Saturday 29 October 2011

Pythagoras' Theorem - From Concrete to Abstract

I have found this the best way to teach Pythagoras' Theorem. The focus of the lesson is actually on showing the students how to go from the concrete to the abstract.
The lesson takes about 100 minutes, a double where I am teaching.
Students have commented on what fun they had during the double.

First, engage the students with Donald Duck in Mathemagic Land.
The first 5 minutes or so talks about the secret sect of Pythagoreans and their contribution to the understanding of music and the golden section.


Don't forget to discuss with your students that the Pythagoreans allowed women in their sect, while women were still not classed as citizens in ancient Greek society. The Pythagoreans were ahead of their times!


Next, introduce the theorem.
It has lasted over 2500 years, although there is evidence that the Babylonian society understood the relationship between the squares of the sides of a right triangle way before the Greeks.
People think that knowing Pythagoras' Theorem makes you smart. Check out the brainless scarecrow in the Wizard of Oz who, when granted a brain, recites a version of Pythagoras' theorem using an isosceles triangle, instead of a right triangle!


Now,

The square on the hypotenuse = the sum of the squares on the other two sides.


Then have your students explore a proof using a great interactive applet available at



Then, give the students a worksheet showing a right triangle and the squares on the three sides. Once they have found a proof using the applet, they cut out the two smaller squares and glue these inside the large square.

Here is the worksheet.

Time to go to the abstract.
Have the students draw a 3cm, 4cm, 5cm triangle as accurately as they can, then draw a square on each side.
Then ask them to find the area of each of these squares.
Now, the abstract connection. Interpret Pythagoras' Theorem and replace the words with the areas of each square. At this point in the lesson, you will here gasps of understanding - the Ah Ha! moments.

Conclusion
I conclude by giving some homework that involves finding the unknown hypotenuse. The worksheet I use is from a set of books called, "Math-O-Magic". The worksheets have corny jokes but the students like doing them. The books are available at http://members.iinet.net.au/~markobri/mom.html

I am sure you will enjoy this lesson. Have fun!