Using Biological Populations to estimate
Pi and Find the Formula for the Area of a
Circle - A Cross Curricular Activity
This is a guide to a simulation using population data from a biology field
trip to estimate pi and to find the formula for the area of a circle.
We can use data from a Biology field trip to the beach, studying snails, crabs, or seaweed deposits at low
tide. (Ebbing or incoming tides will work, but avoid high tide).
While it is possible to look up pi or the formula for the area of a circle it is the mathematical process
that is important in this investigation, as well as the cross curricula connections.
Equipment: for a tightly concentrated population of snails we use a
30cm X 30cm square of acetate, with a 15cm radius circle drawn on the square.
Alternatively, use tape measures or rope to mark out a square and an inscribed
circle (Point to ponder: why does the actual size of the shapes not matter?)
30cm
30cm
Question 1:
Find an area with a fairly large number of snails which appear to be “randomly” distributed. Place your
acetate over the snail sand count the number of snails inside the square (including the circle), and also
the number of snails inside just the circle.
Record your data.
Snails inside the square (including the circle) :_________ Snails inside the circle only: ___________
Question 2:
Repeat the activity for a large number of samples. Ideally you would use several thousand snails so you
might combine with another group or groups. Do not worry if the sample areas overlap.
Sample
Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Snails
Inside
the
circle
Snails
inside
the
square
Sample
Number
Snails
Inside
the
circle
Snails
inside
the
square
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
Sample
Number
Snails
Inside
the
circle
Snails
inside
the
square
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
Totals
Question 3
a) What was the total number of snails inside the circle?
b) What was your total number of snails inside the square?
c) What are the formulae for the area of a circle and the area of a square, both in terms of r?
d) Using your formulae, what is the ratio of snails inside the circle/points inside the square?
What is the significance of this number?
e) Explain how you can use your result to estimate π
f)
Use the previous two answers to determine the area of any circle.
Question 8
Describe other populations you could use to estimate pi and explain why you think they would work.
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
Question 9
Explain what difference it would make, if any, if the population was not randomly distributed. (Is the
population you used likely to have been actually randomly distributed? Why/why not?)
_____________________________________________________________________________________
_____________________________________________________________________________________
_____________________________________________________________________________________
This version of the activity uses
seaweed deposits at low tide in
Porirua Harbour.
Notes To Teacher:
A. Suitable populations
This field trip-based approach will work for any population which is approximately randomly distributed
over a defined area.
B. How the mathematical process works
You should work through the following ideas through discussion and the
use of diagrams:
•
•
•
The area of the square is proportional to r2
The area of the square is 4r2
The area of the circle is proportional to r2 but not by a factor of 4.
Let’s call the factor π. So area = πr2
•
The ratio of the
•
Multiply the ratio by 4 to estimate pi.
•
Use the value of pi to calculate the area of any circle.
C. Links to Key Competencies
Depending on the amount of scaffolding the teacher gives the students in planning and executing this
activity the following key competencies emerge:
•
•
•
•
Managing self if the activity is set as a homework assignment, and resilience in dealing with the
uncertainties of field work.
Relating to others if the students participate in small groups to gather data and make initial
estimates, then combine their data with other groups to address variation.
Participating and contributing in that even the most mathematically challenged student can be
assigned a meaningful role in the activity.
Thinking in
o transferring the activity to populations that are specific to their own environment – e.g.
identifying other populations that would give similar results if a population of snails is
not available
o testing and reflecting on hypotheses
o
making connections between school and the outside environment, across curriculum
areas, and across mathematics (probability, statistics, algebra)
Andrew Tideswell
[email protected]