Aim: To define and discover the Pythagorean Theorem
Students will be able to:
- Identify the right angle, hypotenuse, and legs of a right triangle
- Explore the relationship between the lengths of the three sides of a right triangle to develop the Pythagorean Theorem
- Use the Pythagorean Theorem to determine the unknown length of a side of a right triangle
- Determine whether a given triangle is a right triangle by applying the Pythagorean Theorem and using a calculator
Student Technology: Geometer’s Sketchpad, Google Spreadsheets & any technology tool of their choice to present a proof developed by the student
Teacher Technology: Geometer’s Sketchpad, Google Spreadsheets & projector
Length of Lesson: 5 days about 40 minutes each day
Investigating and Understanding the Pythagorean Theorem
- The Pythagorean Theorem is one of the most useful relations in mathematics.
- In the middle grades, an investigation of the lengths of the sides of right triangles and the area of squares drawn on those sides introduces students to irrational numbers,
- Pythagorean triples (derived from right triangles with integer sides), and methods of indirect measurement used for solving real-life problems.
- Students construct a variety of right triangles using a right-angled set square, cutting corners from pieces of paper or cardboard, or using dynamic geometry software.
- They measure the sides of these various right triangles and record measurements in a spreadsheet.
- Students use the spreadsheet to look for possible patterns in the measurements. They also use the spreadsheet to square the values of each measurement and look for possible relations among squared values.
- Once the Pythagorean relation has been established, students generate visual proofs using duplicate cutouts of right triangles and the dynamic software.
- They search the Web for information on Pythagoras and many different visual proofs. They investigate the possible generalization of the theorem to other similar shapes drawn on the sides of right triangles using dynamic geometry software.
Description of Project or Narrative
1) Working in groups of three or four, students create a variety of right-angled triangles by cutting corners from rectangular sheets of paper or cardboard using a straight edge. Each group measures the three sides of their triangles and enters the measurements into a spreadsheet. Students investigate and discuss relations between the long side of each triangle and the two shorter sides. The teacher then introduces the terms hypotenuse (long side) and legs (shorter sides).
2) Groups share with the whole class the patterns or relations they have found. Most likely, someone in the group suggested squaring the measures and summing the squares of the legs. If not, then suggest this as an exploration.
3) Use The Geometer’s Sketchpad to extend the investigation and form generalizations. Students create a script for constructing right triangles, take measurements using Sketchpad, square these measurements, sum the squared legs, and see how this sum compares to the squared hypotenuse as they dynamically change the side measurements of their right triangles. Using a script for constructing squares, students construct a square on each side of their right triangle, measure the areas of these squares, and investigate relations among the areas. Students should also construct squares on the sides of non-right triangles to see if the relationships hold for any triangle.
Procedures and Learning Activities
4) As an experiment, construct other polygonal figures on the sides of the right triangle. Students create scripts for various polygons (e.g., equilateral triangles and pentagons). With the constraint that the polygons on each side must be similar, have students measure the areas and investigate relations among these areas. The goal is to have students make their own conjectures about these relationships.
5) Building from students’ conjectures about the relationships of the three areas, introduce the Pythagorean Theorem—if students haven’t already mentioned it themselves! Have students research Pythagoras and his contributions to mathematics. Hundreds of Web sites examine Pythagoras. Students can use these to find information.
6) Construct visual proofs of the Pythagorean Theorem using any technology tool of their choice. Have groups find at least four different visual proofs they’ve learned from the Web sites. Students should demonstrate their proof to the rest of the class using their own reasoning from the visual demonstrations.
7) Using Sketchpad, investigate the dynamic proofs illustrated in Pythagoras Plugged In (Bennett, 1995). Have groups brainstorm their own dynamic proofs of Pythagoras’s Theorem using Sketchpad.
8) Using a spreadsheet, investigate integer values for the three sides of a right triangle (these are called Pythagorean Triples).
Day: 1 Duration: 40 minutes - Students will complete activity 1
Teacher will make a class spreadsheet and display it using the overhead projector
Students discuss the relationship between the three sides and try to develop a mathematical solution with their group
Discuss as a class the mathematical solution
Day: 2 Duration: 40 minutes - Review the directions with the class on how to use Geometer’s Sketchpad
Allow time for students to complete the investigations. Monitor students while they are working and assist students as needed
At the end of class, remind students that they will have more time during the next class session to complete the investigations
Day 3: Duration: 40 minutes
Allow time to finish investigations from the previous day
Students are given the rest of class time to research the Pythagorean Theorem Proof using the list of websites under resources
Day 4: Duration 40 minutes
Students use a technology tool of their choice and with their group develop their own proof using what they had researched on the web and using the geometer’s sketchpad about right triangles.
Day 5: Duration 40 minutes
- Students present their proof to the class
Resources Software: Spreadsheet, The Geometer’s Sketchpad (Key Curriculum Press) Book: Bennett, D. (1995). Pythagoras plugged in: Proofs and problems for The Geometer’s Sketchpad . Berkeley, CA: Key Curriculum Press. Other: Rulers, cardboard, straight-edge ruler, scissors, paper, pencil Web Resources Title: Pythagorean Theorem Information URL: http://mathforum.org/dr.math/faq/faq.pythagorean.html Annotation: Basic information about the Pythagorean Theorem. Title: The Pythagorean Theorem URL: http://mathforum.org/library/drmath/view/54713.html Annotation: Explains the Pythagorean Theorem. Title: How Tall is Hal? URL: http://mathforum.org/library/drmath/view/55302.html Annotation: A problem that requires the use of the Pythagorean Theorem. Title: Inverse Pythagorean Theorem URL: http://mathforum.org/library/drmath/view/55309.html Annotation: Discusses the inverse of the Pythagorean Theorem. Title: Pythagorean Theorem URL: http://mathforum.org/library/drmath/view/57232.html Annotation: Explains the Pythagorean Theorem. Title: President Garfield and the Pythagorean Theorem URL: http://mathforum.org/library/drmath/view/57552.html Annotation: Provides information about President Garfield's proof of the Pythagorean Theorem. Title: Converse of the Pythagorean Theorem URL: http://mathforum.org/library/drmath/view/62215.html Annotation: Information about the converse of the Pythagorean Theorem. Title: How Many Proofs of the Pythagorean Theorem URL: http://mathforum.org/library/drmath/view/62539.html Annotation: Provides information about the various proofs of the Pythagorean Theorem.
Associated Technology Standards http://www.iste.org/Content/NavigationMenu/NETS/ForStudents/2007Standards/NETS_for_Students_2007.htm 1) Creativity and Innovation Students demonstrate creative thinking, construct knowledge, and develop innovative products and processes using technology. (Activities 1, 3, 4, 7, & 9) a.) apply existing knowledge to generate new ideas, products, or processes. d.) identify trends and forecast possibilities. 2) Communication and Collaboration Students use digital media and environments to communicate and work collaboratively, including at a distance, to support individual learning and contribute to the learning of others. (Activities 1 & 2) a.) interact, collaborate, and publish with peers, experts, or others employing a variety of digital environments and media. d.) contribute to project teams to produce original works or solve problems. 3) Research and Information Fluency Students apply digital tools to gather, evaluate, and use information. (Activity 5) a.) plan strategies to guide inquiry. b.) locate, organize, analyze, evaluate, synthesize, and ethically use information from a variety of sources and media.
Associated Technology Standards (Continued…) http://www.iste.org/Content/NavigationMenu/NETS/ForStudents/2007Standards/NETS_for_Students_2007.htm 4) Critical Thinking, Problem Solving, and Decision Making Students use critical thinking skills to plan and conduct research, manage projects, solve problems, and make informed decisions using appropriate digital tools and resources. (Activities 2, 3, 6, 7, 9) c) collect and analyze data to identify solutions and/or make informed decisions. 5) Digital Citizenship Students understand human, cultural, and societal issues related to technology and practice legal and ethical behavior. (Activities 1, 2,3) b) exhibit a positive attitude toward using technology that supports collaboration, learning, and productivity. 6) Technology Operations and Concepts Students demonstrate a sound understanding of technology concepts, systems, and operations. (Activity 3) a.) understand and use technology systems. b.) select and use applications effectively and productively.
Associated Curriculum Standards NYS Math Standards Geometry Strand http://www.emsc.nysed.gov/ciai/mst/mathstandards/revisedg7.html Students will identify and justify geometric relationships, formally and informally. Geometric Relationships 7.G.5 Identify the right angle, hypotenuse, and legs of a right triangle 7.G.6Explore the relationship between the lengths of the three sides of a right triangle to develop the Pythagorean Theorem 7.G.8 Use the Pythagorean Theorem to determine the unknown length of a side of a right triangle 7.G.9 Determine whether a given triangle is a right triangle by applying the Pythagorean Theorem and using a calculator
Reserve The Computer Lab for Five Days
Ensure each computer has internet access for research and Google spreadsheet
Ensure each computer has Geometer’s Sketchpad
Ensure the teacher has access to a projector with computer access to the internet and Geometer’s Sketchpad
Create a Rubric
Student Centered Learning Activities
Students investigate relations between the long side of each triangle and the two shorter sides using construction paper and working in groups of three or four.
Students’ use Geometer’s Sketchpad to extend the investigation and form generalizations by creating a script for constructing right triangles.
Students create scripts for various polygons and students make their own conjectures about these relationships.
Students will construct visual proofs of the Pythagorean Theorem after researching the Web and using any technology tool of their choice.
Using a spreadsheet, investigate integer values for the three sides of a right triangle
~ Blooms Taxonomy ~
Comprehension: Students investigate and discuss relations between the long side of each triangle and the two shorter sides.
Application: Students demonstrate their proof to the rest of the class using their own reasoning from the visual demonstrations.
Analysis: As an experiment , students construct other polygonal figures on the sides of the right triangle using Geometer’s Sketchpad
- Students create a variety of right-angled triangles by cutting corners from rectangular sheets of paper or cardboard using a straight edge.
- Students create a script for constructing right triangles, take measurements using Sketchpad, square these measurements, sum the squared legs, and see how this sum compares to the squared hypotenuse as they dynamically change the side measurements of their right triangles.
- Construct visual proofs of the Pythagorean Theorem using any technology tool of their choice.
Evaluation: Students predict the relationship of the three sides of a right triangle using the spreadsheet.
Higher Order /CriticalThinking
Use of Technology In Lesson
- Using The Geometer’s Sketchpad or any other dynamic geometry software, lies in encouraging students to ask “What if?” questions.
The software enables students to test their theories on their own.
This exploration of conjectures and developing informal logical arguments encourages students to develop habits that are mathematically powerful.
In addition, it is the classroom conversation or discourse about these conjectures and informal proofs that show their understanding and provides learning opportunities for the entire class.
The spreadsheet and the dynamic software makes testing multiple theories easier, thus making mathematics thought more intriguing.
The following assessment points appear periodically within the learning activity. They can be used for formative performance assessment.
- Group discussions and teamwork during investigations and research Rubric
- Demonstration of visual proofs with rational explanations ( Rubric )
- The accuracy of the mathematics used in the presentation ( Rubric )
Three Rubrics will be used to assess each of these points and will be averaged together for a final grade. Rubrics were made using teachology.com