let's put history into our mathematics classrooms

Let's Put History into Our Mathematics ClassroomsAuthor(s): Charalampos ToumasisSource: Mathematics in School, Vol. 24, No. 2 (Mar., 1995), pp. 18-19Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30215151 .

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LET' SPUT

history IN OUR

CLASSROOMS

CLASSROOMS

by Charalampos Toumasis

Many people have spoken and written at times about the need to integrate the history of mathematics into mathematics teaching (Jones, 1957; Fauvel, 1991a). Especially in recent years the importance of the history of mathematics in relation to the teaching of mathematics was widely recognized and promoted (Fauvel 1991b). However, while one can find many articles and books on the history of mathematics, one will scarcely find so much as a hint on how to integrate the material into teaching at school. This problem is a general one. In fact it is difficult for the mathematics teacher to find easily accessible historical material and develop activities for classroom use. In this article I shall present an example of such an activity which uses an idea from the past, offering at the same time an excellent opportunity to the students to apply some trigonometric identities and make use of hand-held calculators.

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The trisection of an angle is a problem that frequently arises in geometry classes. Although the impossibility of this construction, using only a compass and straightedge, has long been established, many students and practitioners of mathematics have sought to develop an accurate but approximate construction with Euclidean tools (Eves 1990, p. 12; Yates 1971). One such simple approximate construction is called the Snell approximation (Eves 1990, p. 131). One simple variation of the Snell's approximation is the following:

Suppose we are given an acute angle O to trisect having measure 9. On its sides take two points A and B respectively such as OA=OB. Let C be the midpoint of OA. With B as centre and BC as radius draw a circle. With A as centre and AC as radius draw another circle. These circles intersect at D. Our desired approximation will be angle DOA (figure 1).

B

D

C A

Mathematics in School, March 1995



We can offer a justification for this construction which only uses the law of sines, the identity sin (x -y) = sin x cos y-cos x sin y, and the half-angle formula

9 sin 39 tan - 2 1 + cos 9

(Kaufmann 1989, p. 784).

x ~Y(

0C A

If x represents the measure of the angle DOA we can compare x with 35/3, expressing x as a function of 3. In figure 2, BC = BD and AC = AD as radii of the same circle respectively. Thus, AABC AADB. Moreover AAOB is isosceles and

180x-9 39 m L OBA= m L OAB - 2 - 90x

2 2 2

Also, m L ODA = 180x - 2(90x - 9/2) - x= 9- x.

Using the law of sines in triangle ODA we have

sin x sin (9 - x) sin x sin (9 - x) AD OA AD 2AD

Therefore,

sin (9 - x) = 2 sin x

or

sin 19 cos x-cos 3 sin x = 2 sin x.

Dividing by cos x yields

sin 3 sin 3 - cos 3 tan x= 2 tan x or tan x= (1)

2 + cos 39

From the half-angle formula, we also have

9 sin tan - (2)

2 1 + cos 3

If we solve (2) for sin 3 and substitute this into (1), there follows

(tan 3/2) (1 + cos 3) tanx= (3) 2 + cos xq

But as it is well known, when 9 (Measured in radius) is small, tan 9 39 and cos 9 x 1. Thus, taking 9 (therefore x) to be small (3) becomes

9/2(1 + 1) 89 2+1 3

That explains why we have a reasonable approximate construction for small angles.

In fact, x is always smaller than -. This can be shown as 3

Mathematics in School, March 1995

follows: Let OD cut the circle (A,AC) at E. Then, since OE<AE, y>x (1). Therefore, z=x+y (2) and 9= x+ (-x)=x+mLODA=x+z (3). By (1), (2) and (3) we get

9= x+x+y= 2x+y> 3x and x< - 3

Since L AOB is an acute angle, if we go on a bit we have the more interesting fact that

3 sin 3 tan - > tan x-= 3 2 + cos 9"

A good exercise for the students is to ascertain how close is this approximation to the real value of 39/3. Using a hand-held calculator we can illustrate it better. The following table (in degrees) offers an overview of how close

x2 + cos is to 9/3.

2+o

3 3/3 x

10 3.3333333 3.3295604 20 6.6666666 6.6362698 30 10.000000 9.8960887 40 13.333333 13.082487 50 16.666666 16.164881 60 20.000000 19.106602 70 23.333333 21.862216 89 29.666666 26.362974

As a teacher of high school mathematics I have very often made use of the above approximation in my trigonometry or geometry classes. Students always find it very interesting to deal with this construction, which offers an excellent opportunity to them to apply some trigonometric identities and make use of hand-held calculators. Moreover, it can help them to understand the fact that a problem in mathematics is a problem in mathematics and not in geometry or trigonometry or algebra only. The National Council of Teachers of Mathematics Curriculum and Evaluation Standards for School Mathematics (1989, p. 148) encourages the exploration of interconnections among mathematical ideas, as well as the use of calculators when an approximate answer is adequate. An efficient way to introduce students to the "real world" is to pursure approximate answers that are "close enough". The construction presented in this article, inspired from the past, exhibits some connections between trigonometry and geometry and can stimulate a worthwhile discussion of approximation theory. 1-x

References Eves, H. (1990) An introduction to the history of mathematics, 6th Edition,

Saunders College Publishing. Fauvel, J. (1991a) Using history in mathematics education, For the Learning

of mathematics, 11, 2, 3-6. Fauval, J. (Ed.) (1991b) Special issue on history in mathematics education,

For the Learning of mathematics, 11, 2. Jones, P. (1957) The history of mathematics as a teaching tool, Mathematics

Teacher, 50, 59-64. National Council of Teachers of Mathematics (1989) Curriculum and

Evaluation Standards for School Mathematics, Reston, Va: The Council. Kaufmann, J. (1989) Algebra with trigonometry. Second Edition, PWS-

KENT Publishing Company. Yates, R. (1971) The trisection problem. Reston, Va: National Council of

Teachers of Mathematics.

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let's put history into our mathematics classrooms

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