I Taught It Wrong.

I realized that I taught certain things in the wrong order in Geometry this year.  I usually teach 2 dimensional figures first, followed by volume and surface area of 3 dimensional figures and if I have time, I throw in circles.

After this year of teaching the material, I've come to the conclusion that I need to do 2 D figures such as triangles and rectangles first and include both circumference and area of circles.  This might make the transition into volume and surface area for 3 dimensional figures easier.

I hate to admit I made a mistake when I did not take into account the best order to teach this material.  Ohh well, now to look at how to help reteach the material.  I think I'm going to spend time introducing circles, the formula for area and circumference.

Picture 1 - square

I have this activity to introduce circles to my Geometry students.  I challenge them to create a circle from a piece of paper, scissors and a pen.  They may not use anything else, other than those three items.  A female elder from Dillingham shared it with a class I took. It is the method her mother taught her to make circles as they didn't have compasses to use.

First you cut the piece of paper into a square since a circle is x^2 + y^2 = r^2 and the r is the radius or the distance from the center of the square to the edge.  So you cut a piece of paper that is exactly the length from the center of the square to the edge after folding the square so there are four lines creased into it.  (Picture 1)

Picture 2 - With first set of cuts

Now take the short strip of paper and use it to mark a distance up the diagonals from the center to the vertex.  Mark the distance for each vertex.  Fold over the corners to form a small triangle with the base being perpendicular to the diagonal.  Cut the triangles off so you now have 8 vertex. (Picture 2).

Picture - 3 With second set of cuts

Repeat the process so you've made 8 smaller triangles that you cut off so now you have 16 smaller vertex and your square is starting to resemble a circle.  (Picture 3).  If you repeat the process a couple more times, you have something that looks almost like a circle and with a bit of trimming, you've got a circle. 

This is based on Archimedes theorem that if you have a polynomial with enough sides, you end up with a circle.  Out of all the years, I've done this, only one or two students figured it out.  Usually after 10 minutes of letting them struggle, I sit over at the side and slowly work through the process so students can wonder over and check things out. 

I love challenging them.  Give it a try.