# Get the Knowledge that sets you free...Science and Math for K8 to K12 students

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Bernoulli's principle is a critical aspect of aerodynamic research, which made emergence of the aircraft as a valuable means of transportation.
Math - The language of science

Math is the language of science: be it physical or life. The beauty of the language is its ability to state truths with unique precision and convey volumes of information in brief terms. It is a language that is not far removed from our daily lives. Every time you step out of home or are at home, you do wave parts of the Math into your language.

Math finds application in business, industry, music, sports, medicine, agriculture, engineering, and just about any field you can think of. Take transportation, for instance. The flight of an airplane can neatly be explained by Bernoulli's {P + 1/2(p * v2) = constant} principle. As to why a spaceship follows a trajectory resembling the figure-eight (considered an optimum route), look no further than Newton's elegant equation. It is not a stretch, to say that, remarkable formulae such as E = mc2 have changed our world and lives, for the better.

Now and in the future, the usage of this language will be more extensive. Which is precisely why, you need to know concepts in arithmetic, analytical geometry, calculus, algebra, and probability. You need to know how the concepts are connected and inter-connected. Not only will this help you in Math, but will help you understand concepts in other fields like electromagnetic, hydrodynamic, thermodynamic, quantum mechanics and genomics clearly.

You can learn and practice this language until you get fluent in it with thousands of questions across topics, clear explanation of answers and Worked out solutions.

Number theory is the branch of math concerned with the study of the integers, and of the objects and structures that naturally arise from their study.
Numbers, Numbers Everywhere

What does come to your mind when a friend says, “Let’s do math?” Problems to attend to; equations to be used; or complex numbers to contend with – quite naturally, math is about them all, and much more. Solving a theorem or a problem is like going on a long trek or scaling a mountain. The journey is challenging, adventurous, and often unpredictable. The feeling on arriving at the destination is, however, exhilarating.

Think of a number between 1 and 10. Now go through the following steps:
i. Multiply by the number 3
ii. Add 6 to the product
iii. Triple it

By now you would have figured that any number between 1 and 10 put through the above sequence will eventually be the magic number 9. Now, try figuring out the reason behind this? It would take you into number theory and the world of algebra. We could lay down the answer, but prefer you figure it out.

While the above is a good trick, Math is more than just numbers. Math is the language of nature.

Nature’s numbers Fibonacci pattern (top): Sun flower, Pineapple
Logarithmic spiral (bottom): Milkyway galaxy, Nautilus shell
Nature’s Numbers

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 ..... did you notice anything? Exactly, each number (after the initial 0 and 1) is the sum of the previous two. The Fibonacci sequence is interesting and will find its recurrence in nature. For instance, a lot of common flowers have 1, 3, 5, 8, 13, or 21 petals. Sunflowers, for instance, have 34, 55, or 89 petals. The study of pineapple will indicate that the row of scales on its surface follow two spirals sloping in the opposite directions. Now, if you count the number of scales along the spirals, the numbers exhibit Fibonacci pattern like (8 and 13) with 8 spiraling along the right and 13 to the left.

Spiral is another interesting phenomenon. The logarithmic spiral (angle between the tangent and the radius of the vector is same for all points of the spiral) often appears in nature: Milkyway galaxy, nautilus shell, pine cones, deer antlers, broccoli, human fingerprints, etc. The logarithmic spiral is also remarkable owing to its self-similarity. In other words, after scaling (uniformly increasing or decreasing the size), the spirals can be rotated such that they match the original figure. A Mandelbrot set is a great example of the concept of self-similarity.

Deepest part of space to the seed pattern in a sunflower, nature expresses itself in the language of math.

Auto racing is full of math applications and concepts. For a race team to be successful they must understand math. Math is used in every aspect of a race car’s design and performance.
Applications of Math

The speeds, the maneuvers, and the overtaking – watching a motor race is exciting. Viewing it from a math standpoint can be just as exhilarating. Yes, calculus: the field of study that deals with motion and change.

The math of motion, that allows you to treat a moving object as a point tracing a path through space and by freezing the action, calculate the speed and acceleration at a specific instant.

To the many, the winner of the race is the one with the least time over the distance. As a student of calculus, you know better. You have the advantage of knowing concepts such as functions, approximation, rate of change, and integration.

In fact, the very same concepts when delved deeper can help you figure much more: the wind velocities at specific intervals in an approaching tornado; or functions to use to model the vibrations of drumbeats and cymbals; or calculation of force exerted by water in a dam; or role of differential equations in a baseball game.

Seven Bridges of Konigsberg - How It Led to Modern Network Theory
Seven Bridges of Konigsberg

The river Pregel looped through the Prussian city of Konigsberg dividing it into four areas. Connecting the four areas were seven bridges. The townspeople all knew that the seven bridges could not all be crossed in a continuous walk. Or to rephrase it, the problem was to find a walk through the city that would allow you to cross each bridge once and only once (watch the animation).

The solution? Leonhard Euler proved that the problem had no solution, but realized an important principle in play. Euler reasoned (negative resolution) that in such a network some re-tracing was inevitable whenever there are three or more points at which an odd number of pathways converge. In doing so, he laid the foundations of graph theory, and presaged the development of topology. Euler’s work on re-tracing and observations about the network of lines connecting a number of points in 3-D laid the foundations for the modern network theory. The applications of which include you being able to access WWK and link through the site seamlessly.

The history of math is filled with interesting observations in the physical world leading to the development of applications of great value.

Medical imaging works because of a combination of very careful measurement techniques, sophisticated computer algorithms, and powerful mathematics.

From arithmetic to algebra, geometry to probability, calculus to topology – knowing the underlying concepts and working on a range of problems is important. The rigor of applying logic and techniques to compute, abstract, and manipulate symbols will enable you to see how all of math is tied together.

We understand your need to see the big picture and practice math skills. Which is why, we have paced the Math content to suit your learning style.

• Sequential mastery of skills (adaptive problems)
• Multiple opportunities to practice problem solving
• Showcase of worked out solutions
• Visual explanation of abstract concepts

Our objective is to ensure that whenever a friend says, ‘let’s do math’, you are the first to raise the hand.