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38 At Right Angles | Vol. 3, No. 1, March 2014
In this short note we describe some incidents in mathematics teaching
—as they actually occurred in the classroom.
Finding the sine of 15 degrees
It is class IX. The teacher is teaching trigonometric ratios of 30∘
. Suddenly an idea pops up in his mind. He draws Figure 1 and
asks the students, “Can anybody calculate sin 15∘
This topic is not in the IX-th standard text book. But a few students
accept this challenge, a few more silently appreciate their efforts to
solve the problem, and the rest wait for the period to end! Two days
They write: Let ; then (since sin 30∘ ),
and as well because . Also, √3, so
√3. Let . Using the Pythagorean theorem,
= 1 √ √ √3.
n this short note we describe some incidents in mathematics
teaching— as they actually occurred in the classroom.
in the classroom
Keywords: Exploration, trigonometry, sine, surd, square root, ratio
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At Right Angles | Vol. 3, No. 1, March 2014 39
sin 15∘ = 1 = 1
8 + 4√3
4 + 2√ √ .
sin 15∘ = 1
√ √ .
This can be written this in different ways, e.g.:
sin 15∘ = √3 − 1
2√2 = √ √
4 = √6 − √2
Dividing a line segment in a given ratio
It is class X. The teacher is teaching the chapter on similarity and angle bisectors. A question is posed:
“Draw a line segment o� length 8.3 cm. Using compass and ruler, locate a point on it such that
.” Most students use a method based on parallels, but Figure 2 shows an imaginative
solution, based on a different idea; it is presented without any words. Can you explain why it works?
BHARAT KARMARKAR is a freelance educator. He believes that learning any subject is simply a tool to learn
better learning habits and a better aptitude; what a learner really carries forward after schooling is learning
skills rather than content knowledge. His learning club, located in Pune, is based on this vision. He may be
contacted at email@example.com.
• cm (given)
• is the required point