# Enjoy this Maths Teaser

Sriranjani Ranganathan, IT for Change

Teachers Response Let a=x

a+a=a+x

2a-2x = a+x-2x

2(a-x)= a-x divide bothsides by a-x

2=1

Bindu Thirumalai IT for Change You have said that let a=x in step 1. Therefore this step 2(a-x)= a-x divide both sides by a-x is not possible because a-x = 0 and you cannot divide by zero, its undefined.

Sriranjani Ranganathan, IT for Change Yes, this is the fallacy. Rules of cancellation and algebraic manipulation do not hold when zero is involved...

# CaRMetal is a Geogebra like free software

Last week I installed a new software which was available in Application--> Ubuntu Software Center ---> Education --> CaRMetal. Within minutes the software installed and to my surprise it was just resembled Geogebra. It is easier than Geogebra but I don't know whether it is better than th latter. I found it very easy and I needed some geometric figures to set a question paper for my 10th Boys. I exported it as .SVG file. It was very smart, I could scale it up or down in a Writer file when inserted in to it as picture. The file extension of the CaRMetal file is .zir. Later I found that the software was designed In Germany by Eric Hakenholz and is in use since 1989. [ Quite old] Please follow the Wikipedia @ http://en.wikipedia.org/wiki/CaRMetal

Bindu Thirumalai, IT for Change Dear Tharanath and Radha and All Teachers First of all I am glad you are exploring more tools. Two is always better than one. This is the beauty of Public Software, we always have more variety to choose from. And now that you know GeoGebra, you will easily be able to try and experiment with CaRMetal. Most mathematics tools will follow similar processes.

Where you can use CaRMetal. 1. For pure Geometry constructions 2. For 3D visuals which Geogebra does not have as yet 3. Animation is there, it is not as simple as sliders , we can explore it a little more

Where to use GeoGebra 1. When we combine Algebra and Geometry - this feature is not as good in CaRMetal. 2. When we combine statistics (spreadsheets) , with algebra, charts - Geometry - this feature is not there in CaRMetal.

# P th term of an AP is Q and Q th term of an AP is P . Then find PQ th term ?

Mallikarjun Sudi, Ghs Yelhari All maths teachers pls solve this problem P th term of an AP is Q and Q th term of an AP is P . Then find PQ th term ?

Sneha Titus, University Resource Centre, Azim Premji University This is a very nice problem. Here is the solution that I worked out.

Let the nth term be Tn = a + (n‒1)d, where a is the first term and d is the common difference.

Given: Tp = q = a + (p‒1)d

And Tq = p = a + (q‒1)d

So Tp ‒Tq = q ‒ p = a + (p‒1)d ‒[a + (q‒1)d]

```                   q ‒ p  = pd ‒ d ‒ qd + d
```
```                  q ‒ p  = (p ‒ q) d
```

∴ d = = ‒ 1

And since Tq = p = a + (q‒1)d,

```                            p = a + (q ‒1)(‒1) = a ‒q + 1
```
```               so that a = p + q ‒1
```

Now, Tpq = a + (pq‒1)d = a + (pq – 1)(‒1) = a ‒pq + 1 = p + q ‒1 ‒pq + 1 = p + q ‒pq

Example T2 = 4 and T4 = 2, what is T8

T2 = 4 = a + (2‒1)d = a + d

T4 = 2 = a + (4‒1)d = a + 3d

T2 ‒T4 = 4 ‒ 2 = a + d ‒[a + 3d] = – 2d

```              2 = ‒2d
```

∴ d = ‒1

And T2 = 4 = a + (2‒1)(‒1) = a ‒1

∴ a = 4 + 1 = 5

And T8 = a + (8‒1)d] = a +7 d = a ‒ 7 = 5 ‒ 7 = ‒2

The A.P with a = 5, d = ‒1 is

5, 4, 3, 2, 1, 0, ‒1, ‒2,……….

Notice that T2 = 4 and T4 = 2 and T8 = ‒2

# Which day is Pi Day ?

Mary Shyla, GMPS Begur Which day is Pi Day ?

Sneha Titus, University Resource Centre, Azim Premji University March 14 is called Pi Day (3-14) The date is in American format For more information you could check out [[1]] Incidentally June 28 (6-28) is called 2 pi day!

# The having of wonderful ideas

M N Baig, Director, RMSA, Karnataka Dear Ms Bindu, It is an excellent article. Regards, mnbaig
N J Krishnan Thanks! Definitely worth reading.

Incidentally this provoked me into investigating Ms Duckworth further and found the following further readings.

1. http://en.wikipedia.org/wiki/Eleanor_Duckworth Wikipedia on her. Do read the sections 'The main ideas of teaching/learning research' and 'some teaching/learning examples'

2. http://www.criticalexplorers.org/investigations/ This is a compilation of ongoing investigative approaches to specific topics in History.

3. http://www.criticalexplorers.org/blog/categories/classroom-teaching/ - The blog linked to the above web site. Do read the pieces by Mike Fishback - particularly the one on Khan Academy.

# Trignometric Ratios

Radha Narve, GHS Begur Contrubuted this GeoGebra File

1. The GeoGebra file below is an introduction to basic trignometric ratios
1. consecutive interior angles http://www.karnatakaeducation.org.in/KOER/Maths/trignometry introduction with basic ratios.html