## Friday, September 13, 2013

### Area and circumference of circles

The Khan academy does a wonderful job presenting a single concept and solving  problems that use the concept.  The solution of many problems involves concepts taught in multiple videos.  Teachers can present multiple videos to students and ask them to solve problems that demand knowledge of more than one concept.

The first problem in this post requires knowledge about both the circumference and area of a circle.

Khan academy video on the circumference of a circle

Khan academy video on the area of a circle

Problem One:  A rancher has one hundred meters of fence.   What is the radius of a circle that uses all 100 meters of fence?   What is the area of the circle?

Answer to Problem One:  The circumference for the circle is C=2x Pi x R so  the radius is

R=100/(2x3.1415)

or

R=15.9

The area of the circle is A=pi x R2

Using the numbers in our problem the area of the circle is

A=3.1415*15.92

or

758.4 square meters

Problem Two:  A student solving problem one above found that R=65.8.  He looks at his spreadsheet and can’t find the mistake.  What mistake did this person make?

Answer: It is likely that this student got confused about the rules governing the order of operations.   100 / pi x 2 is not the same as 100 / (pi x 2)

At some point it may be useful to show the student the following videos.

Problem Three:  What is larger a circle with diameter of 10 meters  or a circle with an area of 120 square meters?

Answer:  The circle with a diameter of 10 meters has an area of 78.5 meters square.

(A diameter of 10 meters is the same as a radius of 5 meters.  The area therefore is (3.1415 x  52) or 78.5 square meters, which is smaller than 120 square meters.

it is easy to create more word problems that both build on the videos that I have shown here and require the student look at more videos.

What is larger a circle with a diameter of x meters or a square with a perimeter of y meters?

You would have to show videos on perimeter and areas of squares here.

I need to think about this one.  What is meant by larger -- the object with the largest circumference or the object with the largest area?  Area should be correct but some thought would not hurt and the problem might be reworded to avoid ambiguity.

My main points here are that Khan videos are a great tool and teachers need to help students see the interconnections across concepts by creating problems that rely on the knowledge of multiple videos.

I am continuing my writing on school reforms and teach mathematics at