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PYTHAGOREAN THEOREM - ILLUSTRATIONS & APPLICATIONS

Subject: ALGEBRA AND GEOMETRY

Grade Level: 9 & 10

Materials: See below.

About: The students illustrate the Pythagorean theorem by verifying the area of the square on hypotenuse is equal to the sum of the areas of the squares on the other two sides. They find the unknown side of the right triangle when two sides are given using pythagorean theorem. The students solve the pythagorean puzzle and apply the theorem in the math world as well as in the real world to solve problems. Finally, they recognizes the importance of the pythagorean theorem and its applications in day-to-day life situations.

Students verify the theorem by counting the areas of the three squares and equating the larger one with the sum of the other two smaller ones. They show the concept by drawing the three squares with different colors and also by pasting different-colored chips. They solve the pythagorean puzzle and apply the pythagorean theorem effectively in the daily life situations. They find the length of the diagonal of a square or rectangle and the distance between two points in a coordinate plane, locate the irrational numbers on the number line, find the height of the building, etc.

The illustration process gives the student a clear idea about the theorem. It is a good hands-on experience. Students really enjoy the animated version of the illustrations of the theorem from various websites. Solving the pythagorean puzzle is an enjoyable and exciting adventure/game. The life examples as well as applications in the math world presented in the websites introduce the importance of the theorem and develop a desire to learn more about it.

Especially for the 9th and 10th grade, this unit plan is very helpful in teaching the concept clearly. The illustration process gives a good hands-on experience to the students. The teacher can use the Internet for demonstrating the concept by animation in a powerful and effective manner, and can show real-world examples. The worksheets help the teacher to plan and engage the students effectively. The teacher feels satisfaction after seeing the students' achievement, and the students feel the same!

 Objectives
Understanding the meaning of a^2 + b^2 = c^2 of the Pythagorean theorem.
Experiencing the illustration process of the Pythagorean theorem.
Understanding the concept of the theorem through the animated version of the illustration from the websites.
Developing the skill of solving the Pythagorean Puzzle.
Identifying the different proofs for the same single theorem.
Applying the Pythagorean theorem in the math world to solve problems.
Applying the Pythagorean theorem in the real world to solve problems.
Noticing the different applications of the Pythagorean theorem by visiting several websites.

Websites
This link gives the animated illustration of the Pythagoren theorem. We can see three different types of animated illustrations.
http://mathsisfun.com/pythagoras.html
This link is a PowerPoint presentation that demonstrates the ladder problem and baseball diamond problem.
http://u46.k12.il.us/shs/anthonygregory/userfiles/47242_pythagoreantheorem.ppt#265,14,Sources
This link gives the animated illustration of the Pythagoren theorem.
http://usna.edu/MathDept/mdm/pyth.html
A collection of 76 approaches to proving the theorem. It gives the different ways of thinking to prove the same concept.
http://cut-the-knot.org/pythagoras/index.shtml
After clicking on this link, if we click on the weblink http://regentsprep.org/Regents/math/fpyth/PracPyth.htm we can see the practice page on the Pythagorean theorem.
http://teachersnetwork.org/readysettech/gabrielli/gabrielli.htm
This link gives the brief history of the Pythagorean theorem.
http://geom.uiuc.edu/~demo5337/Group3/hist.html
This link gives the application of the Pythagorean theorem in the Math world, especially in 3-dimensional geometry.
http://demonstrations.wolfram.com/PythagoreanTheorem3D/
This link gives some of the real-life examples of using the Pythagorean theorem.
http://middle-school-curriculum.suite101.com/article.cfm/the_pythagorean_theorem

Standards
Develop, verify, and explain an argument, using appropriate mathematical ideas and language.
9
Integrated Algebra
Present correct mathematical arguments in a variety of forms.
9
Integrated Algebra
Write a verbal expression that matches a given mathematical expression.
9
Integrated Algebra
Determine the measure of a third side of a right triangle using the Pythagorean theorem, given the lengths of any two sides.
9
Integrated Algebra
Understand and use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and geometric diagrams.
10
Geometry
GUnderstand and make connections among multiple representations of the same mathematical idea.
10
Geometry
Recognize and apply mathematics to situations in the outside world.
10
Geometry
Investigate, justify, and apply the Pythagorean theorem and its converse.
10
Geometry
Find the length of a line segment, given its endpoints.
10
Geometry

Day 1: ILLUSTRATION OF PYTHAGOREAN THEOREM
Objectives
Understanding the meaning of a^2 + b^2 =c^2 of the Pythagorean theorem.
Experiencing the illustration as an hands-on activity.
Identifying the Pythagorean triples.
Materials
rulers, protractors, colored sketches
colored markers
colored chips
glue
Procedures
The students draw the right triangle and constructs the squares on the three sides. By using the ruler, they divide each square into unit squares and count the area of each square by writing the numbers on each unit square. Finally, they verify the area of the square on the hypotenuse is equal to the sum of the areas of the two squares on the other two sides. Refer to Illustration 1.
The students draw the right triangle and constructs the squares on the three sides. They color each square with a different color. By using a ruler, they divide each square into unit squares and count the area of each colored square. Finally they verify the area of the square on the hypotenuse is equal to the sum of the areas of the two squares on the other two sides. Refer to Illustration 2.
The students draw the right triangle and constructs the squares on the three sides. By using theruler, they divides each square into unit squares. Then they paste colored chips on each unit square. Three different colored chips have to be pasted on each square. Finally they verify the area of the square on the hypotenuse is equal to the sum of the areas of the two squares on the other two sides. Refer to Illustration 3.
These three activities can be done either by each student or by dividing the class into three groups with each group doing one activity.
The teacher can explain the meaning of the Pythagorean triples with examples and ask the students to write down three more triples as an activity.
Homework
Ask the students to do one more new way of illustrating the Pythagorean theorem at home.
Assessment
The student understands the real meaning of a^2 + b^2 =c^2 of the Pythagorean theorem. The student experieces the illustration process through the hands-on activity. He/she recognizes the Pythagorean triples.

Day 2: MORE ON PYTHAGOREAN THEOREM
Objectives
Understanding the meaning of a^2 + b^2 =c^2 of the Pythagorean theorem visually.
Exploring the Internet for the new idea of illustrating the theorem.
Exploring the Internet for the new idea of proving the theorem.
Identifying the Pythagorean triples.
Developing the skill of solving the problems on the Pythagorean theorem (worksheet).
Materials
Computers with Internet capability/Computer Lab
LCD Projector/Computer Lab with VISION PROGRAM for demonstration
WORKSHEET(given in the documents section of this unit plan)
Procedures
The teacher provides the weblinks on the board and the students view the different animated illustrations of the Pythagoren theorem: http://mathsisfun.com/pythagoras.html; http://usna.edu/MathDept/mdm/pyth.html; http://davis-inc.com/pythagor/proof2.html Otherwise the teacher can show the animated illustrations using the LCD projector and the computer in the classroom.
Teacher explains the process of finding the third unknown side of the right triangle.
The teacher explains the Pythagorean triple.
The student try to solve the problems from the work sheet under the supervision of the teacher.
The teacher introduces the website http://cut-the-knot.org/pythagoras/index.shtml. and explains to view the site for 75 different proofs for the same Pythagorean theorem.
Homework
1.Completion of the remaining part of the work sheet. 2.The student has to browse the website http://cut-the-knot.org/pythagoras/index.shtml and collect one of the interesting proofs and has to copy it to submit the next day.
Assessment
The student understands the real meaning of a^2 + b^2 =c^2 of the Pythagorean theorem through the animated version. He/she finds the third unknown side of the right triangle and identifies the Pythagorean triples.

Day 3: THE PYTHAGOREAN PUZZLE
Objectives
Developing the skill of finding the the third unknown side of the right triangle.
Finding the diagonal of a rectangle/square.
Understanding the fact that the sum of any two sides of a triangle is always greater than the third side.
Materials
THE PYTHAGOREAN PUZZLE (given in the documents section of this unit plan)
Calculators if needed
Chart showing a simple problem with 2 or 3 steps like the one given in the actual puzzle
Procedures
The teacher explains on a chart the procedure of solving the puzzle by taking a similar model with just 2 or 3 steps.
The student then tries to solve the puzzle applying the Pythagorean theorem.
The student verifies the answer by discussing it with the teacher.
The teacher should not give the value of the correct answer till the end of the class.
For the students who complete correctly and quickly, the teacher asks them to create another new puzzle of their choice in the same way.
Homework
Ask the students to create a similar type of puzzle on their own way. Ask them to explore any Pythagorean puzzle that exists and present to the class.
Assessment
Solves the exercise on the Pythagorean theorem. Finds the third unkown side of the right triangle. Develops the skill of creating the Pythagorean puzzles.

Day 4: PYTHAGOREAN THEOREM - APPLICATIONS IN THE MATH WORLD
Objectives
Discovering the irrational numbers on the real number line.
Finding the distance formula in the coordinate plane.
Finding the ratios of the sides of the special right triangles like 30-60-90 and 45-45-90.
Finding the length of the diagonal of a rectangle or a square.
Finding the Pythagorean Identities in the trigonometry that is the brach of mathematics which is developed on a right triangle.
Materials
rulers, compasses, white paper
graph paper and graph charts
geometric instrument boxes for set squares or protractors for drawing the special right triangles like 30-60-90 and 45-45-90; teacher's geometric nstrument box for demonstration on the board/chart.
chart displaying the definitions of the trigonometric ratios
Procedures
This lesson can be divided into 5 sub lessons if we want to extend it. Otherwise: The teacher demonstrates how to locate the square root of 2, 5, and 13 on the number line. The student locates them in the same way using ruler and compass. See the docoment section of this lesson plan for the applications in the math world.
The teacher demonstrates on the graph chart how to find the distance between two points and derives a formula. By using this formula he/she finds the distance between any two testing points and asks the student to measure the distance using the ruler. A small exercise on finding the distances with the given coordinates of two points can be given for the purpose of practice. See the docoment section of this lesson plan for the applications in the math world.
The teacher finds the ratios of the sides of the special right triangles like 30-60-90 and 45-45-90 and asks the students to use these ratios to do the mini exercises. See the document section of this lesson plan for the applications in the math world.
The teacher shows the method of finding the diagonal of a square/rectangle and gives mini exercise for the students' practice. See the docoment section of this lesson plan for the applications in the math world.
The teacher derives the Pythagorean identities on the board/chart and displays the definitions of the trigonometric ratios. The student applies these identities to solve the problems in the mini- exercises provided by the teacher. (See the docoment section of this lesson plan for the applications in the math world.
Homework
The teacher can create a exercise comprising of all the above five applications. Ask the student to collect at least one application where we use the Pythagorean theorem in the math world.
Assessment
Discovers the irrational numbers on the real number line. Finds the distance formula in the coordinate plane. Finds the ratios of the sides of the special right triangles like 30-60-90 and 45-45-90. Finds the length of the diagonal of a rectangle or a square. Finds the Pythagorean Identities in the trigonometry.

Day 5: PYTHAGOREAN THEOREM - APPLICATIONS IN THE REAL WORLD
Objectives
Finding the length of the ladder for a given height and from a given point on the ground.
Finding the distance of a person from the start point after travelling certain distance in one direction and travelling another distance perpendicular to it.
Finding the distance between first base and third base on a baseball diamond.
Finding the vertical height of the ramp when the distance from the bottom of the ramp and to the truck and length of the ramp are given.
Deciding whether the given angle is a right angle or not (converse of the Pythagorean theorem).
Materials
charts
Computer with Internet access
LCD Projector
rulers and calculators
Procedures
By drawing the picture of the ladder problem, it can be explained how to find the length of the ladder.
http://u46.k12.il.us/shs/anthonygregory/userfiles/47242_pythagoreantheorem.ppt#265,14,Sources We can show this PowerPoint presentation to the students to give clear demonstration of the ladder problem and baseball diamond problem.
Finding the lengths of three sides of the triangle using ruler and verifying the square of the longest side is equal to the sum of the squares of the other two sides. If equal, it is a right triangle.
Homework
Collect the problems from the website and solve them: http://middle-school-curriculum.suite101.com/article.cfm/the_pythagorean_theorem
Assessment
Finds the length of the ladder, distance between first and third bases of the baseball diamond, verifies the given triangle is a right triangle or not. Practices exploring the Internet.

chandrasekhar anumapuram

canumapuram@schools.nyc.gov

Academy of Environmental Science
410East 100th Street
New York, NY 10029

Chandrasekhar Anumapuram has been teaching for 20 years. He currently teaches Math B and Math Projects at Academy of Environmental Science Secondary High School in New York City. His top priority is effectively using the computer and technology in teaching mathematics. His goal is to show students that mathematics is everywhere in daily life and that everyday problems can be solved with simple math concepts and formulas.


Important documents for this lesson plan.

PT-ILLUSTRATIONS-1,2,3.doc
PYTHAGOREAN THEOREM-WORK SHEET.doc
PYTHAGOREAN PUZZLE-SEKHAR'S CREATION.doc
PT-IRRATIONAL NUMBERS.doc
PT APPLICATIONS MATH WORLD.doc

 

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