Everyone learns!

At school we have been trying out some of the ‘three act’ lessons by Dan Meyer.  There has been a real buzz from them and, wonderfully, everyone (teachers and pupils) appears to have learned something from them!

A colleague, who hadn’t come across these lessons before learned what a great resource they can be.  He used Best Circle with his Yr 11 group.

The group learned more about circles, specifically that a circle is the shape that bounds the maximum area for a given perimeter.

One pupil in the class is a bit of a programmer: he took things further.  That afternoon he created a webpage (written in javascript) where you can draw a circle and it will tell you how accurate you are.  Brilliantly, this gives a percentage and features sad/happy kittens to show you how impressed the feline world is with your effort.  My colleague put it up on his website.  The page is here.

A couple of weeks on he turned it into an iPad app, which can be downloaded for free here.  

He learned that maths can be rather cool!

And me?  Well I saw yet another example of why sharing is so important, whether it is ideas, experiences, materials or iPad apps!

How do the programs work? 

Well, first of all they join up the line that is drawn so it forms a closed shape.  Then it measures the perimeter and the area of the shape. 

From this it calculates the radius you would get if the perimeter were the circumference of a circle and the radius you would get if the area were the area of a circle.  Then it divides the latter radius by the former one and gives the percentage.

Questions:

1]  Can you get close to 100% ?

2]  Can you get close to 0% ?

3]  What happens to the area if you draw a shape that crosses itself (such as a figure of 8) ?

Here are two alternative methods for giving a percentage score for the accuracy of a circle:

Method A]  Assume the perimeter drawn is that of a circle and work out the radius of the circle.  Use this to calculate the area of the perfect circle.  Compare that to the actual area drawn.

Method B]  Assume the area drawn is that of a circle and work out the radius of the circle.  Use this to calculate the perimeter of the perfect circle.  Compare that to the actual perimeter drawn.

4]  What is the link between the answers the two methods give?

5]  Which one is equivalent to the method used in the app?

6]  Which method is better and why?