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Name: Jaxson J.
Status: student	
Age:  N/A
Location: N/A
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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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