# Postulates and proofs: Let's take it to the courtroom!

In this unit, the process of solving proofs is practiced using the comparison and framework of a courtroom setting. Students will work in groups to solve a proof and then defend it in a class courtroom setting.

**A lesson plan for grades 9–10 Mathematics**

Provided by Kenan Fellows Program

In this unit, the process of solving proofs is practiced using the comparison and framework of a courtroom setting. Students will work in groups to solve a proof and then defend it in “court.” This unit challenges and engages students, while building their confidence as they learn to support their arguments with sound, logical statements and reasons. Students will have both individual and group assessments during these lessons

## Learning outcomes

Students will:

- apply prior knowledge of the definitions, postulates, theorems, lines, angles, and triangles to solve proofs.
- demonstrate the ability to create a sequential flow of reasoning by writing each logical step in a proof as a separate
*statement*in the left column and writing its justification in the right hand*reasoning*column. - demonstrate a conceptual understanding of how to solve a two-column proof by searching for the answers to higher order thinking questions, revealing a more thorough understanding of the content.
- be able to make a convincing presentation to a courtroom that a proof is correctly solved.
- demonstrate proficiency of using inductive and deductive reasoning to solve proofs and problems.

## Teacher planning

### Time required

Seven to eight class periods (on a block schedule) or fourteen class periods (on a regular class schedule)

### Materials needed

- Geometry proofs — each group is assigned a different proof and each member of the group needs their own copy
- Group folder documents:
- Courtroom bellringer activity — one per student
- Problem-solving circle role cards — one per group
- Geometry vocabulary chart — one per student
- Collaborative group work rubric — one per student

- Trial Guide — one per student
- “Let’s take it to the courtroom!” rubric set:
- Teacher observation rubric — one per class
- Proof rubric for judge — one per group
- Trial rubric for jurors — one copy per juror for each trial

- Proofs unit test — one per student
- Pencils

### Technology resources

- Laptops for research and presentations
- Presentation software or program (such as Prezi, SlideRocket, SMART Notebook, Word, PowerPoint)
- Multimedia projector and computer
- Interactive whiteboard
*Courtroom 101*video (optional)- Teen court video (optional)
**Note:**The video is a teen court case in Florida. The content surrounds a teen that made a bad choice, and it involves the topic of drugs. Please watch the video first to decide if it is appropriate for use in your class.

### Handouts

- Geometry proofs
- This packet contains ten proofs (one for each group). It also contains an answer key for each proof.
- Open as PDF (634 KB, 20 pages)

- Group folder documents
- This packet contains the documents needed for each group’s folder, with the exception of the proofs. It contains the bellringer activity, the role cards, the vocabulary chart, and the collaborative group work rubric.
- Open as PDF (441 KB, 12 pages)

- Trial guide
- This guide contains the documents needed for the groups to prepare for trial. It also contains a proposed court schedule.
- Open as PDF (187 KB, 6 pages)

- “Let’s take it to the courtroom!” rubric set
- These are the various rubrics used by both the students and the teacher at different stages throughout the unit.
- Open as PDF (479 KB, 5 pages)

- Proofs unit test
- This document contains the unit test. It includes a modified version of the test for an alternative assessment.
- Open as PDF (220 KB, 20 pages)

### Prior knowledge

This lesson assumes that the following Geometry concepts have already been covered in class:

- tools
- lines
- angles
- logical reasoning
- the congruence of angles
- lines and angles
- solving a two-column proof

## Pre-activities

- Divide the students into groups of three. You can divide the students into groups based on different criteria (such as student work habits or proof-solving ability).
- Place group assignment cards at each table.
- Place a folder at each table. Each group’s folder should contain:
- one copy of the courtroom bellringer activity for each student.
- one set of the problem-solving circle role cards.
- one geometry proof for the court case (there are ten cases from which to choose). Assign a proof for each of the groups and provide a copy for each member of each group.
- one copy of the vocabulary chart for each student.
- one copy of the collaborative group work rubric for each student.

## Activities

### Day one

- As students walk in the classroom, have them find their name and group on a table.
- Have each student get a copy of the bellringer activity from their group folder. Direct them to begin working on it right away.
- Once the students have completed the activity, lead a whole-group discussion about the students’ answers to the activity.
- Explain that they will be working in groups to solve a proof. They will have to defend their solution in “court” in the classroom.
- Discuss the following concepts:
- what a court room looks like and what its purpose is in this classroom
- compare what the students know about activities conducted in court rooms to solving a proof
- what in a court room is similar to a postulate (theorem, statement, reason, etc.)
- how the presentations of the proofs will take place in setting similar to a court room

- Next, have each group take out the problem-solving circle role cards from their folder. Have them assign roles and explain on the sheet why each person was selected for each role.
- Guide the students to take out the proof case that has been assigned specifically to each group. Have each student individually begin to solve the proof.
- Once everyone has attempted to solve their proof on their own, the students can then discretely discuss and complete the proof within their groups. Students will take on the roles they chose from the problem-solving circle role cards and work to solve the proof they have been given. The groups should be discrete because they will be competing with other groups to defend their position in solving the proof in court.
- Each student should use resources and communicate clearly with the other two group members. Students will turn in completed proofs for teacher review.
- Each student should also fill out the group- and self-reflection logs on the collaborative group work rubric.
- Students should then begin working on the vocabulary chart. This should be completed for homework, if necessary.

### Day two

- Secure materials for students to use to create their group presentations (laptops, computer lab, chart paper, markers, etc.). Discuss the software/program options students have for creating and presenting their two-column proofs (such as SMART Notebook, Prezi, PowerPoint, or Slide Rocket)
- Students will spend the class period preparing for the presentation of their proof. This should include finishing their presentation and rehearsing it.
- Review the court schedule (found at the back of the Trial guide) with the class so everyone understands the schedule.
- Any remaining class time may be used by the groups to finalize their presentations.

### Day three

- You may choose to begin class by showing this optional video,
*Courtroom 101*, from TeacherTube. This video describes basic information about courtroom roles and procedures. - Pass out a copy of the Trial guide to each student. Review this guide with the class. Answer any questions the students may have about the information in the guide. The students will use this guide to help them set up roles and prepare for “court,” which will be held over the next several class periods.
- You may also choose to show this video from YouTube.
**Note:**The video is a teen court case in Florida. The content surrounds a teen that made a bad choice, and it involves the topic of drugs. Please watch the video first to decide if it is appropriate for use in your class. It accomplishes the task of modeling the different phases of a trial vividly.

### Days four – seven

- The classroom should already be set up like a courtroom.
- The student groups will take turns defending their proofs in front of the “court,” as outlined in the trial schedule.
- Jurors will use the provided rubric to evaluate each case.
- Refer to the Trial guide as necessary to keep track of procedures and the schedule.

### Day eight

Students will complete a test that includes all the proofs presented in “court” (though they will not know in advance that these proofs will be on the test).

## Assessment

This unit provides multiple opportunities for assessment. Various rubrics for different stages in the unit can be found in the “Let’s take it to the courtroom!” rubric set (see Handouts above).

**Day one:**Students will complete the collaborative group work rubric. You may also choose to use the individual proof solutions to understand where both individual students and groups are in their understanding of concepts.**Day two:**Observe and evaluate students based on participation and content knowledge, using the teacher observation rubric as a guide. Additionally, students’ vocabulary charts will be graded.**Days four – seven:**The teacher/judge will use a rubric to evaluate each group as they present in “court.” Jurors will also use a rubric to evaluate the Defense and Prosecuting teams.**Day eight:**The final test will be used as a summative assessment of the unit.

## Modifications

- A CLOZE test may be used for the final test. A modified version of the test can be found in the “Proofs unit test” document (see Handouts above).
- You may choose to group students based on how well they solve proofs and give groups proofs of varying levels of difficulty.
- You could also include a “statement and reason bank” for the group to use. Another option is to cut the statements and reasons into strips, which makes solving the proof more like solving a puzzle (by putting the statements and reasons in the correct columns and sequence to solve the two column proof).
- Allow students to present proof solutions orally, if necessary.
- Allow students to draw and notate the information on a diagram instead of writing out the words, if necessary.

## Critical vocabulary

### Postulates and theorems

- Triangle Sum Theorem
- The sum of the angle measures of a triangle is 180°.
- Exterior Angle Theorem
- The measure of an exterior angle of a triangle is equal to the sum of the measure of its remote interior angles.
- Third Angles Theorem
- If two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles is congruent.
- Side-Side-Side (SSS) Congruence Postulate
- If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
- Side-Angle-Side (SAS) Congruence Postulate
- If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
- Angle-Side-Angle (ASA) Congruence Postulate
- If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.
- Angle-Angle-Side (AAS) Congruence Postulate
- If two angles and a non-included side of one triangle are congruent to the corresponding angles and non-included side of another triangle, then the triangles are congruent.
- Hypotenuse-Leg (HL) Congruence Theorem
- If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent.
- Isosceles Triangle Theorem
- If two sides of a triangle are congruent, then the angles opposite the sides are congruent.
- Converse of the Isosceles Triangle Theorem
- If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

## Supplemental information

- Geometry the easy way
- This book from Barron’s Educational Series (Third Edition, 1997) contains proof ideas.
- North, South, East, and West Protocol
- This exercise identifies the way individuals prefer to work and what they may bring to a group.
- Collaborative learning: Group work and study teams
- This site from UC-Berkeley contains more ideas on grouping.
- U.S. Department of Justice 101
- This site gives more background information on how courtrooms are set up and the roles various people play in courtrooms in the United States.

- Common Core State Standards
- Mathematics (2010)
#### High School: Geometry

- Congruence
- GEO.C.10Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half...
- GEO.C.9Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of...

- Congruence

## North Carolina curriculum alignment

### Mathematics (2004)

#### Grade 9–12 — Geometry

**Goal 2**: Geometry and Measurement - The learner will use geometric and algebraic properties of figures to solve problems and write proofs.**Objective 2.01**: Use logic and deductive reasoning to draw conclusions and solve problems.**Objective 2.02**: Apply properties, definitions, and theorems of angles and lines to solve problems and write proofs.**Objective 2.03**: Apply properties, definitions, and theorems of two-dimensional figures to solve problems and write proofs:- Triangles.
- Quadrilaterals.
- Other polygons.
- Circles