Pythagorean Theorem Differentiated Instruction for Use in an ...

Pythagorean Theorem
Differentiated Instruction for Use in an
Inclusion Classroom
Grade Level: Seven
Time Span: Four Days
Tools: Calculators, The Proofs of Pythagoras, GSP, Internet
Colleen Parker
1

Objectives
Students will be able to:
identify the hypotenuse and legs of a right triangle
state the formula for the Pythagorean Theorem
use the Pythagorean Theorem to calculate the lengths of the sides of a right triangle
use the Pythagorean Theorem to identify right triangles
New York State Standards
7.G.5 Identify the right angle, hypotenuse, and legs of a right triangle
7.G.6 Explore the relationship between the lengths of the three sides of a right triangle to
develop the Pythagorean Theorem
7.G.8 Use the Pythagorean Theorem to determine the unknown length of a side of a right
triangle
7.G.9 Determine whether a given triangle is a right triangle by applying the Pythagorean
Theorem and using a calculator
7.N.16 Determine the square root of non-perfect squares using a calculator
7.RP.7 Develop, explain, and verify an argument using mathematical ideas and language
7.RP.8 Justify an argument by using a systematic approach
7.CM.9 Increase their use of mathematical vocabulary and language when communicating with
others
7.CM.10 Use appropriate language, representations, and terminology when describing objects,
relationships, mathematical solutions, and rational
NCTM Standards
Compute fluently and make reasonable estimates
Analyze characteristics and properties of two- and three-dimensional geometric shapes and
develop mathematical arguments about geometric relationships
Recognize reasoning and proof as fundamental aspects of mathematics
Use the language of mathematics to express mathematical ideas precisely
2

Resources
Textbook
Prentice Hall
Middle Grades Math Tools for Success, course 2
Chapters 8-6, 8-7 pp 359-367
Other resources
Prentice Hall
Middle Grades Math Tools for Success, course 3
Chapter 5-10 pp 263-268
Prentice Hall
Connected Mathematics Looking for Pythagoras
Chapters 3, 4 pp 27-51
Prentice Hall
Algebra
Chapter 1-8 pp 45-51
ETA Cuisenaire
The Proof of Pythagoras
Proof 1 pp 2-3
3

Materials and Equipment
Scissors (class set)
glue sticks (class set)
copies of Pythagorean Theorem worksheets puzzle proof one and puzzle proof
two
transparency of proof puzzle one
right triangle tray and green and blue tiles (ETA Cuisenaire, The Proof of
Pythagoras, proof 1)
copies of notes for Days 2
transparency of notes page 3
calculators
transparencies of questions from the textbook 8 pp 362, 19 and 20 pp 363, and 20
pp 367
copies of directions for the activity for Day 3
several rectangular boxes and triangles cut from construction paper which fit
diagonally inside the box
string
tape
copies of practice 8-6 from workbook
computer
The Geometer’s Sketchpad
internet connection
copies of activity sheets for each of the 6 additional activities
4

Overview
Introduction
I developed this unit in response my student teaching experience with an
inclusion class. I had students who worked on many different levels. Some students
could complete the lesson with minimal direct instruction while other students needed
careful step by step instruction. I wanted to develop related activities for
mathematically precocious students to explore while I worked with the rest of the
class. I hoped that this would keep them interested in the mathematics and decrease
classroom disruptions.
The alternative activities can be assigned to students based on your experience
with your students. I would require students to complete the class work before
moving on but you may have a gifted student for which this is not necessary. The
activities are not necessarily associated with one particular day and can be used at
each teacher’s discretion. The additional activities are included after Day 4 in the unit
plan.
Day 1
Use activities to help all students discover the relationship between the areas of
squares built on the sides of right triangles. Students will complete two ‘puzzle’
proofs by inspection of the Pythagorean Theorem. Each puzzle is based on a different
right triangle and a different proof.
Day 2
Review right triangles and the area of a square. Introduce right triangle vocabulary,
leg and hypotenuse. Discuss the activities of day one and use what was learned to
develop the Pythagorean Theorem. Review how to use the calculator to estimate
square roots. Introduce using the Pythagorean Theorem as a way to solve problems
finding the side lengths of right triangles.
Day 3
Review formula for the Pythagorean Theorem and how to use a calculator to estimate
square roots. Use the Pythagorean Theorem to solve problems in context. End this
day with an activity that allows the students to apply what they have learned.
Students will work cooperatively to find the longest rod that will fit inside a box.
Day 4
Review leg, hypotenuse, and formula for the Pythagorean Theorem. In Egyptian
Surveying Activity students create a right triangle with a piece of string that has been
partitioned into 12 equal pieces. This activity will reinforce using the Pythagorean
Theorem with right triangles only. Students will practice determining whether or not a
triangle is a right triangle.
5

Additional Activities:
Activity 1: Investigate the area of squares proof using The Geometer’s Sketchpad
Set up this activity on a PC with GSP. Students will be able to alter the size
of the right triangle to investigate whether the area of squares proof works with
any right triangle.
Activity 2: Area proof using half circles
This activity allows students to investigate whether an area proof will work if the
shapes built on the sides of the right triangle are not squares.
Activity 3: Algebraic proof
This activity is for students who have superior algebraic skills. Follow the
directions from The Proofs of Pythagoras page 6.
Activity 4: Internet search for President Garfield’s Proof
Use this activity to make a Social Studies connection. Have students search the
internet to learn more about more about Garfield and find his proof.
Activity 5: Finding the length of a line segment.
This activity encourages students to use the Pythagorean Theorem to determine
the length of line segments drawn on a grid.
Activity 6: Theodosian Spiral
This activity provides an interesting application for the Pythagorean Theorem.
6

Day 1
Lesson – Puzzle Proofs of the Pythagorean Theorem
Objectives:
Students will be able to:
state that the area of squares built on the legs of a right triangle is equal to the area of
the square built on the hypotenuse.
conjecture that the same relationship applies to all right triangles.
Materials:
Scissors, glue sticks, copies of Pythagorean Theorem worksheets puzzle proof one (page 7),
puzzle proof two (page 8) and puzzle pieces (page 9), transparency of proof puzzle one (page 7),
right triangle tray and green and blue tiles
Outline of Activities:
1. Review right triangles. Students should define right angle and find examples in the classroom.
Students should define right triangle. Create a right triangle in corner of the blackboard using a
yard stick.
2. Students should begin puzzle proof one. Their task is to cover the largest square with pieces of
the smaller squares and look for the relationships between the areas of the squares. Read through
the directions with the students. Have then cut out the pieces and cover the two small squares.
Then use the same pieces to cover the largest square. There may be students who can work
ahead and begin puzzle 2 on their own. Demonstrate the solution to the puzzle using the
transparency.
3. Students should begin puzzle two. Their task is to cover the largest square with pieces of the
smaller squares and look for the relationships between the areas of the squares. Read through the
directions with the students. Have then cut out the pieces. They should cover the two small
squares and then use the same pieces to cover the largest square. Students who finish the puzzle
proofs quickly and identify the relationship can begin work on an alternate activity. Demonstrate
the solution to the puzzle proof using the right triangle tray and green and blue tiles.
4. Class discussion. Can the pieces of the two smaller squares be used to cover the larger square?
Invite students to form conjectures about whether they think this will work for all right triangles.
We did two proofs of the Pythagorean Theorem, point out that there are many different proofs
including one by President Garfield.
Assign students additional activities as needed.
7

Name _________________________________
Example 1
1. Cut out pieces 1, 2, 3, 4, and 5.
2. Cover the smaller squares with pieces. Piece 1 covers the smallest square, pieces 2, 3, 4, 5 cover
the medium square.
3. Cover the largest square with the pieces from the smaller two squares (pieces 1, 2, 3, 4, and 5).
Glue the pieces down.
5
2
4
3
1
What kind of triangle is created by the squares?_______________
Can you cover the largest square with pieces from the smaller squares?___________
8

Name _________________________________
Example 2
1. Cut out pieces 6, 7, 8, 9, 10, 11 and 12.
2. Cover the smaller squares with pieces. Pieces 6, 7, 8 covers the smallest square, pieces 9, 10, 11
and 12 cover the medium square.
3. Cover the largest square with the pieces from the smaller two squares (pieces 6, 7, 8, 9, 10, 11 and
12). Glue the pieces down.
6
8
7
13
11
12
10
9
What kind of triangle is created by the squares?_______________
Can you cover the largest square with pieces from the smaller squares?___________
9

pieces for Puzzle Proof One
5
2
4
3
1
6
8
pieces for Puzzle Proof Two
7
13
11
12
10
9
10

Day 2
Lesson – Developing the Pythagorean Theorem
Objectives:
Students will be able to:
identify the legs and hypotenuse of a right triangle
state the Pythagorean Theorem
use the Pythagorean Theorem to find the lengths of the sides of right triangles
Materials:
Right triangle tray and green and blue tiles, copies of notes, transparency of notes page 3,
calculators
Outline of Activities:
1. Review right triangles and the area of a square. Draw a right triangle on the board and use it to
introduce right triangle vocabulary. Ask students what kind of triangle it is? Where is the right
angle? The hypotenuse is across from the right angle, also the longest side. The other sides of a
right triangle are called the legs. Draw a square with side s on the board. Students should be able
to identify the area.
2. Develop the Pythagorean Theorem formally. Ask a student to remind the class of the activity we
did yesterday. What does this tell us about the area of the squares? The area of the small square +
the area of the medium square = the area of the large square. Using the right angle tray, green
and blue tiles follow the directions for proving the Pythagorean Theorem on page 2. Emphasize
that the legs are always a and b and the hypotenuse is always c.
3. What does the Pythagorean Theorem look like with numbers? Add numbers, a = 3 and b = 4.
What will a² be? What will b² be? 9 + 16 = 25 25 is equal to c². As a class discuss the example
on the notes page 3. Ask the students to point out which side are the legs and which side is the
hypotenuse. Review how to find a square root using a calculator. Have students fill in the boxes
at the bottom of page 3. Press the yellow 2 nd key, then x², and be sure to close the parentheses.
4. How can we use the Pythagorean Theorem? The Pythagorean Theorem allows us to find the
length of a side of a right triangle if we know the length of the other two sides. Do example
problems for finding missing side lengths (notes page 4 and 5).
5. Ask a student to restate the Pythagorean Theorem. Ask a student what the Pythagorean Theorem
can be used for.
Assign students additional activities as needed.
11

Name:________________________ Page 1
The Pythagorean Theorem
A right triangle will have one angle measuring 90˚.
Two sides of a right triangle are called legs and one side is called the hypotenuse.
The hypotenuse is the longest side of the triangle. The longest side is always the side
opposite the right angle.
The legs are the two shorter sides of the triangle. The legs are adjacent to the right
angle.
Right Triangle
hypotenuse
leg
right angle
leg
From the activity:
The two squares built on the legs of the right triangle can be cut apart and used
to fill the square built on the hypotenuse of the right triangle.
What does this tell us about the area of the squares?
12

page 2
Count the blocks in the small and
medium squares. What should the
area of the largest square be?
The area of the smallest square is 9 or 3². The area of the medium square is 16 or
4². The area of the largest square is 25 or 5². 9 + 16 = 25 or 3² + 4² = 5².
Pythagorean Theorem: the legs of a right triangle have the lengths a and b and the
hypotenuse has a length of c. Remember that the area of a square = side x side or s².
Then the areas of the squares built on the sides are a², b², and c².
So a² + b² = c².
The Pythagorean Theorem can only be used with right triangles. The legs are
always called a and b. The hypotenuse is always called c.
13

page 3
Check it out!
What is the length of the leg
AC?
What is the length of the leg
CB?
The area of the two smaller squares is
The area of the larger square is
What is the length of the hypotenuse AB?
The length of the hypotenuse is c. Estimate the square root of c² to find c.
c² = 18 The square root of 18 is
The length of the hypotenuse is
Review – How to find a square root on the calculator.
Record the key strokes for finding the square root of 18 below.
Using your calculator find the following (round to the nearest tenth):
√5 = √12 = √22 = √154 = √41 =
14

page 4
How can we use the Pythagorean Theorem? The Pythagorean Theorem allows us to
find the length of a side of a right triangle if we know the lengths of the other two
sides.
Find the length of the hypotenuse of the triangles below.
a = 5
b = 8
c = ?
Substitute into the formula
8 5² + 8² = c²
find the length of the hypotenuse
5
a =
b =
10 Substitute into the formula
Find the length of the hypotenuse
5
The legs of a triangle are 6 and 7.
Label the triangle at the right and
find the hypotenuse. Round to the
nearest 100 th .
15

Find the length of the missing leg on the triangles below.
Page 5
a = 5
9 b = ?
5 c = 9
Substitute into the formula
5² + b² = 9²
Find the length of the missing leg.
a =
b =
c =
20 22 Substitute into the formula
Find the length of the missing leg.
The triangle at the right has a leg
of 12 and a hypotenuse of 23.4.
Label the triangle at the right and
find the missing leg. Round to the
nearest 100 th .
16

Name:___answer key_____________________ Page 1
The Pythagorean Theorem
A right triangle will have one angle measuring 90˚.
Two sides of a right triangle are called legs and one side is called the hypotenuse.
The hypotenuse is the longest side of the triangle. The longest side is always the side
opposite the right angle.
The legs are the two shorter sides of the triangle. The legs are adjacent to the right
angle.
Right Triangle
hypotenuse
leg
right angle
leg
From the activity:
The two squares built on the legs of the right triangle can be cut apart and used
to fill the square built on the hypotenuse of the right triangle.
What does this tell us about the area of the squares?
When the areas of the squares built on the legs are added together, it is equal to the
area of the square built on the hypotenuse.
page 3
17

Check it out!
What is the length of the leg
AC? 3
What is the length of the leg
CB? 3
The area of the two smaller squares is 3²
The area of the larger square is 3² + 3² = c²
9 + 9 = 18 The area of the larger square is 18.
What is the length of the hypotenuse AB?
The length of the hypotenuse is c. Estimate the square root of c² to find c.
c² = 18 The square root of 18 is 4.24
The length of the hypotenuse is 4.24
Review – How to find a square root on the calculator.
Record the key strokes for finding the square root of 18 below.
2 nd x² 18 ) enter
Using your calculator find the following (round to the nearest tenth):
√5 = 2.2 √12 = 3.5 √22 = 4.7 √154 = 12.4 √41 = 6.4
18

page 4
How can we use the Pythagorean Theorem? The Pythagorean Theorem allows us to
find the length of a side of a right triangle if we know the lengths of the other two
sides.
Find the length of the hypotenuse of the triangles below.
a = 5 25 + 64 = 100
b = 8 c²= 89
c = ?
c = √89
Substitute into the formula c = 9.433981132
8 5² + 8² = c²
find the length of the hypotenuse
5
a = 5 25 + 100 = 125
b = 10 c² = 125
10 Substitute into the formula c = √125
5² + 10² = c² c = 11.18033989
Find the length of the hypotenuse
5
The legs of a triangle are 6 and 7.
Label the triangle at the right and
find the hypotenuse. Round to the
nearest 100 th . 9.22
19

Find the length of the missing leg on the triangles below.
Page 5
a = 5 25 + b² = 81
9 b = ? 25 + b² – 25 = 81 – 25
5 c = 9 b² = 56
Substitute into the formula b = √56
5² + b² = 9² b = 7.483314774
Find the length of the missing leg.
a = ? a² + 400 = 484
b = 20 a² + 400 – 400 = 484 - 400
c = 22 a² = 84
20 22 Substitute into the formula a = √84
a² + 20 = 22 a = 9.16515139
Find the length of the missing leg.
The triangle at the right has a leg
of 12 and a hypotenuse of 23.4.
Label the triangle at the right and
find the missing leg. Round to the
nearest 100 th . 20.09
20

Day 3
Lesson – Applying the Pythagorean Theorem
Objectives:
Students will be able to:
state the Pythagorean Theorem
use a calculator to estimate a square root
use the Pythagorean Theorem to solve problems in context
Materials:
Textbook, transparencies of questions 8 pp 362, 19 and 20 pp 363, and 20 pp 367, calculators,
several rectangular boxes and triangles cut from construction paper which fit diagonally inside the
box, copies of activity directions
Outline of Activities:
1. Review the formula for the Pythagorean Theorem and how to use the calculator to estimate a
square root. Have students draw a right triangle on a piece of paper and label the legs. Have
students exchange papers and find the length of the hypotenuse. Students can return the papers
so the originator can determine if the answer is correct.
2. Discuss with class question 8 pp362 in textbook using overhead transparency. Draw a triangle
and label the legs and hypotenuse. Emphasize that the Pythagorean Theorem only works for
right triangles and that the legs are always a and b and the hypotenuse is always c. What length
is missing? A leg. Answer 10 ft.
3. Discuss with class question 19 pp 363 in textbook using overhead transparency. Draw a triangle
and label the legs and hypotenuse. Emphasize that the Pythagorean Theorem only works for
right triangles and that the legs are always a and b and the hypotenuse is always c. What length
is missing? A leg. Answer 12 ft.
4. Discuss with class question 20 pp 363 in textbook using overhead transparency. Draw a
rectangle. Review what a diagonal is with the class and that diagonal divides a rectangle into
two right triangles. Ask students which sides of the newly created triangles are the legs and
which side is the hypotenuse. Discuss why the Pythagorean Theorem can be used to determine
the length of the diagonal. Draw a diagonal and label the legs and hypotenuses of the right
triangles. What length is missing? The hypotenuse. Answer 10 ft.
5. Discuss with class question 20 pp 367 in textbook using overhead transparency. Draw a square.
Review what a diagonal is with the class and that diagonal divides a square into two right
triangles. Ask students which sides of the newly created triangles are the legs and which side is
the hypotenuse. Discuss why the Pythagorean Theorem can be used to determine the length of
the diagonal. Draw a diagonal and label the legs and hypotenuses of the right triangles. What
length is missing? A leg. Answer a 12 in, b about 17 in. Students who work ahead and solve
these problems on their own can work on additional activities.
21

6. Divide students into groups of three or four. Give each group a rectangular box with the
dimensions of the box clearly labeled. Ask students to find the length of the longest rod that will
completely fit inside the box to the nearest quarter of an inch. Some students are going to
require extra help with this activity. Have triangles cut from construction paper which fit
diagonally inside the box to help students visualize the length they need to find.
Answers will vary depending on the box, students will have to first finds the length of the
diagonal of the bottom of the box and then find the distance from a lower corner to the opposite
upper corner.
7. Review the formula for the Pythagorean Theorem. Demonstrate how to find the length of the
longest rod that will completely fit inside the box. First find the length of the diagonal of the
bottom of the box and then find the distance from a lower corner to the opposite upper corner.
Assign students additional activities as needed.
Homework:
In textbook pp 366 14, 15, 16
Answers 14) 671 meters 15) 65 meters 16) .5 meters
22

Name: ______________________________
Activity:
Find the length of the longest rod that will fit inside the box.
Materials: a rectangular box, calculator
1. Examine the box and find its dimensions.
2. Determine where in the box the rod will go.
3. Decide which triangles you will use to determine the length of the rod.
4. Identify the missing side of the triangles and use the Pythagorean Theorem to
determine the side length. Use your calculator as necessary.
5. Label the box below with the dimensions found in your group. Show all your
work on this sheet.
6. Each student from the group must complete and turn in the activity sheet.
Hint: You will need to use the Pythagorean Theorem twice.
23

Day 4
Lesson – Identifying Right Triangles
Objectives
Students will be able to:
identify the legs and hypotenuse of a right triangle
state the Pythagorean Theorem
use the Pythagorean Theorem to determine if a triangle is a right triangle
Materials:
Calculator, string, tape, copies of practice 8-6 from workbook
Outline of Activities:
1. Go over the homework problems. Use this as an opportunity to review how to use the
Pythagorean Theorem to find missing side lengths.
2. Review right triangle vocabulary and the formula for the Pythagorean Theorem. Ask students, if
the side lengths of a triangle do not reflect the Pythagorean Theorem, does this mean that the
triangle is not a right triangle?
3. Students should begin ancient Egyptian surveying activity. It may be necessary to demonstrate
how to divide the string into twelve equal sections. Fold string in half, fold haves in half, divide
quarters into thirds by trial and error. Have a student demonstrate the solution on the overhead.
Ask students again, if the side lengths of a triangle do not reflect the Pythagorean Theorem, does
this mean that the triangle is not a right triangle?
4. Demonstrate using the Pythagorean Theorem to identify a non-right triangle. Is a triangle with
sides 7, 25, 20 a right triangle? Use the Pythagorean Theorem a² + b² = c². Is 7² + 20² = 25² ? 49
+ 400 ≠ 625 This triangle is not a right triangle. Is a triangle with sides 5, 15, 30 a right triangle?
Use the Pythagorean Theorem a² + b² = c². Is 5² + 15² = 18² ? 25 + 225 ≠ 324 This triangle is not
a right triangle.
5. Write the following side lengths on the board and have students determine whether or not the
triangles are right triangles. Which of these triangles are right triangles? 12,16,20 right triangle
8,15,17 right triangle 12,9,16 not a right triangle 4,7,8 not a right triangle 9,8,12 not a right
triangle 20,21,29 right triangle Use this as an opportunity to work with students who are having
difficulties.
6. Summarize Pythagorean Theorem Unit. Ask students to identify the legs and hypotenuse of a
right triangle. Ask students for the formula for the Pythagorean Theorem. Ask students how the
Pythagorean Theorem can be used.
Assign students additional activities as needed.
Homework:
Practice 8-6 Exploring the Pythagorean Theorem form workbook.
24

Answers 1) 22 cm 2) 51 in 3) 16 ft 4) 25 m 5) 111 yds 6) 18 mi
7) yes 8) no 9) yes 10) no 11) no 12) yes
13) 60 ft 14) 10 ft 15) no 16) yes
25

Name: ___________________________________
Activity: Ancient Egyptian Surveying
The Nile River flooded annually destroying property boundaries in ancient Egypt. The
ancient Egyptians used the Pythagorean Theorem to make a land surveying tool which
they used to reestablish boundaries.
1. Use a pen or marker to divide your string into 12 equal segments. Tape the ends of
the string together to form a loop.
2. Try to form a right triangle with the side lengths that are whole numbers.
If you are having difficulty, ask someone to help you hold your string. Use the
corner of a sheet of paper to measure your angle.
What are the side lengths of the triangle you formed?
Do these side lengths reflect the Pythagorean Theorem?
Demonstrate your answer.
Do you think a 6, 8, 10 triangle is a right triangle?
Explain your answer.
Check your answer using the Pythagorean Theorem.
26

Name: _answer key_____________________________
Activity: Ancient Egyptian Surveying
The Nile River flooded annually destroying property boundaries in ancient Egypt. The
ancient Egyptians used the Pythagorean Theorem to make a land surveying tool which
they used to reestablish boundaries.
3. Use a pen or marker to divide your string into 12 equal segments. Tape the ends of
the string together to form a loop.
4. Try to form a right triangle with the side lengths that are whole numbers.
If you are having difficulty, ask someone to help you hold your string. Use the
corner of a sheet of paper to measure your angle.
What are the side lengths of the triangle you formed? 3, 4, and 5
Do these side lengths reflect the Pythagorean Theorem?
yes
Demonstrate your answer.
3² + 4² = 5²
9 + 16 = 25
Do you think a 6, 8, 10 traingle is a right triangle?
yes
Explain your answer. 6,8,10 are multiples of 3,4,5
Check your answer using the Pythagorean Theorem.
6² + 8² = 10²
36 + 64 = 100
27

Additional Activities
Activity 1:
Investigate the area of squares proof using The Geometer’s Sketchpad.
Teacher Directions
Set up a station with a computer and GSP
Build squares on the sides of a right triangle, follow the directions below.
1. construct a line
2. construct a point not on the line
3. construct a line perpendicular to the point and the first line
4. construct the intersection of the lines
5. construct one point on each of the lines and label the points
6. hide the lines and the points used to construct the lines
7. construct segments between the remaining three points
8. construct squares on each side of the triangle
9. construct the interior of each square
10. name the squares A, B, and C
11. measure the area of each square
12. add the areas of the two smaller squares
28

Name: ________________________
Activity 1:
Investigate the area of squares proof using The Geometer’s Sketchpad.
Use the cursor to move point A and point B. Notice that the sizes of the figures change but not
the shapes.
Find where the areas of the squares are located on the computer screen.
Record the areas you found for five different positions of points A and B in the table below.
Area of square A Area of square B Area of square C Area of squares
A + B
What do you notice about the relationships of the areas of the squares?
Does the area of square on the longest side always equal the areas of the squares on the two
shorter sides?
29

Name: _____________________________
Activity 2
We used the relationship between squares built on the sides of a right triangle to discover the
Pythagorean Theorem. Does the same relationship apply if we draw half circles on the sides of
the right triangle?
4
5
3
Find the area of each half circle.
How are the areas of the half circles related?
30

Name:_______________________________
Activity 3
There are many proof for the Pythagorean Theorem. Use the diagram below to develop an
algebraic proof.
b
a
a
c
c
b
a + b
b
c
c
a
a
b
The area of the large square is equal to the area of the small square and the areas of all four
triangles.
Write an equation
Area of the large square = area of the small square + area of the 4 triangles
___________________ = ___________________ + __________________
Multiply
Simplify
31

Name: ____________________________
Activity 4:
Search the internet to find President Garfield’s proof and list the website where you found it.
____________________________________________________________________
What quadrilateral is Garfield’s proof based on?
_________________________________
Search the internet to find out about James Garfield. Write a brief report about what you found
out. Include where and when Garfield was born, how he was educated, how long his
presidency lasted and at least one other thing you found notable. Use complete sentences. List
the website where you found your information.
_____________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
__________________________________________________________________________
32

Name:_____________________________
Activity 5
Use the Pythagorean Theorem to find the lengths of segment AB, segment CD, segment EF
segment GH and segment JK.
B
C
A
F
G
D
E
J
H
K
Length of segment AB
5 10
Length of segment CD
Length of segment EF
Length of segment GH
Length of segment JK
33

Name: _______________________________
Activity 6
The Theodosian Spiral begins with a triangle which has legs measuring 1 unit. Each new triangle is
drawn using the hypotenuse of the previous triangle as one leg with a second leg measuring 1 unit.
This creates a spiral which winds around counterclockwise.
Use the Pythagorean Theorem to find the length of each hypotenuse in the Theodosian Spiral. Label
the spiral below. Express the lengths using √ symbol.
1
1
1
1
1
2
1
1
1
1
Add the next two triangles to the spiral. Label the length of the sides.
Describe the method you used. (Use complete sentences)
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Answer key for the activities.
Activity 1:
Student’s answers will vary.
Activity 2:
Leg of length 3
Area = ½ π(1.5)² = 3.5 sq units
Leg of length 4
Area = ½ π(2)² = 6.3 sq units
Leg of length 5
Area = ½ π(2.5)² = 9.8 sq units
The sum of the areas of the half circles built on the legs is equal to the area of the half circle
built on the hypotenuse.
Activity 3:
(a + b )² = c² + 4(1/2ab)
a² + ab + ab + b² = c² + 4(1/2)ab
a² + 2ab + b² = c² + 2ab
a² + 2ab – 2ab + b² = c² + 2ab – 2ab
a² + b² = c²
Activity 4:
Garfield’s proof can be found: http://mathworld.wolfram.com/PythagoreanTheorem.html
Garfield’s proof is based on a trapezoid.
Information on Garfield can be found: http://www.americanpresident.org/history/jamesgarfield/
Activity 5:
Length of segment AB
2² + 5² = 29
√29 = 5.385164807
Length of segment CD
2² + 3² = 13
√13 = 3.605551275
Length of segment EF
3² + 5² = 34
√34 = 5.830951895
Length of segment GH
1² + 5² = 26
√26 = 5.09901919514
Length of segment JK
2² + 2² = 8
√8 = 2.828427125
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Activity 6:
1
1
1
2
3
1
1
6
5
2
1
7
1
1
8
1
3
10
11
1
1
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