The Pythagorean Theorem

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The Pythagorean Theorem

  1. 1. Student Page Title Introduction Task Process Evaluation Conclusion Credits [ Teacher Page ] A WebQuest for 8th Grade (Mathematics) Designed by Jason Harr [email_address] Based on a template from The WebQuest Page Photo on Flickr by iliasamos
  2. 2. Student Page Title Introduction Task Process Evaluation Conclusion Credits [ Teacher Page ] Hello and welcome to the adventure of discovering the Pythagorean Theorem. The Pythagorean Theorem is one of the greatest theorems known today. This discovery was credited to Pythagoras of Samos. Pythagoras believed that everything was related to mathematics and that numbers were the ultimate reality and, through mathematics, everything could be predicted and measured. So as you embark on your discovery remember that great minds think alike and don’t give up no matter what the cost. The School of Athens by Raffaello Sanzio
  3. 3. Student Page Introduction Task Process Evaluation Conclusion Credits [ Teacher Page ] <ul><li>As you embark on your discovery you will be given three objectives to complete: </li></ul><ul><li>Objective #1: </li></ul><ul><ul><li>Learn about who Pythagoras was and the contributions he made </li></ul></ul><ul><ul><li>Create a handout on the information you have found using Microsoft Publisher </li></ul></ul><ul><ul><li>Objective #2: </li></ul></ul><ul><ul><li>Discover the several different proofs for the Pythagorean Theorem </li></ul></ul><ul><ul><li>Create your own proof for the Pythagorean Theorem using similar triangles </li></ul></ul><ul><ul><li>Objective #3: </li></ul></ul><ul><ul><li>Choose one of four missions that you will be given </li></ul></ul><ul><ul><li>Create a power point presentation displaying how you completed your mission and your findings </li></ul></ul>Title
  4. 4. Student Page Title Introduction Task Process Evaluation Conclusion Credits [ Teacher Page ] <ul><li>Objective #1: </li></ul><ul><li>First we will look at the history of the Pythagorean Theorem. In doing so we will discover the creator of the Pythagorean Theorem, Pythagoras of Samos. So check out these websites to help with your discovery. You may want to create a list of all the interesting facts that you find. </li></ul><ul><li>A website about Pythagoras </li></ul><ul><li>Another website about Pythagoras </li></ul><ul><li>A website about the Pythagorean Theorem </li></ul><ul><li>Now that you have learned about this great mathematician you will use Publisher to create a handout on Pythagoras to show your findings. </li></ul><ul><li>Include: </li></ul><ul><li>Your name </li></ul><ul><li>A picture of Pythagoras </li></ul><ul><li>A list of the interesting facts that you found (At least 10) </li></ul><ul><li>Why you thought these facts were interesting or important </li></ul>Move on to Objective #2
  5. 5. Student Page Title Introduction Task Process Evaluation Conclusion Credits [ Teacher Page ] <ul><li>Objective #2: </li></ul><ul><li>Now your next objective is to discover several different proofs for the Pythagorean Theorem and in doing so will make your own proof using similar triangles. So check out these websites to help in completing objective #2. </li></ul><ul><li> Proofs for the Pythagorean Theorem </li></ul><ul><li>Hands on learning for the Pythagorean Theorem </li></ul><ul><li>An animated proof for the Pythagorean Theorem </li></ul><ul><li>A website about similar triangles </li></ul><ul><li>There’s actually over three-hundred different proofs for the Pythagorean Theorem. Now it’s your turn to discover one of them. Given the figure below find a way to show that a 2 + b 2 = c 2 using similar triangles. Begin by comparing triangles ABC and CBH. Then compare triangles ABC and ACH. (Hint: solve for a 2 and b 2 then add the two equations together.) </li></ul>Move on to Objective #3 x y
  6. 6. Student Page Title Introduction Task Process Evaluation Conclusion Credits [ Teacher Page ] <ul><li>Objective #3: </li></ul><ul><li>Well done! You’re off to a great start. Your next objective is to choose a mission. Your missions are: </li></ul><ul><li>Discovering the height of a ladder (A fireman’s worst nightmare) </li></ul><ul><li>Finding the quickest way across a pond (A Geologist’s best friend) </li></ul><ul><li>Is he safe or not? (A players revenge) </li></ul><ul><li>Creating the roof of a house (A constructor’s job) </li></ul><ul><li>Once you have completed your mission you will create a power point showing your results and how you achieved your results. </li></ul><ul><li>Include: </li></ul><ul><li>Your name </li></ul><ul><li>Your mission and an introduction </li></ul><ul><li>The steps you used in completing your mission (Be specific!) </li></ul><ul><li>The result of your mission and a conclusion </li></ul>
  7. 7. Student Page Title Introduction Task Process Evaluation Conclusion Credits [ Teacher Page ] Discovering the height of a ladder (A fireman’s worst nightmare) Back to Objective #3 022006-02 by SlimWhitman The tallest building in Boonville is 356 feet. The average height for a fire engine is 12 feet. If the fire department is required to be 50 feet away from the building how long (in feet) must the ladder be to reach someone on the roof of the tallest building? Round to two decimal places. If the same fire engine was called to another town who’s tallest building was 515 feet how much longer (in feet) should the ladder be to reach someone on the roof of this building? Round to two decimal places. 50 feet
  8. 8. Student Page Title Introduction Task Process Evaluation Conclusion Credits [ Teacher Page ] Finding the quickest way (A Geologist’s best friend) Back to Objective #3 A geologist is looking for gold. To reach his destination he must traverse around a swamp. He heads south 2.7 miles and then heads east 5.2 miles. If the geologist could cross the swamp how much distance (in miles) would he have saved? Round to one decimal place. 2.7 miles 5.2 miles If the geologist walks a mile in 30 minutes how much time (in hours) would he save? Round to one decimal place. Alabama Swamp Morning by Raggs56
  9. 9. Student Page Title Introduction Task Process Evaluation Conclusion Credits [ Teacher Page ] Is he safe or not? (A players revenge) Back to Objective #3 The average time for a runner to steal second base is 4.5 seconds. The average speed for a pitcher is 90 mph, the average speed for a catcher is 80 mph, and the exchange for a catcher is 2 seconds. If the runner got a second head start would he be safe? Round answers to two decimal places. Hint: Begin by finding out how many seconds it takes the ball to go from the pitcher to the catcher. Then find out how many seconds it takes the ball to go from the catcher to second base. To do this you will need to find the distance from the catcher to second base. 60 feet 90 feet 90 feet 90 feet Wrigley Baseball Diamond by rachelleb
  10. 10. Student Page Title Introduction Task Process Evaluation Conclusion Credits [ Teacher Page ] Creating the roof of a house (A constructor’s job) Back to Objective #3 A constructer is trying to find how much square feet of wood he would need for the roof of his building. If the base of the roof is 40 feet long and the height is 15 feet how much square feet of wood would the constructor need to build his roof? (The roof is 50 feet long) 40 feet 15 feet 20 feet Roof Construction by netopNyrop
  11. 11. Student Page Title Introduction Task Process Evaluation Conclusion Credits [ Teacher Page ] Beginning 1 Developing 2 Accomplished 3 Exemplary 4 Score Mathematical Concepts Explanation shows complete understanding of the mathematical concepts used to solve the problem(s). Explanation shows substantial understanding of the mathematical concepts used to solve the problem(s). Explanation shows some understanding of the mathematical concepts needed to solve the problem(s). Explanation shows very limited understanding of the underlying concepts needed to solve the problem(s) OR is not written. Mathematical Reasoning Uses complex and refined mathematical reasoning. Uses effective mathematical reasoning Some evidence of mathematical reasoning. Little evidence of mathematical reasoning. Mathematical Errors 90-100% of the steps and solutions have no mathematical errors. Almost all (85-89%) of the steps and solutions have no mathematical errors. Most (75-84%) of the steps and solutions have no mathematical errors. More than 75% of the steps and solutions have mathematical errors. Use of Manipulatives Student always listens and follows directions and only uses manipulatives as instructed. Student typically listens and follows directions and uses manipulatives as instructed most of the time. Student sometimes listens and follows directions and uses manipulatives appropriately when reminded. Student rarely listens and often &quot;plays&quot; with the manipulatives instead of using them as instructed. Neatness and Organization The work is presented in a neat, clear, organized fashion that is easy to read. The work is presented in a neat and organized fashion that is usually easy to read. The work is presented in an organized fashion but may be hard to read at times. The work appears sloppy and unorganized. It is hard to know what information goes together. Completion All problems are completed. All but 1 of the problems are completed. All but 2 of the problems are completed. Several of the problems are not completed.
  12. 12. Student Page Title Introduction Task Process Evaluation Conclusion Credits [ Teacher Page ] <ul><li>How about that! You learned about Pythagoras and the proofs for the Pythagorean Theorem. Not only that, but you also applied the Pythagorean Theorem to a specific problem and completed you mission. Well done! </li></ul><ul><li>Additional problems using the Pythagorean Theorem </li></ul>
  13. 13. Student Page Title Introduction Task Process Evaluation Conclusion Credits [ Teacher Page ] Websites: http://en.wikipedia.org/wiki/Pythagoras http://www.utm.edu/research/iep/p/pythagor.htm http://www.geom.uiuc.edu/~demo5337/Group3/hist.html http://www.cut-the-knot.org/pythagoras/index.shtml http://www.frontiernet.net/~imaging/pythagorean.html http://www.usna.edu/MathDept/mdm/pyth.html http://www.onlinemathlearning.com/similar-triangles.html http://www.geom.uiuc.edu/~demo5337/Group3/bball.html http://regentsprep.org/Regents/Math/fpyth/PracPyth.htm Images: http://www.flickr.com/photos/iliasamos/569622949/ http://www.flickr.com/photos/rachelleb/1340876677/ http://www.flickr.com/photos/netopnyrop/16887274/ http://www.flickr.com/photos/slimwhitman/102227628/ http://www.flickr.com/photos/patches56/244883531/ http://en.wikipedia.org/wiki/Image:Sanzio_01_Pythagoras.jpg
  14. 14. [ Student Page ] Title Introduction Learners Standards Process Resources Credits Teacher Page A WebQuest for 8th Grade (Mathematics) Designed by Jason Harr [email_address] Based on a template from The WebQuest Page Evaluation Teacher Script Conclusion Photo on Flickr by iliasamos
  15. 15. [ Student Page ] Title Introduction Learners Standards Process Resources Credits Teacher Page This lesson was developed as part of the Colorado State University curriculum for EDU 331. The lesson for this WebQuest is to discover the Pythagorean Theorem, who Pythagoras was, and the aspects that the Pythagorean Theorem has in the real world. As students go through this lesson they should enhance their knowledge about the Pythagorean Theorem and learn new skills in working in Publisher and PowerPoint. Evaluation Teacher Script Conclusion
  16. 16. [ Student Page ] Title Introduction Learners Standards Process Resources Credits Teacher Page This lesson is anchored in eighth grade mathematics and involves computer science, English, and history to a lesser extent. The lesson can easily be extended to additional grades such as an advanced seventh grade class or ninth/tenth grade mathematics. Students will need to know how to set up comparison ratios, have prior knowledge in how to solve a comparison ratio, and basic knowledge on using PowerPoint and Publisher. Students will also be using similar triangles and may need to be reminded on how to compare similar triangles. So I have included a website to help guide students through this part. Evaluation Teacher Script Conclusion
  17. 17. [ Student Page ] Title Introduction Learners Standards Process Resources Credits Teacher Page <ul><li>Math Standards Addressed </li></ul><ul><li>Standard 4 : Students use geometric concepts, properties, and relationships in problem-solving situations and communicate the reasoning used in solving these problems. </li></ul><ul><li>In order to meet this standard, a student will </li></ul><ul><ul><li>connect various physical objects with their geometric representation; </li></ul></ul><ul><ul><li>connect mathematical concepts from across the standards with their geometric representations; </li></ul></ul><ul><ul><li>recognize, draw, describe, and analyze geometric shapes in one, two, and three dimensions; </li></ul></ul><ul><ul><li>make, investigate, and test conjectures about geometric ideas; and </li></ul></ul><ul><ul><li>Solve problems and model real-world situations using geometric concepts. </li></ul></ul><ul><li>This lesson also helps students build on their critical thinking skills, allows students to be creative in making their PowerPoint and Publisher assignments, and helps build teamwork. </li></ul>Evaluation Teacher Script Conclusion
  18. 18. [ Student Page ] Title Introduction Learners Standards Process Resources Credits Teacher Page To see the process that the students will go through refer to the Student Page. Students Process Page. The lesson is organized into three sections, each section is an objective for the student. The first section is designed to find information about who Pythagoras was and the contributions he made. The second section is designed to discover what the Pythagorean Theorem is and to create a proof for the Pythagorean Theorem. The final section is designed to help students use the Pythagorean Theorem and discover the several different uses the Pythagorean Theorem has. This lesson can take several class periods to complete. I would recommend two or three class periods. This lesson was also designed for students to work in four separate groups based on their interest and skill level in completing objective #3. Some students may have a hard time in applying what they have already learned to what they are learning now. Evaluation Teacher Script Conclusion
  19. 19. [ Student Page ] Title Introduction Learners Standards Process Resources Credits Teacher Page <ul><li>What's needed to implement this lesson: </li></ul><ul><ul><li>A computer lab. </li></ul></ul><ul><ul><li>Computers need to have Publisher and PowerPoint. </li></ul></ul><ul><ul><li>The Internet. </li></ul></ul><ul><ul><li>Lots of Creativity. </li></ul></ul><ul><li>This WebQuest uses several different websites that you can view on the </li></ul><ul><li>Credits Page . </li></ul><ul><li>When implementing this lesson aides or parents are very helpful to help guide students in their process. You may also want to have students go and verify or reenact what their problem is in objective #3. </li></ul>Evaluation Teacher Script Conclusion
  20. 20. [ Student Page ] Title Introduction Learners Standards Process Resources Credits Teacher Page Evaluation Teacher Script Conclusion Beginning 1 Developing 2 Accomplished 3 Exemplary 4 Score Mathematical Concepts Explanation shows complete understanding of the mathematical concepts used to solve the problem(s). Explanation shows substantial understanding of the mathematical concepts used to solve the problem(s). Explanation shows some understanding of the mathematical concepts needed to solve the problem(s). Explanation shows very limited understanding of the underlying concepts needed to solve the problem(s) OR is not written. Mathematical Reasoning Uses complex and refined mathematical reasoning. Uses effective mathematical reasoning Some evidence of mathematical reasoning. Little evidence of mathematical reasoning. Mathematical Errors 90-100% of the steps and solutions have no mathematical errors. Almost all (85-89%) of the steps and solutions have no mathematical errors. Most (75-84%) of the steps and solutions have no mathematical errors. More than 75% of the steps and solutions have mathematical errors. Use of Manipulatives Student always listens and follows directions and only uses manipulatives as instructed. Student typically listens and follows directions and uses manipulatives as instructed most of the time. Student sometimes listens and follows directions and uses manipulatives appropriately when reminded. Student rarely listens and often &quot;plays&quot; with the manipulatives instead of using them as instructed. Neatness and Organization The work is presented in a neat, clear, organized fashion that is easy to read. The work is presented in a neat and organized fashion that is usually easy to read. The work is presented in an organized fashion but may be hard to read at times. The work appears sloppy and unorganized. It is hard to know what information goes together. Completion All problems are completed. All but 1 of the problems are completed. All but 2 of the problems are completed. Several of the problems are not completed.
  21. 21. [ Student Page ] Title Introduction Learners Standards Process Resources Credits Teacher Page <ul><li>The WebQuest model is best suited for learners who can navigate the Web on their own and can read the kinds of material commonly found on the Web. We can stretch the format to reach primary-aged learners, developmental English Language Learners and special populations by creating a facilitated WebQuest, one that requires an adult or older peer to drive things. </li></ul><ul><li>Use this page for that facilitator. The facilitator would print this page out and use it to guide their progress through the WebQuest. </li></ul><ul><li>Read the introduction to the student on the introduction page . </li></ul><ul><li>Read the tasks that the student will be guided through on the task page . </li></ul><ul><li>Guide the student through objective #1 . (You may need to help the student by telling them where they are and what websites they need to discover. Also you may need to help them create their handout, but allow the student to be creative.) </li></ul><ul><li>Guide the student through objective #2 . (Again you may need to help the student by telling them where they are and what websites they need to discover. You may also need to help guide them through the proof.) </li></ul><ul><li>Guide the student through objective #3 . Be sure to tell them to choose one of the missions by clicking on the mission. You may need to help guide the student in how to solve the problem (the mission) and to create their PowerPoint showing how they solved the problem. To get back to objective #3 you need to click on the link that says back to objective #3 . </li></ul><ul><li>This page is linked to the Process segment off of the Teacher Page </li></ul>Evaluation Teacher Script Conclusion
  22. 22. [ Student Page ] Title Introduction Learners Standards Process Resources Credits Teacher Page Thanks for checking out my WebQuest. I hope that it is or was helpful in teaching your students and will help them become more successful in their futures. Feel free to e-mail me any comments or considerations that you have. Evaluation Teacher Script Conclusion
  23. 23. [ Student Page ] Title Introduction Learners Standards Process Resources Credits Teacher Page Evaluation Teacher Script Conclusion Websites: http://en.wikipedia.org/wiki/Pythagoras http://www.utm.edu/research/iep/p/pythagor.htm http://www.geom.uiuc.edu/~demo5337/Group3/hist.html http://www.cut-the-knot.org/pythagoras/index.shtml http://www.frontiernet.net/~imaging/pythagorean.html http://www.usna.edu/MathDept/mdm/pyth.html http://www.onlinemathlearning.com/similar-triangles.html http://www.geom.uiuc.edu/~demo5337/Group3/bball.html http://regentsprep.org/Regents/Math/fpyth/PracPyth.htm Images: http://www.flickr.com/photos/iliasamos/569622949/ http://www.flickr.com/photos/rachelleb/1340876677/ http://www.flickr.com/photos/netopnyrop/16887274/ http://www.flickr.com/photos/slimwhitman/102227628/ http://www.flickr.com/photos/patches56/244883531/ http://en.wikipedia.org/wiki/Image:Sanzio_01_Pythagoras.jpg

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