geometry-labs

Lab 8.6: Pick’s Formula
Prerequisites: Lab 8.4 (Geoboard Area)
provides the necessary foundation.This
lab is also related to Lab 8.3 (A Formula
for Polyomino Perimeter), although that
is a prerequisite only for Question C.
Pick’s formula is a surprising result, and the
search for it is a worthwhile mathematical
challenge at this level.There is no need for
students to memorize it, but if they do, they
may find it useful when working on Lab 9.1
(Taxicab Versus Euclidean Distance).
Answers
1. 4.5
2.
Inside dots Boundary dots Area
3 5 4.5
4 4 5
2 12 7
3. Answers will vary.
4. The area increases by 1.
5. Answers will vary.
6. The area increases by 1 2 .
b
7. A i 1
2
Discussion Answers
A. There is no limit to the area for a given
number of inside dots, because a tail of any
length can be added to the figure without
adding any more inside dots.
b
B. A 2 1
C. It is essentially the same formula, though it is
more general.The perimeter of a polyomino
is equal to its number of boundary dots.
9 Distance and Square Root
Lab 9.1: Taxicab Versus Euclidean Distance
Prerequisites: Lab 8.5 (Geoboard Squares) is
essential, both for its introductory work with
square roots and for laying the visual and
computational groundwork for the calculation
of Euclidean distance.
The purpose of Problem 1 is to clarify
something that Euclidean distance is not
and also to help frame students’ thinking
about Euclidean distance, since in both
cases it is useful to use horizontal and
vertical distance (run and rise, as we say
in another context). If you think students
need to work more examples, those are
easy enough to make up. In fact, you may
ask students to make up some examples
for each other.
For Problem 2, students can use either their
records from Lab 8.5 or Pick’s formula to find the
area of the squares, or the Pythagorean theorem if
they think of it and know how to use it.
Problem 3 is an interesting but optional
extension. For a discussion of a taxicab
geometry problem related to Problem 3,
see Lab 9.6 (Taxicab Geometry).
Question A requires the use of absolute value.
If your students are not familiar with it, you may
use this opportunity to introduce the concept
and notation, or you may skip this problem.
The answer to Question C is a consequence of
the triangle inequality (in Euclidean geometry).
We return to taxi-circles in Lab 9.6.
214 Notes and Answers Geometry Labs
© 1999 Henri Picciotto, www.MathEducationPage.org