Arcs and Sector Areas

Currently in Geometry, the students are working on area, circumference, arcs, area of sectors, etc.  They are struggling to relate the topics and try to look at each topic as a separate topic.  I want to find ways that might help them learn to flow from one topic to the next.

I found this 39 page Pdf which contains a whole unit on arcs and area of sectors. It has everything from the pre-assessment to the lesson plan to the activities and the post assessment.  This is great in that it introduces radians so it would be perfect between circles and the trigonometric ratios.  I wish I'd found this about two weeks ago when I was starting this.  That being said, I could easily use parts of this to help my students learn the material better through the use of activities rather than relying on worksheets.  It is labeled as a concept development packet.

This site has a great pdf of cards with real life problems for finding circumference, area, radius, and other interesting things.  For instance one card has you calculate a sheet of gold to cut coins from but the sheet has to be of the smallest size so there is less waste.  I like the variety of problems since the problems are not all circumference, or area, or diameter.

This slide share shows where circles are found in real life from theater to car tires, to the earth, to manhole covers.  It has some good information and would be a great introduction to the card activity.

The Department of Education in Virginia has a great 5 page lesson plan and activities for arc length.  There are two activities included.  One finds the length between two ends which are actually arcs and requires use of the Pythagorean Theorem while the other is an activity that requires students to cut a 10 inch cake into 12 pieces.  The questions require thought so it will take them a while to complete.

This activity has students working to figure out the area of a large cookie that is at least 8 inches in diameter.  They end up calculating area and circumference of the cookie, the area of their slice and the length of the edge (arc) of the cookie.  When done they can eat it.

Finally, this power point presentation has some great practice problems for students that are done in such as way as to have a friendlier feel than most worksheets.  There are instructional slides scattered through out the problems.  I like the way its organized.

Have fun checking it out.

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.