standard_description: Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.
problem_activity: Activity
The goal of this activity is for students to see the usefulness of the triangle congruence theorems in proving other results. In this proof, students are expected to draw an auxiliary line, find two congruent triangles, determine that they are congruent by the Side-Side-Side Triangle Congruence Theorem, and then use the fact that corresponding parts of congruent triangles are congruent to finish the proof.
Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Arrange students in groups of 2. Remind students of the tips for writing proofs using triangle congruence theorems that are on display.
Representing, Conversing: MLR7 Compare and Connect.
Use this routine to help students develop the mathematical language of geometric proofs. After partners revise their drafts into a clear proof that angle
\(B\)
is congruent to angle
\(D\)
, invite them to create a visual display of their proof. Then ask partners to quietly circulate and read at least two other visual displays in the room. Give students quiet think time to consider what is the same and what is different about their proofs. Next, ask partners to discuss what they noticed. Listen for students who reference the proof from a previous lesson that opposite sides of a parallelogram are congruent. Also, listen for and amplify the language students use to justify why triangle
\(ABC\)
is congruent to triangle
\(CDA\)
and angle
\(B\)
is congruent to angle
\(D\)
.
Design Principle(s): Cultivate conversation
Action and Expression: Provide Access for Physical Action.
Provide access to tools and assistive technologies such as dynamic geometry software. Some students may benefit from a checklist or list of steps to be able to use the software.
Supports accessibility for: Organization; Conceptual processing; Attention
Student Facing
Quadrilateral
\(ABCD\)
is a parallelogram. By definition, that means that segment
\(AB\)
is parallel to segment
\(CD\)
, and segment
\(BC\)
is parallel to segment
\(AD\)
.
Prove that angle
\(B\)
is congruent to angle
\(D\)
.
Work on your own to make a diagram and write a rough draft of a proof.
With your partner, discuss each other’s drafts.
What do you notice your partner understands about the problem?
What revision would help them move forward?
Work together to revise your drafts into a clear proof that everyone in your class could follow and agree with.
Student Response
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Anticipated Misconceptions
If students struggle longer than is productive, direct them to their reference charts. What do they know about parallelograms? (Opposite sides are congruent.)
Activity Synthesis
Select partners who used the same approach and whose work is at similar levels of clarity to work together in groups of 4.
Instruct students to read the other group’s proof and decide if they agree with it, and if it could be improved to convince a skeptic. Invite each partnership to say one thing they noticed and liked about the proof, and one thing a skeptic might wonder about the proof.
If needed, brainstorm some things skeptics might be wondering about, such as:
When you did that rotation, how did you know these points had to line up?
How did you know those would be the same line?
How did you know those lines could only intersect there?
How did you know the Side-Side-Side Triangle Congruence Theorem would apply to these triangles? Did you list all three reasons?
How did you know those sides were congruent?
How does knowing the triangles are congruent help you prove the angles are congruent?
Did you use any information that wasn’t given or that you didn’t have a reason for?
Did what you proved match the problem?


aligns: no