pair's conclusion about their triangles?
- Check the four different triangles in your group. Decide informally whether any of them are not congruent to the others, or whether they can all be translated, rotated, and/or reflected so they map onto each other exactly.
Synthesize
"Based on your work with the triangles, let's make a conjecture about criteria for triangle congruence."
- Again, encourage students to use precise language. You're headed for something like, "If the corresponding sides of triangles are congruent, then the triangles are congruent."
- If students say, "If all the sides of the triangles are the same then the triangles are congruent," ask, "Do you mean the triangles have to be equilateral?"
- Encourage them to use the language of "corresponding sides."
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.
- "The definition of congruence we are going to be working with is "Two shapes are congruent if and only if there exists a sequence of rigid transformations that map one shape onto the other." Explain how knowing that two shapes are congruent by this definition also tells you that the two shapes will have corresponding sides congruent and corresponding angles congruent."
- "How confident are you that if two shapes have all pairs of corresponding sides congruent and all pairs of corresponding angles congruent, that you will definitely be able to make a series of reflections, rotations, and translations that map one onto the other. Why?"
- “When we looked at triangles, we found a simpler criteria that can probably be used to prove two triangles are congruent. Do you think there are other 'shortcuts' we will be able to use to prove that two triangles are congruent, without finding the rotations, reflections, and translations needed to match them up? Give an example of one, or explain why you think the shortcut about side lengths is the only possible shortcut."
- Possible Extension Discussion: "Today's explorations probably convinced you that any two triangles with the same set of side lengths are congruent to each other, but that two quadrilaterals with the same set of side lengths are not necessarily congruent.”
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:
Teacher Reflection
Facilitate meaningful mathematical discourse:
- A lot of focus in this lesson is on students articulating a precise mathematical conjecture. Are there other moments in your curriculum or in earlier courses where students work on formulating a conjecture precisely? If not, how might you add it to other units that precede this one? If so, is it something your students struggled with?
Support productive struggle in learning mathematics:
- We chose Side-Side-Side as the triangle congruence shortcut to focus on in the lesson in part because we felt it was the easiest for students to tackle if they were unfamiliar with geometric constructions, in part because it is often students' naive answer to what you need to know to be sure two triangles are congruent, and in part because it is not always obvious to students that the three sides can't be arranged in different ways that will produce triangles with different angle measurements -- Side-Angle-Side is often seen as a surer bet to definitely guarantee congruence. In the next lesson, technology is used to help students test other sets of shortcuts. Would you start with a different shortcut in this lesson? How would using a different shortcut change the ways students might struggle in this lesson? How would using a different shortcut change the ways students might struggle in this lesson?
Leave your thoughts in the comments below.