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The Pythagorean Theorem

Learning Module

Abstract

This learning module is intended for Grade 8 students. It utilizes "Digits" from Pearson for students' independent work. Through this learning module students learn about Pythagoras's contributions to society, investigate why the Pythagorean Theorem works, and how to find missing sides of a triangle given the other two sides.

Common Core State Standards and Objectives

8.G.6 Explain a proof of the Pythagorean Theorem and its converse.

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.

8.G.8  Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Objectives:

  • Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two dimensions.
  • Explore a proof of the Pythagorean Theorem and its converse.
  • Apply the Converse of the Pythagorean Theorem to determine if a triangle is a right triangle.
  • Apply the Pythagorean Thorem to find the distance between two points in a coordinate system.
  • Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two dimensions.

 

1. Essential Questions

For the Student

By the end of the unit, you should be able to answer the following questions.

  1. If you do not have measurement tools, how can you deduce what the side lengths are of a right triangle?
  2. How can you deduce that a triangle is right?
  3. How are the side lengths of a right triangle and the side lengths of squares related?
  4. When you know the lengths of two sides of a right triangle, how do you find the third?

Assignment: Create a new Google Document. Title it "PT- Essential Questions" and share it with your teacher. Type in the essential questions. As we progress through the unit, you will be answering these questions in your own words.

For the Teacher

Process

Teacher may want to post these questions in the room. These quesitons should be referenced throughout the unit. Students should answer these questions in their own words throughout the unit as a way of assessing their understanding.

2. Readiness

For the Student

During this unit, you be building on skills you have learned in both 7th grade and earlier this year. These skills include:

  • Finding the square of a number.
  • Fiding the square root of a number.

  • Solving one-step algebraic equations.

  • Solving two-step algebraic equations.

  • Solving algebraic equations with variable on both sides.

  • Defining the properties of a square and a triangle.

Assignment: Log in to Digits. Complete the Readiness Assignment that is located in your "To Do" list. Then, complete any intervention lesson that is assigned to you based on your results. Be sure to listen to all parts of the lesson, complete the journal page, and complete the 10 questions online for each lesson assigned. The intervention lessons are due by the end of next week.

18-1 Journal
22-2 Journal
22-3 Journal
23-4 Journal
25-7 Journal

For the Teacher

Purpose

Students will complete a readiness assessment that tests skills that should've already been learned.

Method

Students take the assessment online. A student will answer three questions on a topic. If they answer two of the three correctly, they pass that section. Answering less than two questions correctly will cause a student to be assigned an intervention lesson on that topic. Students will have access to a lesson with which they must fill out the journal page and a ten question practice set for each topic not passed.

Differentiation

Based on the assessment result, students will automatically be placed into two groups. Level G will include students that are struggling with the readiness material. Level K will include all students that have passed their readiness material. These levels will determine types of practice questions asked throughout each lesson in the unit. Level G will include more "think about the process" quesitons. Level K will include more "challenge" questions.

References:  
 Fennell, F., & Education, I. (2012). Digits (Common core ed.). Boston, Mass.: Pearson Education. 

Paper-Pencil version of the Readiness assessment for those students that have a hard time tesing on the computer.

 

3. The Man Behind the Theory

For the Student

Introduction:

The Pythagorean Theorem was applied to Mathematics by the greek philosopher and mathematician, Pythagoras.  You can find information on him through the following links: 

Your Project:

You may work by yourself or with 1-2 other people (no more than 3 in a group).

Choose 1 “box” on the choice board and complete that project.

Projects will be presented to the class.

Choice Board

Comment:  Tell each member in your group (each group only needs to comment once) and the project that you have chosen to complete.

 

For the Teacher

Purpose

Students often ask, "Who comes up with this stuff?" This project gives them a chance to answer this question. By completing their project, students will learn more about Pythagoras's contrabutions to not only the mathematical world, but also in other fields such as art and music.

Method:

Give students 3-5 days in class to research and prepare their presentation.

Differentiation:

Each choice is created based on Gardner's Theory of Multiple Intelligences.

Source

Grading:  At the end of the project, ask students to complete the self-reflection page.  The last column is for teachers to mark what they feel students have accomplished.  

Self-Reflection/Teacher assess

 

4. Investigation

For the Student

In this investigation, we will be using squares to form right triangles.

Process:

  1. Decide with your partner who will create the "even" squares and who will create the "odd" squares. 
  2. If you are creating the "even" squares, you will begin by using your grid paper to cut a square with side length of 2 units, then 4 units, then 6 units, etc. Continue through 14 units.
  3. If you are creating the "odd" squares, you will begin by using your grid paper to cut a square with side length of 1 unit, then 3 units then 5 units, etc. Continue through 15 units.
  4. Inside each square, write the square's area.
  5. Using three of your squares, try to create a triangle by connecting three squares at a corner. One side of the square should create one side of the triangle.
  6. Record your findings.
PT Investigation Record

Assignment:  With your partner, discuss the relationships in the picture at the bottom of this post.  Record your findings  in your notebook. Then, email your teacher. Use PT Investigation as the Subject line. Type the following sentences, but fill in the blank with the answer you believe is correct.

When the sum of the two smaller areas is greater than the larger area then I have   a(n) ______________________ triangle.
When the sum of the two smaller areas is less than the larger area, I have a(n)_______________________________ triangle.
When the sum of the two smaller areas is equal to the larger area, I have a(n) _______________________________ triangle.

 

For the Teacher

Purpose

The purpose of this activity is for students to begin answering the first three essential questions:

  1. If you do not have measurement tools, how can you deduce what the side lengths are of a right triangle?
  2. How can you deduce that a triangle is right?
  3. How are the side lengths of a right triangle and the side lengths of squares related?

Teacher Tips

Check students chart from time to time to make sure they are determing the right type of triangle. Have students go back and review any triangles they have labeled incorrectly. This way they can find accurate patterns.

5. The Pythagorean Theorem (Part 1)

For the Student

Watch the following videos on how to use the Pythagorean Theorem to find a missing side of a right triangle given two sides.

  1. Pythagorean Theorem Cartoon
  2. Kahn Academy: The Pythagorean Theorem

With your partner, answer the following problems in your notebook.

 

 

 

 

Update: Create 2 questions that uses the Pythagorean Theorem to find the length of the hypotenuse. Create 2 questions that uses the Pythagorean Theorem to find the length of a side. Show how to solve each problem. Check two other classmate's work. Comment on their post showing the work you took to solve their problems.

For the Teacher

Purpose:

The purpose of this lesson is to answer the following essential question:

When you know the lengths of two sides of a right triangle, how do you find the third?

Process:

Students watch the videos to learn how to use Pythagorean Theorem to solve for the missing length.  Then, practice this skill in their notebook with a partner.

Teacher Tip:

Check students' notebooks as they solve their independent practice problems. Encourage students to work with their partner to help deepen their understanding of the concept.

References:  
 Fennell, F., & Education, I. (2012). Digits (Common core ed.). Boston, Mass.: Pearson Education. 

6. The Pythagorean Theorem (Part 2)

For the Student

Log in to Digits. In your "To Do" list, complete the following:

  1. Lesson 12-2 Homework ~ Finding the hypotenuse given two legs
  2. Lesson 12-3 Homework ~ Finding the length of the leg given the hypotenuse and the other leg.
  3. Lesson 12-4 Homework ~ The Converse of the Pythagorean Theorem

Remember: You must earn at least a 94% on each assignment. If you answer incorrectly, you can click on "Similar Exercise" to try a new problem.

Email: Email your teacher. Use Pythagorean Theorem in the subject line. Rate yourself on your knowledge of Pythagorean theorem using the following:

1 - "I am just learning this and need more help on this topic."

2 - "I can do this if I look at an example."

3 - "I can do this on my own without any help."

4 - "I can do this on my own and explain it to someone else."

For the Teacher

Purpose:

This gives students the opportunity to practice using Pythagorean Theorem independently.

Teacher Tips:

Watch for students using the "Similar Exercise" tool often. They will need a quick re-teach on the lesson.

Students are required to earn a 94% or better on each assignment.

Remind students that they can also utilize the "Show me an example" and "Help me solve this" options to their advantage.

Check student's self evaluation.

Note:  This assignment is for those classes using the online program, Digits by Pearson.  For a paper-pencil version see the attachment.  Level G includes challenge problems and is for those students on grade level or above.  Level K includes "thinking about the process" questions and is for those students who are on grade level or below.  

12-2 Level G
12-2 Level K
12-3 Level G
12-3 Level K
12-4 Level G
12-4 Level K

References:

 Fennell, F., & Education, I. (2012). Digits (Common core ed.). Boston, Mass.: Pearson Education. 
 

 

7. Careers and the Pythagorean Theorem

For the Student

"Why do we need to know this?" said every math student ever.

Watch the following video:

Media embedded October 11, 2015
  1. Click on the link to find a career that uses The Pythagorean Theorem:  XPMath
  2. Go to the following link:  Occupational Outlook Handbook
  3. Search for the career you chose in step 1.

Comment:  What career did you choose?  How might your career use The Pythagorean Theorem?  (Make sure to write a new way that your career uses the theorem than what has already been posted.)

Update:  Write a summary of your research.  What career did you choose?  What do they do?  What type of environment do they work in?  How does someone become one?  What does it pay?

For the Teacher

Purpose

This lesson is for the student that always says, "When will I ever use this?".  In this lesson, students will research a career that uses the Pythagorean Theorem.

Method:

Students watch a short video on careers that use math.  Then they choose a career that uses the Pythagorean Theorem (according to XPMath).

Extension:  Have students work in groups.  Each group chooses a career that uses Pythagorean Theorem.  Have students create a task that someone in thier career may have to solve using the Pythagorean Theorem.  Have groups trade and try to solve each other's tasks.

8. The Parachute Jump

For the Student

You need:

  1. Coffee filter
  2. a paper clip
  3. a hole punch
  4. 4 pieces of string about 12 inches long

Directions:  

  1. Punch 4 holes around the coffee fileter at equal distances.
  2. Tie the end of each string through the holes on the coffee filter.
  3. Join the string at the bottom and tie the paper clip on the end. 
  4. Hold the parachute in your hand and extend your arm so it is parallel to the floor.  The parachute should be over the coordinate plane that is taped onto the floor.
  5. Drop the parachute.
  6. Your partner needs to do the same.
  7. Record the nearest ordered pair for each parachute (use integers) on the coordinate plane on your paper.
  8. Record the vertical and horizontal distance between the two parachutes.
  9. Find the actual distance using the Pythagorean Theorem.
Parachute Jump Recording Sheet

Update:  Create a problem that a student could solve by using the Pythagorean Theorem to find distance between two points.  Then, solve your problem.  When you are finished, check the work of two of your classmates.  Comment on their post agreeing or disagreeing with them, using your work to support your claim.

For the Teacher

Purpose

To give students real-life practice finding distance between two points using a coordinate plane.

Process

Before this activity, draw coordinate planes on the classroom floor using painter's tape.  There should be enough that two students can share one plane.

Tips

Students must tie all of the string to the paper clip at the same time.  This way the string is the same length all around the parachute.

Students should use the closest integer ordered pair to their parachute landing.  This will take out any confusion or arguement over numbers with decimals or fractions as the goal is to have students understand the process of using Pythagorean Theorem to find distance.

References

 Martin, H. (2006). Differentiated Instruction for Mathematics: Instructions and Activities for the Diverse Classroom. Walch Publishing. 
 

 

9. Finding Distance on the Coordinate Plane

For the Student

Log in to Digits. In your "To Do" list, complete the following:

Lesson 12-5 Homework ~ Finding Distance on the Coordinate Plane

Remember: You must earn at least a 94% on each assignment. If you answer incorrectly, you can click on "Similar Exercise" to try a new problem.

Email: Email your teacher. Use Fiding Distance in the subject line. Rate yourself on your knowledge of finding distance on the coordinate plane using the following:

1 - "I am just learning this and need more help on this topic."

2 - "I can do this if I look at an example."

3 - "I can do this on my own without any help."

4 - "I can do this on my own and explain it to someone else."

For the Teacher

Purpose:

This gives students the opportunity to practice using Pythagorean Theorem to find distance independently.

Teacher Tips:

Watch for students using the "Similar Exercise" tool often. They will need a quick re-teach on the lesson.

Students are required to earn a 94% or better on each assignment.

Remind students that they can also utilize the "Show me an example" and "Help me solve this" options to their advantage.

Note:  This assignment is for those classes using the online program, Digits by Pearson.  For a paper-pencil version see the attachment.  Level G includes challenge problems and is for those students on grade level or above.  Level K includes "thinking about the process" questions and is for those students who are on grade level or below.  

12-5 Level G
12-5 Level K

References:

Fennell, F., & Education, I. (2012). Digits (Common core ed.). Boston, Mass.: Pearson Education. 

10. Final Assessment

For the Student

Log in to Digits. In your "To Do" list, complete the following:

Topic 12 Assessment

Remember: Once you answer a problem, you cannot go back.  Make sure you have the answer you would like to submit before moving on to the next question.

Email: Email your teacher. Use PT Assessment in the subject line.  List three things you learned in this Unit.

For the Teacher

Purpose

The purpose of this activity is to assess each student on what they have learned in this unit.

Method

Students will complete an 8 question test applying the skills they have learned in this unit.  Then, students will self-reflect by emailing the teacher a list of three things they have learned throughout the unit.

Note:  This assignment is for those classes using the online program, Digits by Pearson.  For a paper-pencil version see the attachment.

Topic 12 Assessment

References:

 Fennell, F., & Education, I. (2012). Digits (Common core ed.). Boston, Mass.: Pearson Education.