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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

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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

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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

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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

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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.

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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.

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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.

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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.

What kind of triangle is created by the squares?_______________ Can you cover the largest square with pieces from the smaller squares?___________

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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.

What kind of triangle is created by the squares?_______________ Can you cover the largest square with pieces from the smaller squares?___________

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pieces for Puzzle Proof One

pieces for Puzzle Proof Two

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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 2nd 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.

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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?

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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.

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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 =

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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

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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 100th.

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Page 5 Find the length of the missing leg on the triangles below. 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 100th.

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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

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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.

2nd 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

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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

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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 100th. 9.22

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Page 5 Find the length of the missing leg on the triangles below. 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 100th. 20.09

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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.

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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

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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.

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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.

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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

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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.

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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

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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

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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.

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?

Area of square A Area of square B Area of square C Area of squares A + B

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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? Find the area of each half circle. How are the areas of the half circles related?

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Name:_______________________________ Activity 3 There are many proof for the Pythagorean Theorem. Use the diagram below to develop an algebraic proof. 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

a + b

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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. _____________________________________________________________________

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Name:_____________________________

Activity 5

Use the Pythagorean Theorem to find the lengths of segment AB, segment CD, segment EF segment GH and segment JK. Length of segment AB Length of segment CD Length of segment EF Length of segment GH Length of segment JK

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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. 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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