Title: Circle of Knowledge- Pythagorean Theorem
Type: Lesson Plan
Subject: Geometry
Grade Range: 8th Grade
Duration: 1 Class Period- 50 minutes
Author: Carter Hooks
Instructional Unit Content
Content Area Standard:
TAG Standard
Summary/Over View
Enduring Understanding:
Essential Question(s):
Concepts to Maintain:
Understand and apply the Pythagorean Theorem
MCC.8.G.6 Explain a proof of the Pythagorean Theorem and its converse.
MCC.8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in
right triangles in real-world and mathematical problems in two and three
dimensions.
MCC.8.G.8 Apply the Pythagorean Theorem to find the distance between two
points in a coordinate system.
Solve real-world and mathematical problems involving volume of cylinders,
cones, and spheres.
MCC.8.G.9 Know the formulas for the volume of cones, cylinders, and spheres and
use them to solve real-world and mathematical problems.
Work with radicals and integer exponents.
MCC.8.EE.2 Use square root and cube root symbols to represent solutions to
equations of the form x2 = p and x3 = p, where p is a positive rational number.
Evaluate square roots of small perfect squares and cube roots of small perfect
cubes. Know that √2 is irrational.
Higher Order Critical Thinking Skills
1. The student asks probing, insightful, and relevant questions.
2. The student responds to questions with supporting information that
reflects in-depth knowledge of a topic.
7. The student examines an issue from more than one point of view.
15. The student recognizes that the responsibility to examine and challenge
existing ideas and theories is an ongoing process.
Students explore the side lengths of right triangles by drawing squares
using the sides of the triangle, dividing the two smaller squares into
triangles, cutting out the new triangles and placing them on top of the
larger square. This proves the area of the larger square is equal to the sum
of the areas of the two smaller squares or the square of the hypotenuse is
equal to the sum of the squares of the two legs in a right triangle.
At the end of this lesson the student will understand that
 The Pythagorean theorem can be used to find missing sides lengths
of a right triangle.
 The Pythagorean triangle can be used to prove or disprove that a
triangle is a right triangle.
 The area of the squares made by squaring the side lengths of a
right triangle have a special relationship illustrated by the
Pythagorean theorem (extension – this relationship holds true for
the area of all regular polygons)
How has our understanding about the Pythagorean theorem and its
converse helped people with their working and everyday lives throughout
history?
 That the area of a square is side2
 That the inverse of x2 is x
 That a right triangle has one angle measuring 90 degrees and the side
opposite that angle is called the hypotenuse and the other two sides
are called legs.
Evidence of Learning:
What students should know:


How to apply the Pythagorean theorem to different situations involving
right triangles
How to prove that the Pythagorean theorem is correct by using a model
like blocks, cut out shapes or a geoboard
What students should be able to do:



Materials
Procedure:
Find the length of a side of a right triangle given the length of the other two
sides
Prove whether a triangle is a right triangle using the Pythagorean theory
Use a model to prove that the area of the square created by squaring the
length of the hypotenuse is equal to the sum of the areas of the two squares
created by squaring the legs of a right triangle. Thus proving that a 2+ b2= c2
Handout 1: Practical uses of the Pythagorean theorem
Handout 2: Graph with right triangle and squares drawn from its sides
Handout 3: “Circle of Knowledge” question cubes
Manipulatives like blocks and geoboards, graph paper, scissors, glue sticks
or tape
Phase 1: Hook
1. Have the student answer the following question in their math notebook:
“If we can find unknown lengths of sides of right triangles by using the
Pythagorean Theorem can we use the Pythagorean Theorem to determine
if a triangle is a right triangle?”
2. Give students 1-2 minutes to write it down their reasoning in their math
journal/notebook, and take a few volunteers to share with a class.
Phase 2: Acquiring Content Needed for Discussion
1. Pose the essential question, “How has our understanding about the
Pythagorean theorem and its converse helped people with their working
and everyday lives throughout history?”
2. Distribute handout, “Practical Uses of the Pythagorean theorem”. Each
student from a group of 4 will read about the uses of the theorem. Have
the students highlight and underline important parts from the reading.
Have the students take notes on the different uses the Pythagorean
theorem has.
Phase 3: Kindling the Discussion
1. Have students discuss the reading and go over the examples in the text.
Next have each student complete one problem from page 4 of the
handout.
2. Once the problems are complete, have the students roll question cube #
1 and participate in the discussion to questions such as:
 How does the length of the hypotenuse compare to the length of
the legs?
 How does the measure of an angle in a right triangle relate to the
length of the opposite side?
 How does the length of a side in a right triangle relate to the
measure of the opposite angle?
 How does the sum of the lengths of any two sides of a right triangle
relate to the length of the third side?
Modification: For students that may struggle visualizing the shapes and
ideas that are being discussed, make available manipulatives to help
reinforce the ideas visually and kinesthetically.
Phase 4: Synthesis Activity
In this problem, you will explore an isosceles right triangle. You answer
questions about the completed figure on the following page. A diagonal
of a square is a line segment connecting opposite vertices of the square.
Let’s explore the side lengths of more this triangle.
1. An isosceles right triangle is drawn on the grid shown below.
a. A square on the hypotenuse and the legs have been drawn for you. Use
different colored pencils to shade each small square.
b. Two diagonals in each of the two smaller squares have also been drawn
for you.
c. Cut out the two smaller squares along the legs. Then, cut those squares
into fourths along the diagonals that have been drawn.
d. Arrange the pieces you cut out to fit inside the larger square on the
graphic organizer. Then, tape the triangles on top of the larger square.
Answer these questions in your math notebook/journal.
e. What do you notice?
f. Write a sentence that describes the relationship among the areas of
the squares.
g. Determine the length of the hypotenuse of the right triangle.
Justify your solution.
Once the questions have been completed, have the students roll question
cube # 2 and participate in a discussion to from the second cube.
Closing: Group Reflection (1 per group to be turned in as exit slip) Since this
special relationship exists between the squares of the side lengths of right
triangles, is it possible that other shapes have relationships with the sides
of right triangles? How could you find out if they exist? Explain your
reasoning and develop some ideas to try.
Practical Uses of the Pythagorean Theorem
The Pythagorean Theorem is primarily used in architecture, construction, surveying, and engineering. “Modern
carpentry work is so much easier when the Pythagorean Theorem is applied to the task at hand. Roof framing,
squaring walls, and foundations rely on this basic principle of mathematics.”
Carpentry-Pro-Framer http://www.carpentry-pro-framer.com/pythagorean-theorem.html
Builders often need to construct a square corner. The Pythagorean Theorem tells us that if a corner is square,
then the sides of a triangle built on that corner will satisfy the formula a2 + b2 = c2. But we can also prove that
the converse of the theorem is true. If the sides of a triangle satisfy the formula a2 + b2 = c2, then the triangle
is a right triangle, with a square corner. (The converse switches the “if” “then” clauses of the sentence.) This
fact has been used since ancient times to construct square corners.
To make a square corner, the ancient Egyptians used a rope marked with twelve even segments. That’s
because twelve segments can make a triangle with sides 3, 4, and 5. That triangle satisfies the formula a2 + b2
= c2 (32 + 42 = 52), and so a 3-4-5 triangle has a right angle. The rope-stretchers would stake out the triangle
with the rope and so mark a right angle.
Example: If the framing shown is constructed correctly with
a right angle in the corner, what will the carpenter’s tape measure read on the diagonal?
The diagonal is the hypotenuse of a right triangle with legs 15 inches and 36 inches. According
to the Pythagorean Theorem,
a2 b2 c2
152 362 225 1296 1521
c 1521 39
The tape measure should read 39 inches.
What length should the wires be to hold the pole at a right angle to the roof?
There are two wires, each the hypotenuse of a separate right triangle.
a2 b2 c2
402 902 9700
c 9700 98.489
a2 b2 c2
202 902 8500
c 8500 92.195
One wire should be about 98.5 inches and the other about 92.2 inches.
A carpenter is using a 20 foot ladder against a straight wall.
The top of the ladder rests against the wall.
The ANSI/OHSA safety regulations for ladders instruct:
“Make sure the ladder is about 1 foot away from the vertical support
for every 4 feet of ladder height between the foot and the top support.”
a) How far away from the wall should she place the foot of the ladder?
If the ladder is 1 foot away for every 4 feet of ladder length,
the ratio of distance from wall to ladder length is 1:4.
1 ft
4 ft

=
x ft
20 ft
and cross-multiplying, 4x = 20, so x = 5 ft.
b) How far up the wall will the ladder reach?
On the figure shown, we know that A (ladder length) = 20 feet. The ladder length is the
hypotenuse of the right triangle the ladder makes with the wall, so side c = 20. One leg
of the triangle is 5 feet, so we’ll call that leg b and find the length of the other leg,
labeled h in the diagram.
a2 b2 c2
h2 52 202
h2 25 400
h2 400 -25 375
h 375 18.028 The ladder will reach about 18 feet up the wall.
Example: Find length of pipe needed for the sprinkler system shown.
There are two right triangles shown:
One has legs 8 feet and 12 feet.
a2 b2 c2
82 122 208
c 208 14.422
One has legs 6 feet and 8 feet.
a2 b2 c2
62 82 100
c 100 10
Adding all the lengths of pipe, we get
6 ft + 14.4 ft + 12 ft + 10 ft + 18 ft = 60.4 ft
Example: The pilot knows she has descended 1000 feet and that she has traveled 18,000 feet through
the air. How far has she traveled in horizontal distance along the ground?
a2 b2 c2
10002 b2 180002
1,000,000 b2 324,000,000
b2 323,000,000
b 323,000,000 17,972.201
The plane has traveled about 17,972 feet
along the ground.
Worksheet Name _______________________________________
1. A carpenter measured the diagonal of a 48 inch by 60 inch gate to the nearest sixteenth of an inch and
found it to be 771/16 inches. Is the gate out of square?
2. Gutters are to be installed along the roofline and extend another 6 inches past the end of the roof. How
many feet of gutter are needed?
3. Bailey planned this garden using an online tool. The length along the bottom fence is 22 feet and the length
along the right hand fence to the end of the path at the gate is 12 feet. How long is the diagonal path?
4. A painter is using a 30 foot extension ladder. He follows the safety regulations, making sure the base of the
ladder is 1 foot from the wall for every 4 feet of ladder height.
How high up the wall will the ladder reach?
Question Cube # 1
How does the length
of a side in a right
triangle relate to the
measure of the
opposite angle?
How does the sum
of the lengths of
any two sides of a
right triangle relate
to the length of the
third side?
How does the
measure of an angle
in a right triangle
relate to the length
of the opposite side?
In an isosceles right
triangle how are the
legs of the triangle
related? And how
does this change the
Pythagorean
Theorem?
How can you use the
Pythagorean
Theorem to prove
whether or not a
triangle is a right
triangle?
How does the
length of the
hypotenuse
compare to the
length of the legs?
Question Cube # 2
How do the areas of
the two
smaller squares
compare to
each other?
How do the areas of
the eight
small triangles
compare to
each other?
If the sum of the
areas of both small
squares is 50 square
units, what is the
area of the large
square?
If all eight triangles fit
exactly inside the largest
square, what does this
imply about
the sum of the areas of the
eight small triangles and
the area of the largest
square?
If all the eight
triangles have
equal areas, what
is the area of one
small triangles?
If the areas of the two
smaller squares are the
same, is the sum of the
areas of the 4 triangles
in one small square
equal to the sum of the
areas of the 4 triangles
in the other small
square? Explain.