Lesson Plan for the Proof of the Pythagorean Theorem
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Dawn Wardner
Geometry A (Math A)
Salamanca High School
9th/10th Grade Non-Tenured Teacher
Lesson: Proof of the Pythagorean Theorem
November 28, 2007 Lesson Plan for the Proof of the Pythagorean Theorem
I. Instructional Goals:
The goal of this lesson is to verify the proof of the Pythagorean Theorem both visually and algebraically, and to find distance using the Pythagorean Theorem.
II. Performance Objectives:
The students will demonstrate their knowledge of the Pythagorean by completing a hands- on lab that allows them to see the proof of the theorem.
III. Learning Standards: Students will be supporting the following standards.
Geometry
Use visualization, spatial reasoning, and geometric modeling to solve problems
. use geometric models to gain insights into, and answer questions in, other areas of mathematics.
Problem Solving
. build new mathematical knowledge through problem solving.
. solve problems that arise in mathematics and in other contexts.
Reasoning & Proof
. develop and evaluate mathematical arguments and proofs.
. select and use various types of reasoning and methods of proof.
Communication
. use the language of mathematics to express mathematical ideas precisely.
IV. Materials
For this lab the following material are needed: lab, pencil, yellow square, blue square, scissors, glue, ruler, overhead proof of the Pythagorean Theorem, dry erase board and markers. V. Instructional Procedure
A. Initiating Strategy
Collect any homework from the previous day. Begin by asking the students what they remember about the Pythagorean Theorem and if they have ever seen the proof. Then show the picture that they are going to be presented with and explain where the right triangle is and what the squares represent.
B. Teaching Procedure
This lab is designed to enforce student based learning, so the direct instruction is very limited. First, the teacher with complete the initiating strategy and pass out the lab and the required materials. Students will work through the lab in groups of two. When the majority of the class have reached part 3 of the lab the teacher will stop the class and have a student show how the pieces from square A and square B fit into square C. This way the teacher will be sure that everyone is further then step 3. After about 10 minutes the teacher will stop the class again to go over questions number 8 – 10 with the class if needed. The students will be given the rest of the class period to finish the follow-up questions.
C. Closure
The lesson will come to closure by reviewing the importance of the Pythagorean Theorem and it use to finding distance. Any student who has not completed the lab during class will be asked to finish the lab for the following day.
VI. Adaptations
The teacher will be walking around the room to help any student who needs assistance with their assignment. Students will also be able to work with their peers to obtain the needed information.
VII. Assessment of student performance
The students will assessed during class, the teacher will be checking to see what they have done and what they need help on. The students will be farther assessed on this material at a later date by analyzing their exams.
VIII. Teacher self-reflection
If the students need additional help with their lab the lesson may need to be followed by a discussion of the material the following day. A further reflection will be done by analyzing the upcoming exams Name ______Date ______Pythagorean Theorem Lab
Directions: In groups of two please work through the following steps and answer the provided questions along the way.
Objective: During this lab you will prove the Pythagorean Theorem visually and algebraically.
Proving the Theorem Visually
1. Cut out square A (blue) and square B (yellow), then cut along bold lines. (Square A should be in 3 pieces and square B should be in 4 pieces.)
2. Now arrange the squares so they fit into the proper locations, square A and square B.
C
A a c
b
B 3. According to the Pythagorean Theorem the area of square A plus the area of square B should equal to area of square C. Use you cut outs of the areas squares A and B and paste them onto square C to show this is true.
Does the area of square C equal the sum of the areas of square A and B?
______
So, what is the Pythagorean Theorem? (Remember area is squared)
______
Proving the Theorem Algebraically
4. Find the area of square A and the area of square B in square units (square centimeters). You can find the length of each side by counting the square units.
2 2 A(square A) = s A(square B) = s A = A = A = A =
5. Now, that you have found the areas of square A and square B. What is the sum of the these two areas?
______
6. Now, in order to show that the sum of area A and B is equal to the area of square C, we will need to find the area of square C. First, we will need to find a side of square C.
You will find that it is difficult to find the length of a side on square C by counting units? Why is this?
______
Instead you will need to use a ruler and measure, c, the longest side of the triangle on the first page. What is the length a side of square C (the longest side of ABC )?
______
7. We just measure distance to help us prove that the Pythagorean Theorem is true. Now, what is the Area of square C? Please show your work and the area formula.
______8. Now your answer from step 5 and step 7, should be the same. If they are not please go back and fix them now. This proves that the theorem is correct!! Please verify this check by substituting in the values (a, b, and c) into the formula of the Pythagorean Theorem.
9. Please remember that the Pythagorean Theorem only applies to right triangles. The length of side c is called the hypotenuse and a and b are called the legs. The hypotenuse is always opposite the right angle. Please label the legs and the hypotenuse on the triangles below.
10. Also remember that the Pythagorean Theorem can be used to calculate distance of a given line that is not vertical or horizontal. It is used by creating a right triangle from the given line. Please find the distance of the example below in simplest radical form.
? Please complete the following questions. As always, remember to show all work! When necessary, express your answers in simplest radical form
Find the missing lengths of the right triangles below.
11. 12. 15 in 13.
6ft 16cm 8 ft 9in 7 cm
Find the length of the following segments, by creating right triangles and using the Pythagorean Theorem.
14. 15.
16. Find the distance between the points (-4, 1) and (3, 2).
2
-5 5
-2 Name ______Date ______Pythagorean Theorem Lab – Answer Key
Directions: In groups of two please work through the following steps and answer the provided questions along the way.
Objective: During this lab you will prove the Pythagorean Theorem visually and algebraically.
Proving the Theorem Visually
1. Cut out square A (blue) and square B (yellow), then cut along bold lines. (Square A should be in 3 pieces and square B should be in 4 pieces.)
2. Now arrange the squares so they fit into the proper locations, square A and square B.
C
A a c
b
B 3. According to the Pythagorean Theorem the area of square A plus the area of square B should equal to area of square C. Use you cut outs of the areas squares A and B and paste them onto square C to show this is true.
Does the area of square C equal the sum of the areas of square A and B?
______Yes______
So, what is the Pythagorean Theorem? (Remember area is squared)
______a 2 + b 2 = c 2______
Proving the Theorem Algebraically
4. Find the area of square A and the area of square B in square units (square centimeters). You can find the length of each side by counting the square units.
2 2 A(square A) = s A(square B) = s A = 32 A = 42 A = 9 cm2 A =16 cm2
5. Now, that you have found the areas of square A and square B. What is the sum of the these two areas?
9 cm 2 + 16 cm 2 = 25 cm 2______
6. Now, in order to show that the sum of area A and B is equal to the area of square C, we will need to find the area of square C. First, we will need to find a side of square C.
You will find that it is difficult to find the length of a side on square C by counting units? Why is this?
_The length of the side is not vertical or horizontal.______
Instead you will need to use a ruler and measure, c, the longest side of the triangle on the first page. What is the length a side of square C (the longest side of ABC )?
______5 cm 2______
7. We just measure distance to help us prove that the Pythagorean Theorem is true. Now, what is the Area of square C? Please show your work and the area formula.
A = s2 A = 52 A = 25 cm2 _____25 cm 2______8. Now your answer from step 5 and step 7, should be the same. If they are not please go back and fix them now. This proves that the theorem is correct!! Please verify this check by substituting in the values (a, b, and c) into the formula of the Pythagorean Theorem. a2 + b2 = c2 32 + 42 = 52 9 + 16 = 25 25 = 25
9. Please remember that the Pythagorean Theorem only applies to right triangles. The length of side c is called the hypotenuse and a and b are called the legs. The hypotenuse is always opposite the right angle. Please label the legs and the hypotenuse on the triangles below.
Leg Hypotenuse
Leg Leg Hypotenuse
Leg
10. Also remember that the Pythagorean Theorem can be used to calculate distance of a given line that is not vertical or horizontal. It is used by creating a right triangle from the given line. Please find the distance of the example below in simplest radical form.
?
a2 + b2 = c2 45 32 + 62 = c2 9 5 9 + 36 = c2 45 = c2 3 5 cm 45 = c Please complete the following questions. As always, remember to show all work! When necessary, express your answers in simplest radical form
Find the missing lengths of the right triangles below.
11. 12. 15 in 13.
6ft 16cm 8 ft 9in 7 cm
a2 + b2 = c2 a2 + b2 = c2 a2 + b2 = c2 62 + 82 = c2 a2 + 92 = 152 a2 + 72 = 162 36 + 64 = c2 a2 + 81 = 225 a2 + 49 = 256 100 = c2 144 = a2 207 = a2 100 = c 144 = a 207 = a 10 ft 12 in 3 23 cm
Find the length of the following segments, by creating right triangles and using the Pythagorean Theorem.
14. 15.
a2 + b2 = c2 a2 + b2 = c2 52 + 72 = c2 22 + 92 = c2 25 + 49 = c2 4 + 81 = c2 84 = c2 84 = c2 84 = c 85 = c 2 21 units 85 units
16. Find the distance between the points (-4, 1) and (3, 2).
a2 + b2 = c2 2 72 + 12 = c2 49 + 1 = c2 50 = c2 -5 5 50 = c 5 2 units
-2 Geometer’s Sketchpad squares (B – yellow) Geometer’s Sketchpad squares (A – blue)