Download Bloom`s Taxonomy applied to understanding the Pythagorean

Bloom’s Taxonomy applied to understanding the Pythagorean Theorem
Knowledge
1. State the Pythagorean Theorem
2. True/False: The Pythagorean Theorem only works for right triangles.
3. Using the Pythagorean Theorem, find the length of the hypotenuse of a right triangle whose legs
are of length 6 cm and 8 cm.
4. Using a,b,c to represent the lengths of the sides of a right triangle (with c representing the length
of the hypotenuse), define the Pythagorean Theorem.
5. True/False: The Pythagorean Theorem states that the square of the length of the hypotenuse of a
right triangle is equal to the sum of the lengths of the two sides.
Comprehension
1.
2.
3.
4.
Give an example of a Pythagorean Triple and show that it is indeed an example.
Explain how you would use the Pythagorean Theorem to find the height of an equilateral triangle.
Given a right triangle whose legs are of length 3 and 4, what is the length of the hypotenuse?
Explain, in detail, the process you would use to find the hypotenuse of a right triangle, given the
legs of the triangle.
5. Use the Pythagorean Theorem in order to determine whether a triangle with side lengths 9, 12 and
15 is a right triangle.
6. Give an example of a triangle where the lengths of two sides are known and the third side can be
found by using the Pythagorean Theorem.
7. Label the sides of this triangle, using “a”, “b” and “c”, according to this formula: b 2  c 2  a 2 .
(insert a generic right triangle)
Application
1. Find the height of an equilateral triangle with side length 9.
2. If I need to reach a window 12 ft. off the ground and I only have 5 ft. of room fro the base of the
wall, at most how long does my ladder need to be?
3. Find the length of the diagonal of a rectangle with side lengths 4 and 5.
4. Jim drove 400 miles due east and then 30 miles due south to get from his house to his
grandmother’s house. His brother Joe flew straight to their grandmother’s house from their house.
Who traveled more miles and how many more miles did they travel?
5. On a hot summer day, 4 foot tall Sally walks outside and looks at her shadow. She notices that her
shadow stretches across 4 squares in the sidewalk. Given that a sidewalk square is 3 feet long,
what is the distance from the top of Sally’s head to the tip of her shadow? Explain how you arrive
at your answer.
Analysis
1. Use the Pythagorean Theorem to prove the existence of irrational numbers. Provide an example
with a diagram.
2. If the hypotenuse of a right triangle measures 169 cm, what are the possible whole number lengths
of the other two legs?
3. Prove the Pythagorean Theorem.
4. The law of cosines states that, for any triangle with sides lengths a, b, and c,
a 2  b2  c 2  2ab cos A (were A is the measure of the angle opposite side a). Use this law to
derive the Pythagorean Theorem.
5. An equilateral triangle has vertices at (0,0) and (6,0) in a coordinate plane. What are the
coordinates of the third vertex?
6. We have a wooden box that measures 4 ft by 3 ft by 2 ft. What is the longest straight pole you can
fit inside the box?
7. Explain how the Pythagorean Theorem and the distance formula (for a Cartesian coordinate
system) are related.
8. Does the Pythagorean Theorem apply to isosceles triangles?
9. Draw a picture that illustrates the validity of the Pythagorean Theorem.
Synthesis
1. Prove sin 2 x  cos2 x  1
2. Would the Pythagorean Theorem hold true in three dimensions? If so, what would it say and how
would it work? If not, why?
3. Use the Pythagorean Theorem to explain why the graph of x 2  y 2  r 2 is a circle.
4. Create a story problem the solution of which involves use of the Pythagorean Theorem.
5. Describe how you would teach a younger sibling the concept and use of the Pythagorean Theorem.
Communicate your understanding through exploratory examples and sample problems.
6. Create a pictorial model that illustrates the Pythagorean Theorem in terms of “squares”.
7. Construct a situation in which the Pythagorean theorem would be used to solve a mathematical
problem that musicians may have.
Evaluation
1. The Pythagorean Theorem has been proved in over 100 different ways. What is the value in
examining more than one proof of a Theorem?
2. How could you effectively assess someone’s understanding of the Pythagorean Theorem? What
might be a “not so effective” way of assessing their understanding?
3. Persuade me of the usefulness of the Pythagorean Theorem in a career area of interest to you.
4. To “solve a triangle” means to determine the lengths of each side and the measure of each angle.
Given a right triangle, what is the minimum amount of information needed in order to solve it?
Justify your conclusions.
5. What is your favorite proof of the Pythagorean Theorem? Why do you prefer it over the others
with which you are familiar?
6. What is the purpose of including the Pythagorean Theorem in the mathematics curriculum?
7. Why is the Pythagorean Theorem one of the most fundamental theorems in mathematics?
8. You are asked to teach the Pythagorean Theorem to a group of 8th graders.