Short mathematics task for K9-K12

The concept of convergence and the area of a funny figure

Build a robot that can use the laser pointer to point on a blackboard.

Draw a rectangle on the blackboard. Calculate the area of A_{rectangle} of the rectangle.

Draw a circle on the rectangle. Calculate the area of A_{circle }of the circle by using the well-known formula.

Now we want to decide the area of the circle by counting marks.

Count how many times the mark lands within the circle, and how many times the mark lands outside the circle, but still within the frame of the rectangle.

Let A_{1} be the first approximation to the area of the circle.

A_{1}=

Compare the A_{1} which has been calculated above to A_{circle}.

Run the program again and decide A_{2,} which is the next approximation to the area of the circle, there are now 50 marks in total on the wall (not all of them will be inside the frames of the rectangle)

Compare the A_{2, }which has been calculated above, to A_{circle}.

Now calculate the corresponding A_{3 }and A_{4 }for when there are respectively 75 and 100 marks on the wall.

You should now be able to decide the sequence of A_{1,}A_{2,} A_{3,} A_{4}… They are nearing A_{circle}, this is the process of converging.

Now draw a funny figure on a rectangle. Use a piece of A4 paper or a piece of carton.

By using the method above, decide the area of this funny figure.

If you want an answer to compare to, you can weigh the rectangle then cut out the funny figure and weigh that too.

You will now be able to use the formula