Case Study Y

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Below is an HTML copy of the teacher's guide to the LOGO activity featured in Case Study Y.

Objectives

To give an understanding of

1. the fact that polygons are an approximation to circles as the number of sides increases
2. that there is a relationship between the perimeter of the polygon (cf. circumference) and the distance across the centre of the polygon (cf. diameter).

Some reminders about using LOGO may be necessary. Although the routines for drawing the polygon are provided, it might be profitable if you have time to consider how they work. It might also be a good idea to try some empirical investigation on paper of the relationship between the circumference and diameter of circles, to give an idea where the investigation might be heading.

Lesson Ideas

Open the file P:\MATHS\PI_INVST.LOG. This will give access to two new routines:

scale n l : draws a scale below the current cursor position with n divisions each l units wide.
polygon n s : draws a polygon with n sides each of length s.

The polygon routine is used to draw a polygon, leaving the pointer facing across the centre of the polygon. If the scale routine is to be used, then it should be used before the polygon routine.

Once a polygon has been drawn, a combination of the forward (fd) and backward (bk) commands can be used to determine the distance across the polygon. Note that it is the back of the triangle pointer which denotes the current position.

It is a good idea to try two or three different side lengths for each polygon to show that the ratio of perimeter to ‘diameter’ remains roughly constant for the same number of sides. A possible table of values to try could be:

Number Of Sides (N)	Length Of Sides (S)	Perimeter Of Polygon (P)	Distance Across Centre Of Polygon (D)	Ratio Of Perimeter To Distance Across Centre (P/D)
5	50	250	...	...
5	60	300	...	...
5	70	350	...	...
6	40	240	...	...
6	50	300	...	...
6	60	360	...	...
7	40	280	...	...
7	50	350	...	...
8	30	240	...	...
8	40	320	...	...
9	30	270	...	...
9	40	360	...	...
etc.	...	...	...	...

The pupils should notice that the odd-numbered sides give lower values than the even-numbered sides. This is because the even-numbered polygons consistently over-estimate the value of PI. This can be a useful discussion point.

The end result of this investigation should be an understanding that there is a relationship between the circumference and diameter of circles, but reinforcement in lessons will be necessary.