1. Topic-
Pythagorean's Theorem
 
2. Content-
theorem has numerous proofs, possibly the most of any mathematical theorem. These are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The theorem can be generalized in various ways, including higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and indeed, to objects that are not triangles at all, but n-dimensional solids. The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power.
The longest side of the triangle is called the "hypotenuse", so the formal definition is:

Definition of Pythagorean's Theorem: In a right angled triangle:
the square of the hypotenuse is equal to
the sum of the squares of the other two sides.


Key Vocabulary:

Right Angle
Right Angled Triangle (Right Triangle)
Hypotenuse
Legs of a Triangle
Square
Exponent
Square Root
Perfect Square
Sum
 
3. Goals: Aims/Outcomes-
1. Students will gain knowledge about triangles and understand the relationship between the 3 sides of a right triangle by applying Pythagorean's Theorem.
 
4. Objectives-
1.Students will identify a Right Triangle
2.Students will locate the two Legs of a Right Triangle and located the Hypotenuse.
3. Students will calculate the length of either leg or the hypotenuse when given the length of two sides.
4. Students will apply new knowledge to other applications.
5. Students will create their own example of a use (example) of Pythagorean's Theorem.
 
5. Materials and Aids-
Pencil
Paper
Worksheets
Calculator
 
6. Procedures/Methods-

A. Introduction-

In this lesson, we examine a very powerful relationship between the three lengths of a right triangle. With this relationship, we will find the exact length of any side of a right triangle, provided we know the lengths of the other two sides. We also study right triangles where all three sides have whole number lengths.

Proving the Pythagorean Theorem
The Pythagorean theorem is this relationship between the three sides of a right triangle. Actually, it relates the squares of the lengths of the sides. The square of any number (the number times itself) is also the area of the square with that length as its side. The longest side of a right triangle, the side opposite the right angle, is called the hypotenuse, and the other two sides are called legs. Suppose a right triangle has legs of length a and b, and a hypotenuse of length c. The Pythagorean theorem states that a2 + b2 = c2, This means that the area of the squares on the two smaller sides add up to the area of the biggest square.

This proof of the Pythagorean theorem has been adapted from a proof developed by the Chinese about 3,000 years ago. With the Pythagorean theorem, we can use any two sides of a right triangle to find the length of the third side.
 

B. Development-

Pythagorean Word Problems:
Many word problems involve finding a length of a right triangle. Identify whether each given length is a leg of the triangle or the hypotenuse. Then solve for the third length with the Pythagorean theorem.

Example 1. A diagonal board is needed to brace a rectangular wall. The wall is 8 feet tall and 10 feet wide. How long is the diagonal?

Having a rectangle means that we have a right triangle, and that the Pythagorean theorem can be applied. Because we are looking for the diagonal, the 10-foot and 8-foot lengths must be the legs.

Example 2
A 100-foot rope is attached to the top of a 60-foot tall pole. How far from the base of the pole will the rope reach?

We assume that the pole makes a right angle with the ground, and thus, we have a right triangle. The hypotenuse is 100 feet.

Example 3: Check whether the following lengths of the triangles are using the Pythagorean rule:
8m, 17m 15m
4m, 12m, 13m
20in., 21in., 27in.
45ft., 53ft., 28ft.

Example 4: Pythagorean Triples

Since the Pythagorean theorem was discovered, people have been especially fascinated by right triangles with whole number sides. The most famous one is the 3-4-5 right triangle, but there are many others, such as 5-12-13 and 6-8-10. A Pythagorean triple is a set of three whole numbers a-b-c with a2 + b2 = c2. Usually, the numbers are put in increasing order.

Experiment with the numbers given until you find all Pythagorean Triples listed.


( 8, 15, 17) ( 7, 24, 25)
(20, 21, 29) (12, 35, 37)
( 9, 40, 41) (28, 45, 53)
 

C. Practice-

1.
2.A Fun Way to Calculate a Square Root
There is a fun method for calculating a square root that gets more and more accurate each time around:

a) start with a guess (let's guess 4 is the square root of 10)
b) divide by the guess (10/4 = 2.5)
c) add that to the guess (4 + 2.5 = 6.5)
d) then divide that result by 2, in other words halve it. (6.5/2 = 3.25)
e) now, set that as the new guess, and start at b) again

3. 19. How long is the diagonal of a standard piece of paper that measures by 11 inches?
20. A 20-foot ladder must always be placed 5 feet from a wall it is leaned against. What is the highest spot on a wall that the ladder can reach?
21. Televisions and computer screens are always measured on the diagonal. If a 14-inch computer screen is 12 inches wide, how tall is the screen?
22. The bottom of a cardboard box is a 20-inch square. What is the widest object that could be wedged in diagonally?
 

D. Independent Practice-

1.Write two word problems that will use Pythagorean's Theorem. Include the answers to both problems. You will hand this in to the teacher, who will in turn give to a fellow student to calculate.

Draw an illustration to represent the following:

Years ago, a man named Pythagoras found an amazing fact about triangles:

If the triangle had a right angle (90°) ...

... and you made a square on each of the three sides, then ...

... the biggest square had the exact same area as the other two squares put together!

Example: Does the triangle with lengths (10,24,26) have a Right Angle?
 

E. Accommodations (Differentiated Instruction)-

Accommodations vary depending upon the student and the specific Learning Disability. Some accommodations include: additional time to complete activities, calculator for students with dysgraphia, and handwriting assistance for those with deficits in fine motor skills.
 

F. Checking for understanding-

Assessments will be given at the end of the lesson/unit to ensure understanding of the lesson.
 

G. Closure-

Most students ask the same question all the time. "When will I ever use it or need this information?"

You have just learned of several well-known examples using Pythagorean's Theorem. If you begin to look at your surroundings, you will probably find a few real-life examples!
 
7. Evaluation-
Student overall achieved a 6% increase in Test Scores as indicated by the Post Test given at the end of the instructional unit.
 
8. Teacher Reflection-
More practice should have been given with exponents, squaring a number, and finding the length of the hypotenuse or 3rd side.
 

This Lesson Plan is available at (www.teacherjet.com)