Proof Parties
Purpose: To allow students to develop, solidify, and practice their ability to formally justify a
mathematical conjecture. The emphasis is upon the geometric reasoning and care in communicating
that reasoning—not upon the specific format of that reasoning.
Background: Proof Parties are designed to work like Writer’s Workshops work in language arts. Each
student will not only be responsible for developing, refining, and finishing his or her own proof, but also
for acting as a peer editor/coach for their partner and as an “outside reviewer” for other students’
proofs as well. Thus, the mathematical proof is analogous to a persuasive essay. Like persuasive essays,
mathematical proofs can be accomplished in multiple ways, although there is usually some familiar and
common structural elements. They are a specific kind of mathematical writing.
Launch:
For Proof Party 1: Begin by explaining the analogy between mathematical proofs and persuasive essays
and briefly develop the rationale for using a rough draft-internal review-second draft-external reviewfinal draft model for developing a proof.
For all Proof Parties: Arrange the students into pairs. Assign each pair a code (say, letters “A”, “B”, etc.)
so that it will be easier later to have pairs exchange papers. Students should be given the list of 4 to 6
conjectures to be used during this particular Proof Party (see Proof Party Topics below), from which
they should choose one conjecture to work on. Each member of a pair should choose different
conjectures, so that each pair is responsible for crafting two proofs (1 minute). Then have each student
take another minute to verbally explain to their partner however they can why they believe their chosen
conjecture is true. They should then write down as quickly as they can what they said (or pointed at, or
drew). They will use these notes as a starting point for writing a proof.
Explore:
Students proceed to develop their proofs, working as partners on each partner’s proof, following the
Proof Party Protocol. The teacher should circulate using good questioning to help students past stuck
points and, above all, helping students to express their reasoning and make explicit assumptions they
may be making without realizing it. A list of questions that can support thinking for students in the early
creation phase of a proof is included in Proof Questions.docx.
Discuss: Ideas for students sharing their work with the full class are included in the Proof Party Protocol
as well.
Proof Party Protocol
1. First Draft. Each student prepares a rough draft of a written proof of a theorem chosen from a
pool of options.
2. Internal Review. In pairs, students conference about the rough draft proofs, using the following
protocol:
a. Reviewing student reads the proof and, without any clarifying comments from the
author student, verbally summarizes the proof that they’ve just read.
b. Author student takes notes about where the reviewer is confused (or not convinced).
c. Reviewing student asks a question about the proof and author student answers.
d. Author student a question about the reviewer’s response to the proof and the reviewing
student answers.
e. Repeat c and d until questions are exhausted.
f. Each pair conference about BOTH rough draft proofs, focusing first on one of them, and
then on the other.
3. First Revision. Individually, students re-write their proofs on a new page.
4. External Review: Students exchange their first revisions with a different student than they
participated with during the internal review step. Each student annotates the draft they receive
with marks indicating arguments and justifications they:
a. Find completely convincing mathematically (*)
b. Feel that need more justification (#)
c. Don’t understand well enough to decide whether it’s convincing or not (?)
Finish the review with a short written recommendation for how the author can improve the
proof.
5. Peer Collaboration. In their original pairs, students work together to revise and rewrite both of
their proofs, in preparation for possibly sharing them with the whole class.
6. Public Exhibition. This can take several forms depending on the goals of the lesson. Possibilities
include:
a. Gallery walk. Students post their final revisions together with those of others working
on the same theorem. Different theorems are arranged around the room. Students
view their own “theorem-group” and write down any approaches or justifications or
communication ideas they saw that they felt were particularly effective or high quality.
Students can then rotate in groups to the other “theorem-groups” areas and continue
to take notes on anything that seemed particularly excellent. Once students have had a
little bit of time observing each “theorem-group”, reconvene as a whole class and share
out the Items of Excellence that they observed in the proofs.
b. Whole-group sharing around a single theorem. Student papers are shared, in an order
strategically selected and sequenced by the teacher, with the whole class (using a
document viewer, ideally). Student authors do not necessarily have to present to the
class. Because the goal of this task is focused on formal, written, mathematical
arguments and justification, it is probably appropriate to let the written proof speak for
itself. Observing students are invited to contribute to a growing list of features of the
proof that were particularly effective.
c. Whole-group sharing around multiple theorems. Student papers are shared, in an order
strategically selected and sequenced by the teacher—ideally sharing most, if not all, of
the theorems in the pool of options and illustrating a variety of approaches to proof.
Observing students are invited to contribute to a growing list of features of the proof
that were particularly effective.
d. Theorem-group sharing followed by whole-group shareout. Students who’ve worked on
the same theorem form sharing groups to read each other’s proofs and, collaboratively,
compile a list of effective features of the proofs. Whole class then compiles a class list
with ideas offered by the groups.
Be sure to keep the emphasis on what is correct, effective, and excellent about the proofs and,
not upon what is missing, weak, or wrong about the proofs during the public exhibition phase.
Proof Party Topics
Proof Party 1
1)
2)
3)
4)
Vertical angles are congruent
When Parallel lines are cut by a transversal congruent angle pairs are created.
When Parallel lines are cut by a transversal supplementary angle pairs are created.
Points on the perpendicular bisector of a line segment are equidistant from the segment’s
endpoints.
Proof Party 2
1)
2)
3)
4)
Opposite sides of a parallelogram are congruent.
Opposite angles of a parallelogram are congruent.
The diagonals of a parallelogram bisect each other.
Rectangles are parallelograms with congruent diagonals.
Proof Party 3
1)
2)
3)
4)
Prove that the sum of the interior angles of a triangle = 180°.
Prove that the base angles of an isosceles triangle are congruent.
Prove that if two angles of a triangle are congruent, the triangle is isosceles.
Prove the segment joining midpoints of two sides of a triangle is parallel to the third side and
half the length.
5) Prove the medians of a triangle meet at a point.
Proof Party 4
Prove any of the revised conjectures that were generated during the Circle Explorations.
Proof Questions: Guiding students in creating proofs
General:
What information am I given?
Would drawing a diagram be helpful? Does it have useful labels you can use?
Are there any terms in this problem that you don’t understand?
How can we write this in geometric notation?
What are we trying to prove specifically?
What are some theorems or definitions relating to this problem that may be useful?
What could we try to get started?
Are we communicating everything we need to to be clear?
Can someone else follow this logic?
Can we find a counterexample?
If Applicable to Conjecture being Proven:
Can we create similar triangles?
Would extending a segment help?
Could we construct an altitude? a perpendicular bisector? an angle bisector? a median?
Could we construct parallel lines?
Can we prove any triangles are congruent?