Maps, Routes and Mathematics

What is a Map? And when did you use a route map lastly? oh, and what does math has to do with maps? Well, lets see.

Long long ago, actually not so long ago, before the google map and GPS we still travelled.👍
We found our way through, in unknown places and yes People discovered places...
The most important tools which we used were The Sun, Moon and the Stars and of course the Cardinal Directions (East, West, North and South). Simple right!.


Focus Points

·       Visualizing objects/space and representing them in a paper

This is a beautiful concept to develop the observational skills among children. I always use the “count and recollect the objects game” for kids. In this game a specific number of things (say 20) are displayed in a room. Kids can observe the object for 10mts and to list them down in a paper, without seeing. Well, one must be conscious about their surrounding right! Now in this concept, you do not just list them down. You draw them out!!! Excellent.

·       Orientation – Relative position of objects in space w.r.t cardinal directions

Age old traditions use nature as relative constant. The Sun, The Moon and Yes “The Pole Star”, which has been the sailors’ compass to reach shores safely (That is opportunity for an interesting narration/story). Where is the north of your classroom and how would you identify that? Once you have done that you can draw the floor map of your classroom, then your school, your community, state, country and what not! But if you do a map without marking your north, what can go wrong?

·       Idea of dimensions (representing big/small objects in picture)

Ok, I got my North right. Its time to put down things in paper. There are different things here. Like the door which is bigger than the window. The benches and chairs and things big and small. How does my eyes perceive them? How can I help other see this in a drawing? Yup get your creative skills do the task. There is no right or wrong, good or bad… just the intention to communicate through picture. Imagine you are helping a new student to navigate your class, now the challenge is you should do this only through drawings.

·       Idea of distance (and scale)

It’s a drawing right, so we put down biiiiiiiiiiiiiiig spaces (I mean length and breadth) in small paper. But be careful with which one should be long/short… have you ever thought, Why are rooms invariably rectangular spaces? (Golden Ratio)


So, Did you discover the “Still Life Artist” in you today?

The key factors to logically approach a destination in the shortest possible time (yup the math of movement) lies in positioning things relatively, with symbols for reference. When kids are taught with these two factors and use natural reference, we are teaching them a survival skill.
Key points to focus
1. Compass rose/Cardinal directions (East, West, North and South)

The figure used to represent the orientation of the cardinal directions, in a map. It can be as simple as just an arrow marking the North, or a elaborate one like this giving all the eight directions. Without a compass rose, your map will not be valid.

2. Symbols (Pictorial representation of Physical Landmarks)

3. Map Key or Legend (helps to understand the map)

(https://clipground.com/images/key-mapping-clipart-3.jpg)

Doing a class activity on map symbols will be fun. Visualisation is a very important skill to understand the language of Nature.


Food for Thought!!!
Indian topography is very ancient. The main sources of ancient Indian geographical works are the Vaidikas, the Ramayana, the Mahabharata, the work of Buddhists, Jains and the Puranas.
The rishis of Rigveda initially formulated the principle of four directions, i.e., Purva (east), Paschima (west), Uttar (north) and Dakshina (south).
By adding Zenith (Meru) and Nadir (Badavanala), it was raised to six. Afterwards eight and ten directions are frequently mentioned in the Puranic literature.

Wondering about Squares and Cubes

G is my friend who comes home, to get help in mathematics. He is extremely passionate and crazy about cricket. He spends close to 5hrs in the ground with his district team. Well, he is in class 8. When we met for the first time, he said he is not able to follow anything in math class, so he usually shuts off. He would be glad if I can help. "My pleasure". I love one-one time specifically for learning. G doesn’t talk much. I must be very sensitive to his very subtle facial expressions to know what he feels. And most of the times he is tired after his cricket practice.


It was a sunny evening and G walked in at 6pm after his field practice. We sat down and after over a cup of fresh mint juice, started with cubes and cube roots (chapter7 NCERT). I was telling him about cubes what they are… And G gave me this very deep thoughtful look!. “why am I learning this?” he slowly asked, Is it really important to know this???????????. I was not sure how answer that question. To me numbers are always exciting, as exciting as playing cricket was for G.

There are just 4 perfect cubes from 1-100. 
1  =1x1x1
8  =2x2x2
27=3x3x3
64=4x4x4

Guess what? cubes were in news recently with a major breakthrough. Diophantus of Alexandria, a 3rd century mathematician had posed the problem of sum of 3 cubes.

x³+y³+z³=k, with k being all the numbers from one to 100.

Looks simple?, well not really. Enjoy reading
This was in 1994
http://www.asahi-net.or.jp/~KC2H-MSM/mathland/math04/matb0100.htm 

Note the "???" for some numbers 

In March 2019 Andrew Booker solved the sum of 3 cubes for 33In September 2019 Professors Andrew Booker and Andrew Sutherland found the solution for 42 by using Charity Engine
https://phys.org/news/2019-09-sum-cubes-solvedusing-real-life.html

Watch them in action!!!
for 42
for 33
for 74

Area and the space within....

Yesterday my daughter came up with the classic doubt, "I don't understand what Area is?"... can you help?

Well to begin with a story, 

That was the time of river valley civilisations. Man started growing his own food by the banks of rivers as it was convenient source of water for your plants. Now, the ‘societies’ or ‘living in groups’ was just beginning and each one chose a place for themselves. Eventually, the question of "which pace belongs to whom?" cropped up. Now to measure a piece of land, it is not enough if you just measured the length! oh there are two sides... 

Ok then you measure the length and breath. Is that enough? No... the land belonging to a person is a closed space you see, with 4 sides. All this while people had to measure only length or breath. (We say one dimension). Now they must measure a closed space, which has both length and breath. So, you need a "New Unit", of measurement (two dimension). That's how they came up with the concept of "Area of a Space".

Now imagine, this is a piece of land belonging to you.

How do you create a unit of measurement, with what you already know? Let’s try. 

The smallest unit for measuring length which you use is 1cm? Ok. Now consider a closed space with all 4 sides=1cm. What shape will it form? It will be a square. Let’s call this 'a unit square with area 1square cm'. 

 

 Area 1 cm² = 100 mm²

Like your 1cm which is smallest unit for measuring length, this 1sq cm is the smallest unit for measuring area. (No, you don’t have a physical object like your ruler/scale for this). Area of your piece of land will now be equal to the number of unit squares it occupies…

 

which is 3sq cm in this case. You say the area of your piece of land is 3sq cm. (pictures not to scale)

The unit can be cm, m or any unit you choose; accordingly, the area will be in sq cm, sq m etc.

We have learnt another measurement today and this is for 2d spaces. The unit of Area is sq. units 

Let us list the measurements we know - 

One Dimension Measures: Length, breath, height

Two Dimension Measures: Area, Perimeter, circumference (for circle)

Inclined Measures: Angle

Revising Metric units of length:

10 mm = 1 cm

10 cm = 1 dm (decimeter)

10 dm = 1m 

(100 cm = 1 m)

1000 m = 1 km

That's about it!!!