Why does a2+b2=c2?

We plan to learn...

•What the Pythagorean Theorem (a2+b2=c2) means.

•How to "prove" (convince someone through mathematical reasoning) that this equation is true for any right triangle.[ Geometry Standard 14.0: Students prove the Pythagorean theorem. ]

a

b c

Why does a2+b2=c2?

Making Congruent Right Triangles:

•You have received orange paper, white paper, and scissors.

•You will be making 4 congruent (equal) right triangles.

•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.

•With the scissors, cut off a large triangle containing the "loose" corner. Watch my example.

•You now have 4 right triangles that are exactly the same. Label the short leg "a", the long leg "b", and the hypotenuse (the longest side) "c".

a

b c

You will be working with a partner. You may help each other, but each of you will do your own shapes:

Making a big square:

•Get a full piece of scratch paper.

•On the paper, arrange your triangles like this:

•Trace around the big square on the outside.

•Trace around the little square on the inside.

•Show me the result.

Why does a2+b2=c2?

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b c

Why does a2+b2=c2?

Changing the square:

•Take the triangles out of the big square.

•Put them back in like this:

•Trace the shapes you got.

•Show me the result.

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b c

Why does a2+b2=c2?

Compare & Discuss with your partner:

•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.

•What are the areas of the various shapes you traced?

•Do a2, b2, or c2 show up?

•If so, how are they related.

•What does this have to do with the Pythagorean Theorem?

•Be prepared to include me in your discussion as I circulate.

•Be prepared to share what you found with the class.

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b c

Why does a2+b2=c2?

What did we learn?

•What does a2+b2=c2 means in terms of areas?

•When is the Pythagorean Theorem useful in math class?

•When is the Pythagorean Theorem useful outside of class?

•How did we show that it works with the triangles we cut out?

•Did it work for everybody?

•Should it work for any right triangle? Why or why not?

•More thoughts?

•Can you think of other ways to prove that a2+b2=c2?

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b c

Why does a2+b2=c2?

Please put your names on your tracings and your triangles. I will collect them at the end.

You are each receiving a small "closure" worksheet. Please complete the worksheet, then ask me to check it.

Once I check and collect your work (tracings, triangles and closure), you may leave. See you Monday.

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b c