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Hey Pythagoras! Help Me Understand Your Theorem!

Lesson Plan Information | Lesson Plan Activities | Printable version (including handouts) (PDF)

Pre-Teaching Considerations
Standard Use Math to Solve Problems and Communicate

Student goals Students may have heard of Pythagorean Theorem but are uncertain how to solve these kinds of problems or why they would be important in their everyday lives.

Materials
Graph paper (white)
graph paper (colored)
square tiles or 1" paper squares
rulers
colored paper
calculators
Pythagoras Vocabulary Self-Inventory Chart
Parts of a Right Triangle Handout
Pythagorean Triples Handout
Pythagoras Learning Objects

Assessment/Evidence
Pythagorean Triples Handout
Two real-life problems submitted to teacher for quiz development
Completion of student-generated quiz

Learner prior knowledge Previous experience with square numbers (can be represented by dots in a square array) and square roots (a number when multiplied by itself equals a given number) and the representation of each. Students recognize a right triangle (triangle with a 90° angle). The lesson Quilting Geometry provides students with a background of identifying angles.


Activities/Curricular Resources (Real-Life Applications)
Each student is given the Pythagoras Vocabulary Self-Inventory chart with a list of words discussed in this lesson. Students identify "I know the word, heard of it, or have no idea," then discuss words already known and words that need to be defined before the lesson. Be sure to review what squaring a number means (5² = 5 times 5.)

TEACHER NOTE Briefly discuss the Greek mathematician, philosopher and religious leader Pythagoras. Pythagoras, who died about 475 BC, was the first to prove a theorem about right triangles that was known to the Babylonians 1000 years earlier. Check out these web sites for some background information on Pythagoras: Pythagorean Theorem or Biography of Pythagoras

Identify the parts of a right triangle. The handout Parts of a Right Tiangle has an illustration of a right triangle that students can label. Review the mathematical meaning of the Pythagorean Theorem.
In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle.)
Ask the students to translate the math into plain English. It can be summarized as follows:
The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.


"Prove" Pythagorean Theorem by completing the following activity:
  1. With a sheet of colored paper, construct a right triangle with the following measurements. Use a corner of the paper for the right angle. One leg will be 3 inches (use a ruler to measure this length) and the other leg will be 4 inches. Remember the legs are the sides of the triangle that form the sides of the right angle. Join the end points of the two legs to form the hypotenuse. Check to be sure it is 5 inches long.
  2. Carefully cut out the triangle and glue it in the center of a sheet of paper.
  3. Square the length of each side of the triangle (3² = 9, 4² = 16, and 5² = 25). Use 9 square tiles to form a square next to the side which is 3 inches long. Use 16 tiles to construct a square next to the side that is 4 inches long. Finally, use 25 tiles to construct a square next to the side which is 5 inches long.
  4. Using the Pythagorean Theorem, we can see that 16+9 = 25. Using our model we can physically see that, too.


Discuss with the students that a triangle with sides of 3, 4, and 5 is a right triangle and called a Pythagorean triple. This next activity will help the students find more Pythagorean triples.
  1. Working with a partner, have the students construct (with colored graph paper - centimeter grid paper works well for this) as many perfect squares with areas from 1 to 225 as possible. This may seem like a huge task, so remind the students what a perfect square (an integer that is the square of an integer) is. Have each group cut out and label the perfect squares they have constructed. (i.e. 2² = 4, on a square of 4; 3²= 9, on a square of 9, etc.)
  2. Using the perfect squares, the students will construct right triangles. Review the (3, 4, 5) triangle they used in the previous activity. Lay the squares out on grid paper of the same size and a different color. Match up the side of the square to the side of the right triangle.
  3. Experiment with the perfect squares to discover as many Pythagorean triples / right triangles as possible. Tape the perfect squares next to the sides of the right triangle.
  4. Construct a chart to record data about the right triangles using the handout Pythagorean Triples.


Provide practice using the Pythagorean Theorem in real world problems. There are 2 problems included at the end of the lesson that provide good practice. In addition, investigate Pythagorean Theorem problems on the GED test. Have pairs of students solve each problem. Discuss as a class the clues that "told" them the problem required using the Pythagorean Theorem.

Brainstorm with the students situations where right triangles occur:
  • Amount of wire needed to run from top of a pole to a point 6 feet from the base of the pole
  • Straight line distance between locations on roads that are perpendicular to one another
  • Diagonal distance of a rectangular picture frame or TV -- how screens are measured
  • Length of a ramp when you know the height and linear distance it covers
  • Diagonal distance across a park
  • Area of a lot
  • A ladder against the side of a house
  • Length of the ramp for a moving van.

Be sure to discuss clues that might indicate the problem requires using the Pythagorean Theorem [right angle (square corner, perpendicular to the ground), finding a length of a side of a "triangle" (it might be an imaginary triangle), and diagonals].

Students will create two illustrated problems with solutions that require the Pythagorean Theorem. Below each problem, students will detail a solution that shows in words and numbers how the answer was attained. After assessing each student's problem for situational accuracy, use the problems to create a worksheet for the students to complete.

TEACHER NOTE Since the Pythagorean Theorem is one of the most missed questions on the GED test, it is important that students get lots of practice with problems using the theorem. The next activity gives students an opportunity to explore ways the Pythagorean Theorem might be presented on the GED.

Students can choose to practice Pythagorean Theorem problems in a GED textbook or by completing the Pythagorean Theory Webquest. Additional practice can be found at Pythagorean Theorem.


Follow-Up
Reflection/Evaluation
not yet completed

Next Steps
Look at other proofs of the Pythagorean Theorem. Pythagoras Learning Objects will give students additional practice with the theorem, right angles and using a calculator to find solutions.


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