### Few Questions to Make Learning Effective

Here, we are taking an example to demonstrate how students are answering the questions right, getting good marks, letting the concept level problem continue and getting stuck in their studies or career at later stage. We also discuss, what approach can help in making learning right.

I hope, readers are aware of linear equations in one variable. Something like 15x+5=90, or 3x+5=2x+9.

Let, a student is asked to solve the 2nd equation 3x+5=2x+9. It's easy for most of the students. He solves this equation using right method and gets the answer as 4. Correct and full marks. Does it mean that he knows the concept of linear equations in one variable good enough?

Why should he solve such equations? Where can these concepts be used? Can he frame a question based on real life scenario where such equation may be needed? Can he visualize the impact of change in coefficients or constant values on the nature of the equation? If he is not clear about all these, then solving large number of such equations makes no sense. But we are doing that only. What percent of students who are able to solve the equation comfortably are able to answer these questions?

Let us ask the student to change the coefficient of x on either the right-side expression or the left side expression in the above equation so that a negative value of x satisfies the equation. That is, he is required to modify the equation by modifying one of the coefficients of x so that when the modified equation is solved, the answer is a negative value. Guess and check may be one technique to do this. But he should think of some better ways of doing this.

Consider this. Two friends Ram and Shyam went to a local farmer's market where locally produced potatoes were being sold in one of the stalls. There was an offer on purchase of potatoes. Buy any quantity of potatoes and get all-you-can-eat pani-puri at Rs 9 or all you can eat pizza at Rs 5. Ram purchased 3 Kg. of potatoes and chose to have pizza. Whereas Shyam purchased 2 Kg. of potatoes and went for pani-puri. Both of them paid same amount in that stall. Based on this information, find the cost of potatoes per Kg.

How will a student answer this question? Can knowledge of linear equations be useful here? Is the equation 3x+5=2x+9 of any use here?

Suppose the student assumes the cost of potatoes as x Rs./Kg. So, he finds that amount paid by Ram is 3x+5 and the amount paid by Shyam is 2x+9. Since, both have paid the same amount, both these expressions are equal. Hence, 3x+5=2x+9. Same equation which has been solved earlier. This gives x=4. So, the price of potatoes is 4 Rs./Kg.

Linear equations and inequalities are extensively used in mathematical modelling. These models are made to represent real life situations and conduct various experiments with them to study the behavior of the system better, quicker and with lesser cost and risk.

Let us end this article with some more questions that make one understand and use linear equations better. Think about these easy questions:

-- Whose decision can change the coefficient of x in the left-side expression (Ram, Shyam or Shopkeeper)?

-- Whose decision can change the coefficient of x in the right-side expression (Ram, Shyam or Shopkeeper)?

-- Whose decision can change the value of constant terms (Ram, Shyam or Shopkeeper)?

-- If the price of potatoes is increased, what happens to the equation? Can Ram, Shyam or the shopkeeper do something to bring back the equality?

-- Can you find the cost of all-you-can-eat pizza so that Ram and Shyam both pay the same amount for their present purchases irrespective of the price of the potatoes?

Practice, apply and create variations to understand the concepts well rather than just solving several questions of varied difficulty levels.