Discovering the Pythagorean Theorem

Geometry, according to Wikipedia, is “Earth-measuring” and a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of shape.

Geometry, according to some students, is completing worksheets that have “diagrams are not to scale” written on them, solving for unknown sides of meaningless triangles, and being told formulas created by some guy named Pythagoras.

In the past, I contributed to the second definition of geometry by telling students the Pythagorean Theorem and not giving them a chance to discover it themselves.  After telling them this theorem, I destroyed any interest they might have had by handing out a plethora of worksheets.  We need to stop this destruction of geometry and start showing students the true power and capabilities that geometry opens up.

How can this be done? 

First, stop telling students what the Pythagorean Theorem is and let them discover it for themselves.  If this theorem is an outcome of your course, but you are unfamiliar with how to show it, click here for 92 PROOFS OF THE PYTHAGOREAN THEOREM.

I have not integrated all 92, but here is one discovery method:

Put students in small groups and give each student a stick that is 5 feet in length (or some smaller ladders might work, and you might want to go outside for this activity).  Instruct students to lean the stick against a wall and have them measure the distance the bottom is from the wall, how far up the wall the stick reaches and long the stick is.  Have them complete this for different distances.  The challenge is to eventually be able to determine how high the stick reaches up the wall by only measuring the distance from the wall the stick is, and by knowing the length.

 Eventually, some groups might come up with the relationship and some groups might struggle with it.  For the groups that are having troubles, guide them to put the stick 4 feet away from the wall.  This way the stick will reach 3 feet up the wall.  You can also give them hints as squaring the distances.  Guide them towards the theorem, but don’t just give up and tell them the theorem.

As more and more groups are discovering the relationship, you can ask them to try their theory against walls that are not exactly vertical.  This will show them that this theorem only works for right angled triangles.

The important aspect of this one activity is the meaning created behind the activity.  You will witness that before the theorem is even talked about, students will start to see that by changing one length the other two lengths also change.  They will also realize that the stick will always be the longest side. Two ideas we sometimes skip over or take for granted that students understand.  This activity can also be used to introduce the TAN, SINE, and COSINE ratios.