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Pythagorean Theorem

A hands-on way for students to learn how to prove the Pythagorean Theorem.

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Pythagorean Theorem

  1. 1. Why does a 2 +b 2 =c 2 ? <ul><li>We plan to learn... </li></ul><ul><li>What the Pythagorean Theorem (a 2 +b 2 =c 2 ) means. </li></ul><ul><li>How to &quot;prove&quot; (convince someone through mathematical reasoning) that this equation is true for any right triangle. </li></ul><ul><li>[ Geometry Standard 14.0: Students prove the Pythagorean theorem. ] </li></ul>a b c
  2. 2. Why does a 2 +b 2 =c 2 ? <ul><li>Making Congruent Right Triangles: </li></ul><ul><li>You have received orange paper, white paper, and scissors. </li></ul><ul><li>You will be making 4 congruent (equal) right triangles. </li></ul><ul><li>To do this, fold your paper in half one way, then in half the other way. Watch my example to see how. Be careful to line up the corners exactly. </li></ul><ul><li>With the scissors, cut off a large triangle containing the &quot;loose&quot; corner. Watch my example. </li></ul><ul><li>You now have 4 right triangles that are exactly the same. Label the short leg &quot;a&quot;, the long leg &quot;b&quot;, and the hypotenuse (the longest side) &quot;c&quot;. </li></ul>a b c
  3. 3. <ul><li>You will be working with a partner. You may help each other, but each of you will do your own shapes: </li></ul><ul><li>Making a big square: </li></ul><ul><li>Get a full piece of scratch paper. </li></ul><ul><li>On the paper, arrange your triangles like this: </li></ul><ul><li>Trace around the big square on the outside. </li></ul><ul><li>Trace around the little square on the inside. </li></ul><ul><li>Show me the result. </li></ul>Why does a 2 +b 2 =c 2 ? a b c
  4. 4. Why does a 2 +b 2 =c 2 ? <ul><li>Changing the square: </li></ul><ul><li>Take the triangles out of the big square. </li></ul><ul><li>Put them back in like this: </li></ul><ul><li>Trace the shapes you got. </li></ul><ul><li>Show me the result. </li></ul>a b c
  5. 5. Why does a 2 +b 2 =c 2 ? <ul><li>Compare & Discuss with your partner: </li></ul><ul><li>Label (in terms of a, b, and c) the lengths of all the sides of the shapes you traced. Use your (labelled) triangles to help. </li></ul><ul><li>What are the areas of the various shapes you traced? </li></ul><ul><li>Do a 2 , b 2 , or c 2 show up? </li></ul><ul><li>If so, how are they related. </li></ul><ul><li>What does this have to do with the Pythagorean Theorem? </li></ul><ul><li>Be prepared to include me in your discussion as I circulate. </li></ul><ul><li>Be prepared to share what you found with the class. </li></ul>a b c
  6. 6. Why does a 2 +b 2 =c 2 ? <ul><li>What did we learn? </li></ul><ul><li>What does a 2 +b 2 =c 2 means in terms of areas? </li></ul><ul><li>When is the Pythagorean Theorem useful in math class? </li></ul><ul><li>When is the Pythagorean Theorem useful outside of class? </li></ul><ul><li>How did we show that it works with the triangles we cut out? </li></ul><ul><li>Did it work for everybody? </li></ul><ul><li>Should it work for any right triangle? Why or why not? </li></ul><ul><li>More thoughts? </li></ul><ul><li>Can you think of other ways to prove that a 2 +b 2 =c 2 ? </li></ul>a b c
  7. 7. Why does a 2 +b 2 =c 2 ? <ul><li>Please put your names on your tracings and your triangles. I will collect them at the end. </li></ul><ul><li>You are each receiving a small &quot;closure&quot; worksheet. Please complete the worksheet, then ask me to check it. </li></ul><ul><li>Once I check and collect your work (tracings, triangles and closure), you may leave. See you Monday. </li></ul>a b c