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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∘

, 45∘

,

60∘

. 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

∘

∘ ∘

Figure 1.

They write: Let ; then (since sin 30∘ ),

and as well because . Also, √3, so

√3. Let . Using the Pythagorean theorem,

= 1 √ √ √3.

2

I

n this short note we describe some incidents in mathematics

teaching— as they actually occurred in the classroom.

in the classroom

Explorations in

Teaching

Bharat Karmarkar

Keywords: Exploration, trigonometry, sine, surd, square root, ratio

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At Right Angles | Vol. 3, No. 1, March 2014 39

Hence:

sin 15∘ = 1 = 1

8 + 4√3

.

4 + 2√ √ .

Hence:

sin 15∘ = 1

√

= 1

√ √ .

This can be written this in different ways, e.g.:

sin 15∘ = √3 − 1

2√2 = √ √

4 = √6 − √2

4 .

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?

Try it!

3

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 learningclubpune@gmail.com.

• cm (given)

• cm

• cm

• is the required point

Figure 2:

4

Figure 2.