Proving the Pythagorean Theorem

I taught the Pythagorean Theorem last week for the first time in a while.  The common core states that students will:

  • CCSS.Math.Content.8.G.B.6 Explain a proof of the Pythagorean Theorem and its converse.
  • CCSS.Math.Content.8.G.B.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

So, I found a great proof that I thought the students would be able to understand.  The first day we discussed the theorem and what it is used for, vocabulary, a few problems and pythagorean triples.

The next day we discussed the vocabulary again but we focused on a^2, b^2 and c^2 being areas.   I asked them to draw a triangle with legs of 3 and 4.   I had students use the square of 3 and the square of 4 to make the hypotenuse squared (Making sure they used all of the pieces).  They were successful at finding out the new square to be 5 x 5 or 25 units.  We then discussed the fact that the Pythagorean Theorem works for ALL right triangles even if their sides are not whole numbers.  I had students draw another triangle with legs of their choice and create two squares, each with sides of their legs. Then I had them duplicated those squares and use one set to cut apart and make a new larger square that would fit on the hypotenuse side.  I was interested in what they would find out.  Students quickly figured out that they hypotenuse what not going to be a whole number.  (they picked their legs and it worked out in all but one case).  They worked at cutting down those perfect units and created pieces to add on to the wholes.  I was excited to hear the conversations as the learning was happening!   It was a great hands-on learning day with the students leading the learning.

Here are some pictures:

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Exploring the MathTwitterBlogosphere

One of my favorite problems that I start the school year with is Crossing the River.

There are 8 and 2 children that have to cross the river but only one adult or one child or two children can be in the boat.  How many one way trips does it take to get everyone to the other side?

The students share their solutions but more importantly their thinking about how they solved the problem, we discuss many different ways to solve the problem.  We even act it out.  I extend their thinking with more adults and the same amount of children to try to get them to think about patterns and then a function.  I love the way they all become so invested in this problem.  It is a problem that all students, no matter what level, can solve and feel like they have conquered it!  My students also get a kick out of spending 1-2 days on one problem.

Here are some more problems like it:

http://pbskids.org/cyberchase/math-games/crossing-the-river/

Crossing the River (with a Wolf, a Goat, and a Cabbage http://www.mathcats.com/explore/river/crossing.html

http://www.mathsisfun.com/chicken_crossing_solution.html

Who am I?

I am a mother, wife, teacher and mentor.  I am also usually a lurker…one that reads other blogs, tweets and websites but doesn’t share too much.  I take it all in, hoping to become better by picking up great ideas from others.  I thought it was time for me  to share my world….