Real World Mathematics Collaboration

• Content objectives – Prove theorems about right triangles (specifically the Pythagorean theorem)
• Literacy/language objectives – Justify and explain problem-solving decisions to peers; develop the use of academic language by using content-specific vocabulary to communicate ideas orally and in writing
• Engagement/interaction objectives – Work collaboratively in small groups to construct viable arguments and critique the reasoning of others; look for and make use of structure; produce a better product as a group than as an individual

Common Core State Standards for Mathematics

• CCSS.MATH.CONTENT.HSG.SRT.C.8
  Use trigonometric ratios and the Pythagorean theorem to solve right triangles in applied problems.

Common Core State Standards for English Language Arts

• CCSS.ELA-Literacy.W.9-10.1
  Write arguments to support claims in an analysis of substantive topics or texts, using valid reasoning and relevant and sufficient evidence.

The Unit
This five-day unit on proving why the Pythagorean theorem works fell in the middle of the curriculum year. During this unit, students looked at five different ways to prove the Pythagorean theorem. The featured lesson focused on Bhaskara’s proof and occurred on the second day of the unit.

Before the Video
Students worked on other proofs to understand the function of a geometric proof and how to construct one. Ms. Brown gave students a pre-lesson quiz to assess their current understanding of proofs.

During the Video
Upon arrival, students were given a warm-up to address the problem area that Ms. Brown had seen in quiz results—students were weak in developing a system of equations to prove that a large polygon was a square. Students worked with partners in the warm-up to take on the same challenge. A few students were called to the board to write out their proofs. The quiz became known as a “rough draft,” as Ms. Brown had students read and write responses to the comments that she had given on their individual quizzes. Students then worked with partners to edit their rough drafts to create stronger responses.

After the Video
Students were assessed on what they learned. Their proof work was displayed in the classroom.

Teacher Prep
Ms. Brown consulted with other teachers about various proofs of the Pythagorean theorem. Teaching this lesson was a first for her, so she did the work as if she were a student to be sure she understood the direction of the lesson and to anticipate any areas in which students might have difficulty.

Prior Knowledge
To participate in this lesson, students needed to know how to use a system of equations and the triangle angle sum theorem.

Differentiated Instruction
Ms. Brown provided students with sentence stems to help them with their proof writing. To help visual and kinesthetic learners see how the proof worked, she had students cut out, label, and move around the shapes for each.

Group Interaction
Ms. Brown emphasized the importance of collaboration in mathematical work. She had students work with partners so that they could think things through together, ask each other questions, and clarify things for each other. To manage group interaction, Ms. Brown walked around the room listening, checking work, noticing conversations, reading facial cues, and asking probing questions when necessary.

• Mathematics Assessment Project: Classroom Challenges: A Formative Assessment Lesson: Proofs of the Pythagorean Theorem
• Graph paper
• Straight edge
• Individual whiteboards
• Promethean interactive whiteboard

Formative Assessment
Ms. Brown used formative assessment practices to gather information about student understanding and skills. As she walked around the classroom, she checked students’ work for evidence of what they knew. She used the pre-lesson quiz to provide feedback and created an opportunity for students to make improvements in their work.

Summative Assessment
Students earned a grade for the classroom work. They also took a test at the end of the unit in which they wrote formal proofs using three different ways to show why the Pythagorean theorem works.