P-Numbers
Introduction
In today's lesson, we will be investigating the relationship between the perimeter and the M-value and P-value as the number of sides of the regular polygon increases.
Play around with this.
What do you think the orange line is? What about the red line?
What have you noticed about the number of sides it is increasing by? Why do you think that is the case?
How do you think the P-value is calculated? What about the M-value?
What happens to the shape as the number of sides increases? What does it start to look more and more like?
What happens to the P-Value as the number of sides increases? What about the M-value?
Let's now collect some data and check and see if your observations are right!
You can either use this data collection sheet and this graph paper (PRINT US)
Or make a copy of this spreadsheet (you may wish to turn this into a Google Sheet)
Fill in the white columns.
Can you work out what the formulas of the grey columns are and what they do?
Can you present your findings on a graph? Hint: p-number on the vertical axis against number of sides on the horizontal axis.
Does it match with your expectation?
Further Questions and Challenges
When n=6 the P-Value and the M-Value both start with a ‘3’, but the numbers after the decimal point are different.
How big does n have to be for the first number after the decimal point to be the same for both the P-Value and the M-Value? That is, they are both “three-point-something” where the first digit of the “something” is the same.
How big does n have to be before the second digit after the decimal point is the same for both the P-Value and the M-Value?
What about for the third digit after the decimal point?
If you could add more sides, do you think the P-Value and the M-Value would ever be the same? Why/ why not?
As the number of sides increases, the P-Value and the M-Value become closer. Try to find out more about the number that they are heading towards. Does it look like a number that you are familiar with? What is it? Find out what the answer is here.
Further Practice
Some of the essential skills introduced in this lesson are π (pi) and the circumference of a circle. You may also wish study ahead and learn about the area of a circle. Practise these relevant skills on DrFrostMaths, CorbettMaths, MyiMaths and Eedi.
Transum
Circles - try this self-marked exercise. There are altogether 6 levels. Only the first two are relevant, but you will learn more about circles in the following lesson.
In Terms of π - try this self-marked exercise. There are altogether 2 levels.
Pi-mon - want to memorise the digits of pi? Do it with help from some musical notes!
For more goodies on cicrles on transum, click on the hyperlink.
Area and Circumference of Circles
Ken Stafford: Past Paper Qs : Solutions
Dr Frost : Questions (Ans. inc.)
For some textbook practice, see below, answers are here:
10 Ticks - Circle Perimeter Problems.pdfExtension
There are some excellent videos about π including A Brief History of Pi and here are some more advanced ones: Pi, the "celebrity number' and 5 Facts You Should Know About Pi and for something fun Why is The Simpsons not in Base 8? In this video Simon Singh talks about Pi and Maths in The Simpsons cartoon.
Try this circumfraction puzzle which is loosely related to circumference.