## Pieces of Proof

- Lesson

There is a leap to be made from understanding postulates and theorems in geometry to writing proofs using them. This lesson offers an intermediate step, in which students put together the statements and reasons to build a formal proof.

The following is a list of postulates, properties and definitions used in the proofs in this lesson:

- Addition postulate
- Vertical angles are congruent
- Reflexive property
- Substitution property
- All sides of a square are congruent
- Definition of
*angle bisector* - Definition of
*midpoint* - Proving triangles congruent using Side, Side, Side postulate, SSS
- Proving triangles congruent using Side, Angle, Side postulate, SAS
- Proving triangles congruent using Angle, Side, Angle postulate, ASA

### Prepare Materials for Class

Print copies of each proof on a different color of paper to help keep track of what goes together. Cut the proof page into two pieces as shown, keeping the diagram, Given, and Prove together. Then cut the statements and reasons into individual strips, separating the statements from the reasons. Put the statement and reason strips and the diagram in an envelope for each proof. Make enough of these proof kits so that each student group will have a copy of each proof.

### With Students

Divide students into groups and provide each group with a proof kit envelope for Proof 1.

Encourage students to clear a space on their table or cluster of desks so they have room to spread out all the strips and the diagram for the proof.

- Determine which strips are
*statements*, and which are*reasons*. - Determine which statements are
*given*to start constructing the proof. - Allow the groups to work through the proof by matching
statements and reasons until they have built a proof that the group
agrees on. On their own paper, each student draws the diagram and
writes out what is given and what is to be proven. Have each student
then write out the proof that their group has agreed on before putting
the strips back in the envelope. The group then produces 1 poster of
their complete proof. They should re-create the diagram, list the given
statements, and re-create the proof on the poster. If each group has
done the same proof, then put a few of the posters up at the front of
the room to compare.
When looking at the posters as a class, as the following questions:

- How accurate is the diagram? Are the segments and angles that look congruent in the diagram actually congruent?
- Are the corresponding parts labeled in the correct order?
- Did any of the groups put markings in the diagram?
- If there are different correct ways to order the statements in the proof, what are they?
- Which statements must come ahead of other statements? Why?
- For which statements or reasons can you predict the ordering?
- How does this predictable order help you write a proof?

After discussion of the first proof, the groups are ready to move on and tackle the other proofs. The proofs are as follows:

- Proof 1: congr. triangles by ASA and the addition postulate
- Proof 2: congr. triangles by SAS and the addition postulate
- Proof 3: congr. triangles by SAS and the transitive property
- Proof 4: congr. triangles by SAS and the angle bisector
- Proof 5: congr. triangles by SSS and the properties of a square
- Proof 6: congr. triangles by ASA and the conguence of vertical angles

This lesson offers proof kits for straight-forward congruent triangle proofs. This format can be useful with other proofs, including theorems.

- Proof kits consisting of envelopes with the statement and reason strips and diagrams inside. If you use all the proofs, you'll want to print copies of each proof on different colored paper to help keep them separate.
- Scissors
- Poster paper and markers (optional)

**Assessment Options**

- Have students make up their own proof kits and exchange with their peers.
- Have student pairs use a given proof kit to build a proof. The pairs will write an essay justifying the order in which they have placed the statements and reasons.

**Extensions**

- Provide proof kits that contain only statements. Students have to determine the reasons as well as write out the proof.
- Provide students with proof kits that contain extraneous statements or reasons. Students have to identify which statements or reasons are not needed for the proof as well as write out the proof.
- Provide students with proof kits that have some statements or reasons missing. Students have to supply the missing pieces as well as write out the proof.

**Questions for Students**

- Explain how working backwards from what you are trying to prove can be helpful in developing a proof.
- Explain why it is necessary to label corresponding parts of congruent figures in the same order.
- How does having each statement and each reason written on separate strips of paper help you build the proof?
- If you had to choose between being given just the statements or just the reasons, which would you prefer? Why?

**Teacher Reflection**

- What advantages and disadvantages do you see in using separate strips for the statements and reasons as students learn to build a proof?
- How did you structure the transition from using strips for the statements and reasons to just writing the proof?
- What would be the benefits of students using proof kits to help build a more complex proof?
- What other intermediate steps can you take students through to help them learn the skills needed to build a proof?

### Learning Objectives

By the end of this lesson students will:

- Identify the properties that exist in a given figure
- Apply postulates and theorems to a figure to build a formal proof

### NCTM Standards and Expectations

- Explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them.

- Establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others.