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Toothpicks and Geometry
Name: Jaxson J.
Status: student
Age: N/A
Location: N/A
Country: N/A
Date: N/A
Question:
My students in my class have just completed a series of
puzzles using toothpicks. However, in the teacher's workbook, there was a
question asked by one of my students, that it did not answer. The student
asked what was the relevance of this activity to geometry. As I am just a
first- year teacher I would like some help. Thank you for your time and effort.
Replies:
Jaxson,
As I have not seen the puzzles, I cannot provide a definite answer. Still,
I can say what I expect is true.
Puzzles with toothpicks, or any other combination of straight-line objects,
gives a real-object introduction to geometry. Many things can be discovered
about geometry with toothpicks. Three toothpicks together makes an
equilateral triangle. Twelve toothpicks arranged correctly can demonstrate
the Pythagorean Theorem (3-4-5 triangle). By building a hexagon from
equilateral triangles, you can demonstrate that the three equal angles are
each 60 degrees. Such puzzles, used wisely, can demonstrate geometry beyond
lines drawn on a piece of paper.
Dr. Ken Mellendorf
Illinois Central College
Tooth picks represent straight line segments of equal length, say L. You
could of course change that by lining them up to get L, 2L, 3L and so on. Or
you could break them and get roughly 1/2 L, 1/3 L, and so on. But assume for
the moment you do not do this.
Line segments are the "stuff" of which Euclidean geometry is made so there
are a lot of relevance. Here are just a few that come to mind:
You can illustrate the Pythagorean theorem for a right triangle of sides: 3,
4, and hypotenuse 5 and lay out squares on each side and demonstrate that
S3^2 + S4^2 = S5^2
Construct a rectangle of sides S3 and S4 with diagonal S5 and show that the
area of the triangle is 1/2 the area of the rectangle.
Use some glue to make a tetrahedron using six toothpicks.
Use 12 toothpicks to make a cube and illustrate that the volume is V = area
of the base * height.
Lay one toothpick vertically. At its base, lay one toothpick horizontally
(to the right). At the left put another toothpick vertical, and lay 2
toothpicks horizontally. Repeat the process so that there are
0,1,2,3,5,8,13, ..., (n-1)+(n-2) horizontal toothpicks separated by a single
toothpick laid vertically. This is the famous Fibonacci sequence. Have the
students calculate the area and the perimeter of the ever increasing sized
right triangles.
Try to construct an isosceles triangle. What is its perimeter, and area?
Back to Pythagoras: Make various right triangles with different numbers of
toothpicks for each of the sides: e.g. (1,1), (1,2), (1,3),(2,2), (2,3) and
so on. Have the students use other toothpicks to measure the hypotenuse,
and challenge them to come up with the result:
(S1)^2 + (S2)^2 = (S3)^2 by inductive proof! That will take care of any
smart alecks.
Use the toothpicks to make a horizontal x-axis and vertical y-axis and
illustrate the number line and the concept of ordered pairs, and negative
numbers (a concept the ancients had a really tough time accepting -- how can
something less than nothing have any meaning??). Show things like (+2) -
(-3) = 5, (-4) - (-2) = -2 etc.
Computer talk: How many toothpicks does it take to make all the digits 0-9
like on a calculator?
I hope these are enough examples to establish relevance.
Vince Calder
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Update: June 2012
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