After working on a math problem for several hours, when you find its solution, which you think is correct, it can be rechecked by using reasonableness. In simple words, reasonableness gets used when you need to verify a math solution to find out whether it is correct or not. There are different methods which you can use to verify your solutions; students must use these methods to recheck their solutions so that they won't lose any mark in the final exams.
[Image will be uploaded soon]
When a student is solving a solution, the one thing that can boost his/her confidence is the feeling of doing the sum correctly. Reasonableness acts as a saviour here because it conveys whether the solution is correct or wrong. Once the students are confident that they are solving the problems correctly, then they will efficiently solve the rest problems in the exam paper.
Some students feel like they are doing double work by using this method, but if their dream is to get an A in the maths exam, then this method is their lifesaver. They shouldn't hesitate to use it whenever required.
For maintaining accuracy, students can make use of this method to verify their multiplications for corrections. Making mistakes in multiplication will ruin the solution, so students need to be careful in this part. This method also helps to multiply two digits numbers or larger without even using a calculator. Students who learn this method properly will completely understand the mathematical process of multiplication.
For example, choose two numbers, such as 21 and 31; in the first step, round 21 to 20 and 31 to 30, then multiply 20 and 30, you will get your answer as 600. Then multiply 21 and 31, you will get your answer as 651. Now subtract 600 from 651, the answer will be 51; this shows that the difference is very less; thus, your answer is correct. If you would have got a more considerable difference, then you had to consider it wrong.
While solving division problems, you have to be reasonable because students who take unreasonable steps can never solve the questions correctly. There are certain deductively valid arguments that the students need to use to verify their solutions.
Division problems are straightforward to solve, they seem harder, but they are not. They become easier when you start solving these problems with estimation. The divisor and dividend can get rounded or simply examined in both short and long division problems. They are rounded in order to get an approximate value that is closer to the correct answer. Students who have understood the method can easily find out the quotient of a division problem.
For example, let's divide 43879 / 2847. Firstly round up the dividend to 44000 and the divisor to 3000. Here you can estimate that the solution to 44000 / 3000 will be close to 15 because 45 / 3 = 15. After doing the calculation, the answer is 15.41.
Sometimes try to look for patterns that are familiar to you and can help you in solving different problems. For example, to solve 35428 / 5928, you can recollect the square of 6, which is 36; this will allow you to estimate the approximate value quickly. Here the actual answer is 5.98, which is very close to 6.
Students can examine the divisor, simply it and check for multiples that are close to the value of the dividend. For example, in 74833 / 77, there are two options: if you are rounding the divisor and making it 80, then your estimate will be 9 based on 8 × 9 = 72, and if you are rounding it to 70, then the estimate will be 10 on the basis of 10 × 7 = 70. The quotient is 971.86, so both the estimates are valid; a student can use either.
There are two types of arguments which are:
Deductively Valid Argument
A deductive argument is an argument that is considered deductively valid by the arguer. It is proved valid by providing a guarantee of the truth of the conclusion providing that the argument premises are true.
Valid Inductive Argument
An inductive argument is considered to be an argument that the arguer intends to be so much stronger than if the premises become true, then there is no chance that the conclusion will become false. So the success of inductive argument depends on the strength of the matter of degree.
1. What is Rationalism?
Ans: Rationalism is the belief that knowledge is something that is acquired through reason without any need for senses. Mathematical knowledge is considered to be the best example of rationalism because only in maths, we can use rational thought to find out the depths of different problems, proofs for different Questions, etc. With the help of rationalism, students can acquire the ability to deduce complex problems. A student who knows how to think rationally can secure good marks in the maths exam. Many mathematical concepts are confusing and are difficult to understand, but with rationalism, all these complex concepts seem easy.
2. How Reasonableness Helps a Student?
Ans: The suspicion among children that whether they are solving the math problems correctly or not can become the main reason for committing mistakes. So to avoid this, students should use reasonableness for verification of their solved problems. Once they are aware that they are solving the problems in the right way, their confidence gets boosted up. Thus, they will do much better in their exams. This method had helped many students to secure good marks in the exam. Don't worry about the double work because with regular practice, you can verify each question in seconds using this method.